K3 PROJECTIVE MODELS IN SCROLLS

TRYGVE JOHNSEN AND ANDREAS LEOPOLD KNUTSEN

Abstract. We study the projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls.

We use this study to classify and describe all projective models of K3 surfaces of genus g ≤ 10.

1. Introduction

It is well known that all K3 surfaces are diffeomorphic, and that there is a 20-dimensional family of analytical isomorphism classes of K3 surfaces. Moreover there is a countable union of 19-dimensional families of K3 surfaces with base point free line bundles on them.

A pair (S, L) of a K3 surface S and a base point free line bundle L with L^{2} = 2g − 2 will be called a polarized K3 surface of genus g. The sections of L give a map ϕ_{L} of S into P^{g}, and the image is called a projective model of S. If ϕ_{L} is birational, the image is a surface of degree 2g − 2 in P^{g}.

The following is well known: For g = 3 the birational projective model is a quartic surface and for g = 4 a complete intersection of a quadric and a cubic hypersurface. For g = 5 the general model is a complete intersection of three hyperquadrics. For 6 ≤ g ≤ 10 and g = 12 it is shown by Mukai in [Mu1] and [Mu2] that the general projective model is a complete intersection in certain homogeneous varieties contained in projective spaces of larger dimension than g.

We will study the projective models of a particular kind of polarized K3 surfaces. Recall the result of Green and Lazarsfeld [G-L2], which states that if L is a base point free line bundle on a K3 surface S, then Cliff C is constant for all smooth irreducible C ∈ |L|, and if Cliff C < ⌊\frac{g−1}{2}⌋, then there exists a line bundle M on S such that M_{C} := M ⊗O_{C} computes the Clifford index of C for all smooth irreducible C ∈ |L|.

It therefore makes sense to define the Clifford index Cliff L of a base point free line bundle, or the Clifford index Cliff_{L}(S) of a polarized K3 surface (S, L) as the Clifford index of the smooth curves in |L|.

By the Existence Theorem 3.8 below, for any pair of integers (g, c) such that g ≥ 2 and 0 ≤ c ≤ ⌊\frac{g−1}{2}⌋, there exists an 18-dimensional family of polarized K3 surfaces of genus g and of Clifford index c.

In this paper we study and classify projective models of non-Clifford general polarized K3 surfaces of any genus larger than two.

The central point is that by the result of Green and Lazarsfeld, there exists in these cases a divisor class D on S computing the Clifford index of L. We will show that 0 ≤ D^{2} ≤ c+2. We can choose such a divisor that is base point free, so that in particular the general member of |D| is a smooth curve. Such a divisor (class) will be called a free Clifford divisor for L. (The definition only depends on the class of D.)

The images of the members of |D| by ϕ_{L} span sublinear spaces inside P^{g}. Each subpencil {D_{λ}} within the complete linear system |D| then gives rise to a pencil of sublinear spaces.

For each fixed pencil the union of these spaces will be a rational normal scroll T . We will investigate under which conditions this scroll is smooth. These scrolls are the natural ambient spaces for non-Clifford general K3 surfaces.

In the cases c = 1 and 2 with D^{2} = 0, the description of the projective models is particularly nice, since they are then complete intersections in their corresponding scrolls.

If T is smooth, we will be able to find a resolution (up to certain invariants) of ϕ_{L}(S) inside T for arbitrary c.

If T is singular, we take the blow up f : S̃ → S at the D^{2} base points of the pencil {D_{λ}} and show that the projective model ϕ_{H}(S̃) of S̃ by the base point free line bundle H := f^{∗}L + f^{∗}D − E, where E is the exceptional divisor, is contained in a smooth rational normal scroll T_{0} which is a smoothing of T . We find a resolution of ϕ_{H}(S̃) inside T_{0} and investigate the possibility of pushing down this resolution to a resolution of ϕ_{L}(S) inside T .

We also give a description of those projetive models for g ≤ 10 that are Clifford general, but still not general in the sense of Mukai (i.e. they are not complete intersections in homogeneous spaces). These models are also contained in scrolls, and can be analysed in a similar manner. This can also be done for g = 12, which we leave to the reader.

Together with Mukai’s results this will then give a complete picture of the birational projective models for g ≤ 10 (and g = 12). For g = 11 and g ≥ 13 our description of non-Clifford general projective models is not supplemented by any description of general projective models at all. We hope, however, that our description of the non-general models may have some interest in themselves.

The paper is organized as follows.

In Section 2 we recall some basic facts about rational normal scrolls, and how to obtain such scrolls from surfaces with pencils on them. In Section 3 we define the essential concept of a free Clifford divisor. We end the section by proving the above mentioned existence theorem.

In Section 4 we study in detail the singular loci of the projective model ϕ_{L}(S) and the scroll T in which we choose to view this model as contained. We show that we can always find a free Clifford divisor D such that the singular locus of T is “spanned” by the images of the base points of the pencil {D_{λ}} and the contractions of smooth rational curves across the members of the pencil. We also include a study of the projective model if c = 0 (the hyperelliptic case), which is due to Saint-Donat [SD]. Some of the longer proofs of the results in this section are postponed until Section 5.

In Section 6 we study the resolution of ϕ_{L}(S) inside its scroll T when T is smooth. In this case a general hyperplane section of T is a scroll formed in a similar way from a pencil computing the gonality on a canonical curve C of genus g (the gonality is c+2). Such scrolls were studied in [Sc], and our results for K3 surfaces in smooth scrolls are quite parallel to those of [Sc].

In Section 7 we treat the case when the scroll T is singular. The approach is to study the blow up f : S̃ → S at the D^{2} base points of the pencil {D_{λ}} and the projective model S′′ := ϕ_{H}(S̃) of S̃ by the base point free line bundle H := f^{∗}L + f^{∗}D − E, where E is the exceptional divisor. The pencil |f^{∗}D − E| defines a smooth rational normal scroll T_{0} that contains S′′ and is a smoothing of T . We use results from [Gr] to show how resolutions of ϕ_{L}(D_{λ}) in its linear span in P^{g}, for each D_{λ} (from the chosen subpencil of |D|), can be used to obtain a resolution of S′′ in T_{0}. We prove that S′′ is normal, and use this to give more details about the resolution. We give conditions under which we can push down the resolution to one of ϕ_{L}(S) in T . Here we use results from [Sc]. We end the section by investigating some examples for low genera.

In Section 8 we study the set of projective models in smooth scrolls for c = 1, 2 and 3 (< ⌊\frac{g−1}{2}⌋). We study the sets of projective models in (c + 2)-dimensional scrolls of given types. Since the scroll type is dependent on which rational curves that exist on S, and therefore on the Picard lattice, it is natural that the dimension of the set of models in question in a scroll as described, is dependent on the scroll type. We study this interplay, and obtain a fairly clear picture for c = 1 and 2. For c = 3 we study a Pfaffian map of the resolution of ϕ_{L}(S) in the scroll. We give a conjecture for the dimension of the set of projective K3 models inside a smooth scroll of a given type, for arbitrary c < ⌊\frac{g−1}{2}⌋).

In Section 9 we study the projective models for g ≤ 10 that are Clifford general but not general in the sense of Mukai (i.e. they are not complete intersections in homogeneous spaces). By the concrete description in [Mu2] of such surfaces, it follows that their projective models are also contained in scrolls. We analyse them in a similar manner.

We conclude by giving a complete list and descripton of all birational projective models of K3 surfaces for g ≤ 10.

1.1. Acknowledgements. The authors thank Kristian Ranestad for suggesting the problem. This paper was written while the authors were visitors at the Department of Mathematics, University of Utah, and at the Max-Planck-Institut für Mathematik, Bonn. We thank both institutions for their hospitality.

We are also grateful to S. Ischii, S. A. Strřmme, B. Toen and J. E. Vatne for useful conversations.

1.2. Notation and conventions. We use standard notation from Algebraic Geometry. The ground field is the field of complex numbers. All surfaces are algebraic surfaces. By a curve on a surface S is always meant an irreducible curve (possibly singular), i.e. a

prime divisor. Line bundles and divisors are used with little or no distinction, as well as the multiplicative and additive notation. Linear equivalence of divisors is denoted by ∼, and numerical equivalence by ≡. Note that on a K3 surface linear and numerical equivalence is the same.

If L is any line bundle on a surface, L is said to be numerically effective, or simply nef, if L.C ≥ 0 for all curves C on S. In this case L is said to be big if L^{2} > 0.

If F is any coherent sheaf on a variety V, we shall denote by h^{i}(F) the complex dimension of H^{i}(V, F), and by χ(F) the Euler characteristic ∑(−1)^{i}h^{i}(F).

If D is any divisor on a normal surface S, Riemann-Roch for D is χ(O_{S}(D)) = \frac{1}{2}D.(D + K_{S}) + χ(O_{S}).

By a K3 surface is meant a smooth surface S with trivial canonical bundle and such that H^{1}(O_{S}) = 0. In particular h^{2}(O_{S}) = 1 and χ(O_{S}) = 2.

We will make use of the following results of Saint-Donat on line bundles on K3 surfaces. The first result will be used repeatedly, without further mention.

Proposition 1.1. [SD, Cor. 3.2] A complete linear system on a K3 surface has no base points outside of its fixed components.

Proposition 1.2. [SD, Prop. 2.6] Let L be an invertible sheaf on a K3 surface S such that |L| ≠ ∅ and such that |L| has no fixed components. Then either

(i) L^{2} > 0 and the general member of |L| is an irreducible curve of arithmetic genus L^{2}/2 + 1. In this case h^{1}(L) = 0, or

(ii) L^{2} = 0, then L ≃ O_{S}(kE), where k is an integer ≥ 1 and E is an irreducible curve of arithmetic genus 1. In this case h^{1}(L) = k − 1 and every member of |L| can be written as a sum E_{1} + ⋯ + E_{k}, where E_{i} ∈ |E| for i = 1, …, k.

Note that by Bertini the general members in |L| and |E| are smooth.

Lemma 1.3. [SD, 2.7] Let L be a nef line bundle on a K3 surface S. Then |L| is not base point free if and only if there exist smooth irreducible curves E, Γ and an integer k ≥ 2 such that

L ∼ kE + Γ, E^{2} = 0, Γ^{2} = −2, E.Γ = 1.

In this case, every member of |L| is of the form E_{1} + ⋯ + E_{k} + Γ, where E_{i} ∈ |E| for all i.

To show the existence of K3 surfaces with certain divisors on it, a very useful result is the following by Morrison:

Proposition 1.4. [Mo, Cor. 2.9(i)] Let ρ ≤ 10 be an integer. Then every even lattice of signature (1, ρ − 1) occurs as the Picard group of some algebraic K3 surface.

Consider now the group generated by the Picard-Lefschetz reflections

ϕ_{Γ} : Pic S ⟶ Pic S
D ↦ D + (D.Γ)Γ

where Γ ∈ Pic S satisfies Γ^{2} = −2. Note that a reflection leaves the intersections between divisors invariant.

The following result will also be useful for our purposes:

Proposition 1.5. [B-P-V, VIII, Prop. 3.9] A fundamental domain for this action is the big-and-nef cone of S.

This means that given a certain Picard lattice, we can perform Picard-Lefscetz reflections on it, and thus assume that some particular line bundle chosen (with positive selfintersection) is nef.

2. Surfaces in scrolls

In the beginning of this section we briefly review some basic facts that can be found in [Sc].

Definition 2.1. Let E = O_{P^{1}}(e_{1}) ⊕ ⋯ ⊕ O_{P^{1}}(e_{d}), with e_{1} ≥ … ≥ e_{d} ≥ 0 and f = e_{1} + ⋯ + e_{d} ≥ 2. Consider the linear system L = O_{P(E)}(1) on the corresponding P^{d−1}- bundle P(E) over P^{1}. We map P(E) into P^{r} with the complete linear system H^{0}(L). Then r = f + d − 1. The image T is by definition a rational normal scroll of type e= (e_{1}, …, e_{d}). The image is smooth, and isomorphic to P(E), if and only if e_{d} ≥ 1.

Remark 2.2. Some authors, like in [P-S], use the term rational normal scroll only if e_{d} ≥ 1 (so that T is smooth), but for our purposes it will be more convenient to use the more liberal definition above.

Let L be a spanned line bundle on a smooth surface S. We denote by ϕ_{L} the natural morphism

ϕ_{L} : S ⟶ P^{h^{0}(L)−1} := P^{g}

defined by the complete linear system |L|.

Assume that L can be decomposed as

L ∼ M + N,

where M ≃ O_{S}(D) and N ≃ O_{S}(F) are line bundles on S (with D and F their corresponding divisors) satisfying h^{0}(M), h^{0}(N) ≥ 2. Choose a 2-dimensional subspace W ⊆ H^{0}(S, D), which then defines a pencil

{D_{λ}}_{λ∈P^{1}} ⊆ |D|.

Each ϕ_{L}(D_{λ}) will span a (h^{0}(L) − h^{0}(L − D) − 1)-dimensional subspace of P^{g}, which is called the linear span of ϕ_{L}(D_{λ}) and denoted by \overline{D_{λ}}. It turns out that the variety swept out by these linear spaces,

T = ∪_{λ∈P^{1}} \overline{D_{λ}} ⊆ P^{g},

is a rational normal scroll:

Proposition 2.3. [Sc] The multiplication map

W ⊗ H^{0}(S, F) ⟶ H^{0}(S, L)

yields a 2 × h^{0}(F) matrix with linear entries whose 2 × 2 minors vanish on ϕ_{L}(S). The variety T defined by these minors contains ϕ_{L}(S) and is a rational normal scroll of degree f := h^{0}(F) and dimension d := h^{0}(L) − h^{0}(L − D). In particular d + f = g + 1.

Decomposing the pencil {D_{λ}} into its moving part {D′_{λ}} and fixed part Δ,

D_{λ} ∼ D′_{λ} + Δ,

the type (e_{1}, …, e_{d}) of the scroll T is given by

e_{i} = #{j | d_{j} ≥ i} − 1, (1)

where

d = d_{0} := h^{0}(L) − h^{0}(L − D),
d_{1} := h^{0}(L − D) − h^{0}(L − 2D′ − Δ),
⋮
d_{i} := h^{0}(L − iD′ − Δ) − h^{0}(L − (i + 1)D′ − Δ),
⋮

Conversely, if ϕ_{L}(S) is contained in a scroll T of degree f, the ruling of T will cut out on S a pencil of divisors (possibly with base points) {D_{λ}} ⊆ |D| with h^{0}(L − D) = f ≥ 2, whence inducing a decomposition

L ∼ M + N,

as above.

We will study these scrolls more thoroughly in the case where S is a K3 surface. We will see that in the K3 case natural decompositions of L as above will occur when the smooth curves in |L| do not have the general Clifford index. We will turn back to this in the next section. First we get some general results.

For any decomposition L ∼ D + F, with h^{0}(D), h^{0}(F) ≥ 2, write D.F = c + 2, for some integer c. Assuming D.L ≤ F.L, or equivalently D^{2} ≤ F^{2}, we have by the Hodge index theorem that D satisfies the numerical conditions below:

2D^{2} ≤^{(i)} L.D = D^{2} + c + 2 ≤^{(ii)} 2c + 4

with equality in (i) or (ii) if and only if L ≡ 2D and L^{2} = 4c + 8.

Indeed, the condition D.L ≤ F.L can be rephrased as 2D.L ≤ L^{2}, and by the Hodge index theorem 2D^{2}(D.L) ≤ D^{2}L^{2} ≤ (D.L)^{2}, with equalities only if L ≡ 2D.

If the set

A(L) := {D ∈ Pic S | h^{0}(D), h^{0}(L − D) ≥ 2}

is nonempty, define the integer μ(L) as

μ(L) := min{D.F − 2 | L ∼ D + F and D, F ∈ A(L)}
= min{D.L − D^{2} − 2 | D ∈ A(L)}

and set

A^{0}(L) := {D ∈ A(L) | D.(L − D) = μ(L) + 2}

For K3 surfaces we have the following result:

Proposition 2.4. Let L be a spanned and big line bundle on a K3 surface S such that A(L) ≠ ∅. Then μ(L) ≥ 0 and any divisor D in A^{0}(L) will have the following properties:

(i) the (possibly empty) base locus Δ of D satisfies L.Δ = 0, (ii) h^{1}(D) = 0.

Proof. The first statement follows from the fact that any member of the complete linear system of a spanned and big line bundle on a K3 surface is numerically 2-connected (see [SD, (3.9.6)], or [Kn1, Thm. 1.1] for a more general statement).

We first show (i).

If D is nef but not base point free, then by Lemma 1.3, D ∼ kE + Γ, for an integer k ≥ 2 and divisors E and Γ satisfying E^{2} = 0, Γ^{2} = −2 and E.Γ = 1. Since L is spanned, we must have E.L ≥ 2 [Kn1, Thm. 1.1], so D.L − D^{2} = (kE + Γ).L − (2k − 2) ≥ kE.L − 2k − 2 ≥ E.L+2(k −1)−2(k −1) = E.L, which implies E.L = 2, μ(L) = 0, and as asserted Γ.L = 0.

If D is not nef, there exists a smooth rational curve Γ such that Γ.D < 0. Letting D′ := D−Γ and we have D′ ∈ A(L) and L.D′−D′^{2}−2 = D.(L−D)−L.Γ+2Γ.D+2 ≥ D.(L−D), whence L.Γ = 0, Γ.D = −1, D′^{2} = D^{2}, L.D′ = L.D and D′(L − D′) = D.(L − D) = c + 2. Continuing inductively, we get that Δ.L = 0, as desired.

Since Δ.L = 0 and (D −Δ).L−(D −Δ)^{2} ≥ D.L−D^{2}, we must have D^{2} ≥ (D −Δ)^{2} ≥ 0. We now prove (ii). If h^{1}(D) ≠ 0, there exists by Ramanujam’s lemma an effective decomposition D ∼

D_{1} + D_{2} such that D_{1}.D_{2} ≤ 0. By the Hodge index theorem (and the fact that D^{2} ≥ 0) we can assume D_{1}^{2} ≥ 0 and D_{2}^{2} ≤ 0, with equalities occurring simultaneously. The divisor D_{1} is in A(L), and writing F := L − D we get D_{1}.(F + D_{2}) = D.F + D_{1}.D_{2} − D_{2}.F ≥ F.D, whence D_{2}.F ≤ D_{1}.D_{2} ≤ 0. But L is nef, so D_{2}.L = D_{2}.D + D_{2}.F ≥ 0, which implies D_{2}.F = D_{1}.D_{2} = D_{1}^{2} = D_{2}^{2} = 0. Now the same argument works for D_{1}, so D_{1}.F = 0 and we get the contradiction D.F = (D_{1} + D_{2}).F = 0.

Writing L ∼ D + F, the above result is of course symmetric in D and F. It turns out that we can choose one of them to have an additional property. More precisely, we have :

Proposition 2.5. Let L be a spanned line bundle on a K3 surface S such that A(L) ≠ ∅. We can find a divisor D in A^{0}(L) such that either |D| or |L − D| (but not necessarily both at the same time) is base point free and its general member is smooth and irreducible. If L is ample, then for any divisor D in A^{0}(L) the above conditions will be satisfied for both |D| and |L − D|.

Proof. Let D ∈ A^{0}(L). Denote its base locus by Δ and assume it is not zero. Then L.Δ = 0 by the previous proposition.

If D is nef but not base point free, then D ∼ kE + Γ as above and the smooth curve E will satisfy the desired conditions.

If D is not nef, there exists a smooth rational curve Γ such that Γ.D < 0. Letting D′ := D − Γ, we can argue inductively as above until we reach a divisor which is base point free or of the form kE + Γ.

This procedure can of course not be performed on both D and L−D simultaneously, but if L is ample, they are both automatically base point free.

The fact that the general member of |D| (or |L − D|) is a smooth curve now follows from Proposition 1.2, since h^{1}(D) = 0.

Remark 2.6. Note that by the proofs of the two previous propositions, if D is not nef, then D′^{2} = D^{2}. This means that given a divisor D ∈ A(L), we can find a divisor D_{0} ∈ A(L) satisfying the additional conditions in Proposition 2.5 and such that D_{0}^{2} ≤ D^{2}.

In the next section we will study the perhaps most important example of such decompositions of the line bundle L, namely the case where all smooth curves in |L| have a non-general Clifford index.

3. The Clifford index of smooth curves in |L| and the definition of the scrolls T (c, D, {D_{λ}})

We briefly recall the definition and some properties of gonality and Clifford index of curves.

Let C be a smooth irreducible curve of genus g ≥ 2. We denote by g_{d}^{r} a linear system of dimension r and degree d and say that C is k-gonal (and that k is its gonality) if C posesses a g_{k}^{1} but no g_{k−1}^{1}. In particular, we call a 2-gonal curve hyperelliptic and a 3-gonal curve trigonal. We denote by gon C the gonality of C. Note that if C is k-gonal, all g_{k}^{1}’s must necessarily be base point free and complete.

If A is a line bundle on C, then the Clifford index of A is the integer

Cliff A = deg A − 2(h^{0}(A) − 1).

The Clifford index of C itself is defined as

Cliff C = min{Cliff A | h^{0}(A) ≥ 2, h^{1}(A) ≥ 2}.

Clifford’s theorem then states that Cliff C ≥ 0 with equality if and only if C is hyperelliptic and Cliff C = 1 if and only if C is trigonal or a smooth plane quintic.

At the other extreme, we get from Brill-Noether theory (cf. [A-C-G-H, V]) that the gonality of C satisfies gon C ≤ ⌊\frac{g+3}{2}⌋, whence Cliff C ≤ ⌊\frac{g−1}{2}⌋. For the general curve of genus g, we have Cliff C = ⌊\frac{g−1}{2}⌋.

We say that a line bundle A on C contributes to the Clifford index of C if h^{0}(A), h^{1}(A) ≥ 2 and that it computes the Clifford index of C if in addition Cliff C = Cliff A.

Note that Cliff A = Cliff ω_{C} ⊗ A^{−1}.

The Clifford dimension of C is defined as

min{h^{0}(A) − 1 | A computes the Clifford index of C}.

A line bundle A which achieves the minimum and computes the Clifford index, is said to compute the Clifford dimension. A curve of Clifford index c is (c + 2)-gonal if and only if it has Clifford dimension 1. For a general curve C, we have gon C = c + 2.

Lemma 3.1. [C-M, Theorem 2.3] The gonality k of a smooth irreducible projective curve C satisfies

Cliff C + 2 ≤ k ≤ Cliff C + 3.

The curves satisfying gon C = Cliff C + 3 are conjectured to be very rare and called exceptional (cf. [Ma, (4.1)]).

Recall the following result of Green and Lazarsfeld:

Theorem 3.2. [G-L2] Let L be a base point free line bundle on a K3 surface S with L^{2} > 0. Then Cliff C is constant for all smooth irreducible C ∈ |L|, and if Cliff C < ⌊\frac{g−1}{2}⌋, then there exists a line bundle M on S such that M_{C} := M ⊗ O_{C} computes the Clifford index of C for all smooth irreducible C ∈ |L|.

Note that since (L − M) ⊗ O_{C} ≃ ω_{C} ⊗ M_{C}^{−1}, the result is symmetric in M and L − M. Based on this result, we make the following definition:

Definition 3.3. Let L be a base point free and big line bundle on a K3 surface. We define the Clifford index of L to be the Clifford index of all the smooth curves in |L| and denote it by Cliff L.

Similarly, if (S, L) is a polarized K3 surface we will often call Cliff L the Clifford index of S and denote it by Cliff_{L}(S).

It turns out that we can choose the line bundle M appearing in Theorem 3.2 above so that it satisfies certain properties. We will need the following result in the sequel.

Lemma 3.4. [Kn1, Lemma 8.3] Let L be a base point free line bundle on a K3 surface S with L^{2} = 2g − 2 ≥ 2 and Cliff L = c.

If c < ⌊\frac{g−1}{2}⌋, then there exists a smooth curve D on S satisfying 0 ≤ D^{2} ≤ c + 2, 2D^{2} ≤ D.L (either of the latter two inequalities being an equality if and only if L ∼ 2D) and

Cliff C = Cliff(O_{S}(D) ⊗ O_{C}) = D.L − D^{2} − 2

for any smooth curve C ∈ |L|.

It is also known (see e.g. [Ma]) that D satisfies h^{0}(D ⊗ O_{C}) = h^{0}(D) and h^{0}((L − D) ⊗ O_{C}) = h^{0}(L − D) = h^{1}(D ⊗ O_{C}) for any smooth curve C ∈ |L|.

From the results in the previous section, it is also clear that

Cliff C = min\{μ(L), ⌊\frac{g − 1}{2}⌋\}

for any smooth C ∈ |L|.

Summarizing, and using Propositions 2.4 and 2.5, we have that if L is a spanned line bundle on a K3 surface S, of sectional genus

g = g(L) = \frac{1}{2}L^{2} + 1,

and the smooth curves in |L| have Clifford index

c < ⌊\frac{g − 1}{2}⌋ = ⌊\frac{L^{2}}{4} ⌋,

(which in particular implies L^{2} ≥ 4c + 4), then there exists a divisor (class) D on S with the following properties (with F := L − D): (C1) c = D.L − D^{2} − 2 = D.F − 2, (C2) D.L ≤ F.L (eqv. D^{2} ≤ F^{2}) and if equality occurs, then either L ∼ 2D or h^{0}(2D−L) =

0, (C3) h^{1}(D) = h^{1}(F) = 0.

A divisor (class) D with the properties (C1) and (C2) above will be called a Clifford divisor for L. This means in other words that D and L − D compute the Clifford index of all smooth curves in |L|. The property (C2) can be considered an ordering of D and L − D. Any Clifford divisor will automatically fulfill property (C3) and the (possibly empty) base loci Δ of |D| and Δ′ of |L − D| will satisfy L.Δ = L.Δ′ = 0 by Proposition 2.4.

By Proposition 2.5 we can find a Clifford divisor D satisfying the properties (C4) the (possibly empty) base locus Δ of F satisfies L.Δ = 0, (C5) |D| is base point free and its general member is a smooth curve,

A divisor D satisfying all properties (C1)-(C5) will be called a free Clifford divisor for L. Since this definition only depends on the class of D, we will by abuse of notation never distinguish between D and its divisor class. Hopefully, this will not cause any confusion.

We will now take a closer look at two particular kinds of free Clifford divisors, namely:

(a) D^{2} = c + 1, or

(b) D^{2} = c, L ∼ 2D + Δ, with Δ > 0.

It turns out that these free Clifford divisors are of a particular form.

Proposition 3.5. Let L be a spanned and big line bundle of non-general Clifford index c on a K3 surface, and let D be a free Clifford divisor.

If D is as in (a) above, then L^{2} = 4c + 6 and (E0) L ∼ 2D + Γ, where Γ is a smooth rational curve satisfying Γ.D = 1.

If D is as in (b) above, then L^{2} = 4c + 4 and (letting all Γ_{i} be smooth rational curves) either (E1) L ∼ 2D + Γ_{1} + Γ_{2}, D^{2} = c, D.Γ_{1} = D.Γ_{2} = 1, Γ_{1}.Γ_{2} = 0, or (E2) L ∼ 2D+2Γ_{0}+2Γ_{1}+⋯+2Γ_{N} +Γ_{N+1}+Γ_{N+2}, D^{2} = c, and the following configuration:

[figure: D; Γ_{0}; Γ_{N}; Γ_{N+1}; Γ_{N+2}]

Furthermore, there can only exist Clifford divisors of one of the three types (E0)-(E2) (on the same surface), and all such are linearly equivalent.

Proof. If D is as in case (a), the proof of the first statement is similar to the proof of [Kn1, Lemma 7.4] and is left to the reader.

Now assume D is as in case (b). Then we have L^{2} = 2D.L+Δ.L = 2(D^{2} +c+2) = 4c+4. This gives Δ^{2} = (L − 2D)^{2} = −4 and D.Δ = \frac{1}{2}(L.Δ − Δ^{2}) = 2. By Proposition 4.3 below, any smooth rational curve Γ in Δ such that Γ.D > 0, satisfies Γ.D = 1, and any two such curves are disjoint. So we have to distinguish between two cases.

If there exist two distinct rational curves Γ_{1} and Γ_{2} in Δ such that Γ_{1}.D = Γ_{2}.D = 1 and Γ_{1}.Γ_{2} = 0, write L ∼ 2D + Γ_{1} + Γ_{2} + Δ′, for some Δ′ ≥ 0. Then 0 = Γ_{i}.L = 2 − 2 + Γ_{i}.Δ′ gives Δ′.Γ_{i} = 0, for i = 1, 2. Clearly D.Δ′ = 0, so

(2D + Γ_{1} + Γ_{2}).Δ′ = 0,

whence Δ′ = 0, since L is numerically 2-connected, and we are in case (E1).

If there exists a rational curve Γ occurring with multiplicity 2 in Δ such that Γ.D = 1, write L ∼ 2D + 2Γ + Δ′, for some Δ′ ≥ 0. Since 0 = Γ.L = 2 − 4 + Δ′.Γ, we get Δ′.Γ = 2. Iterating the process, we get case (E2).

Assume now D is given and B is any free Clifford divisor as in (E0)-(E2). We want to show that B ∼ D.

If D is of type (E0), we must have L^{2} = 4c + 6, so B must also be of type (E0), which means that L ∼ 2B + Γ_{0}, where Γ is a smooth rational curve such that B.Γ_{0} = 1.

Since

0 = Γ_{0}.L = 2D.Γ_{0} + Γ.Γ_{0},

and D is nef, we get the two possibilities

(i) D.Γ_{0} = Γ.Γ_{0} = 0 or (ii) D.Γ_{0} = 1, Γ = Γ_{0}.

If (i) were to happen, we would get

4D.B = (L − Γ)(L − Γ_{0}) = L^{2} = 4c + 6,

which is clearly impossible.

Hence Γ = Γ_{0} and D ∼ B.

If D is of type (E1) or (E2), we have L^{2} = 4c + 4, so B must also be of type (E1) or (E2) and will therefore satisfy either

1) L ∼ 2B + Γ′_{1} + Γ′_{2}, or

2) L ∼ 2B + 2Γ′_{0} + ⋯ + 2Γ′_{N} + Γ′_{N+1} + Γ′_{N+2}, where the Γ′_{i} are smooth rational curves with configurations as in the cases (E1) and (E2).

Assume now that D is of type (E1). The proof if D is of type (E2) is similar.

If 2) holds, we get from

0 = Γ′_{i}.L = 2D.Γ′_{i} + Γ_{1}.Γ′_{i} + Γ_{2}.Γ′_{i},

and the fact that D is nef, that

Γ_{j}.Γ′_{i} = 0 or Γ_{j} = Γ′_{i}, i = 0, …, N + 2, j = 1, 2.

This gives

4D.B = (L − Γ_{1} − Γ_{2})(L − 2Γ′_{0} − ⋯ − 2Γ′_{N} + Γ′_{N+1} − Γ′_{N+2}) ≤ L^{2} = 4(c + 1),

whence D.B ≤ c + 1 and (D − B)^{2} ≥ −2, and by Riemann-Roch, if D ≁ B, either D − B or B − D is effective. The argument below is symmetric in those two cases, so assume D ∼ B + Σ, for Σ effective and Σ^{2} ≥ −2. Then Σ.L = 0 and Σ^{2} = −2 by the Hodge index theorem. Furthermore,

L ∼ 2D + Γ_{1} + Γ_{2} ∼ 2B + Γ_{1} + Γ_{2} + 2Σ ∼ 2B + 2Γ′_{0} + ⋯ + 2Γ′_{N} + Γ′_{N+1} + Γ′_{N+2},

whence

2Σ ∼ 2Γ′_{0} + ⋯ + 2Γ′_{N} + Γ′_{N+1} + Γ′_{N+2} − Γ_{1} − Γ_{2},

and 2Σ.B = 2 − (Γ_{1} + Γ_{2}).B ≤ 2 (since B is nef). By

0 = Σ.L = 2B.Σ + (Γ_{1} + Γ_{2}).Σ + 2Σ^{2},

we get (Γ_{1} + Γ_{2}).Σ ≥ 2, and

2Σ.D = (L − Γ_{1} − Γ_{2}).Σ ≤ −2,

contradicting the nefness of D. So we are in case 1) above and again from

0 = Γ′_{i}.L = 2D.Γ′_{i} + Γ_{1}.Γ′_{i} + Γ_{2}.Γ′_{i},

and the fact that D is nef, we get the three possibilities:

(i) D.Γ′_{1} = 1, Γ′_{1} = Γ_{1}, D.Γ′_{2} = Γ_{2}.Γ′_{2} = 0, (ii) D.Γ′_{i} = Γ_{1}.Γ′_{i} = Γ_{2}.Γ′_{i} = 0, i=1,2, (iii) D.Γ′_{i} = 1, Γ′_{i} = Γ_{i}, i = 1, 2.

In case (i) we get the absurdity 4D.B = (L − Γ_{1} − Γ_{2}).(L − Γ′_{1} − Γ′_{2}) = 4(c + 1) − 2. In case (ii) we get 4D.B = 4(c + 1), whence D.B = c + 1. We calculate (D − B)^{2} ≥ −2,

and by Riemann-Roch, if D ≁ B, either D − B or B − D is effective. Writing D ∼ B + Σ, for Σ effective and Σ^{2} = −2, we get the same contradiction as above.

So we are in case (iii) and B ∼ D.

Note that any Clifford divisor D will satisfy the numerical conditions:

2D^{2} ≤^{(i)} L.D = D^{2} + c + 2 ≤^{(ii)} 2c + 4

(∗) with equality in (i) or (ii) if and only if L ∼ 2D and L^{2} = 4c + 8.

In particular,

(2) D^{2} ≤ c + 2, with equality if and only if L ∼ 2D and L^{2} = 4c + 8,

and by the Hodge index theorem

D^{2}L^{2} ≤ (L.D)^{2} = (D^{2} + c + 2)^{2}. (3)

It turns out that free Clifford divisors of self-intersection c + 1 will play a particular role. The following proposition describes this case.

Proposition 3.6. Let L be a spanned and big line bundle on a K3 surface and let c be the Clifford index of all smooth curves in |L|. Then the following conditions are equivalent:

(i) all smooth curves in |L| are exceptional (i.e. have gonality c + 3), (ii) there is a free Clifford divisor of type (E0), (iii) all free Clifford divisors are linearly equivalent and of type (E0).

Proof. The equivalence between (i) and (iii) follows from the proof of [Kn1, Prop. 8.6]. We now show that if D is a free Clifford divisor of type (E0) and D′ is any other free Clifford divisor, then D′ ∼ D.

So let B := D − D′. Define R′ := L − 2D′ and note that R′ ∼ 2B + Γ. We have

c + 2 = D.(L − D) = (D′ + B).(D′ + B + Γ)
= D′^{2} + (2B + Γ).D′ + B.(B + Γ) = c + 2 + B.(B + Γ),

whence B^{2} + B.Γ = 0. Combined with Γ.D = Γ.(D′ + B) = 1, and since Γ.D′ = 0 or 1 by Lemma 5.3(c), we get

Γ.D′ = 1, B^{2} = 0, B.Γ = 0,

whence

B.R′ = B.(2B + Γ) = 0 and R′.D′ = 2B.D′ + 1.

This gives

L.B = 2D′.B + R′.B = 2D′.B = R′.D′ − 1.

But this implies

B.L − B^{2} − 2 = R′.D′ − 3 < D′^{2} + R′.D′ − 2 = c,

whence we must have B ∼ 0, as desired.

Remark 3.7. This result can be seen as a generalization of [SD, Rem. 7.13] and [E-L-M-S, Thm. 4.3].

Now we return to the theory of scrolls. Let D be a free Clifford divisor.

If D^{2} = 0, then |D| = {D_{λ}}_{λ∈P^{1}} is a pencil, which defines in a natural way a scroll containing ϕ_{L}(S).

If D^{2} > 0, then dim |D| = \frac{1}{2}D^{2} + 1 > 1, and we choose a subpencil \{D_{λ}\}_{λ∈P^{1}} ⊆ |D| as follows: Pick any two smooth members D_{1} and D_{2} ∈ |D| intersecting in D^{2} distinct points and such that none of these points belong to the union of the finite set of curves

(4) {Γ | Γ is a smooth rational curve, Γ.L ≤ c + 2}.

Then

{D_{λ}}_{λ∈P^{1}} := the pencil generated by D_{1} and D_{2}.

The pencil {D_{λ}} will be without fixed components (but with D^{2} base points) and define in a natural way a scroll containing ϕ_{L}(S) and of type determined as in equation (1) by the integers

d_{i} = h^{0}(L − iD) − h^{0}(L − (i + 1)D), i ≥ 0. (5)

Since all choices of subpencils of |D| will give scrolls of the same type, and scrolls of the same type are isomorphic, the scrolls arising are up to isomorphism only dependent on D. We denote these scrolls by T = T (c, D, {D_{λ}}).

Since h^{1}(L) = 0, we get by the conditions (C1) and (C5) that T has dimension

dim T = d_{0} = h^{0}(L) − h^{0}(F) = c + 2 + \frac{1}{2}D^{2}, (6)

and degree

deg T = h^{0}(F) = g − c − 1 − \frac{1}{2}D^{2}. (7)

Furthermore, each D_{λ} ∈ {D_{λ}} has linear span

\overline{D_{λ}} = P^{c+1+ \frac{1}{2} D^{2}}. (8)

Given integers g ≥ 2 and 0 ≤ c ≤ ⌊\frac{g−1}{2}⌋, one may ask whether there actually exists a pair (S, L), where S is a K3 surface, L^{2} = 2g − 2 and all smooth curves in |L| have Clifford index c.

The following result gives a positive answer to this question. We will conclude this section by giving its proof.

Theorem 3.8. Let g and c be integers such that g ≥ 3 and 0 ≤ c ≤ ⌊\frac{g−1}{2}⌋. Then there exists a polarized K3 surface of genus g and Clifford index c.

The theorem is an immediate consequence of the following

Proposition 3.9. Let d and g be integers such that g ≥ 3 and 2 ≤ d ≤ ⌊\frac{g−1}{2}⌋ + 2, and let S be a K3 surface with Pic S = ZL ⊕ ZE, where L^{2} = 2(g − 1), E.L = d and E^{2} = 0. Then L is base point free and

c := Cliff L = d − 2 < ⌊\frac{g − 1}{2}⌋

Furthermore, E is the only Clifford divisor for L (modulo equivalence class).

To prove this proposition, we first need the following basic existence result:

Lemma 3.10. Let g ≥ 3 and d ≥ 2 be integers. Then there exists a K3 surface S with Pic S = ZL ⊕ ZE, such that L is base point free and E is a smooth curve, L^{2} = 2(g − 1), E.L = d and E^{2} = 0.

Proof. By Propositions 1.4 and 1.5, we can find a K3 surface S with Pic S = ZL ⊕ ZE, with intersection matrix

[ \begin{matrix}L^{2} & L.E \\ E.L & E^{2} \end{matrix} ] = [ \begin{matrix}2(g − 1) & d \\ d & 0 \end{matrix} ]

and such that L is nef. If L is not base point free, there exists by Proposition 1.3 a curve B such that B^{2} = 0 and B.L = 1. An easy calculation shows that this is impossible. By [Kn2, Proposition 4.4], we have that |E| contains a smooth curve.

Proof of Proposition 3.9. Let S, L and E be as in Lemma 3.10, with d ≤ ⌊\frac{g−1}{2}⌋ + 2. Note that since E is irreducible, we have h^{1}(E) = 0. By the cohomology of the short exact sequence

0 ⟶ O_{S}(E − L) ⟶ O_{S}(E) ⟶ O_{C}(E) ⟶ 0,

where C is any smooth curve in |L|, we find that h^{0}(O_{C}(E)) ≥ h^{0}(E) = 2 and h^{1}(O_{C}(E)) = h^{0}(L − E) ≥ 2, so O_{C}(E) contributes to the Clifford index of C and

c ≤ Cliff O_{C}(E) ≤ E.L − E^{2} − 2 = d − 2 < ⌊\frac{g − 1}{2}⌋.

Now since c < ⌊\frac{g−1}{2}⌋, then there has to exist an effective divisor D on S satisfying

c = Cliff O_{C}(D) = D.L − D^{2} − 2.

Since both D and L − D must be effective and E is nef, we must have

D.E ≥ 0 and (L − D).E ≥ 0.

Writing D ∼ xL + yE this is equivalent to

dx ≥ 0 and d(1 − x) ≥ 0,

which gives x = 0 or 1. These two cases give, respectively, D = yE and L − D = yE. This shows that c = E.L − E^{2} − 2 = d − 2. This argument also shows that there are no other Clifford divisors but E.

This concludes the proof of Theorem 3.8.

Also note that the proof of Proposition 3.9 gives the following result, which is of its own interest:

Theorem 3.11. Let g and k be integers such that g ≥ 2 and 2 ≤ k ≤ ⌊\frac{g+3}{2}⌋. Then there exists a K3 surface containing a smooth curve of genus g and gonality k.

The surfaces constructed in Proposition 3.9 all have the property that the only free Clifford divisor (modulo equivalence class) is a smooth elliptic curve E. One could also perform the same construction with lattices of the form

[ \begin{matrix}L^{2} & L.D \\ D.L & D^{2} \end{matrix} ]

with D^{2} > 0 (and satisfying the constraints given by equations (2) and (3)), but for each pair (g, c) there might be values of D^{2} that cannot occur. See Proposition 10.1 for a result concerning low values of c. We will in Sections 9 and 10 perform more such constructions, also with lattices of higher ranks.

4. The singular locus of the surface S′ and the scroll T

We start this section by describing the image S′ := ϕ_{L}(S) by the complete linear system |L| on the K3 surface S.

Proposition 4.1. Let L be a spanned and big line bundle on a K3 surface S, and denote by ϕ_{L} the corresponding morphism and by c the Clifford index of the smooth curves in |L|.

(i) If c = 0, ϕ_{L} is 2 : 1 onto a surface of degree \frac{1}{2}L^{2}, (ii) If c > 0, then ϕ_{L} is birational onto a surface of degree L^{2} (in fact it is an isomorphism

outside of finitely many contracted smooth rational curves), and S′ := ϕ_{L}(S) is normal and has only double rational singularities. In particular K_{S′} ≃ O_{S′}, and p_{a}(S′) = 1.

Proof. These are well-known results due to Saint-Donat [SD] (see also [Kn1] for further discussions).

Let D be a free Clifford divisor and {D_{λ}} a subpencil of |D| chosen as described in the previous section.

Define the subset D of the pencil {D_{λ}} by

D := {D_{λ} ∈ {D_{λ}} | ϕ_{L} does not contract any component of D_{λ}}.

We then have

Lemma 4.2. If c > 0, then L_{D_{λ}} is very ample for all D_{λ} ∈ D.

Proof. By [C-F, Thm. 3.1] it is sufficient to show that for any subdivisor A of D_{λ} we have L.A ≥ A^{2} + 3.

If A^{2} ≥ 0, then we have L.A ≥ A^{2} + c + 2 ≥ A^{2} + 3 (which actually holds for any divisor A on S). If A^{2} ≤ −2, then L.A ≥ 1 ≥ A^{2} + 3, unless L.A = 0, which proves the lemma.

In the rest of this section we focus on the singular locus of the rational normal scroll T = T (c, D, {D_{λ}}) constructed as in the previous section.

It is well-known that the singular locus of a rational normal scroll of type (e_{1}, …, e_{d}) has dimension r − 1, where

r := #{e_{i} | e_{i} = 0}. (9)

From equation (1) we have

r = d_{0} − d_{1}.

By property (C2), when L ≁ 2D,we have for R := L − 2D by Riemann-Roch

h^{0}(R) = \frac{1}{2}R^{2} + 2 + h^{1}(R).

Note that we have h^{0}(R) > 0 if L^{2} ≥ 4c + 6, and h^{0}(R) = 0 if and only if L^{2} = 4c + 4 and h^{1}(R) = 0.

If L ∼ 2D, we have D^{2} = c + 2 and h^{0}(L − 2D) = h^{0}(O_{S}) = 1. Combining this with the lemma above, we get the following expression for r:

(10) r = \{ \begin{matrix}D^{2} + h^{1}(L − 2D) & if L ≁ 2D (equiv. D^{2} ≠ c + 2), \\ D^{2} − 1 & if L ∼ 2D (equiv. D^{2} = c + 2) \end{matrix}

The next results will show that the term D^{2} (or D^{2} − 1) can be interpreted geometrically as follows: The pencil {D_{λ}} has n = D^{2} distinct base points, denote their images by ϕ_{L} by x_{1}, …, x_{n}. The linear spaces \overline{D_{λ}} that sweap out the scroll T will intersect in the linear space spanned by these points, which we denote by < x_{1}, …, x_{n} >. This is a P^{n−1} when L ≁ 2D and a P^{n−2} when L ∼ 2D.

Define the set

(11) R_{L,D} := {Γ | Γ is a smooth rational curve, Γ.L = 0 and Γ.D > 0}.

The members of ϕ_{L}({D_{λ}}) will intersect in the points {ϕ_{L}(Γ)}_{Γ∈R_{L,D}} in addition to the images of the D^{2} base points of {D_{λ}}. If these extra points pose new independent conditions, they will contribute to the singular locus of T . We will show below that among all free Clifford divisors, we can choose one such that the term h^{1}(L − 2D) will correspond exactly to the singularities of the scroll arising from the contractions of the curves in R_{L,D}.

The contraction of smooth rational curves Γ which are not in R_{L,D}, will occur in some fiber. Indeed, since D.Γ = 0 one calculates h^{0}(D − Γ) = h^{0}(D) − 1, whence Γ will be a component of a unique reducible member of {D_{λ}}. Clearly, such contractions which occur in some fiber, and not transversally to the fibers, will not influence the singularities of T .

The proofs of the next three propositions are rather long and tedious, and will therefore be postponed until the next section.

Proposition 4.3. Let D be a free Clifford divisor for L and Γ a curve in R_{L,D}.

Then D.Γ = 1, F.Γ = −1 and Γ is contained in the base locus Δ of F. As a consequence, Δ.D = #R_{L,D}, where the elements are counted with the multiplicity they have in Δ.

Furthermore, if γ is any reduced and connected effective divisor such that γ.L = 0 and γ.D > 0, then D.γ = 1.

In particular, the curves in R_{L,D} are disjoint.

We defined the cases (E0)-(E2) in the previous section. We also need to define the following two cases for c = 0: (E3) L ∼ 3D + 2Γ_{0} + Γ_{1}, Γ_{0} and Γ_{1} are smooth rational curves, c = D^{2} = 0, L^{2} = 6,

D.Γ_{0} = 1, D.Γ_{1} = 0, Γ_{0}.Γ_{1} = 1. (E4) L ∼ 4D + 2Γ, Γ is a smooth rational curve, c = D^{2} = 0, L^{2} = 8, D.Γ = 1.

Note that in all cases (E0)-(E4) we have h^{1}(L − 2D) = Δ.D − 1. More precisely we have: (E0) Δ = Γ, Δ.D = 1, h^{1}(L − 2D) = 0, (E1) Δ = Γ_{1} + Γ_{2}, Δ.D = 2, h^{1}(L − 2D) = 1, (E2) Δ = 2Γ_{0} + 2Γ_{1} + ⋯ + 2Γ_{N} + Γ_{N+1} + Γ_{N+2}, Δ.D = 2, h^{1}(L − 2D) = 1, (E3) Δ = 2Γ_{0} + Γ_{1}, Δ.D = 2, h^{1}(L − 2D) = 1, (E4) Δ = 2Γ, Δ.D = 2, h^{1}(L − 2D) = 1.

Remark 4.4. If D is any free Clifford divisor not of type (E0)-(E4), then it will follow from equation (28) in Section 5 that h^{1}(L − 2D) ≥ Δ.D.

Proposition 4.5. Among all free Clifford divisors for L there is one, call it D, with the following property (denoting by Δ the base locus of F := L − D):

If D is not of type (E0)-(E4), then

h^{1}(L − 2D) = Δ.D.

We will also need the following bound on Δ.D:

Proposition 4.6. We have for D a free Clifford divisor

D.Δ ≤ \frac{1}{2}c + 1 − D^{2},

except for the case L^{2} ≤ 4c + 6 and Δ = 0, the cases (E0)-(E4) above, and the case

L^{2} = 4c + 4, D.Δ = 1, Δ^{2} = −2. (12)

In this latter case, D^{2} < c.

We now study the singular locus V of the scroll T . By equation (10) we know its dimension r − 1, and in the following results we will see which points in ϕ_{L}(S) that span V and how ϕ_{L}(S) intersects V. We will divide the treatment into the two cases c = 0 and c > 0. We recall from Proposition 4.1 that these two cases are naturally different.

We will now treat the case c > 0. Since we choose the base points of the pencil {D_{λ}} to be distinct and to lie outside of the finitely many curves in R_{L,D}, the images by ϕ_{L} of these points will be n = D^{2} distinct points in ϕ_{L}(S), denote them by x_{1}, …, x_{n}, and their preimages by p_{1}, …, p_{n}. Let m = D.Δ and let

R_{L,D} = {Γ_{1}, …, Γ_{r}}, (13)

and define

(14) m_{i} := multiplicity of Γ_{i} in Δ.

Then m = ∑_{i=1}^{r} m_{i}. Denote by y_{1}, …, y_{r} the images (distinct from x_{1}, …, x_{n}) of the contractions of the curves in R_{L,D}, and by q_{1,λ}, …, q_{r,λ} their corresponding preimages in each fiber. So q_{i,λ} = Γ_{i} ∩ D_{λ}.

In the cases (E0)-(E2) of Proposition 4.5, we use the following notation: (E0) y = ϕ_{L}(Γ), (E1) y_{1} = ϕ_{L}(Γ_{1}), y_{2} = ϕ_{L}(Γ_{2}), (E2) y_{0} = ϕ_{L}(Γ_{0}).

We will denote by q_{λ}, q_{1,λ}, q_{2,λ} and q_{0,λ} their respective preimages in the fiber D_{λ}.

Also, denote the special case L ∼ 2D by (Q).

For each D_{λ} ∈ D, we can identify D_{λ} with its image D′_{λ} := ϕ_{L}(D_{λ}) on S′ by Lemma 4.2. Moreover, we clearly have that the multiplicities of the points p_{1}, …, p_{n}, q_{1,λ}, …, q_{r,λ} on each D_{λ} is one, hence these points are all smooth points of D_{λ}, and consequently all x_{1}, …, x_{n}, y_{1}, …, y_{r} are smooth points of D′_{λ}.

For any D_{λ} ∈ D, we define Z_{λ} to be the zero-dimensional subscheme of length n + m of D_{λ} defined by

Z_{λ} := p_{1} + ⋯ + p_{n} + m_{1}q_{1,λ} + ⋯ + m_{r}q_{r,λ}. (15)

In particular

O_{D_{λ}}(Z_{λ}) ≃ O_{D_{λ}}(D + ∑_{i=1}^{r} m_{i}Γ_{i}). (16)

This zero-dimensional scheme can, by the isomorphism between D_{λ} and D′_{λ}, be identified with the following zero-dimensional subscheme of D′_{λ}, which we by abuse of notation denote by the same name:

Z_{λ} = x_{1} + ⋯ + x_{n} + m_{1}y_{1,λ} + ⋯ + m_{r}y_{r,λ}.

Note that in the case (Q) all the Z_{λ} are equal to p_{1} + ⋯ + p_{n} and will be denoted by Z.

In the special cases (Q), (E0)-(E2) we will also define the following zero-dimensional subschemes of Z_{λ} (which we again will identify to their corresponding subschemes of D′_{λ}): (Q) Z^{i} := p_{1} + ⋯ + p̂_{i} + ⋯ + p_{n}, (E0) Z_{0,λ} := p_{1} + ⋯ + p_{n}, (E1) Z_{i,λ} := p_{1} + ⋯ + p_{n} + q_{i,λ}, i = 1, 2, (E2) Z_{0,λ} := p_{1} + ⋯ + p_{n} + q_{0,λ}.

By < Z > we will mean the linear span of the zero-dimensional scheme Z on S′.

The following is the main result of this section:

Theorem 4.7. Assume c > 0. Among all free Clifford divisors for L there is one, call it D, satisfying the property in Proposition 4.5 and with the following three additional properties:

(a) If D is not of type (Q), (E0), (E1) or (E2), then for all D_{λ} ∈ D we have

V =< Z_{λ} >≃ P^{n+m−1},

and if D is of one of the particular types above, then:

(Q) V =< Z^{i} >=< Z >≃ P^{n−2}, (E0) V =< Z_{0,λ} >=< Z_{λ} >≃ P^{n−1}, (E1) V =< Z_{1,λ} >=< Z_{2,λ} >=< Z_{λ} >≃ P^{n}, (E2) V =< Z_{0,λ} >=< Z_{λ} >≃ P^{n}.

(b) V does not intersect S′ (set-theoretically) outside the points in the support of Z_{λ}.

(c) For any irreducible D_{λ}, we have

V ∩ D_{λ} = Z_{λ}.

In the theorem above, the following convention is used: P^{−1} = ∅ (which happens if and only if n = m = 0 and implies that the scroll is smooth).

Remark 4.8. If D is any free Clifford divisor, we have V ⊇< Z_{λ} >≃ P^{n+m−1}, except in the cases (Q), (E0)-(E2), where the property (a) is automatically fulfilled.

If D is not of type (E1) or (E2), the properties (b) and (c) automatically hold. If D is of type (E1) or (E2), then it might be that V intersects S′ outside of the support of < Z_{λ} >.

The proof of Theorem 4.7 will be divided in the general case and in the special cases (Q), (E0)-(E2). We will only prove the two first properties. The last one will be left to the reader.

In this section, we give the proofs for the general case and the case (Q). The proofs for the other cases are postponed until the next section.

We will write λ ∈ D for a λ such that D_{λ} ∈ D.

Proof of Theorem 4.7 in the general case. Let s := n+m. To prove that < Z_{λ} >≃ P^{n+m−1} it suffices to prove that the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{Z_{λ}})

is surjective for all λ ∈ D.

So assume this map is not surjective for some λ. Then there exists a subscheme Z′ ⊆ Z_{λ} of length s′ ≤ s, for some integer s′ ≥ 2 (since L is base point free), such that the map H^{0}(L) → H^{0}(L ⊗ O_{Z′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{Z′′}) is surjective for all proper subschemes Z′′ ⫋ Z′. We now use Proposition 4.6.

If Δ = 0 and L^{2} ≤ 4c + 6, we have n = D^{2} ≤ c and m = 0, so

L^{2} ≥ 4(c + 1) ≥ 4(n + 1) = 4(s + 1).

If we are in the case given by (12), we have

L^{2} ≥ 4(c + 1) ≥ 4(n + 2) = 4(s + 1).

In all other cases, we have s = m + n ≤ ⌊\frac{1}{2}c⌋ + 1 by Proposition 4.6, so

L^{2} ≥ 4(c + 1) ≥ 4(⌊\frac{1}{2}c⌋ + 2) ≥ 4(s + 1).

Therefore, by [Kn1], there exists an effective divisor B passing through Z′ and satisfying B^{2} ≥ −2, h^{1}(B) = 0 and the numerical conditions

2B^{2} < B.L ≤ B^{2} + s′ < 2s′.

If B^{2} ≥ 0, then B would induce a Clifford index c_{B} ≤ s′ − 2 ≤ n + m − 2 on the smooth curves in |L|. If Δ = 0 and L^{2} ≤ 4c + 6, we get the contradiction c_{B} ≤ n − 2 ≤ c − 2. If we are in the case given by (12), we get the contradiction c_{B} ≤ n − 1 ≤ c − 2. Finally, in all other cases, we have c_{B} ≤ n + m − 2 < ⌊\frac{1}{2}c⌋, again a contradiction.

Hence B^{2} = −2 and B is supported on a union of smooth rational curves. Furthermore, B.L ≤ s′ − 2 and B.D ≥ s′ (the last inequality follows since D_{λ} passes through Z_{λ}).

We now consider the effective decomposition

L ∼ (D + B) + (F − B).

Firstly note that L.(D + B) ≤ n + s′ + c and (D + B)^{2} ≥ n + 2s′ − 2, whence (F − B)^{2} = (L − D − B)^{2} ≥ 2c − n + 2 ≥ c + 2 > 0, so that h^{0}(F − B) ≥ 2.

Secondly, L.(D + B) − (D + B)^{2} − 2 ≤ c − s′ < c, a contradiction.

For the second statement, it suffices to show that there is no point x_{0} ∈ S′ − {x_{1}, …, x_{n}, y_{1}, …, y_{r}} such that S′ has an (s + 1)-secant (s − 1)-plane through Z_{λ} and x_{0} for all λ.

Assume, to get a contradiction, that there is such a point x_{0}. Choose any preimage p_{0} of x_{0}, and denote by X_{λ} the zero-dimensional scheme defined as the union of Z_{λ} and p_{0}. Fix any λ such that D_{λ} is irreducible.

In these terms we have that the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{X_{λ}})

is not surjective.

Then there exists a subscheme X′ ⊆ X_{λ} of length s′ + 1 ≤ s + 1, for some integer s′ ≥ 1, such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′′}) is surjective for all proper subschemes X′′ ⫋ X′.

Since L^{2} ≥ 4(s + 1) by the above, there exists by [Kn1] again an effective divisor B passing through X′ and satisfying B^{2} ≥ −2, h^{1}(B) = 0 and the numerical conditions

2B^{2} ≤ B.L ≤ B^{2} + s′ + 1 ≤ 2s′ + 2.

As above, if B^{2} ≥ 0, we would get a contradiction on the Clifford index c. Hence B^{2} = −2 and B is supported on a union of smooth rational curves. Furthermore, B.L ≤ s′ − 1 and B.D ≥ s′ (the last inequality follows since D_{λ} is irreducible).

As above, the effective decomposition

L ∼ (D + B) + (F − B)

induces a Clifford index < c on the smooth curves in |L|, unless s′ = 1, B.L = 0 and B.D = 1. This means that p_{0} lies in some divisor which is contracted to one of the points y_{1}, …, y_{r}. Hence x_{0} is one of these points, a contradiction.

Proof of Theorem 4.7 in the case L ∼ 2D. It suffices to prove that if there is a point x_{0} ∈ S′ − {x_{1}, …, x̂_{i}, …, x_{n}} for some i, such that S′ has an n-secant (n − 2)-plane through x_{0} and Z^{i}, then x_{0} = x_{i}.

Choose any preimage p_{0} of x_{0}, and denote by X_{i} the zero-dimensional scheme defined by p_{0} and Z^{i}. We will show that if the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{X_{i}})

is not surjective, then p_{0} = p_{i}.

Let X′ ⊆ X_{i} be a subscheme of length n′ ≤ n, for some integer n′ ≥ 2, such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′′}) is surjective for all proper subschemes X′′ ⫋ X′.

By assumption, we have n = D^{2} = c + 2 and L^{2} = 4c + 8 = 4n. Hence, by [Kn1], there exists an effective divisor B passing through X′ and satisfying B^{2} ≥ −2 and the numerical conditions

2B^{2} ≤^{(a)} L.B ≤ B^{2} + n′ ≤^{(b)} 2n′,

with equality in (a) or (b) implying L ∼ 2B.

Since B passes through X′, we have B.D ≥ n′ − 1, whence B.L ≥ 2n′ − 2. From the inequalities above, we get B^{2} ≥ n′ − 1 ≥ 0, so we have n′ = n and B.L = B^{2} + n, since otherwise B would induce a Clifford index < n − 2 = c on the smooth members of |L|. This leaves us with the two possibilities:

(i) B^{2} = n and L ∼ 2B, or (ii) B^{2} = n − 1.

In the second case, one easily finds L ∼ 2B + Γ, for Γ a smooth rational curve, which is impossible, since L ∼ 2D. So we are in case (i), and B ∈ |D|.

Now recall by [Kn1] how the divisor B is obtained: It sits in an exact sequence

0 ⟶ O_{S}(B) ⟶ E ⟶ O_{S}(B) ⟶ 0,

where E is a rank two bundle given as an extension

0 ⟶ O_{S} ⟶ E ⟶ L ⊗ J_{X′} ⟶ 0.

Tensoring both sequences with O_{S}(−B) and taking cohomology, one gets

h^{0}(B ⊗ J_{X′}) = h^{0}(E − B) = 2,

so there is a pencil P of divisors in |D| passing through X′.

We claim that any divisor D_{0} ∈ |D| passing through n − 1 of the points p_{1}, …, p_{n}, will also pass through the last one. Indeed, by the surjectivity of the map H^{0}(D) → H^{0}(O_{D_{0}}(D)), we reduce to the same statement for O_{D_{0}}(D). By Riemann-Roch, this is equivalent to h^{0}(O_{D_{0}}(Z)) ≥ 2 and O_{D_{0}}(Z) base point free, which are both satisfied since O_{D_{0}}(Z) ≃ O_{D_{0}}(D), and O_{S}(D) is base point free.

Since Z contains the points p_{0}, …, p̂_{i}, …, p_{n}, we have that all the members in P contain all the points p_{0}, …, p_{n}. Therefore, P is the pencil {D_{λ}}, whose general member is smooth and irreducible. Since all the members intersect in n points, we have p_{0} = p_{i}, as asserted.

It will be convenient to make the following definition:

Definition 4.9. A free Clifford divisor satisfying the properties described in Proposition 4.5 and Theorem 4.7 will be called a perfect Clifford divisor.

In the next section we will prove Proposition 4.5 and Theorem 4.7, thus proving that we can find a perfect Clifford divisor.

The main advantage of choosing a perfect Clifford divisor is that we then get a nice description of the singular locus of T and how it intersects S′ as in Theorem 4.7 above. This theorem states that Sing T is “spanned” by the images of the base points of the chosen subpencil of |D| and the contracted curves, and moreover that it intersects S′ in only these points. If D is not perfect, then Sing T ⫌ Z_{λ}, as seen in Remarks 4.4 and 4.8. In Proposition 7.31 below we will see an example where this occurs.

It will also be practical, for classification purposes, to restrict the attention to perfect Clifford divisors, as we will do in Section 10.

Apart from this, any free Clifford divisor will be equally fit for our purposes. We include an additional description of the case (Q):

Proposition 4.10. Assume D is a free Clifford divisor of type (Q) and c ≥ 2. Then ϕ_{L}(S) is the 2-uple embedding of ϕ_{D}(S), unless c = 2 and there exists a smooth elliptic curve E such that E.D = 2 (in which case D is hyperelliptic).

Proof. By [SD, Thm. 6.1] ϕ_{L} is the 2-uple embedding of ϕ_{D}(S), when D is not hyperelliptic.

Conversely, if D is hyperelliptic, then ϕ_{D} is not birational, so ϕ_{L} cannot be the 2-uple embedding of ϕ_{D}(S).

If D is hyperelliptic, then there either exists a smooth curve E satisfying E^{2} = 0 and E.D = 2, or a smooth curve B such that B^{2} = 2 and D ∼ 2B. In the first case, we compute

E.L − E^{2} − 2 = 2E.D − 2 = 2,

whence c = 2, and in the second we get the contradiction

D.L − D^{2} − 2 = 6 > 4 = B.L − B^{2} − 2.

The special case appearing in the proposition will be thouroughly described in Proposition 7.31 below.

If c = 0, there exist two kinds of (free) Clifford divisors for L, namely:

1. D^{2} = 0, D.L = 2 and

2. D^{2} = 2, L ∼ 2D. In both these cases ϕ_{L}(S) is 2 : 1 on each fiber. In the case c = 0 we have the following result:

Proposition 4.11. Assume c = 0. Let D be a free Clifford divisor for L. Then D^{2} = 0 and V = ∅ except for the following cases: (Q) L ∼ 2D, D^{2} = 2, V = {x}, where x is the common image of the two base points of

the chosen pencil {D_{λ}}, (E1) D^{2} = 0, L ∼ 2D + Γ_{1} + Γ_{2}, V = {ϕ_{L}(Γ_{1})} = {ϕ_{L}(Γ_{2})}, (E2) D^{2} = 0, L ∼ 2D + 2Γ_{0} + 2Γ_{1} + ⋯ + 2Γ_{N} + Γ_{N+1} + Γ_{N+2}, V = {ϕ_{L}(Γ_{0})}, (E3) D^{2} = 0, L ∼ 3D + 2Γ_{0} + Γ_{1}, V = {ϕ_{L}(Γ_{0})}, (E4) D^{2} = 0, L ∼ 4D + 2Γ, V = {ϕ_{L}(Γ)}.

Proof. For D^{2} = 0, this follows from the fact that except for the cases (E1)-(E4), the base locus Δ of L − D is zero, which is shown in the proof of [SD, Prop. 5.7]. In the other case, it follows from the equation (9).

All these cases have been completely described in [SD, Prop. 5.6 and 5.7]. When V = ∅, then ϕ_{L}(S) is a rational ruled surface. The cases where there are contractions across the fibers, are the cases (E1)-(E4). In these

cases ϕ_{L}(S) is a cone.

In the case (Q), ϕ_{L}(S) is the Veronese surface in P^{5}.

5. Postponed proofs

In this section we will give the proofs omitted in the previous section. Throughout this section L will be a spanned and big line bundle of non-general Clifford

index c. In particular, this implies L^{2} ≥ 4c + 4.

Also we write F := L − D and R := L − 2D = F − D, and denote the (possibly zero) base divisor of |F | by Δ. Recall that L.Δ = 0 and that we have h^{0}(R) = 0 if and only if L^{2} = 4c + 4 and h^{1}(R) = 0. In particular, h^{1}(R) > 0 implies that R > 0.

Furthermore, we have

Lemma 5.1. If h^{0}(R) = 0, then Δ = 0.

Proof. We have L^{2} = 4c + 4, so we cannot be in the cases (Q) or (E1), whence D^{2} ≤ c. Choose any smooth curve D_{0} ∈ |D| and let F_{D_{0}} := F ⊗ O_{D_{0}}. Then deg F_{D_{0}} = c + 2 ≥ D^{2} + 2 = 2g(D_{0}), whence F_{D_{0}} is base point free.

We first will show that this implies that F is nef.

Taking cohomology of the short exact sequence

0 ⟶ R ⟶ F ⟶ F_{D_{0}} ⟶ 0,

and of the same sequence tensored with −Δ, we get the following two exact sequences (using h^{0}(R) = h^{0}(R − Δ) = h^{1}(R) = 0)

[figure: 0; H^{0}(F); H^{0}(F_{D_{0}}); 0; 0; H^{0}(F − Δ); H^{0}((F − Δ)_{D_{0}}).]

This gives h^{0}((F − Δ)_{D_{0}}) ≥ h^{0}(F_{D_{0}}), whence Δ.D = 0, since F_{D_{0}} is base point free. This means that for any smooth rational curve Γ in the support of Δ, we have Γ.D = Γ.F = 0. Hence F is nef.

By Lemma 1.3 it now suffices to show that F is not of the type F ∼ kE + Γ, for E a smooth elliptic curve and Γ a smooth rational curve satisfying E.Γ = 1 and an integer k ≥ 2. But if this were the case, we would have E.L = 2 + c/k. If c ≠ 0, this would mean that E induces a lower Clifford index than c on the smooth curves in |L|, a contradiction. If c = 0, we get D.F = 2 and D^{2} = 0. But this would give R^{2} = (F − D)^{2} ≥ −2 and by Riemann-Roch, we would then get the contradiction h^{0}(F − D) ≥ 1.

By this lemma, if h^{0}(R) = 0, the Propositions 4.3, 4.5 and 4.6 will automatically be satisfied. So for the rest of this section, we will assume R > 0.

Let F_{0} be the moving component of |F |. Since R > 0, we can write F_{0} ∼ D + A for some divisor A ≥ 0. Thus we have

F ∼ D + R ∼ F_{0} + Δ ∼ D + A + Δ, (17)

and

L ∼ 2D + A + Δ. (18)

We will first study the divisors above more closely.

Lemma 5.2. Except for the cases (E3) and (E4), the general member of |F_{0}| is smooth and irreducible.

Proof. Since |F_{0}| is base point free, by Proposition 1.2 we only need to show that F_{0} ≁ kE, for E a smooth elliptic curve and an integer k ≥ 2.

Assume, to get a contradiction, that F_{0} ∼ kE, then by (17), we have D ∼ E and A ∼ (k − 1)E. Let d := c + 2 = E.L ≥ 2.

Since L ∼ (k + 1)E + Δ, we get L^{2} = d(k + 1), so h^{0}(L) = d(k + 1)/2 + 2, and h^{0}(F) = h^{0}(kE) = k + 1. On the other hand, by equation (7), we have h^{0}(F) = h^{0}(L) − d.

Combining the last three equations, we get

k + 1 = d(k − 1)/2 + 2,

which is only possible if d = 2, i.e. c = 0. A case by case study as in the proof of [SD, Prop. 5.7] establishes the lemma in this latter case.

We gather some basic properties of R.

Lemma 5.3. (a) If R = R_{1} + R_{2} is an effective decomposition, then R_{1}.R_{2} ≥ 0

(b) If γ is an effective divisor satisfying γ^{2} = −2 and γ.R < 0, then γ.R = −1 or −2.

(c) If γ is an effective divisor satisfying γ^{2} = −2 and γ.L = 0, then either γ.D = γ.F = γ.R = 0 or γ.D = 1, γ.F = −1 and γ.R = −2.

(d) If Γ is a smooth rational curve, then Γ ∈ R_{L,D} if and only if Γ.R = −2 and Γ.L = 0.

Proof. To prove (a), one immediately sees that if R_{1}.R_{2} < 0, then the effective decomposition L ∼ (D + R_{1}) + (D + R_{2}) would induce a Clifford index < c.

The other assertions are immediate consequences of (a).

This concludes the proof of Proposition 4.3.

Lemma 5.4. Except for the cases (E3) and (E4), the following holds:

Δ^{2} = −2D.Δ and Δ.A = 0.

Proof. By Lemma 5.2, we have h^{1}(F_{0}) = 0. From 0 = Δ.L = 2Δ.D + Δ.A + Δ^{2}, we get

Δ^{2} = −2Δ.D − Δ.A. (19)

Furthermore, we also have

h^{0}(F_{0}) = h^{0}(F) = \frac{1}{2}F_{0}^{2} + F_{0}.Δ + \frac{1}{2}Δ^{2} + 2 = h^{0}(F_{0}) + (D + A).Δ + \frac{1}{2}Δ^{2},

which implies

Δ^{2} = −2Δ.D − 2Δ.A. (20)

Combining equations (19) and (20), we get Δ.A = 0 and Δ^{2} = −2D.Δ.

We have seen in Proposition 3.6, that if there exists a free Clifford divisor of type (E0), then all free Clifford divisors are linearly equivalent and of type (E0).

We now take a closer look at the types (E1) and (E2).

Proposition 5.5. Let L be a spanned and big line bundle of non-general Clifford index c on a K3 surface and let D be a free Clifford divisor of type (E1) or (E2).

If D′ ≁ D is any other free Clifford divisor, then B := D − D′ > 0 and

Δ.D′ = 0, Δ.B = 2, B^{2} = −2. (21)

Proof. Let R′ := L − 2D′ as usual, and note that R′ ∼ 2B + Δ.

Since R′^{2} = L^{2} − 4(c + 2) = Δ^{2} = −4, we get B^{2} + B.Δ = 0. Combined with Δ.D = Δ.(D′ + B) = 2, we get the two possibilities

(a) Δ.D′ ≥ 2, Δ.B ≤ 0, B^{2} ≥ 0,

(b) Δ.D′ = 0, Δ.B = 2, B^{2} = −2.

Using D′^{2} ≤ c, we calculate

B.L = \frac{1}{2}(R′ − Δ).L = \frac{1}{2}R′.L = \frac{1}{2}(L − 2D′).L
= \frac{1}{2}(L^{2} − 2(D′^{2} + c + 2)) ≥ 2c + 2 − c − c − 2 = 0.

In case (a) we then must have B.L > 0 by the Hodge index theorem, so B ≥ 0 by Riemann-Roch. We get

B.R′ = B.(2B + Δ) = B^{2} ≥ 0 and R′.D′ = 2B.D′ + Δ.D′ ≥ 2B.D′ + 2,

which gives

L.B = 2D′.B + R′.B = 2D′.B + B^{2} ≤ R′.D′ + B^{2} − 2.

But this implies

B.L − B^{2} − 2 ≤ R′.D′ − 4 < D′^{2} + R′.D′ − 2 = c,

whence we must have B ∼ 0.

So we must be in case (b), and by Riemann-Roch we have either B > 0 or −B > 0. We see that B.L > 0 unless D′^{2} = D^{2} = c. But if the latter holds, since both D′ and D are assumed to be smooth Clifford divisors (so that h^{1}(D) = h^{1}(D′) = 0), we have h^{0}(D) = h^{0}(D′), whence D ∼ D′ and B ∼ 0, a contradiction. Hence D.L > 0, so B > 0 and we are done.

As seen below, we will distinguish between inclusions D′ < D as in Proposition 5.5 with Δ′ = 0 and Δ′ ≠ 0 (where Δ′ is the base divisor of |L − D′|).

By Propositions 3.5 and 3.6 it is clear that we can choose a free Clifford divisor D with the two additional properties (recall that Δ as usual denotes the base divisor of |L − D|): (C6) If D′ is any other free Clifford divisor such that D′ > D, then Δ ≠ 0 and D′ is of type

(E1) or (E2). (C7) If D is of type (E1) or (E2) above, and D′ is any other free Clifford divisor satisfying

(C6), then D′ ∼ D. Property (C6) is a maximality condition: it means that we choose a free Clifford divisor

which is not contained in any other free Clifford divisor, unless possibly when Δ ≠ 0 and it is contained in some free Clifford divisor of type (E1) or (E2).

Property (C7) means that if we can, we will choose among all free Clifford divisors satisfying (C6), one that is not of type (E1) or (E2).

It turns out that free Clifford divisors satisfying the additional properties (C6) and (C7) will be perfect, i.e. they will satisfy Propositions 4.5 and 4.7.

Now assume R = R_{1} + R_{2} is an effective decomposition such that R_{1}.R_{2} = 0. Then L ∼ (D + R_{1}) + (D + R_{2}) is an effective decomposition satisfying

(D + R_{1}).(D + R_{2}) = D^{2} + D.(R_{1} + R_{2}) = D.F = c + 2,

so this decomposition induces the same Clifford index c. This means that either D + R_{1} or D + R_{2} is a Clifford divisor. This enables us to prove the following:

Proposition 5.6. Assume D is not of type (E3) or (E4) and satisfies (C6) and (C7). Assume furthermore that there exists an effective decomposition R = R_{1} + R_{2} such that R_{1}.R_{2} = 0 and such that D + R_{1} is a Clifford divisor.

Then either Δ ≠ 0 and D + R_{1} is of type (E1) or (E2), or there exists a smooth rational curve Γ satisfying either (I) Γ.D = Γ.F = Γ.L = 0, Γ.R_{1} = −1, Γ.R_{2} = 1, or (II) Γ.D = 1, Γ.F = −1, Γ.L = 0, Γ.R_{1} = −2, Γ.R_{2} = 0.

Proof. Let D_{1} := D + R_{1} and D_{2} := D + R_{2}. Since D_{1} is a Clifford divisor containing D, we have by condition (C6) that either D_{1} is not a free Clifford divisor, or Δ ≠ 0 and D_{1} is of type (E1) or (E2).

So we can assume D_{1} is not a free Clifford divisor, which means that D_{1} is not base point free.

If D_{1} is nef, then by Lemma 1.3 it is of the form

D_{1} ∼ lE + Γ_{0},

for some smooth elliptic curve E and smooth rational curve Γ_{0} satisfying E.Γ_{0} = 1, and some integer l ≥ 2. This gives

R_{1} ∼ (l − 1)E + Γ_{0} and D ∼ E.

Write

D_{2} = D + R_{2} ∼ D + M + B,

where B ≥ 0 is the base divisor of D_{2} and M ≥ 0. Note that M + B ∼ R_{2}.

We have

0 = R_{1}.R_{2} = ((l − 1)D + Γ_{0}).(M + B) = (l − 1)D.M + (l − 1)D.B + Γ_{0}.M + Γ_{0}.B.

Also, we have an effective decomposition

R ∼ ((l − 1)D + M + B) + Γ_{0},

such that

((l − 1)D + M + B).Γ_{0} = l − 1 + Γ_{0}.M + Γ_{0}.B = (l − 1)(1 − D.M − D.B).

By [SD, Lemma 3.7], if M ≠ 0, either M ∼ kD, for some integer k ≥ 1, or D.M ≥ 2. In this latter case, the latter product would be negative, contradicting Lemma 5.3. So we must have M ∼ kD, for some integer k ≥ 0 and D.B = 0 or 1.

So R ∼ R_{1} + R_{2} ∼ (l − 1)D + Γ_{0} + M + B ∼ (k + l − 1)D + Γ_{0} + B and

c + 2 = D.F = D.(D + R) = ((k + l)D + Γ_{0} + B).D ≤ 2,

which gives c = 0 and B.D = 1. A short analysis as in part (b) of the proof of [SD, Lemma 5.7.2] shows that D is then of type (E3) or (E4).

So D_{1} is not nef, which means that there exists a smooth rational curve Γ such that Γ.D_{1} < 0, whence Γ is fixed in |D_{1}| and Γ.L = 0, by Proposition 2.4. Combining Γ.D_{1} = Γ.D + Γ.R_{1} ≤ −1 and 0 = Γ.L = 2Γ.D + Γ.R_{1} + Γ.R_{2}, we get

1 − Γ.R_{2} ≤ Γ.D ≤ −1 − Γ.R_{1}. (22)

Furthermore, by Lemma 5.3(b), we have

Γ.R = Γ.R_{1} + Γ.R_{2} ≥ −2. (23)

If R_{1} = Γ, we are done by Lemma 5.3(c), so we can assume that R_{1} − Γ > 0. Then by Lemma 5.3(a) we have (R_{1} − Γ).(R_{2} + Γ) = R_{1}.R_{2} + Γ.R_{1} − Γ.R_{2} + 2 ≥ 0, which implies

Γ.R_{1} − Γ.R_{2} ≥ −2. (24)

Combining (23) and (24), we get

−2 − Γ.R_{1} ≤ Γ.R_{2} ≤ 2 + Γ.R_{1} and Γ.R_{1} ≥ −2. (25)

Combining (25) with (22) and Lemma 5.3(c), we end up with the two possibilities given by (I) and (II) above.

We now need a basic lemma about A.

Lemma 5.7. If A = 0, then D is of one of the types (E0)-(E2).

If A^{2} ≤ −2, then one of the following holds:

(a) A^{2} = −4, Δ = 0, L^{2} = 4c + 4, (b) A^{2} = −2, Δ = 0, L^{2} = 4c + 6,

(c) A^{2} = −2, Δ^{2} = −2, D.Δ = 1, L^{2} = 4c + 4. Moreover, in case (c) we have D^{2} < c.

Proof. If A = 0, we must have −4 ≤ Δ^{2} = R^{2} ≤ −2, whence Δ^{2} = −4 or −2, D.Δ = 2 or 1 respectively (by Lemma 5.4), and L^{2} = 4c + 4 or 4c + 6 respectively. An analysis as in Proposition 3.5 now gives that D is as in one of the cases (E0)-(E2).

If A^{2} ≤ −2, we have by R^{2} = A^{2} + Δ^{2} = L^{2} − 4(c + 2) (where we have used Lemma 5.4) that either Δ = 0 and we are in case (a) or (b) above, or that A^{2} = −2, Δ^{2} = −2, D.Δ = 1 (by Lemma 5.4) and L^{2} = 4c + 4, i.e. case (c).

In this latter case, we have

c + 2 = D.F = D^{2} + D.A + D.Δ = D^{2} + D.A + 1,

whence D^{2} = c + 1 − D.A. Since D + A ∼ F_{0} is base point free, we have D.A ≥ 2 by [SD, (3.9.6)], whence D^{2} < c.

We can now prove Proposition 4.6.

Indeed, when we are not in the exceptional cases of the proposition (which are the cases (E0)-(E4) and the cases (a)-(c) of the last lemma), we have A ≠ 0 and A^{2} ≥ 0.

We calculate

c + 2 = F.D = (D + A + Δ).D = D^{2} + A.D + Δ.D (26)

and

A.(2D + Δ) = 2A.D (27)

(by Lemma 5.4). Since D is a Clifford divisor, we must have A.D ≥ D^{2} + Δ.D.

We also have from (26) that A.D = c + 2 − D^{2} − Δ.D. Combining this with the last inequality, we get

c/2 + 1 − D^{2} ≥ Δ.D.

This concludes the proof of Proposition 4.6.

Before proving the next result, we will need the following easy lemma.

Lemma 5.8. Assume D is not as in (E3) or (E4). If A ∼ A_{1} + A_{2} is an effective decomposition such that A_{1}.A_{2} ≤ 0, then

A_{1}.A_{2} = A_{1}.Δ = A_{2}.Δ = 0,

and either D + A_{1} or D + A_{2} is a Clifford divisor.

Proof. By Lemma 5.4 we have Δ.A = 0, so we can assume (possibly after interchanging A_{1} and A_{2}) that A_{1}.Δ ≤ 0 and A_{2}.Δ ≥ 0. Then R ∼ A_{1} + (A_{2} + Δ) is an effective decomposition of R such that

A_{1}.(A_{2} + Δ) = A_{1}.A_{2} + A_{1}.Δ ≤ 0.

By Lemma 5.3(a) we must have equality, whence A_{1}.A_{2} = A_{1}.Δ = A_{2}.Δ = 0.

If A_{1}.L > A_{2}.L (resp. A_{2}.L > A_{1}.L), then clearly D + A_{1} (resp. D + A_{2}) is a Clifford divisor by condition (C2).

If A_{1}.L = A_{2}.L, then D + A_{i} is not a Clifford divisor if and only if h^{0}((D + A_{i}) − (D + A_{3−i} + Δ)) = h^{0}(A_{i} − A_{3−i} − Δ) > 0. Clearly this condition cannot hold for both i = 1 and 2. So we are done.

The next result is the crucial one to prove Proposition 4.5.

Proposition 5.9. If D satisfies (C6) and (C7), then H^{1}(A) = 0 except for the case (E4).

Proof. The result is trivial if A = 0. So we will assume A > 0. Also, the result is fulfilled in the case (E3), so we can assume D is not as in (E3) or (E4). In particular, we can use the Lemmas 5.4 and 5.8.

If h^{1}(−A) = h^{1}(A) > 0, then A cannot be numerically 1-connected, whence there exists an effective decomposition A ∼ A_{1} + A_{2} such that A_{1}.A_{2} ≤ 0. By the previous lemma, we have A_{1}.A_{2} = A_{1}.Δ = A_{2}.Δ = 0, and (possibly after interchanging A_{1} and A_{2}) we can assume that D + A_{1} is a Clifford divisor.

Assume first that D and D +A_{1} are as in the special case where Δ ≠ 0 and D′ := D +A_{1} is a free Clifford divisor of type (E1) or (E2), so the base divisor Δ′ of

F′ := L − D′ ∼ D + A_{2} + Δ ∼ D + A_{1} + Δ′

satisfies Δ′^{2} = −4. Furthermore, by Proposition 5.5, Δ′.D = 0, A_{1}^{2} = −2 and A_{1}.Δ′ = 2. Also, since Δ′.L = 0 and D′L = F′.L, we must have A_{1}.L = A_{2}.L. Also note that A_{1} ≁ A_{2}, since A_{1}.A_{2} = 0.

Since (A_{1}−A_{2}).L = 0, we must by the Hodge index theorem have A^{2} = (A_{1}−A_{2})^{2} ≤ −2. By Lemma 5.7 and the fact that Δ ≠ 0, this gives us

A_{1}^{2} = −2, A_{2}^{2} = 0, Δ^{2} = −2,

We then get from L^{2} = 4c + 4 = (2D + A + Δ).L = 2D.L + A.L = 2D^{2} + 2c + 4 + A.L, that

A.L = 2(c − D^{2}).

Since A_{1}.L = A_{2}.L, we have

A_{1}.L = A_{2}.L = c − D^{2}.

So A_{2} would induce a Clifford index ≤ L.A_{2} − A_{2}^{2} − 2 = c − D^{2} − 2 < c on the smooth curves in |L|, a contradiction.

So we can now use Proposition 5.6 and find a smooth rational curve satisfying one of the two conditions: (I) Γ.D = 0, Γ.A_{1} = −1, Γ.(A_{2} + Δ) = 1, (II) Γ.D = 1, Γ.A_{1} = −2, Γ.(A_{2} + Δ) = 0. In case (I) we get Γ.A = Γ.F_{0} − Γ.D ≥ 0, whence Γ.A_{2} ≥ 1. Since Γ.A_{1} = −1, we have

A_{1} − Γ > 0, and we get an effective decomposition A ∼ (A_{1} − Γ) + (A_{2} + Γ) such that

(A_{1} − Γ).(A_{2} + Γ) = A_{1}.A_{2} − Γ.A_{2} + Γ.A_{1} − Γ^{2} ≤ 0,

so by Lemma 5.8, we must have Γ.A_{2} = 1 and (A_{1} −Γ).(A_{2} +Γ) = 0. Obviously, D +A_{1} −Γ is a Clifford divisor, and we can now repeat the process with A_{1} and A_{2} replaced by A_{1} − Γ and A_{2} + Γ. This will eventually bring us in case (II) after a finite number of steps.

So we can assume that A_{1} and A_{2} are as in case (II). Again, by Γ.A = Γ.F_{0} − Γ.D ≥ −1, we have Γ.A_{2} ≥ 1, whence Γ ≠ A_{1} and A_{1} − Γ > 0. Since

(A_{1} − Γ).(A_{2} + Γ) = A_{1}.A_{2} − Γ.A_{2} + Γ.A_{1} − Γ^{2} ≤ −1,

we have a contradiction by Lemma 5.8.

This concludes the proof of the proposition.

We can now prove Proposition 4.5.

In the cases (E3) and (E4) the proposition is clear, so by Lemma 5.4, we can assume A.Δ = 0 and Δ^{2} = −2D.Δ.

One easily sees that the base divisor of R must contain Δ, so h^{0}(A) = h^{0}(R) = h^{0}(A+Δ).

If A > 0, we have

h^{0}(A) = h^{0}(A + Δ) = \frac{1}{2}A^{2} + 2 + \frac{1}{2}Δ^{2} + h^{1}(R) = h^{0}(A) − h^{1}(A) + D.Δ + h^{1}(R), (28)

whence h^{1}(R) = D.Δ + h^{1}(A). If we choose D such that it satisfies (C6) and (C7), then h^{1}(A) = 0 by Proposition 5.9.

If A = 0, then R = Δ and D is of one of the types (E0)-(E2) by Lemma 5.7. This gives

1 = h^{0}(R) = \frac{1}{2}R^{2} + 2 + h^{1}(R) = −D.Δ + 2 + h^{1}(R), (29)

whence h^{1}(R) = D.Δ − 1.

This concludes the proof of Proposition 4.5.

We now give the proofs of Theorem 4.7 in the cases (E0)-(E2)

Proof of Theorem 4.7 in the case (E0). We first prove that < Z_{0,λ} >≃ P^{n−1} for all λ. If this were not true, the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{Z_{0,λ}})

would not be surjective for some λ.

As usual let Z′ ⊆ Z_{0,λ} be a subscheme of length n′ ≤ n, for some integer n′ ≥ 2, such that the map H^{0}(L) → H^{0}(L ⊗ O_{Z′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{Z′′}) is surjective for all proper subschemes Z′′ ⫋ Z′.

Since L^{2} = 4n + 2 = 4(n − 1) + 6, we get by [Kn1] that there exists an effective divisor B passing through Z′ such that B^{2} ≥ −2, h^{1}(B) = 0 and

2B^{2} < B.L ≤ B^{2} + n′ < 2n′.

If B^{2} ≥ 0, we would get that B induces a Clifford index c_{B} ≤ n′ − 2 ≤ c − 1 on the smooth curves in |L|, a contradiction.

So B^{2} = −2, and B is necessarily supported on a union of smooth rational curves, since h^{1}(B) = 0. But B.L ≤ n′ − 2 ≤ n − 2 = c − 1 and Z′ has length ≥ 2, so it necessarily contains some of the base points of {D_{λ}}. This means that B passes through some of these base points, which contadicts the fact that we have chosen these base points to lie outside of smooth rational curves of degree ≤ c + 2 with respect to L.

So < Z_{0,λ} >≃ P^{n−1}, and by equation (10) and Proposition 4.5 we know that V ≃ P^{n−1}, so the point y does not pose any additional conditions.

To prove the last assertion, assume to get a contradiction that there exists a point x_{0} ∈ S′ − {x_{1}, …, x_{n}, y} such that S′ has an (n + 1)-secant (n − 1)-plane through x_{0} and Z_{0,λ}. Choose any preimage p_{0} of x_{0} and denote by X_{λ} the zero-dimensional scheme defined by p_{0} and Z_{0,λ}. We then have that the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{X_{λ}})

is not surjective. Fix a λ.

Again let X′ ⊆ X_{λ} be a subscheme of length n′ + 1 ≤ n + 1, for some integer n′ ≥ 1, such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′′}) is surjective for all subschemes X′′ ⊆ X′.

Since L^{2} = 4n + 2 and L is not divisible by assumption, we get by [Kn1] again that there exists an effective divisor B passing through X′ satisfying B^{2} ≥ −2, h^{1}(B) = 0 and the numerical conditions

2B^{2} < B.L ≤ B^{2} + n′ + 1 < 2n′ + 2.

If B^{2} = −2 we argue as above, using B.L ≤ n′ − 1 ≤ n − 1 = c, and get a contradiction to our choice of the base points of {D_{λ}} to lie outside of smooth rational curves of degree ≤ c + 2.

So B^{2} ≥ 0, n = n′ and X′ = X_{λ}, so B is a Clifford divisor (the condition L − B ≥ B will be fullfilled by [Kn1]) and by Proposition 3.6, we have D ∼ B. By the proof of Proposition 7.5 in [Kn1], we then have Γ ∩ X_{λ} ≠ ∅, whence we conclude that p_{0} ∈ Γ. This gives the desired contradiction x_{0} = y.

Proof of Theorem 4.7 in the case (E1). We first show that < Z_{i,λ} >≃ P^{n} for i = 1, 2 and any λ. If this were not true, the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{Z_{i,λ}})

would not be surjective.

Let Z′ ⊆ Z_{i,λ} be a subscheme of length n′+1 ≤ n+1, for some integer n′ ≥ 1, such that the map H^{0}(L) → H^{0}(L⊗O_{Z′}) is not surjective, but such that the map H^{0}(L) → H^{0}(L⊗O_{Z′′}) is surjective for all proper subschemes Z′′ ⫋ Z′.

Since L^{2} = 4c+4 = 4(n+1), we have by [Kn1] again that there exists an effective divisor B passing through Z′ such that B^{2} ≥ −2, h^{1}(B) = 0 and

2B^{2} ≤ B.L ≤ B^{2} + n′ + 1 ≤ 2n′ + 2.

If B^{2} ≥ 0, we would get that B induces a Clifford index c_{B} ≤ n′ − 1 ≤ c − 1 on the smooth curves in |L|, a contradiction.

So B^{2} = −2, and B is necessarily supported on a union of smooth rational curves, since h^{1}(B) = 0. But B.L ≤ n′ − 1 ≤ n − 1 = c − 1 and Z′ has length ≥ 2, so B passes through some of the base points of {D_{λ}}. This contadicts the fact that we have chosen these base points to lie outside of smooth rational curves of degree ≤ c + 2 with respect to L.

To prove the second assertion, we will show that if there is a point x_{0} ∈ S′−{x_{1}, …, x_{n}, y_{1}} such that S′ has an (n+2)-secant n-plane through x_{0} and Z_{1,λ}, then x_{0} = y_{2}. By symmetry, this will suffice.

As usual choose any preimage p_{0} of x_{0} and denote by X_{1,λ} the zero-dimensional scheme defined by p_{0} and Z_{1,λ}. We then have that the natural map

H^{0}(L) ⟶ H^{0}(L ⊗ O_{X_{1,λ}})

is not surjective for any λ.

As usual let Let X′_{1,λ} ⊆ X_{1,λ} be a subscheme of length n′_{1,λ} + 2 ≤ n + 2, for some integer n′_{1,λ} ≥ 0, such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′_{1,λ}}) is not surjective, but such that the map H^{0}(L) → H^{0}(L ⊗ O_{X′′_{1,λ}}) is surjective for all proper subschemes X′′_{1,λ} ⫋ X′_{1,λ}.

Since n = D^{2} = c ≥ 2 and L^{2} = 4c + 4 = 4(n + 1), we again have by [Kn1] that there for each λ exists an effective divisor B_{1,λ} passing through X_{1,λ} and satisfying B_{1,λ}^{2} ≥ −2, h^{1}(B_{1,λ}) = 0 and the numerical conditions

2B_{1,λ}^{2} ≤^{(a)} L.B_{1,λ} ≤ B_{1,λ}^{2} + n′_{1,λ} + 2 ≤^{(b)} 2n′_{1,λ} + 4,

with equality in (a) or (b) implying L ∼ 2B_{1,λ}.

Assume first that B_{1,λ}^{2} = −2 for some λ. We then get the same contradiction on the choice of the base points of {D_{λ}}, since B_{1,λ}.L ≤ n′_{1,λ} ≤ n = c.

So we must have B_{1,λ}^{2} ≥ 0 for all λ. Then n′_{1,λ} = n, X′_{1,λ} = X_{1,λ}, L.B_{1,λ} = B_{1,λ}^{2} + n + 2, and B_{1,λ} is a Clifford divisor. The moving part B′_{1,λ} of |B_{1,λ}| is then a free Clifford divisor, so by condition (C7) we have that either B′_{1,λ} ∼ D or there exists a free Clifford divisor P_{1,λ} such that B′_{1,λ} ≤ P_{1,λ} < D with the last inclusion as described in Proposition 5.5, with the additional property that |L − P_{1,λ}| has no fixed divisor, by the conditions (C6) and (C7).

We will show that this latter case cannot occur.

We have that B_{1,λ} passes through X_{1,λ}. Now a (possible) base divisor in |B_{i,λ}| cannot pass through any of the points p_{0}, …, p_{n}, since these points lie outside all the rational curves contracted by L. So we must have B′_{i,λ}.D ≥ n.

In addition, by Proposition 5.5 we must have

D ∼ B′_{1,λ} + γ_{1,λ},

for some γ_{1,λ} > 0 satisfying γ_{1,λ}^{2} = −2. Moreover, B′_{1,λ}.Δ = 0. Hence

B′_{1,λ}.L = B′_{1,λ}.(L − Δ) = 2B′_{1,λ}.D ≥ 2n,

so that

B′_{1,λ}^{2} ≥ n − 2 = D^{2} − 2.

Since h^{1}(B′_{1,λ}) = h^{1}(P_{1,λ}) = h^{1}(D) = 0, we clearly must have B′_{1,λ} ∼ P_{1,λ}.

This means that B′_{1,λ} ∼ B_{1,λ}, so B_{1,λ} < D and B_{1,λ} is a free Clifford divisor. Since h^{0}(B_{1,λ} ⊗ J_{X_{1,λ}}) > 0, there must exist an element of |D| of the form B_{1,λ} + A_{1,λ} passing through Z_{1,λ}, for A_{i,λ} > 0. But since there is only one element of |D| passing through p_{1}, …, p_{n}, q_{1,λ}, which we called D_{λ} and which is smooth and irreducible, we have B_{1,λ} = D_{λ}, a contradiction.

So we must have B′_{1,λ} ∼ D. Recall by [Kn1] that either L − B_{1,λ} ≥ B_{1,λ}, or we are in a case where a certain exact sequence splits (namely the sequence (3.2)), in which case we have both h^{0}(B_{1i,λ} ⊗ J_{X_{1,λ}}) and h^{0}((L − B_{1,λ}) ⊗ J_{X_{1,λ}}) ≠ 0. This gives us the two possibilities:

1. B_{1,λ} ∈ |D|,

2. B_{1,λ} ∈ |D| + Γ_{j(λ)}, for j(λ) = 1 or 2, and there exists an F_{1,λ} ∈ |D| + Γ_{3−j(λ)} passing through X_{1,λ}.

In case 1., since there is only one member of |D| containing p_{1}, …, p_{n}, q_{1,λ}, which we called D_{λ}, we have B_{1,λ} = D_{λ}. But this would mean that p_{0} ∈ D_{λ} for all λ, a contradiction.

In case 2. one easily sees that the only option is p_{0} ∈ Γ_{2}, which means that x_{0} = y_{2}, as desired.

The proof of Theorem 4.7 in the case (E2) is similar, and therefore left to the reader. Since we have seen that the crucial point in proving Propositon 4.5 is to prove that

h^{1}(A) = 0, we get the following result (by checking that the proof of Theorem 4.7 goes through):

Lemma 5.10. Let D be a free Clifford divisor, not of type (E1) or (E2). If h^{1}(A) = 0, then D is perfect.

6. Projective models in smooth scrolls

In this section we will study the case when the scroll T associated to S and L as above is smooth. When D is perfect, this happens if and only if D^{2} = 0 and R_{L,D} = ∅. We recall that ϕ_{L}(S) is denoted by S′.

Since T is smooth, it can be identified with the P^{1}-bundle P(E), where E = O_{P^{1}}(e_{1}) ⊕ O_{P^{1}}(e_{2}) ⊕ ⋯ ⊕ O_{P^{1}}(e_{c+2}), and (e_{1}, e_{2}, …., e_{c+2}) is the type of the scroll.

We will construct a resolution of the structure sheaf O_{S′} as an O_{T} -module.

The contents in this section will be very similar to that in [Sc], where canonical curves of genus g are treated. This is quite natural, since a general hyperplane section of S′ is indeed such a canonical curve.

The following are well-known facts about T in P^{g} (see [Ha] and [E-H]):

(1) deg T = g − c − 1.

(2) dim T = c + 2.

(3) The Chow ring of T is Z[H, F]/(F^{2}, H^{c+3}, H^{c+2}F, H^{c+2} − (g − c − 1)H^{c+1}F), where H is the hyperplane section, and F is the class of the ruling.

(4) The canonical class of T is −(c + 2)H + (g − c − 3)F.

(5) The class of S′ in the Chow ring of T is (c + 2)H^{c} + (c^{2} + 3c − cg)H^{c−1}F.

We will need the betti-numbers of the ϕ_{L}(D_{λ}) in P^{c+1}. These can be found also when T is singular, and will be needed in this case later on.

Lemma 6.1. Let (S, L) be a polarized K3 surface of genus g and of non-general Clifford index c > 0. Let D be a free Clifford divisor satisfying D^{2} = 0. For c ≥ 2, all the ϕ_{L}(D_{λ}) in P^{c+1} have minimal resolutions

0 ⟶ O_{P^{c+1}}(−(c + 2)) ⟶ O_{P^{c+1}}(−c)^{β_{c−1}} ⟶ O_{P^{c+1}}(−(c − 1))^{β_{n−3}} ⟶ ⋯
⟶ O_{P^{c+1}}(−3)^{β_{2}} ⟶ O_{P^{c+1}}(−2)^{β_{1}} → O_{P^{c+1}} → O_{ϕ_{L}(D_{λ})} ⟶ 0,

where

β_{i} = i\binom{c + 1}{i + 1} − \binom{c}{i − 1}.

For c = 1 all the ϕ_{L}(D_{λ}) in P^{2} have the resolution

0 ⟶ O_{P^{2}}(−3) ⟶ O_{P^{2}} ⟶ O_{ϕ_{L}(D_{λ})} ⟶ 0.

Proof. In Proposition 7.6 below we will show that in general (not only for D^{2} = 0) the betti-numbers of all ϕ_{L}(D_{λ}) are constant. Granted this, and the fact that L is very ample on D_{λ} for general λ by Lemma 4.2, the result follows since any smooth ϕ_{L}(D_{λ}) is an elliptic normal curve in P^{c+1}, with a resolution as above. Even so, we will now give a special and different proof for the present case D^{2} = 0.

Pick any D′_{λ} := ϕ_{L}(D_{λ}). Its betti-numbers are equal to those of a general hyperplane section of it. It is sufficient that the linear term defining the hyperplane is not a zero divisor in its coordinate ring R_{λ}. This is essentially [Na, Theorem 27.1].

Now choose a sufficiently general hyperplane H_{λ} in P^{g} so that C_{λ} := H_{λ} ∩ S′ is a smooth canonical curve, H_{λ} does not contain any of the linear spaces \overline{D_{λ}}, and the hyperplane section A_{λ} := H_{λ} ∩ D_{λ} is not a zero divisor of R_{λ}.

We can identify C_{λ} with an element in |L|, and by abuse of notation write O_{C_{λ}}(A_{λ}) = O_{C_{λ}}(D_{λ}) = O_{C_{λ}}(D). This linear system is complete and base point free (in fact it is a pencil computing the gonality) of degree c + 2 on C_{λ}. By [Sc, Lemma p.119] (where there is a misprint) and [Sc, Proposition 4.3] the zero-dimensional scheme A_{λ} then has the betti-numbers β_{i,i+1} = β_{i} = i\binom{c+1}{i+1} − \binom{c}{i−1}.

In particular, these numbers are independent of λ.

The following result is analogous to [Sc, Corollary (4.4)].

Proposition 6.2. Let S be a polarized K3 surface of non-general Clifford-index c > 0, whose associated scroll T as above is smooth.

(a) O_{S′} has a unique O_{T} -resolution F_{∗} (up to isomorphism). If c = 1, the resolution is:

0 ⟶ O_{T} (−3H + (g − 4)F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

If c ≥ 2, the resolution is of the following type:

0 ⟶ O_{T} (−(c + 2)H + (g − c − 3)F) ⟶ ⊕_{k=1}^{β_{c−1}}O_{T} (−cH + b_{c−1}^{k}F) ⟶ ⋯
⟶ ⊕_{k=1}^{β_{1}}O_{T} (−2H + b_{1}^{k}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

where β_{i} = i\binom{c+1}{i+1} − \binom{c}{i−1}.

(b) F_{∗} is self-dual: Hom(F_{∗}, O_{T} (−(c + 2)H + (g − c − 3)F)) ≃ F_{∗}.

(c) If all b_{i}^{k} ≥ −1, then an iterated mapping cone

[[C^{g−c−3}(−(c + 2)) ⟶ ⊕_{k=1}^{β_{c−1}}C^{b_{c}^{k}}(−c)] …] ⟶ C^{0}

is a (not necessarily minimal) resolution of O_{S′} as an O_{P^{g}}-module.

(d) The b_{i}^{k} satisfy the following polynomial equation in n if c ≥ 2:

\binom{n + c + 1}{c + 1}(\frac{n(g − c − 1)}{c + 2} + 1) − n^{2}(g − 1) − 2 =

∑_{i=1}^{c−1}((−1)^{i+1}\binom{n − i + c}{c + 1}(\frac{((n − i − 1)(g − c − 1) + (c + 2))β_{i}}{c + 2} + ∑_{k=1}^{β_{i}} b_{i}^{k}) +

(−1)^{c+1}\binom{n − 1}{c + 1}(\frac{(n − c − 2)(g − c − 1)}{c + 2} + g − c − 2).

Proof. We start by proving (a). We have \overline{D_{λ}} ≃ P^{c+1} by (8). The ϕ_{L}(D_{λ}) have bettinumbers β_{i,j,λ} = dim(T or_{i}^{R_{λ}}(R, k)_{j}), where R is the homogeneous coordinate ring of P^{c+1}, and R_{λ} the coordinate ring R/I_{λ} of ϕ_{L}(D_{λ}). Following [Sc], for c ≥ 2 it is enough to prove:

(1) For fixed i, j the β_{i,j,λ} are the same for all λ.

(2) If c ≥ 2, then β_{i,j,λ} = 0, unless j = i + 1 and i ≤ c − 1, or (i, j) = (c, c + 2).

(3) The common value β_{i,i+1} = β_{i,i+1,λ} is β_{i} = i\binom{c+1}{i+1} − \binom{c}{i−1} for i ≤ c − 1, and β_{c,c+2} = 1. This follows immediately from the lemma above. The easier case c = 1 is dealt with in an analogous manner. The proof of (b) is almost identical to that of [Sc, Corollary 4.4(b)]. In our case we have

Ext_{T}^{i} (O_{S′}, ω_{T}) = ω_{S′} if i = c, and zero otherwise, ω_{S′} = O_{S′}, and ω_{T} = O_{T} (−(c + 1)H + (g − c − 3)F).

The proof of (c) is identical to that of [Sc, Prop. 4.4(c)].

Denote the term i places to the left of O_{T} in the resolution F_{∗} by F_{i}. The proof of (d) then follows from the identity

χ(O_{T} (nH)) − χ(O_{S′}(nH)) = ∑_{i}(−1)^{i}χ(F_{i}(nH)).

The contribution from the F_{c}-term is written out separately. Moreover it is clear that for all large n, we have χ(F_{i}(nH)) = h^{0}(F_{i}(nH), for all i, and χ(O_{T} (nH)) = h^{0}(O_{T} (nH)) since H is (very) ample on T . Then one uses the following well-known fact for a ≥ 0:

h^{0}(P(E), O_{P(E)}(aH + bF)) = h^{0}(P^{1}, Sym^{a}(E) ⊗ O_{P^{1}}(b)). (30)

Remark 6.3. Part (d) of the proposition only gives us the sums of the b_{i}^{k} for each fixed i. The values n = 2, 3, …, c give enough equations to determine these sums. The duality of part (c) gives β_{i} = β_{c−i}, for i = 1, …, c − 1, and after a possible renumeration of the b_{i}^{k}, for k = 1, …, β_{i}, we also have b_{c−i}^{k} = g − c − 3 − b_{i}^{k} for these k. In particular this enables us to identify the sums of the b_{i}^{k} with those of the g − c − 3 − b_{c−i}^{k}. To obtain more information about the individual b_{i}^{k} a more refined study is necessary.

7. Projective models in singular scrolls

In this section we will study more thouroughly the case where the scroll T is singular. As seen above, this necessarily happens if D^{2} > 0 or the set R_{L,D} is non-empty.

We will always assume c > 0, so that ϕ_{L} : S → S′ is birational.

The type (e_{1}, …, e_{d}) of the scroll, where d = \frac{1}{2}D^{2} + c + 2, is such that the last r of the e_{i} are zero, where r is defined as in equation (9) and can be computed as in equation (10).

As we have seen, when D is perfect we have

(31) r = \{ \begin{matrix}D^{2} − 1 & if D is of type (Q), \\ D^{2} + D.Δ − 1 & if D is of one of the types (E0)-(E2), \\ D^{2} + D.Δ & otherwise. \end{matrix}

We will however not assume that D is perfect, unless explicitly stated.

Let n := D^{2} and denote by p_{1}, …, p_{n} the n base points of the pencil {D_{λ}}. Let

\xymatrix{S̃ \ar[r]^{f} & S}

be the blow up of S at p_{1}, …, p_{n}. Denote by E_{i} the exceptional line over p_{i} and let

E := ∑_{i=1}^{n} E_{i}

denote the exceptional divisor. Define

H := f^{∗}L + f^{∗}D − E.

The first observation is:

Lemma 7.1. H is generated by its global sections, h^{1}(H) = 0 and ϕ_{H} is birational; in fact ϕ_{H} is an isomorphism outside of finitely many smooth rational (−2)-curves.

Moreover, a smooth rational curve γ is contracted by H if and only if γ = f^{∗}Γ, for some smooth rational curve Γ on S such that Γ.L = Γ.D = 0.

Proof. Since H − E ∼ (f^{∗}L − E) + (f^{∗}D − E) is clearly nef, we have h^{1}(H) = 0. Since furthermore (H − E)^{2} ≥ 10, it follows from the well-known results of Reider [R, Thm. 1(i) and Cor. 2] that if |H| is not base-point free or ϕ_{H} is not birational, then there exists an effective divisor B on S̃ such that B^{2} = −1 and B.(H − E) = 0, or B^{2} = 0 and B.(H − E) ≤ 2.

Assume there exists such a divisor B.

If B^{2} = −1, then B = E_{i} for some i, say i = 1, and B.(H −E) = E_{1}(H −E) = −2E_{1}^{2} = 2, a contradiction.

If B^{2} = 0, write B = f^{∗}A − rE for some divisor A > 0 on S and integer r ≥ 0. Then A^{2} = r^{2} and by the Hodge index theorem we get A.L > 2r. Furthermore, A.D ≥ 2, since D is base point free. This yields the contradiction

B.(H − E) = (f^{∗}A − rE)(f^{∗}L + f^{∗}D − 2E) = A.(L + D) − 2r > 2r + 2 − 2r ≥ 2.

The last statement is easily checked.

We have h^{0}(H) = \frac{1}{2}H.(H − E) + 2 = \frac{1}{2}L^{2} + \frac{1}{2}D^{2} + c + 4 = g + \frac{1}{2}D^{2} + c + 2 = g + d + 1. Denote by S′′ the surface ϕ_{H}(S̃) in P^{g+d}.

One easily obtains deg S′′ = 2g + 2c + 2 + 2D^{2}.

Proposition 7.2. The surface S′′ is normal, p_{a}(S′′) = 1, and K_{S′′} ≃ O_{S′′}(E′), where E′ is the sum of D^{2} lines that are (−1)-curves on S′′.

Proof. The two last assertions are immediate consequences of S′′ being normal, by [Ar]. Consider the surface S̃ and the commutative diagram in the previous lemma. Denote by E_{H} the set of irreducible curves Γ̃ on S̃ such that Γ̃.H = 0. From the Hodge

Index theorem it follows that such a curve has negative self-intersection. Moreover, by Lemma 7.1

Γ̃ = f^{∗}Γ,

for some smooth rational curve Γ on S such that Γ.L = Γ.D = 0. Thus we can write

E_{H} = f^{∗}(E_{L} − R_{L,D}).

Now let δ̃ be the fundamental cycle of a connected component of E_{H}, p the image of δ̃ on S′′ and U the inverse image of an affine open neighborhood of p. To prove the normality of p it will be sufficient to prove the surjectivity of

H^{0}(U, O_{U}(H − δ̃)) ⟶ H^{0}(δ̃, Oδ̃(H − δ̃)),

hence of

H^{0}(S̃, OS̃(H − δ̃)) ⟶ H^{0}(δ̃, Oδ̃(H − δ̃)).

To show the latter, it will suffice to show

H^{1}(S̃, OS̃(H − 2δ̃)) = 0.

By the degeneration of the Leray spectral sequence

0 ⟶ H^{1}(S, f_{∗}(H − 2δ̃)) ⟶ H^{1}(S̃, H − 2δ̃) ⟶ H^{0}(S, R^{1}f_{∗}(H − 2δ̃))

it will suffice to show that

h^{1}(S, f_{∗}(H − 2δ̃)) = h^{0}(S, R^{1}f_{∗}(H − 2δ̃)) = 0.

Denote by δ the divisor on S such that f^{∗}δ = δ̃. Then δ is connected and δ^{2} = −2 (δ is in fact a fundamental cycle for a connected component of E_{L} minus a curve Γ that is a tail of δ and is such that Γ.D = 1. The fact that δ^{2} = −2 can be checked by inspection for each of the five platonic configurations [Ar]). We then have

f_{∗}(H − 2δ̃) = (L + D − 2δ) ⊗ J_{Z},

where Z is the zero-dimensional scheme corresponding to the n blown up points, and

R^{1}f_{∗}(H − 2δ̃) = R^{1}f_{∗}(−E) ⊗ (L + D − 2δ).

Since f_{∗}O_{E} ≃ O_{Z}, we have R^{1}f_{∗}(−E) = 0, whence we are reduced to proving the vanishing of H^{1}((L + D − 2δ) ⊗ J_{Z}). This will be proved in Lemma 7.3 below.

Lemma 7.3. With the notation as above, H^{1}((L + D − 2δ) ⊗ J_{Z}) = 0.

Proof. We will first need the following fact:

h^{1}(L + D − 2δ) = 0.

The proof for this is rather long and tedious, but does not involve any new ideas and is similar in principle to the proof of [Co, Lemma 5.3.5]. We therefore leave it to the reader.

Note that if D^{2} = 0, then Z = ∅, and we are done. So we will from now on assume that n = D^{2} > 0.

Because of the vanishing of H^{1}(L + D − 2δ), the vanishing of H^{1}((L + D − 2δ) ⊗ J_{Z}) is equivalent to the surjectivity of the map

H^{0}(L + D − 2δ) ⟶ H^{0}((L + D − 2δ) ⊗ O_{Z}).

Assume, to get a contradiction, that this map is not surjective. Let Z′ ⊆ Z be a subscheme of length l + 1 ≤ n = D^{2}, for some integer l ≥ 1, such that H^{0}(L + D − 2δ) ⟶ H^{0}((L + D − 2δ) ⊗ O_{Z′}) is not surjective, but H^{0}(L + D − 2δ) ⟶ H^{0}((L + D − 2δ) ⊗ O_{Z′′}) is for all proper subschemes Z′′.

Since (L + D − 2δ)^{2} > 4l + 4, we can argue as in [Kn1, Lemma 3.5], with the assumption that (L + D − 2δ) be big and nef (which it is not) replaced by the weaker condition h^{1}(L + D − 2δ) = 0, and get the following:

There is an effective decomposition L + D − 2δ ∼ A + B such that A ≥ B, A.B ≤ l + 1, h^{1}(B) = 0, B^{2} ≥ −2 and B passes through Z′.

If B^{2} = −2 (so that B is necessarily supported on a union of smooth rational curves), then we use the fact that we have chosen Z to lie outside of any rational curve Γ such that Γ.L ≤ c + 2 by (4) and

L.B ≤ (L + D).B ≤ l − 1 + 2δ.B ≤ D^{2} − 2 + 2δ.B ≤ c + 2δ.B,

to conclude that we must have δ.B ≥ 2. Hence (δ + B)^{2} ≥ 0.

This yields that we in all cases have

h^{0}(δ + B) ≥ 2.

We now want to show that also

h^{0}(δ + A − D) ≥ 2.

We can write

F ∼ A + B + 2δ − 2D ∼ (A + δ − D) + (B + δ − D) := F_{1} + F_{2}.

This is not necessarily an effective decomposition, but we have F_{1} ≥ F_{2}, since A ≥ B. We can easily calculate

F_{1}.F_{2} = A.B − D^{2} − c ≤ −c < 0,

and since F^{2} = F_{1}^{2} + F_{2}^{2} + 2F_{1}.F_{2} ≥ D^{2} ≥ 2, we must have F_{1}^{2} ≥ 2 or F_{2}^{2} ≥ 2.

If F_{1}^{2} ≥ 2, then either h^{0}(F_{1}) ≥ 2 or h^{0}(−F_{1}) ≥ 2 by Riemann-Roch. Since L.F_{1} = L.A − L.D ≥ \frac{1}{2}(L^{2} + L.D) − L.D = \frac{1}{2}(c + 2 + F^{2}) > 0, we must have h^{0}(F_{1}) ≥ 2, and we are done.

If F_{2}^{2} ≥ 2, then either h^{0}(F_{2}) ≥ 2 or h^{0}(−F_{2}) ≥ 2. In the first case, we get h^{0}(F_{1}) ≥ h^{0}(F_{2}) ≥ 2. In the second, we get F_{1} ∼ F − F_{2} > F, since −F_{2} is effective, whence h^{0}(F_{1}) ≥ h^{0}(F) ≥ 2 again.

So we have an effective decomposition of L as

L ∼ (B + δ) + (A + δ − D),

such that both h^{0}(B + δ) and h^{0}(A + δ − D) ≥ 2 and such that

(B + δ).(A + δ − D) = A.B − D.B + 2 ≤ l − D.B + 3.

Since l + 1 ≤ D^{2} ≤ c + 2, and D.B ≥ 2, since D is base point free and h^{0}(B + δ) ≥ 2, we must have D.B = 2 and l + 1 = n = D^{2} = c + 2. But since B passes through Z, we must have D.B ≥ n, whence the contradiction c = 0.

This concludes the proof of the lemma and hence of Proposition 7.2.

Define the following line bundle on S̃:

D̃ := f^{∗}D − E.

The members of |D̃| are in one-to-one correspondence with the members of the pencil {D_{λ}}. One computes D̃^{2} = 0, so |D̃| is a pencil of disjoint members. Furthermore

h^{0}(H − D̃) = h^{0}(f^{∗}L) > 2,

so |D̃| defines a rational normal scroll T_{0} containing S′′.

Proposition 7.4. T_{0} has dimension d and degree g + 1 and is of type (e_{1} + 1, …, e_{d} + 1).

Proof. The two firts assertions are easily checked.

We have to calculate the numbers h^{0}(S̃, H − iD̃) = h^{0}(S̃, f^{∗}(L − (i − 1)D + (i − 1)E) for all i ≥ 0.

One easily sees that (i − 1)E is a fixed divisor in |f^{∗}(L − (i − 1)D + (i − 1)E| for all i ≥ 1, so we get for all i ≥ 1:

h^{0}(S̃, f^{∗}(L − (i − 1)D + (i − 1)E) = h^{0}(S̃, f^{∗}(L − (i − 1)D) = h^{0}(S, L − (i − 1)D). (32)

We also have

h^{0}(H) − h^{0}(H − D̃) = d. (33)

Defining d′_{i} := h^{0}(S̃, H − iD̃) − h^{0}(S̃, H − (i + 1)D̃), we get by combining (32) and (33) that

d′_{0} = d_{0} and d′_{i} = d_{i−1} for i ≥ 1.

It follows immediately that the type of T_{0} is as claimed.

This means that T_{0} is smooth, so that T_{0} ≃ P(E), where E = ⊕_{i=1}^{d}O_{P^{1}}(e_{i} + 1). Also, we have the maps

\xymatrix{P(E) \ar[r]^{j} \ar[d]^{π} & T_{0} ⊆ P^{g+d} \\ P^{1}}

where j is an isomorphism. Then the Picard group of P(E) satisfies

Pic P(E) ≃ ZH_{0} ⊕ ZF,

where H_{0} := j^{∗}O_{P^{g+d}}(1) and F := π^{∗}O_{P^{1}}(1).

Furthermore, the Chow ring of P(E) is

Z[H_{0}, F]/(F^{2}, H_{0}^{c+3+ \frac{1}{2} D^{2}}, H_{0}^{c+2+ \frac{1}{2} D^{2}}F, H_{0}^{c+2+ \frac{1}{2} D^{2}} − (g + 1)H_{0}^{c+1+ \frac{1}{2} D^{2}}F). (34)

Consider now the morphism i given by the base point free line bundle H := H_{0} − F, where H_{0} = H + F:

i : P(E) ⟶ P^{g}.

One easily sees that i maps P(E) onto a rational normal scroll of dimension d and type (e_{1}, …, e_{d}), whence isomorphic to T . So we can assume that i maps P(E) onto T . By abuse of notation we write

i : T_{0} ⟶ T ,

and this is a rational resolution of singularities of T . Furthermore one easily sees that by construction i restricts to a map

g : S′′ ⟶ S′

which is a resolution of some singularities of S′ (precisely the singularities of S′ arising from the contractions of rational curves across the fibers in S, i.e. the curves in R_{L,D}) and a blow up at the images of the base points of {D_{λ}}.

We get the following commutative diagram:

[figure: P(E); j; T_{0}; i; T; P^{g}; S′′; g; S′; S̃; ϕ_{H}; f; S.; ϕ_{L}]

By construction, one has g ∘ ϕ_{H} = ϕ_{f^{∗}L}.

We would like to study the resolution of S′′ in P(E) ≃ T_{0}. By [Sc], if S′′ has constant betti-numbers β_{ij} = β_{ij}(λ) over P^{1}, we can (at least in principle) find a resolution of O_{S′′} by free O_{P(E)}-modules which restricts to the minimal resolution of O_{S′′_{λ}} on each fiber

P(E)_{λ} ≃ P^{c+1+ \frac{1}{2} D^{2}}.

Clearly, since the map i is the identity on each fiber, the betti-numbers of S′′_{λ} are the same as the betti-numbers of ϕ_{L}(D_{λ}).

Recall that a projective variety V is called arithmetically normal if the natural map

S_{k}H^{0}(V, O_{V} (1)) ⟶ H^{0}(V, O_{V} (k))

is surjective for all k ≥ 0.

We start by showing that the ϕ_{L}(D_{λ}) are all arithmetically normal.

Proposition 7.5. All the ϕ_{L}(D_{λ}) are arithmetically normal in \overline{D_{λ}} = P^{c+1+ \frac{1}{2} D^{2}}.

Proof. We can easily show that

h^{1}(O_{S}(qL − D)) = 0 for all q. (35)

Furthermore, by [SD, Thm. 6.1], we have that

(36) S_{k}H^{0}(S, L) ⟶ H^{0}(S, kL) is surjective for all k ≥ 0.

We have a commutative diagram

\xymatrix{H^{0}(O_{P^{g}}(q)) \ar[r] \ar[d]^{α_{1}} & H^{0}(O\overline{D_{λ}}(qL)) \ar[d]^{α_{3}} \\ H^{0}(O_{S}(qL)) \ar[r]^{α_{2}} & H^{0}(O_{D_{λ}}(qL)).}

Now α_{2} is surjective by (35) and α_{1} is surjective by (36). Hence α_{3} is surjective and ϕ_{L}(D_{λ}) is arithmetically normal.

For each λ ∈ P^{1} define

B^{λ} := ⊕_{q∈Z}H^{0}(D_{λ}, qL) and V^{λ} := H^{0}(D_{λ}, L).

Then B^{λ} is a V^{λ}-module in a natural way.

The symmetric algebra S(V^{λ}) of V^{λ} satisfies

S(V^{λ}) ≃ R,

where R is the homogeneous coordinate ring of P^{c+1+ \frac{1}{2} D^{2}}, and B^{λ} is a graded R-module. We have the Koszul complex

\xymatrix{⋯ \ar[r] & ∧^{i+1}V^{λ} ⊗ B_{j−1}^{λ} \ar[r]^{d_{i+1,j−1}^{λ}} & ∧^{i}V^{λ} ⊗ B_{j}^{λ} \ar[r]^{d_{i,j}^{λ}} & ⋯}

with the Koszul cohomology groups defined by

K_{i,j}(B^{λ}, V^{λ}) := \frac{ker d_{i,j}^{λ}}{im d_{i+1,j−1}^{λ}} .

For each λ we have a minimal free resolution of B^{λ} as an R-module:

\xymatrix{⋯ \ar[r] & ⊕_{i,j}R(−j)^{β_{i,j}} \ar[r] & ⋯ \ar[r] & ⊕_{i,j}R(−j)^{β_{1,j}} \ar[r] & ⊕_{i,j}R(−j)^{β_{0,j}} \ar[r] & B^{λ} \ar[r] & 0,}

and the β_{i,j} are the (graded) betti-numbers for ϕ_{L}(D_{λ}) (since ϕ_{L}(D_{λ}) is arithmetically normal).

By the well-known Syzygy Theorem [Gr, Thm. (1.b.4)], we have

β_{i,j} = dim K_{i,j}(B^{λ}, V^{λ})

(where the dimension is as vector space over C).

So to show that the ϕ_{L}(D_{λ}) all have the same betti-numbers, it is necessary and sufficient to show that the K_{i,j}(B^{λ}, V^{λ}) are isomorphic (as vector spaces) for all λ. This is the contents of the following

Proposition 7.6. The groups K_{i,j}(B^{λ}, V^{λ}) are isomorphic as vector spaces over C for all λ.

Proof. Let D_{λ_{1}} and D_{λ_{2}} be two distinct members of the pencil and define B^{λ_{1}}, B^{λ_{2}}, V^{λ_{1}} and V^{λ_{2}} as above. Since H^{1}(qL − D) = 0 for all q ∈ Z, we have a commutative diagram

[figure: 0; ⊕H^{0}(S, L^{q} ⊗ J_{D_{λ_{1}}} ); ⊕H^{0}(S, L^{q}); B^{λ_{1}}; ψ; 0; 0; ⊕H^{0}(S, L^{q} ⊗ J_{D_{λ_{2}}} ); ⊕H^{0}(S, L^{q}); B^{λ_{2}}; 0,]

where the isomorphism ψ is induced by the isomorphism on the left and the equality in the middle. By restriction, ψ also induces an isomorphism V^{λ_{1}} → V^{λ_{2}}.

We want to show that the map

\xymatrix{∧^{i}V^{λ_{1}} ⊗ B_{j}^{λ_{1}} \ar[r]^{∧^{i}ψ⊗ψ_{j}} & ∧^{i}V^{λ_{2}} ⊗ B_{j}^{λ_{2}}}

induced by ψ descends to Koszul cohomology and gives an isomorphism

\xymatrix{K_{i,j}(B^{λ_{1}}, V^{λ_{1}}) \ar[r] & K_{i,j}(B^{λ_{2}}, V^{λ_{2}}).}

For this it suffices to show that

ψ(v) · ψ(b) = ψ(v · b) for all v ∈ V^{λ_{1}} and b ∈ B^{λ_{1}}.

This follows from the commutative diagram above.

Corollary 7.7. The betti-numbers of the ϕ_{L}(D_{λ}) are the same for all λ.

Example 7.8. As an example we look at the case where D^{2} = 0 and T is singular (i.e. D is not perfect or R_{L,D} is non-empty). In this case the scroll T_{0} can be analyzed with the techniques of Section 6. Proposition 7.2 gives that the canonical sheaf on S′′ is trivial. Lemma 6.1 gives us the betti-numbers of all the D_{λ}. Hence the analogue of Proposition 6.2 goes through completely (we need the triviality of the canonical sheaf to prove the analogue of part (b)) to give a resolution of O_{S′′} as an O_{T_{0}}-module. Set g_{0} = g + d = g + c + 2. Since T_{0} has degree g_{0} − c − 1, dimension c + 2, and spans P^{g_{0}}, we only need to replace g by g_{0} in Theorem 6.2.

Unfortunately, finding the betti-numbers β_{ij} for the ϕ_{L}(D_{λ}) when D^{2} > 0 is not as easy as in the case D^{2} = 0. In fact, we are not able to compute all of them.

Since the betti-numbers are the same for all λ ∈ P^{1} we can choose to work with any smooth irreducible D_{λ}. We recall from Lemma 4.1 that ϕ_{L}(D_{λ}) is isomorphic to D_{λ} and also smooth.

Recall that the line bundle L_{D_{λ}} := L ⊗ O_{D_{λ}} on D_{λ} is said to satisfy property N_{p} if the betti-numbers satisfy the following:

(37) β_{0,j} = \{ \begin{matrix}1 & if j = 0, \\ 0 & if j ≠ 0 \end{matrix} and β_{i,j} ≠ 0 if and only if j = i + 1, for 0 < i ≤ p.

This means that B^{λ} has a resolution of the form

\xymatrix{⋯ \ar[r] & R(−p − 1)^{β_{p,p+1}} \ar[r] & ⋯ \ar[r] & R(−3)^{β_{2,3}} \ar[r] & R(−2)^{β_{1,2}} \ar[r] & R \ar[r] & B^{λ} \ar[r] & 0.}

In our case, we have

Proposition 7.9. Assume c > 0. Then L_{D_{λ}} satisfies property N_{c−1} but not N_{c}.

Proof. The first statement is a direct consequence of [Gr, Thm. (4.a.1)], since deg L_{D_{λ}} = 2g(D_{λ}) + c and h^{1}(L_{D_{λ}}) = 0.

For the second statement, we have by [Kn1] and the conditions (∗) that L_{D_{λ}} fails to be (c + 1)-very ample, and the result follows from [G-L1, Thm. 2].

From [Gr, Thm. (4.a.1)] it also follows that

K_{i,j}(D_{λ}, L) = 0 for j ≥ 3,

whence

β_{ij} = 0 for j ≥ i + 3.

Of course we also have

β_{ij} = 0 for i ≥ h^{0}(L_{D_{λ}}) − 1 = c + 1 + \frac{1}{2}D^{2}.

Also, by the Theorem in [G-L3], we have that for D^{2} > 0:

β_{c,c+1} ≠ 0.

Indeed, L_{D_{λ}} ≃ F_{D_{λ}} + ω_{D_{λ}}, and D^{2} ≤ 2c for c > 0, and we calculate

h^{0}(F_{D_{λ}}) = h^{0}(F) − χ(F − D) = c + 2 − \frac{1}{2}D^{2} ≥ 2,

h^{0}(ω_{D_{λ}}) = \frac{1}{2}D^{2} + 1 ≥ 1

and

h^{0}(F_{D_{λ}}) + h^{0}(ω_{D_{λ}}) − 3 = c.

Summing up, we have Proposition 7.10. For c > 0, the betti-numbers of the ϕ_{L}(D_{λ}) satisfy:

(a) β_{0,j} = \{ \begin{matrix}1 & if j = 0, \\ 0 & if j ≠ 0 \end{matrix} .

(b) For 0 < i ≤ c − 1, β_{i,j} ≠ 0 if and only if j = i + 1,

(c) β_{ij} = 0 for i ≥ c + 1 + \frac{1}{2}D^{2}.

(d) β_{ij} = 0 for j ≥ i + 3.

(e) β_{c,c+1} ≠ 0 for D^{2} > 0.

(f) β_{c,c+2} ≠ 0.

So for D^{2} > 0, the ϕ_{L}(D_{λ}) all have a resolution of the form:

0 ⟶ R(−\frac{1}{2}D^{2} − c − 1)^{β_{\frac{1}{2} D^{2}+c, \frac{1}{2} D^{2}+c+1}} ⊕ R(−\frac{1}{2}D^{2} − c − 2)^{β_{\frac{1}{2} D^{2}+c, \frac{1}{2} D^{2}+c+2}} ⟶ ⋯
⋯ ⟶ R(−c − 2)^{β_{c+1,c+2}} ⊕ R(−c − 3)^{β_{c+1,c+3}}
⟶ R(−c − 1)^{β_{c,c+1}} ⊕ R(−c − 2)^{β_{c,c+2}} ⟶ R(−c)^{β_{c−1,c}} ⟶ ⋯
⋯ ⟶ R(−3)^{β_{2,3}} ⟶ R(−2)^{β_{1,2}} ⟶ R ⟶ B^{λ} ⟶ 0,

where all the betti-numbers in the last two lines are nonzero, whereas the betti-numbers in the upper two lines might be zero.

Example 7.11. As an example we study the case (E0) with c = 1, D^{2} = 2 and g = 6. Then the smooth curves ϕ_{L}(D_{λ}) are (hyperelliptic) of genus 2 in P^{3}. The minimal resolutions of such curves are well known (see [Si]) to be

0 ⟶ R(−4)^{2} ⟶ R(−2) ⊕ R(−3)^{2} ⟶ R ⟶ B_{λ} ⟶ 0.

Remark 7.12. We are not able to find the β_{i,j} directly in all cases, but if we twist the resolution above with n and use the additivity of the Euler characteristic, we obtain the following polynomial identity in the variable n:

(c + 2 + D^{2})n − \frac{1}{2}D^{2} =

\binom{n + c + 1 + \frac{1}{2}D^{2}}{c + 1 + \frac{1}{2}D^{2}} + ∑_{j=2}^{\frac{1}{2} D^{2}+c+2}(−1)^{j}\binom{n + c + \frac{1}{2}D^{2} + 1 − j}{c + 1 + \frac{1}{2}D^{2}}(β_{j−2,j} − β_{j−1,j}).

It is easy to see that this identity determines the β_{j−2,j} − β_{j−1,j} uniquely. Since β_{j−2,j} = 0, for j ≤ c + 1, the β_{j−1,j}, for j = 2, …, c + 1 are determined uniquely.

Setting n = 2, we obtain

β_{1,2} = \frac{1}{8}(4c^{2} + 4c − 2D^{2} + 4cD^{2} − 8 + (D^{2})^{2}).

Setting n = 0, we obtain

β_{\frac{1}{2} D^{2}+c, \frac{1}{2} D^{2}+c+2} = \frac{1}{2}D^{2} + 1.

Example 7.13. As another example we study the case when D^{2} = 2 and c ≥ 2. Then any smooth D′ in {ϕ_{L}(D_{λ})} has genus 2 and degree c + 4 in P^{c+2}. In this P^{c+2}, we have h^{1}(J_{D′}(2)) = 0 by [G-L-P, Theorem p.492]. This gives β_{1,2} = h^{0}(J_{D′}(2)) = \frac{c^{2}}{2} + \frac{3}{2}c − 1, using the exact sequence

0 ⟶ H^{0}(J_{D′}(2)) ⟶ H^{0}(O_{P^{c+2}}(2)) ⟶ H^{0}(O_{D′}(2)) ⟶ 0.

We see that this is the same as we get by using the last part of Remark 7.12. From Remark 7.12 we also get β_{c+1,c+3} = 2.

To be more explicit, we will assume c = 2 in the rest of the example. Then the resolution above reduces to:

0 ⟶ R(−4)^{β_{3,4}} ⊕ R(−5)^{2} ⟶ R(−3)^{β_{2,3}} ⊕ R(−4)^{β_{2,4}} ⟶ R(−2)^{4} → R ⟶ B^{λ} ⟶ 0.

The identity in Remark 7.12 reduces to:

\binom{n + 4}{4} − 6n + 1 = ∑_{j=2}^{5}(−1)^{j}\binom{n + 4 − j}{4}(β_{j−1,j} − β_{j−2,j}),

where β_{1,3} = 0. We have already used β_{3,5} = 2 and β_{1,2} = 4 from inserting n = 0 and n = 2, respectively. Inserting n = 1 gives nothing, while n = 3 gives β_{2,3} = 2. Inserting n = 4 we get β_{2,4} − β_{3,4} = 3. Inserting other values of n or looking at the coefficients in the polynomial identity gives nothing more than this.

Hence the resolution looks like:

0 ⟶ R(−4)^{a} ⊕ R(−5)^{2} → R(−3)^{2} ⊕ R(−4)^{a+3} ⟶ R(−2)^{4} ⟶ R ⟶ B^{λ} ⟶ 0,

for a non-negative integer a.

We now return to the general resolution, following Proposition 7.10. From Corollary 7.7, Example 7.8 for the special case D^{2} = 0 and [Sc, Thm. (3.2)], we obtain the following:

Proposition 7.14. If D^{2} = 0 and c = 1, then the O_{T_{0}}-resolution F_{∗} of O_{S′′} is

0 ⟶ O_{T_{0}}(−3H_{0} + (g − 1)F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

If D^{2} = 0 and c ≥ 2, the resolution is of the following type:

0 ⟶ O_{T_{0}}(−(c + 2)H_{0} + (g − 1)F) ⟶ ⊕_{k=1}^{β_{c−1}}O_{T_{0}}(−cH_{0} + b_{c−1}^{k}F) ⟶ ⋯
⟶ ⊕_{k=1}^{β_{1}}O_{T_{0}}(−2H_{0} + b_{1}^{k}F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0,

where β_{i} = i\binom{c+1}{i+1} − \binom{c}{i−1}.

If D^{2} ≥ 2, then O_{S′′} has a O_{T_{0}}-resolution F_{∗} of the following type:

0 ⟶ F_{\frac{1}{2} D^{2}+c} ⋯ ⟶ F_{c+1} ⟶ F_{c}
⟶ ⊕_{k=1}^{β_{c−1}}O_{T_{0}}(−cH_{0} + b_{c−1}^{k}F) ⟶ ⋯ ⟶ ⊕_{k=1}^{β_{2}}O_{T_{0}}(−3H_{0} + b_{2}^{k}F)
⟶ ⊕_{k=1}^{β_{1}}O_{T_{0}}(−2H_{0} + b_{1}^{k}F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

Here β_{i} = β_{i,i+1}, for i = 1, …, c, and F_{c} is an extension of the non-zero term

⊕_{k=1}^{β_{c,c+2}}O_{T_{0}}(−(c + 2)H_{0} + b_{c,c+2}^{k}F)

by the non-zero term

⊕_{k=1}^{β_{c,c+1}}O_{T_{0}}(−(c + 1)H_{0} + b_{c,c+1}^{k}F).

Moreover F_{i} is an extension of the term

⊕_{k=1}^{β_{i,i+2}}O_{T_{0}}(−(i + 2)H_{0} + b_{i,i+2}^{k}F)

by the term

⊕_{k=1}^{β_{i,i+1}}O_{T_{0}}(−(i + 1)H_{0} + b_{i,i+1}^{k}F)

for i = c + 1, …, \frac{1}{2}D^{2} + c.

Proof. Since the betti-numbers are the same for all λ, this is a direct application of [Sc, Thm. (3.2)].

Proposition 7.15. Let J_{S′′/T_{0}} denote the ideal sheaf of S′′ in T_{0} and J_{S′/T} the ideal sheaf of S′ in T .

We have J_{S′/T} = i_{∗}J_{S′′/T_{0}}.

Proof. This follows since i_{∗}O_{T_{0}} = O_{T} and i_{∗}O_{S′′} = O_{S}. The latter fact is a consequence of g being a birational map of normal surfaces.

We recall that the Chow ring of T_{0} is given by (34). Define H_{T} and F_{T} to be the push-down of cycles by i of H and F respectively.

We have the following description:

Proposition 7.16. (a) The class of S′′ in the Chow ring of T_{0} is

(D^{2} + c + 2)H_{0}^{d−2} + (c − cg − D^{2}(g − 1))H_{0}^{d−3}F.

(b) The class of S′ in the Chow group of T is

(D^{2} + c + 2)(H_{T})^{d−2} + (D^{2}(d − 1 − g) − 4 − cg − c + cd + 2d)(H_{T})^{d−3}F_{T} .

Proof. The class of S′′ is of the type mH_{0}^{d−2} + nH_{0}^{d−3}F, for two integers m and n. To determine m and n one has the equations S′′H_{0}^{2} = deg S′′ = 2g + 2c + 2 + 2D^{2} and S′′H_{0}F = deg(ϕ_{L}(D)) = c + 2 + D^{2}.

Statement (b) is an immediate consequence of i being birational by using the cap product map A^{∗}(P^{g}) ⊗ A_{∗}(T ) → A_{∗}(T ).

We recall the definition β_{i} = β_{i,i+1}, for i = 1, …, c − 1, and d = c + 2 + \frac{1}{2}D^{2} = dim T .

Proposition 7.17. The b_{i,j}^{k} and b_{i}^{k} in Proposition 7.14 satisfy the following polynomial equation in n (set b_{i,i+2}^{k} = β_{i,i+2}^{k} = 0 for all i and k if D^{2} = 0, and set b_{i,i+1}^{k} = b_{i}^{k}, for i = 1, …, c − 1 for all values of D^{2}):

\binom{n + d − 1}{d − 1}(\frac{n(g + 1)}{d} + 1) − n^{2}(g + 1 + c + D^{2}) + \frac{1}{2}nD^{2} − 2 =

∑_{i=1}^{c+ \frac{1}{2} D^{2}}(−1)^{i+1}\binom{n + d − i − 2}{d − 1}(\frac{((n − i − 1)(g + 1) + d)β_{i,i+1}}{d} + ∑_{k=1}^{β_{i,i+1}} b_{i,i+1}^{k}) +

∑_{i=c}^{c+ \frac{1}{2} D^{2}}(−1)^{i+1}\binom{n + d − i − 3}{d − 1}(\frac{((n − i − 2)(g + 1) + d)β_{i,i+2}}{d} + ∑_{k=1}^{β_{i,i+2}} b_{i,i+2}^{k}).

Proof. Denote the term i places to the left of O_{T_{0}} in the resolution F_{∗} by F_{i}. The result follows, similarly as in the proof of [Sc, Prop. 4.4(c)] from the identity

χ(O_{T_{0}}(nH_{0})) − χ(O_{S′′}(nH_{0})) = ∑_{i}(−1)^{i+1}χ(F_{i}(nH_{0})).

To calculate χ(O_{S′′}(nH_{0})) one uses Riemann-Roch on S′′ and deg S′′ = 2g + 2c + 2 + 2D^{2}.

Moreover it is clear that for all large n, we have χ(F_{i}(nH)) = h^{0}(F_{i}(nH), for all i, and χ(O_{T_{0}}(nH_{0})) = h^{0}(O_{T_{0}}(nH_{0})) since H_{0} is (very) ample on T_{0}. Then one uses (30) again.

Remark 7.18. From the last result it is clear that the sums ∑_{k=1}^{β_{i,j}} b_{i,j}^{k} are uniquely determined, but this does not necessarily apply to the b_{i,j}^{k} individually. If D^{2} > 0, it is not even a priori clear that the b_{i,j}^{k} are independent of the choice of pencil inside |D| giving rise to T (c, D, {D_{λ}}).

Corollary 7.19. (a) We have

∑_{k=1}^{β_{1,2}} b_{1,2}^{k} = (\frac{1}{2}D^{2} + c − 1)g + (1 − c − D^{2}).

(b) If D^{2} > 0, then

∑_{k=1}^{β_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2}} b_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2}^{k} = g(\frac{1}{2}D^{2} + 1) + 1.

Proof. We insert n = 2 in Proposition 7.17. That gives part (a) directly. Then we insert n = 0 in Proposition 7.17. That gives

−1 = gβ_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2} − ∑_{k=1}^{β_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2}} b_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2}^{k}.

This immediately gives the statement of part (b), since we have shown in Remark 7.12 that

β_{c+ \frac{1}{2} D^{2},c+ \frac{1}{2} D^{2}+2} = \frac{1}{2}D^{2} + 1.

Example 7.20. We return to the situation studied in Example 7.13, with D^{2} = c = 2, and we will apply Proposition 7.17. From that example and Corollary 7.19 we see that β_{3,5} = 2 and

∑_{k=1}^{β_{3,5}} b_{3,5}^{k} = 2g − 1.

Setting n = 1 in Proposition 7.17, we get nothing, but setting n = 2 we obtain

∑_{k=1}^{β_{1}} b_{1}^{k} = 2g + 1 − β_{1,2} = 2g − 3.

Setting n = 3 we obtain

∑_{k=1}^{β_{2,3}} b_{2,3}^{k} = 2g − 3 − β_{2,3} = 2g − 5.

Continuing this way, we find the difference of the b_{3,4}^{k} and the b_{2,4}^{k} in terms of β_{1}, β_{2,3}, β_{2,4}, β_{3,4}, by setting n = 4. This gives

∑_{k=1}^{β_{3,4}} b_{3,4}^{k} − ∑_{k=1}^{β_{2,4}} b_{2,4}^{k} = (β_{2,3} − 4)g + (β_{2,3} + β_{3,4} − β_{2,4} − 6) = −2g − 7.

This is also the general picture. One uses the values n = 2, 3, …, c to obtain the sums of the corresponding b_{n−1,n}^{k} in terms of the corresponding and previous β_{i,i+1}, and one uses the values n = c + 1, …, d − 1 = c + 1 + \frac{1}{2}D^{2} to obtain the differences of the sums of the b_{n−2,n} and b_{n−1,n} (and n = 0 as in Corollary 7.19) in terms of their corresponding and previous β_{i,i+1} and β_{i,i+2}.

Setting n = 5 in our concrete example above, we get

∑_{k=1}^{β_{3,5}} b_{3,5}^{k} = (β_{2,3} + β_{2,4} − β_{3,4} − 3)g + (β_{2,3} + β_{2,4} − β_{3,4} − 4) − β_{3,5} = 2g − 1.

But this gives nothing new.

Definition 7.21. We define, for integers a and b,

O_{T} (aH + bF) := i_{∗}O_{T_{0}}(aH + bF).

In particular, by the projection formula,

i_{∗}O_{T_{0}}(aH_{0} + bF) = i_{∗}O_{T_{0}}(aH + (a + b)F) = O_{T} (a) ⊗ i_{∗}((a + b)F).

We now return to the general situation. As a consequence of Proposition 7.14 we have the following two results:

Proposition 7.22. Assume b_{i}^{k} ≥ i, for i = 1, …, c − 1 and all k and b_{i,j}^{k} ≥ j − 1 for j = i + 1, i + 2, i = c, …, \frac{1}{2}D^{2} + c, and all k.

If D^{2} = 0 and c ≥ 2, then

0 ⟶ O_{T} (−(c + 2)H + (g − 1)F) ⟶ ⊕_{k=1}^{β_{c−1}}O_{T} (−cH + (b_{c−1}^{k} − c)F) ⟶ ⋯

⟶ ⊕_{k=1}^{β_{1}}O_{T} (−2H + (b_{1}^{k} − 2)F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

is an O_{T} -resolution of O_{S′}.

If D^{2} ≥ 2, then O_{S′} has a O_{T} -resolution F′_{∗} of the following type:

0 ⟶ F′_{\frac{1}{2} D^{2}+c} ⟶ ⋯ ⟶ F′_{c+1} ⟶ F′_{c}
⟶ ⊕_{k=1}^{β_{c−1}}O_{T} (−cH + (b_{c−1}^{k} − c)F) ⟶ ⋯ ⟶ ⊕_{k=1}^{β_{2}}O_{T} (−3H + (b_{2}^{k} − 3)F)
⟶ ⊕_{k=1}^{β_{1}}O_{T} (−2H + (b_{1}^{k} − 2)F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

Here F′_{i} = i_{∗}(F_{i}), for all i.

Proof. See [Sc, p.117]. The essential fact is that the map i : T_{0} ≃ P(E) → T is a rational resolution of singularities, and that we therefore have R^{1}i_{∗}O_{T_{0}} = 0. Moreover i_{∗}O_{S′′} = O_{S′}, and i_{∗}O_{T_{0}} = O_{T}. The condition on the b_{i} and the b_{i,j} gives that each term (except O_{S′′}) in the resolution of O_{S′′} in Proposition 7.14 is an extension of terms of the form O_{T_{0}}(aH+bF), with b ≥ −1. As in [Sc, (3.5)] we then conclude that the resolution therefore remains exact after pushing down.

Remark 7.23. By Proposition 7.15 we already know that the ideal of S′ in T is the pushdown by i of the ideal of S′′ in T_{0}.

If b_{2}^{k} ≥ 2 for all k when c ≥ 3 (resp. b_{2,3}^{k} ≥ 2 and b_{2,4}^{k} ≥ 3 for all k when c ≤ 2), then it automatically follows that R^{1}i_{∗}F_{2} = R^{1}i_{∗}F_{1} = 0, so that we get an exact pushed-down right end

i_{∗}F_{1} ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

This means that the ideal of S′ in T is generated by the push-down by i of the generators of the ideal of S′′ in T_{0}.

The next two results give the first examples of applications of the proposition.

Corollary 7.24. Assume D^{2} = 0.

(a) If c = 1 then O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + (g − 4)F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

(b) If c = 2, then O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + (g − 1)F) ⟶
O_{T} (−2H + a_{1}F) ⊕ O_{T} (−2H + a_{2}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

for two integers a_{1} and a_{2} such that a_{1} ≥ a_{2} ≥ 0 and a_{1} + a_{2} = g − 5.

Proof. Set g_{0} = g + c + 2.

If c = 1, then by Example 7.8 a resolution of O_{S′′} as an O_{T_{0}}-module is:

0 ⟶ O_{T_{0}}(−3H_{0} + (g_{0} − 4)F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

Here g_{0} − 4 ≥ 5, whence (a) follows.

If c = 2, we have g_{0} ≥ 10 and Proposition 6.2(a), gives that a resolution of O_{S′′} as an O_{T_{0}}-module is:

0 ⟶ O_{T_{0}}(−4H_{0} + (g_{0} − 5)F) ⟶
O_{T_{0}}(−2H_{0} + b_{1}F) ⊕ O_{T_{0}}(−2H_{0} + b_{2}F) ⟶ O_{T_{0}} ⟶ O_{S′} ⟶ 0,

for two integers b_{1} and b_{2} such that b_{1} ≥ b_{2} ≥ 0 and b_{1} + b_{2} = g_{0} − 5. From [Br, Thm. 5.1] we have that S′′ is singular along a curve if b_{1} > \frac{2(e_{1}+e_{2}+e_{3})}{3}, where (e_{1}, e_{2}, e_{3}) denotes the type of T_{0}. This is equivalent to b_{2} < \frac{g_{0}−9+2e_{4}}{3} . Hence b_{2} ≥ 1 and (b) follows.

Corollary 7.25. Let c = 1, D^{2} = 2 and g = 6 as in Example 7.11. Then O_{S′′} has the following O_{T_{0}}-resolution:

0 ⟶ O_{T_{0}}(−4H_{0} + 6F) ⊕ O_{T_{0}}(−4H_{0} + 5F)
⟶ O_{T_{0}}(−2H_{0} + 4F) ⊕ O_{T_{0}}(−3H_{0} + 4F) ⊕ O_{T_{0}}(−3H_{0} + 3F)
⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

In particular, O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + 2F) ⊕ O_{T} (−4H + F)
⟶ O_{T} (−2H + 2F) ⊕ O_{T} (−3H + F) ⊕ O_{T} (−3H)
⟶ O_{T} ⟶ O_{S′} ⟶ 0.

Proof. Corollary 7.19 gives b_{1,2}^{1} = 4 and b_{2,4}^{1} +b_{2,4}^{2} = 11, while inserting n = 3 in Proposition 7.17 gives b_{1,3}^{1} + b_{1,3}^{2} = 7. Other values of n gives no more information. Hence we have a resolution

0 ⟶ O_{T_{0}}(−4H_{0} + b_{2,4}^{1}F) ⊕ O_{T_{0}}(−4H_{0} + (11 − b_{2,4}^{1})F) ⟶ F_{1} ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0, (38)

where F_{1} is an extension

0 ⟶ O_{T_{0}}(−2H_{0} + 4F) ⟶ F_{1} ⟶ O_{T_{0}}(−3H_{0} + b_{1,3}^{1}F) ⊕ O_{T_{0}}(−3H_{0} + (7 − b_{1,3}^{1})F) ⟶ 0. (39)

Without loss of generality we assume b := b_{1,3}^{1} ≥ 4, and a := b_{2,4}^{1} ≥ 6. The type of T_{0} is (3, 2, 1, 1).

Look at the composite morphism given by (38) and (39)

α : O_{T_{0}}(−4H_{0} + aF) ⊕ O_{T_{0}}(−4H_{0} + (11 − a)F) ⟶ O_{T_{0}}(−3H_{0} + bF) ⊕ O_{T_{0}}(−3H_{0} + (7 − b)F).

Now α can be expressed by a matrix

[ \begin{matrix}α_{1} & α_{2} \\ α_{3} & α_{4} \end{matrix} ] ,

with

α_{1} ∈ H^{0}(H_{0} + (b − a)F)
α_{2} ∈ H^{0}(H_{0} + (a + b − 11)F)
α_{3} ∈ H^{0}(H_{0} + (7 − a − b)F)
α_{4} ∈ H^{0}(H_{0} + (a − b − 4)F),

whose determinant gives a section g ∈ H^{0}(2H_{0} − 4F) whose zero scheme contains S′′. If (a, b) ≠ (6, 4), we have

H^{0}(H_{0} + (7 − a − b)F) = H^{0}(P^{1}, O_{P^{1}}(10 − a − b) ⊕ O_{P^{1}}(9 − a − b) ⊕ O_{P^{1}}(8 − a − b)^{2}) = 0,

whence α_{3} = 0 and g is a product of two sections of H_{0} + (b − a)F and H_{0} + (a − b − 4)F respectively. But then S′′ would have degenerate fibers S′′_{λ}, contradicting Proposition 7.10(b).

So (a, b) = (6, 4) and we compute

Ext^{1}(O_{T_{0}}(−3H_{0} + 4F) ⊕ O_{T_{0}}(−3H_{0} + 3F), O_{T_{0}}(−2H_{0} + 4F) =
H^{1}(H_{0}) ⊕ H^{1}(H_{0} + F) =
H^{1}(P^{1}, O_{P^{1}}(3) ⊕ O_{P^{1}}(2) ⊕ O_{P^{1}}(1)^{2}) ⊕ H^{1}(P^{1}, O_{P^{1}}(4) ⊕ O_{P^{1}}(3) ⊕ O_{P^{1}}(2)^{2}) = 0,

whence the sequence (39) splits and the first assertion follows.

The second is then an immediate consequence of Proposition 7.22.

Note that by this result, S′′ is cut out in T_{0} by three sections q, c_{1} and c_{2} of O_{T_{0}}(2H_{0}−4F), O_{T_{0}}(3H_{0} − 4F) and O_{T_{0}}(3H_{0} − 3F) respectively.

Now look at the three dimensional subvariety V of T_{0} defined by q ∈ O_{T_{0}}(2H_{0} − 4F). Arguing as in the proof of Proposition 7.16 we find that the class of i(V) in the Chow group of T is 2H_{T} − 2F_{T}, whence i(V) has degree 4 and dimension 3 in P^{6}. As in [SD, (7.12)] we have that i(V) is a cone over the Veronese surface (whose vertex is the image of Γ) and that this variety is the (reduced) intersection of all quadrics containing S′.

A very useful result is the following, involving so called “rolling factors” coordinates (see for example [Ha, p.59], [St, p.3] or [Re]):

Lemma 7.26. The sections of aH−bF on a smooth rational normal scroll of type (e_{1}, …, e_{d}) can be identified with weighted-homogeneous polynomials of the form

P = ∑ P_{i_{1},... ,i_{d}}(t, u)Z_{1}^{i_{1}} … Z_{d}^{i_{d}},

where i_{1} + ⋯ + i_{d} = a, and P_{i_{1},... ,i_{d}}(t, u) is homogeneous of degree −b + (i_{1}e_{1} + ⋯ + i_{d}e_{d}).

If we multiply P by a homogeneous polynomial of degree b in t, u, then we get a defining equation of the zero scheme of the section, in term of homogeneous coordinates of the projective space, within which the scroll is embedded. Here X_{k,j} = t^{j}u^{d_{k}−j}Z_{k}, for k = 1, …, d, and j = 0, …, d_{k}, are coordinates for this space. The equation is uniquely determined modulo the homogeneous ideal of the scroll.

As a first application, we prove the analogous of Corollary 7.24 for c = 3.

Corollary 7.27. Let D^{2} = 0 and c = 3. Then a resolution of O_{S′} as an O_{T} -module is:

0 ⟶ O_{T} (−5H_{0} + (g − 6)F) ⟶ ⊕_{i=1}^{5}O_{T_{0}}(−3H_{0} + a_{i}F) ⟶
⊕_{i=1}^{5}O_{T} (−2H_{0} + b_{i}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

where all the a_{i} and b_{i} ≥ −1 and satisfy ∑_{i=1}^{5} b_{i} = 2g − 12 and a_{i} = g − 6 − b_{i}.

Proof. First recall that g ≥ 9.

As a special case of Proposition 7.14 we obtain that the resolution of O_{S′′} as an O_{T_{0}}- module is:

0 ⟶ O_{T_{0}}(−5H_{0} + (g − 1)F) ⟶ ⊕_{k=1}^{5}O_{T_{0}}(−3H_{0} + b_{2}^{k}F) ⟶
⊕_{k=1}^{5}O_{T_{0}}(−2H_{0} + b_{1}^{k}F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

From Corollary 7.19 we get ∑_{i=1}^{5} b_{1}^{k} = 2g − 2. The self-duality of the resolution in this particular case gives b_{2}^{k} = g − 1 − b_{1}^{k}, if we for example order the b_{1}^{k} in a non-increasing way, and the b_{2}^{k} in a non-decreasing way.

We will show that for all g ≥ 9 all the b_{1}^{k} ≥ 1 and all the b_{2}^{k} ≥ 2, so that we can push down the resolution to one of O_{S′} as an O_{T} -module.

Look at the map

Φ : ⊕_{k=1}^{5}O_{T_{0}}(−3H_{0} + b_{2}^{k}F) ⟶ ⊕_{k=1}^{5}O_{T_{0}}(−2H_{0} + b_{1}^{k}F).

Just like in the analysis of pentagonal curves in [Sc], it follows from [B-E] that the map Φ is skew-symmetrical and that its Pfaffians generate the ideal of S′′ in T_{0}. See also [Wa].

Let the type of T_{0} be (e_{1}, …, e_{5}), where e_{5} = 1 and ∑_{i=1}^{4} e_{i} = g.

A key observation is the following: b_{1}^{1} ≤ e_{1} + e_{3} and b_{1}^{1} ≤ 2e_{2}. The first inequality holds, since otherwise we would have a quadric relation of the form f(t, u, Z_{1}, Z_{2}) = 0 in each fiber. Hence the general fiber would be reducible, a contradiction. The second inequality follows since its negation implies that Z_{1} is factor in one quadric relation satisfied by the points of S′′, a contradiction. This gives b_{1}^{1} ≤ \frac{2(e_{1}+e_{2}+e_{3})}{3} = \frac{2(g−e_{4})}{3}. Hence

b_{2}^{1} = g − 1 − b_{1}^{1} ≥ g − 1 − \frac{2g}{3} + \frac{2e_{4}}{3} = \frac{g − 3 + 2e_{4}}{3} ≥ 2,

since g ≥ 9. Hence b_{2}^{k} ≥ b_{2}^{1} ≥ 3 for all k.

Another key observation is the following: b_{1}^{2} ≤ e_{1} + e_{4} and b_{1}^{2} ≤ e_{2} + e_{3}. The first inequality holds, since otherwise the two-step projection of the general fiber D′′ of S′′ into the Z_{1}, Z_{2}, Z_{3}-plane from P = (0, 0, 0, 0, 1) and Q = (0, 0, 0, 1, 0) would be contained in 2 quadrics. This in only possible if the projected image is a line, and in that case the general fiber would be degenerate (contained in the P^{3} spanned by this line and P and Q). This is impossible. The second inequality holds, since otherwise there would be two independent relations of the form

Z_{1}f(t, u, Z_{1}, …, Z_{5}) + aZ_{2}^{2} = 0

for each fiber. In that case we could eliminate the Z_{2}^{2}-term and obtain one relation with Z_{1} as a factor, a contradiction.

These two inequalities for b_{1}^{2} imply

b_{1}^{2} ≤ \frac{e_{1} + e_{2} + e_{3} + e_{4}}{2} = \frac{g}{2}.

Now we assume for contradiction that b_{1}^{5} ≤ 0. Then we get

b_{2}^{5} − b_{1}^{k} ≥ b_{2}^{5} − b_{1}^{2} = (g − 1 − b_{1}^{5}) − b_{1}^{2} ≥ g − 1 − \frac{g}{2} = \frac{g}{2} − 1,

for k = 2, 3, 4. In the matrix description of the map Φ there is one submaximal minor with one column consisting of zero and sections of H_{0} − (b_{2}^{5} − b_{1}^{k})F, for k = 2, 3, 4. If all entries of this column have Z_{1} as a factor, that would lead to a contradiction, since the minor is the square of one of the generators of the (Pfaffian) ideal of S′′ on T_{0}. To avoid that Z_{1} is a factor in each such entry, we must have e_{2} ≥ \frac{g}{2} − 1. This gives e_{1} + e_{2} ≥ g − 2, and e_{3} + e_{4} ≤ g − (e_{1} + e_{2}) ≤ g − (g − 2) = 2. Hence e_{3} = e_{4} = e_{5} = 1. But this implies that R_{L,D} has at least 3 members, whence D.Δ ≥ 3. But this is impossible by Proposition 4.6.

Hence the assumption b_{1}^{5} ≤ 0 leads to a contradiction, and b_{1}^{k} ≥ 1, for all k. Hence the entire resolution can be pushed down to one of O_{S′} as an O_{T} -module and the result follows.

Remark 7.28. We know by Proposition 7.14 that a finite set of sections of line bundles of type aH_{0} − bF generate the ideal of the surface S′′ on the smooth rational normal scroll T_{0} of type (e_{1}, …, e_{d−r−1}, e_{d−r}, …, e_{d}), where e_{d−r} = ⋯ = e_{d} = 1, e_{d−r−1} ≥ 2, and V = Sing T ≃ P^{r} for some r ≥ 0. Let W = i^{−1}(V). This is a subscroll of T_{0} of type (1, …, 1), that is P^{r} × P^{1}. The ideal generators can be classified into 3 types:

(a) Those that are sections of aH_{0} − bF = aH − (b − a)F, with b > a.

(b) Those that are sections of aH_{0} − bF = aH − (b − a)F, with b = a.

(c) Those that are sections of aH_{0} − bF = aH − (b − a)F, with b < a. For those of type (a) it is clear from Lemma 7.26 that their zero scheme contains W. Likewise one sees that those of type (b) can be written as

f(t, u, Z_{1}, …, Z_{d}) + g(Z_{s+1}, …, Z_{d}),

where Z_{1} or Z_{2} or … or Z_{s} is a factor in every monomial of f(t, u, Z_{1}, …, Z_{d}), while g is homogeneous of degree a = b. Those of type (c) can be written as

f(t, u, Z_{1}, …, Z_{d}) + h(t, u, Z_{s+1}, …, Z_{d}),

where f is as described for type (b), while h is bihomogeneous, of degree a − b > 0 in t, u and degree a in Z_{s+1}, …, Z_{d}.

There is one fundamental difference between the sections of types (a) and (b) on one hand, and those of type (c) on the other. Those of types (a) and (b) are “constant” on the fibers of i, their zero scheme contains either the whole fiber, or no point on the fiber, for each P^{1}, which is a fiber of i. For the sections of type (c) this is only true if h is the zero polynomial, and then its zero scheme contains all of W.

We therefore see that if all b_{1,j}^{k} of the resolution of O_{S′′} as an O_{T_{0}}-module are at least j, then the ideal of S′′ is generated by “fiber constant” equations, and if Q is a point on T_{0} not on S′′, then there is a fiber constant section of the type described, which does not contain Q in its zero scheme. In short, fiber constant equations cutting out S′′ in T_{0} are also equations of S′ in T .

In Section 10 we will classify the possible projective models for g ≤ 10, and in particular the singular scrolls T appearing as T (c, D, {D_{λ}}) in the various cases, and we will also show that projective models givings scrolls of all these types exist. In Corollaries 7.24, 7.25 and 7.27 we showed that in some particular cases we can push down the entire resolution of O_{S′′} to one of O_{S′}. In the rest of this section, through a series of additional examples, we take a closer look at the rest of the singular scrolls appearing for g ≤ 10, and using Lemma 7.26 we will find restrictions on the b_{1}^{k}. (See Remark 7.18.)

Example 7.29. We return to the situation studied in Example 7.13 and Example 7.20. We recall c = 2, D^{2} = 2 and g ≥ 7. From Example 7.20 we see that the ideal of S′′ in O_{T_{0}} is generated by four sections of the type 2H_{0} − b_{k}F, where ∑_{k=1}^{4} b_{k} = 2g − 3.

The type of T_{0} is (e_{1} + 1, e_{2} + 1, e_{3} + 1, 1, 1), where all e_{i} ≥ 0 and e_{1} + e_{2} + e_{3} = g − 4. Let Q be the subscroll of T_{0} formed by the two last directrices, so Q is the inverse image by i of the line in P^{g} spanned by the images by ϕ_{L} of the basepoints of D. We see that Q is a quadric in P^{3}. All the four sections of type 2H_{0} − b_{k}F must intersect Q in the two lines that form the inverse image by i in T_{0} of the images by ϕ_{L} of the basepoints of D . But this is simply the two last directrices. The intersection with Q for one such section is obtained by using Lemma 7.26 to express each of the sections, and set Z_{1} = Z_{2} = Z_{3} = 0. What remains must be a term of the type P_{2−b_{k}}(t, u)Z_{4}Z_{5}, where P_{2−b_{k}} is zero if b_{k} ≥ 3, and a polynomial of degree 2 − b_{k} otherwise.

We order the b_{k} as b_{1} ≥ b_{2} ≥ b_{3} ≥ b_{4}. We see that if b_{4} ≤ 1, in particular if b_{4} ≤ 0, then b_{3} ≤ 2, since otherwise the total intersection of K with the four sections will consist of 2 − b_{4} lines transversal to the two directrices in addition to the two directrices.

If g = 7, it is clear that T_{0} has type (2, 2, 2, 1, 1) or (3, 2, 1, 1, 1). Then Z_{1} is a factor in all sections of 2H_{0} − bF for b ≥ 5 for both scroll types. Hence b_{1} ≤ 4. If b_{1} = 3, then the only possible combination is (b_{1}, …, b_{4}) = (3, 3, 3, 2), since ∑ b_{k} = 11. If b_{1} = 4, then the only a priori possibilities are (4, 4, 2, 1) and (4, 3, 2, 2). But (4, 4, 2, 1) is impossible for type (2, 2, 2, 1, 1), since we then have two quadric relations between Z_{1}, Z_{2}, Z_{3} only. To see this, let D′′ be any smooth curve in |f^{∗}D − E|, which can be identified with its image under ϕ_{H}. The variables Z_{1}, Z_{2}, Z_{3} restricted to D′′ correspond to sections of (H − E)_{D_{0}}. Since this line bundle has degree 2g(D′′) = 4, it is base point free and its sections map D′′ into P^{2} by a one-to-one or two-to-one map. This means that there is at most one quadric relation between Z_{1}, Z_{2}, Z_{3}, whence b_{2} ≤ 3. So for type (2, 2, 2, 1, 1) the only possibilities for (b_{1}, …, b_{4}) are

(3, 3, 3, 2) and (4, 3, 2, 2).

For the type (3, 2, 1, 1, 1), for each fiber of T_{0}, the equations with b_{k} ≤ 2, restricted to the subscroll Z_{1} = Z_{2} = 0 with plane fibers, must cut out a subscheme of length 4, such that each subscheme of length 3 spans a P^{2} (these are the cases (E1) and (E2)). It takes 2 equations to do this. Hence b_{3} ≤ 2. Moreover b_{2} ≤ 3. Assume b_{2} ≥ 4. Then we would have two independent equations of type

Z_{1}f(t, u, Z_{1}, Z_{2}, Z_{3}, Z_{4}, Z_{5}) + Z_{2}^{2}

for general (t, u). From these equations we can eliminate the Z_{2}^{2}-term, and derive one equation with Z_{1} as a factor. This is a contradiction.

Hence the only possibility for (b_{1}, …, b_{4}) for the type (3, 2, 1, 1, 1) is (4, 3, 2, 2).

If g = 8, then T_{0} has type (3, 2, 2, 1, 1). A similar argument as for g = 7, gives that the only possible combinations for (b_{1}, …, b_{4}) are

(5, 4, 2, 2), (5, 3, 3, 2) and (4, 4, 3, 2).

.

For g = 9 the type of T_{0} is a priori either (3, 3, 2, 1, 1) or (4, 2, 2, 1, 1). In Section 10 the type (4, 2, 2, 1, 1) is ruled out when D is perfect. For the type (3, 3, 2, 1, 1) we see that b_{1} ≥ 6 is impossible, since 2H_{0} − 6F only has sections of the form f(Z_{1}, Z_{2}). Moreover b_{4} ≤ 2, since we need to cut out the exceptional fibers. Hence b_{1} = 5, otherwise the sum of the b_{i} would be at most 14, and it is 15. Any section of 2H_{0} − 5F can be written in terms of t, u, Z_{1}, Z_{2}, Z_{3} only. As for one case with g = 7 we see that we cannot have two quadric relations between Z_{1}, Z_{2}, Z_{3} for general fixed (t, u), so b_{2} ≤ 4. We then see that the only possible combination is (b_{1}, b_{2}, b_{3}, b_{4}) = (5, 4, 4, 2).

For g = 10 the type of T_{0} is a priori (3, 3, 3, 1, 1), (4, 3, 2, 1, 1) or (5, 2, 2, 1, 1). In Section 10 it is shown that only the type (4, 3, 2, 1, 1) occurs when D is perfect. In this case a more detailed analysis gives that the only possible combinations for (b_{1}, …, b_{4}) are

(6, 5, 4, 2) and (5, 5, 5, 2).

Example 7.30. Let us study the case c = 2, D^{2} = 4 and g = 9. This gives a non-primitive projective model, with L ≃ 2D (it is the case (Q) described in the text above Theorem 4.7). The scroll T necessarily has type (2, 1, 1, 0, 0, 0) and T_{0} has type (3, 2, 2, 1, 1, 1). By Proposition 7.10 we have β_{1,3} = 0, and by Remark 7.12 we have β_{1,2} = 7 and ∑_{k=1}^{7} b^{k} = 3g − 5 = 22.

We will now find the b_{k}.

Lemma 7.26 gives b_{k} ≤ 4, for all k, since Z_{1} is factor in every section of 2H_{0} − bF, for b ≥ 5. The complete intersection Z_{1} = Z_{2} = Z_{3} = 0 in T_{0} is a subscroll N of type (1, 1, 1) with a plane in each fiber. The 7 equations cutting out S′′ in T_{0} must together cut out four points in each plane fiber, such that no three of these points are collinear, by Theorem 4.7. It is clear that such a configuration of points is contained in exactly two quadrics in each plane. All sections of 2H_{0} − bF with b ≥ 3 vanish on N, so we must have at least two of the b_{k} less than 3. Moreover, every section of 2H_{0} − 4F can be written

Z_{1}f(t, u, Z_{1}, …, Z_{5}) + aZ_{2}^{2} + bZ_{2}Z_{3} + cZ_{3}^{2}.

If b_{4} ≥ 4, there are four independent equations of this kind, so we could eliminate the three last terms and obtain one relation with Z_{1} as a factor. This is a contradiction, so b_{4} ≤ 3.

This leaves the unique possibility (b_{1}, …, b_{7}) = (4, 4, 4, 3, 3, 2, 2).

The following proposition describes the particular case in Proposition 4.10 above, that is we have L ∼ 2D, D^{2} = 4 and c = 2 and D is hyperelliptic, which is also a particular case of the last example. In this case there is a smooth curve E (which is a perfect Clifford divisor for D) satisfying E^{2} = 0 and E.D = 2. Since (D − E)^{2} = 0 and (D − E).L = 4, we have D > E, so E does not satisfy the conditions (C6) and (C7). We will also see that E is not a perfect Clifford divisor for L.

Proposition 7.31. Let L and D be as in the particular case of Proposition 4.10 (where S′ is not the 2-uple embedding of ϕ_{D}(S)).

Then we are in one of the following three cases:

(i) R_{L,E} = ∅ and D ∼ E + E′, where E′ is a smooth elliptic curve such that E.E′ = 2. (ii) R_{L,E} = {Γ_{1}, Γ_{2}} and D ∼ E + Γ_{1} + Γ_{2}. (iii) R_{L,E} = {Γ_{0}} and D ∼ E + Δ_{0}, where Δ_{0} has a configuration with respect to E as in

(E2). Moreover, in case (i), O_{S′} has one of the two following O_{T} -resolutions for a scroll T of

type (2, 2, 2, 0) in P^{9}:

0 ⟶ O_{T} (−4H + 4F) ⟶ O_{T} (−2H + 2F)^{2} ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

or

0 ⟶ O_{T} (−4H + 4F) ⟶ O_{T} (−2H + 4F) ⊕ O_{T} (−2H) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

The latter case occurs only if Sing T ∩ S′ = ∅, so S′ ≃ S′′ where S′′ sits in the smooth scroll T_{0} of type (3, 3, 3, 1). In the cases (ii) and (iii), O_{S′} has the following O_{T} -resolutions for a scroll T of type (4, 2, 0, 0) in P^{9}, whose singular locus is spanned by < Z_{λ} > (using the same notation as in Theorem 4.7):

0 ⟶ O_{T} (−4H + 4F) ⟶ O_{T} (−2H + 4F) ⊕ O_{T} (−2H) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

Proof. The three cases follow from Proposition 4.11, by noting that we clearly have R_{L,E} = R_{D,E}.

We have h^{0}(L) = 10 and h^{0}(L − E) = 6. We leave it to the reader to verify that in case (i) we have

h^{0}(L − 2E) = 3 and h^{0}(L − 3E) = 0,

and that we in case (ii) have

h^{0}(L − 2E) = 3, h^{0}(L − 3E) = 2, h^{0}(L − 4E) = 1 and h^{0}(L − 5E) = 0.

This yields the two scroll types (2, 2, 2, 0) and (4, 2, 0, 0) respectively.

In the cases (ii) and (iii), one can show as in the proof of Theorem 4.7 that Sing T =< Z_{λ} >.

In Subsection 8.4 we will study projective models with c = 2 and the statements about the resolutions will follow from Remarks 8.13 and 8.14.

Example 7.32. Let us study the case c = 3, D^{2} = 2 and g = 9. In Section 10 we will show that projective models with such invariants occur, and that the scroll type of T is either (1, 1, 1, 1, 0, 0) or (2, 1, 1, 0, 0, 0) when D is perfect. By Proposition 7.14 we have β_{1,3} = 0, and by Remark 7.12 we have β_{1,2} = 8 and ∑_{k=1}^{8} b^{k} = 3g − 4 = 23.

Assume first that the type is (1, 1, 1, 1, 0, 0), which implies that T_{0} has type (2, 2, 2, 2, 1, 1). Lemma 7.26 gives b_{k} ≤ 4, for all k, since h^{0}(2H_{0} − bF) = 0 for b ≥ 5. The complete intersection Z_{1} = Z_{2} = Z_{3} = Z_{4} in T_{0} is a subscroll Q of type (1, 1) with a line in each fiber. The 8 equations cutting out S′′ in T_{0} must together cut out two points in each fiber of Q (the inverse image in S′′ of Sing T ∩ S′). For a general fiber, call these points P_{1} and P_{2}. Order the b_{k} as in the previous example. To cut out the two points we must have b_{8} ≤ 2, since all sections of 2H_{0} − bF vanish on Q for b ≥ 3.

For general (t, u), where the fiber D′′ of S′′ is smooth, D′′ is a smooth curve of degree 7 and genus 2, which can be identified with a smooth curve in |f^{∗}D − E|, mapped by the complete linear system |H_{D′′}|, which is of degree 2g(D′′) + 1 = 5 and in particular very ample. Now Z_{1}, Z_{2}, Z_{3}, Z_{4} (restricted to D′′) span H^{0}(H_{D′′}), which embeds D′′ as a curve of degree 5 and genus 2 in P^{3}. As in Example 7.11 we conclude from [Si] that this curve is contained in only one quadric surface. On the other hand all sections of 2H_{0} − 4F can be expressed in terms of Z_{1}, Z_{2}, Z_{3}, Z_{4} only. Hence no more than one of the b_{k} can be 4. This leaves us with only two possible cases:

(b_{1}, …, b_{8}) = (4, 3, 3, 3, 3, 3, 2, 2) or (3, 3, 3, 3, 3, 3, 3, 2).

Assume now that the type is (2, 1, 1, 0, 0, 0), which implies that T_{0} has type (3, 2, 2, 1, 1, 1). Lemma 7.26 gives b_{k} ≤ 4, for all k, since all sections of 2H_{0} − bF have Z_{1} as factor if b ≥ 5. The complete intersection Z_{1} = Z_{2} = Z_{3} in T_{0} is a subscroll N of type (1, 1, 1) with a plane in each fiber. The 8 equations cutting out S′′ in T_{0} must together cut out three independent points in each fiber of N (the inverse image in S′′ of Sing T ∩ S′).

Order the b_{k} as in the previous example. To cut out the three points we must have b_{6} ≤ 2, since all sections of 2H_{0} − bF vanish on N for b ≥ 3, and a net of three quadrics is needed to cut out three independent points in a plane. On the other hand every section of 2H_{0} − 4F can be written

Z_{1}f(t, u, Z_{1}, …, Z_{5}) + aZ_{2}^{2} + bZ_{2}Z_{3} + cZ_{3}^{2}.

This gives b_{4} ≤ 3 as in Example 7.30. This leaves

(b_{1}, …, b_{8}) = (4, 4, 4, 3, 2, 2, 2, 2) or (4, 4, 3, 3, 3, 2, 2, 2),

as the only possibilities.

Example 7.33. The case c = 3, D^{2} = 2 and g = 10 is very similar to the analogous one for g = 9, treated in Example 7.32 and one can show in a similar way that

(b_{1}, …, b_{8}) = (4, 4, 4, 3, 3, 3, 3, 2).

We show in Section 10 that the only possible scroll type for T is (2, 1, 1, 1, 0, 0) when D is perfect, corresponding to the type (3, 2, 2, 2, 1, 1) for T_{0}.

Example 7.34. Similarly, in the case c = 3, D^{2} = 4 and g = 10, which is of type (E0) (with β_{1,2} = 12 and ∑ b_{k} = 4g − 6 = 34), one can show that

(b_{1}, …, b_{12}) = (4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2).

In Section 10 it is shown that the only scroll type occurring for T is (2, 1, 1, 0, 0, 0, 0), which means (3, 2, 2, 1, 1, 1, 1) for T_{0}.

Example 7.35. Let D^{2} = 0 and c = 3, as in Corollary 7.27.

We will show in Section 10 that for g = 9 the only smooth scroll occurring as T = T (3, D) is of type (1, 1, 1, 1, 1), and the singular scrolls occurring are of types (2, 1, 1, 1, 0), (2, 2, 1, 0, 0) and (3, 1, 1, 0, 0), corresponding to the smooth types (3, 2, 2, 2, 1), (3, 3, 2, 1, 1) and (4, 2, 2, 1, 1) for T_{0}. We also show that all these occur. By using similar techniques as in the previous examples, one can show that (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 1, 1, 1) for the type (1, 1, 1, 1, 1), (b_{1}, b_{2}, b_{3}, b_{4}) = (4, 3, 3, 3, 3), (4, 4, 3, 3, 2) or (4, 4, 4, 2, 2) for the type (3, 2, 2, 2, 1), and (b_{1}, b_{2}, b_{3}, b_{4}) = (4, 4, 3, 3, 2) or (4, 4, 4, 2, 2) for the types (3, 3, 2, 1, 1) and (4, 2, 2, 1, 1).

For g = 10 we will show in Section 10 that the only smooth scroll occurring as T = T (3, D) is of type (2, 1, 1, 1, 1), and the singular scrolls occurring are of types (2, 2, 1, 1, 0), (2, 2, 2, 0, 0), (3, 2, 1, 0, 0), corresponding to the types (3, 3, 2, 2, 1), (3, 3, 3, 1, 1) and (4, 3, 2, 1, 1) for T_{0}. We also show that all these occur. Again one can show that (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 2, 2, 1, 1) for the type (2, 1, 1, 1, 1), (b_{1}, b_{2}, b_{3}, b_{4}) = (4, 4, 4, 3, 3) or (4, 4, 4, 4, 2) for the scroll type (3, 3, 3, 1, 1), and (b_{1}, b_{2}, b_{3}, b_{4}) = (5, 5, 4, 2, 2), (5, 5, 3, 3, 2), (5, 4, 4, 3, 2), (4, 4, 4, 3, 3) or (4, 4, 4, 4, 2) for the scroll types (3, 3, 2, 2, 1) and (4, 3, 2, 1, 1).

8. More on projective models in smooth scrolls of K3 surfaces of low Clifford-index

In this section we will have a closer look at the situation described in Section 6 for c = 1, 2 and 3. We recall that T (c, D, {D_{λ}}) is smooth, which is equivalent to the conditions D^{2} = 0 and R_{L,D} = ∅ when D is perfect. The resolution of O_{S′} as an O_{T} -module is given in Proposition 6.2.

From the proof of Theorem 3.8 it is clear that for each of the possible combinations of c and g there is an 18-dimensional family of isomorphism classes of polarized K3 surfaces with smooth scroll T (c, D, {D_{λ}}). Moreover it follows that there will be an 18+dim(Aut(P^{g})) = 17 + (g + 1)^{2}-dimensional family of such projective models of K3 surfaces. This is true, simply because there is only a finite number of linear automorphisms that leaves a smooth polarized K3 surface invariant.

For each value of c (and g) one can pose several questions about the set (or subscheme of the Hilbert scheme) of projective models S′ of K3 surfaces S with elliptic free Clifford divisor D and such that T is smooth.

All scrolls of the same type are projectively equivalent, and hence the configuration of projective models of K3 surfaces in one such scroll is a projectively equivalent copy of that in another. Some questions one can pose, are: How many scrolls are there of a given type?

How many projective models S′ are there within each scroll? In how many scrolls of a given type is a given S′ included?

The answer to the first question is well-known, the remaining ones we will study more closely.

We start with the following well-known result from [Ha]:

Proposition 8.1. The dimension of the set of scrolls of type e and dimension d in P^{g} is

dim(Aut(P^{g})) − dim(Aut(P(E))) = (g + 1)^{2} − 3 − d^{2} − δ_{1},

where δ_{1} = ∑_{i,j} max(0, e_{i} − e_{j} − 1).

If D^{2} = 0, we recall that d = c + 2 for the scroll T (c, D, {D_{λ}}).

8.1. Projective models with c = 1. We have g ≥ 5. Let the projective model S′ and the smooth scroll T be given. As is seen from Proposition 6.2 the surface S′ corresponds to the divisor class 3H − (g − 4)F on T . Moreover, part (c) of the proposition can be applied so we can obtain a resolution in P^{g}. This is even minimal, by the comment in [Sc, Example 3.6].

Assume T has scroll type (e_{1}, e_{2}, e_{3}), with e_{1} ≥ e_{2} ≥ e_{3}

Proposition 8.2. The dimension of the set of sections of divisor type 3H − (g − 4)F in T is equal to 29 + δ_{2}, where δ_{2} = ∑ max(0, g − 5 − ∑_{i=1}^{3} a_{i}e_{i}). Here the first summation is taken over those triples (a_{1}, a_{2}, a_{3}) such that a_{i} ≥ 0, for i = 1, 2, 3, and ∑_{i=1}^{3} a_{i} = 3. If S′ is smooth, a general section is a smooth projective model of a K3 surface.

Proof. We use the formula h^{0}(P(E), (aH + bF)) = h^{0}(P^{1}, Sym^{a}(E) ⊗ O_{P^{1}}(b)), with a = 3 and b = g − 4. This gives 29 + δ_{2}. Being a smooth model of a K3 surface is an open condition on the set of sections of 3H − (g − 4)F, and since one section, the one giving S′, is smooth, a general section of the linear system is so, too.

We also have:

Proposition 8.3. Each projective model S′ of a K3 surface S of Clifford index 1 in P^{g} for g ≥ 5, with T (c, D, {D_{λ}}) smooth, is contained in only one smooth rational normal scroll of dimension 3.

Proof. In [SD, Part (7.12)] it is shown that the scroll T is the intersection of all quadric hypersurfaces containing S′. Moreover, any other smooth scroll T_{2} containing S′ is an intersection of quadric hypersurfaces ([SD, Prop. 1.5(ii)]), each of course containing S′. Hence T_{2} contains T . Since the two scrolls have the same dimension and degree, they must be equal.

From this we conclude:

Corollary 8.4. There is a set of dimension

(g + 1)^{2} + 17 + δ_{2} − δ_{1} = dim(Aut(P^{g})) + 18 + δ_{2} − δ_{1},

where δ_{1} ≥ δ_{2}, parametrizing projective models of K3 surfaces in smooth scrolls of the given type.

Proof. From Proposition 8.1 we see that there is a ((g + 1)^{2} − 12 − δ_{1})-dimensional set of scrolls of the same type as T in P^{g}. We know that each S′ in each such scroll is contained in only one scroll. In each scroll there is a (29 + δ_{2})-dimensional set of projective models of K3 surfaces as described. We have δ_{1} ≥ δ_{2}, since otherwise there would be too many models with Clifford index 1.

Remark 8.5. For c = 1 we have

δ_{1} = max(0, e_{1} − e_{2} − 1) + max(0, e_{1} − e_{3} − 1) + max(0, e_{2} − e_{3} − 1),

and

δ_{2} = max(0, e_{1} − e_{2} − 3) + max(0, e_{1} − e_{3} − 3) + max(0, e_{2} − e_{3} − 3) +
max(0, e_{1} + e_{2} − 2e_{3} − 3) + max(0, e_{1} − 2e_{2} + e_{3} − 3).

Moreover δ_{1} = 0 if and only if the scroll type is maximally balanced, and δ_{2} = 0 if the scroll type is “reasonably well balanced”. It is clear that δ_{1} = 0 implies δ_{2} = 0. We also see that if 5 ≤ g ≤ 8, then δ_{2} = 0. Hence the cases (e_{1}, e_{2}, e_{3}) = (3, 1, 1) or (3, 2, 1) are cases where the number (g+1)^{2}+17+δ_{2}−δ_{1} is strictly less than (g+1)^{2}+17 = dim(Aut(P^{g}))+18, and it is clear that scrolls of these types cannot represent the general projective models with c = 1 and fixed g since by the construction as in Proposition 3.8 we get an 18-dimensional family of such models.

The inequality δ_{1} ≥ δ_{2} does not follow directly from the formulas in Remark 8.5, for example since δ_{1} < δ_{2}, for scroll types (g − 4, 1, 1), when g ≥ 11. This enables us to conclude that these and other scroll types with δ_{1} < δ_{2} do not occur for the scrolls formed from projective models of K3 surfaces as described. This statement will be strengthened to apply for g ≥ 8 below. On the other hand the mentioned type (3, 1, 1) does occur for g = 7. This can be seen by using Lemma 7.26.

For g = 7 and type (3, 1, 1) one then gets a polynomial P of the form:

P_{6,1}(t, u)Z_{1}^{3} + P_{4,1}(t, u)Z_{1}^{2}Z_{2} + P_{4,2}Z_{1}^{2}Z_{3} + P_{2,1}(t, u)Z_{1}Z_{2}^{2} +
P_{2,2}(t, u)Z_{1}Z_{2}Z_{3} + P_{2,3}(t, u)Z_{1}Z_{3}^{2} + c_{1}Z_{2}^{3} + c_{2}Z_{2}^{2}Z_{3} + c_{3}Z_{2}Z_{3}^{2} + c_{4}Z_{3}^{3},

where the P_{i,j} are homogeneous of degree i, and the c_{k} are constants. For any fixed (t, u) and any fixed point in the P^{2} thus obtained, we see that we can avoid that point lying on the zero scheme of P by choosing the P_{i,j} and c_{k} properly, so we conclude that the general section of 3H − (g − 4)F = 3H − 3F is base point free, and hence smooth, by Bertini. Irreducibility also follows by a similar argument.

Using Lemma 7.26, we see that for g = 8 any section of 3H − (g − 4)F = 3H − 4F on a scroll of type (3, 2, 1) can be identified with a P of the form

P_{5,1}(t, u)Z_{1}^{3} + P_{4,1}(t, u)Z_{1}^{2}Z_{2} + P_{3,1}Z_{1}^{2}Z_{3} + P_{3,2}(t, u)Z_{1}Z_{2}^{2} +
P_{2,1}(t, u)Z_{1}Z_{2}Z_{3} + P_{1,1}(t, u)Z_{1}Z_{3}^{2} + P_{2,2}Z_{2}^{3} + P_{1,2}Z_{2}^{2}Z_{3} + c_{1}Z_{2}Z_{3}^{2}.

So here there is no Z_{3}^{3}-term, and from that we conclude that any section of 3H − 4F on a scroll of type (3, 2, 1) must have the directrix line, say Γ, corresponding to e_{3} = 1 as a part of its zero scheme. The fact that δ_{2} < δ_{1} indicates that if we can form smooth projective models of K3 surfaces this way, the surface must have a Picard lattice of higher rank than two. We may check this. If L, E and Γ sit inside a lattice of rank 2, then we can write Γ = aL + bE, for rational numbers a, b. In addition we must have Γ^{2} = −2 and L.Γ = E.Γ = 1. It is easy to check that this is impossible.

It is clear that the set of base points of the linear system 3H − 4F is just the directrix line. This is true since for a fixed value of (t, u) (each fixed fiber) and a point Q outside (0, 0, 1), we can just change the P_{6,1}(t, u)Z_{1}^{3}-term or the P_{2,2}(t, u)Z_{2}^{3}-term, if we want to avoid Q. If we choose c_{1} ≠ 0, then we obtain that the zero scheme of the corresponding section intersects all fibers in curves, smooth at (0, 0, 1). (Here we set Z_{3} = 1 in order to write the equation of the curve in affine coordinates around (0, 0, 1). The existence of the non-zero linear term c_{1}Z_{2} gives smoothness at this point.) Hence the zero scheme of a general section is smooth on all of T . We have basically used the identities 3e_{2} ≥ g − 4 and e_{2} + 2e_{3} = g − 4, to conclude as we do. See Remark 8.8 for references to other authors who have already used this kind of reasoning.

Using Lemma 7.26 again we see that if g ≥ 8, then any section of 3H − (g − 4)F on a scroll of type (g − 4, 1, 1) corresponds to a polynomial P with Z_{1} as a factor, which means that its zero scheme must be reducible as a sum of sections H − (g − 4)F (a subscroll of type (1, 1)) and 2H. Hence these scroll types cannot occur for T (c, D, {D_{λ}}). In a similar way one can draw conclusions about sections on other scroll types. The observation above also has an interesting consequence for the types of singular scrolls T (c, D, {D_{λ}}) arising for the case c = 1, D^{2} = 0, g ≥ 5.

Corollary 8.6. The type of T (c, D, {D_{λ}}) is never (g − 2, 0, 0) for c = 1 and D^{2} = 0.

Proof. Assume the type of T (c, D, {D_{λ}}) is (g − 2, 0, 0). Then the type of the associated scroll T_{0} is (g − 1, 1, 1) = (g_{0} − 4, 1, 1), and the divisor type of S′′ in T_{0} is 3H_{0} − (g_{0} − 4)F (see Example 7.8 and the proof of Corollary 7.24). But we just observed that this is impossible.

In general we conclude in the same way:

Proposition 8.7. If a type (a, b, c) is impossible for a smooth scroll T (c, D, {D_{λ}}) in P^{g} with c = 1 and D^{2} = 0, then the type (a − 1, b − 1, c − 1) is impossible for any scroll T (c, D, {D_{λ}}) in P^{g−3} with c = 1 and D^{2} = 0.

We will make a list including all possible scroll types for smooth T (c, D, {D_{λ}}), for g ≤ 13, with c = 1 and D^{2} = 0. By the previous lemma, this will give a list including all scroll types of T (c, D, {D_{λ}}), smooth or singular, for g ≤ 10 and c = 1 and D^{2} = 0. In the column with headline “# mod.” we give the value of 18 − δ_{1} + δ_{2}.

The information in the list is essentially contained in [Re] and [St, p.8-10]. We include it for completeness, and for the benefit of the reader we also include, in Remark 8.8 below, a few words about how the information can be obtained.

g scroll type # mod. g scroll type # mod. g scroll type # mod.
5 (1, 1, 1)  18  9 (3, 2, 2)  18  12 (5, 3, 2)  16
6 (2, 1, 1)  18  10 (5, 2, 1)  16  12 (4, 4, 2)  17
7 (3, 1, 1)  16  10 (4, 3, 1)  17  12 (4, 3, 3)  18
7 (2, 2, 1)  18  10 (4, 2, 2)  16  13 (7, 3, 1)  18
8 (3, 2, 1)  17  10 (3, 3, 2)  18  13 (6, 3, 2)  16
8 (2, 2, 2)  18  11 (5, 3, 1)  17  13 (5, 4, 2)  17
9 (4, 2, 1)  16  11 (4, 3, 2)  17  13 (5, 3, 3)  16
9 (3, 3, 1)  17  11 (3, 3, 3)  18  13 (4, 4, 3)  18
12 (6, 3, 1)  17

This gives the following possibilities for singular types:

g singular scroll types
5  (2, 1, 0)
6  (3, 1, 0), (2, 2, 0)
7  (4, 1, 0), (3, 2, 0)
8  (4, 2, 0)
9  (5, 2, 0)
10  (6, 2, 0)

The dimensions of the families on the singular scrolls of type (e_{1}, e_{2}, e_{3}) in P^{g} are equal to those of type (e_{1} + 1, e_{2} + 1, e_{3} + 1) on the corresponding smooth scrolls in P^{g+3}.

Remark 8.8. Among the smooth scroll types listed above, we may immediately conclude that a general section of 3H − (g − 4)F is smooth, and hence a smooth projective model of a K3 surface, for the types

(1, 1, 1), (2, 1, 1), (3, 1, 1), (2, 2, 1), (2, 2, 2), (3, 2, 2),
(4, 2, 2), (3, 3, 2), (3, 3, 3), (4, 3, 3), (5, 3, 3), (4, 4, 3).

These are the ones with 3e_{3} ≥ g − 4. The last inequality implies that the complete linear system 3H − (g − 4)F has no base points, and hence a Bertini argument gives smoothness of the general section. The remaining types have the third directrix (of degree e_{3}) as base locus. We have seen above that for the type (3, 2, 1) the zero scheme of the general section of 3H − (g − 4)F is smooth, since 3e_{2} ≥ g − 4 and e_{2} + 2e_{3} = g − 4. The same identities hold for the types (4, 3, 2), (4, 4, 2) and (5, 4, 2) also, so the zero scheme of a general section of 3H − (g − 4)F is smooth for these types too.

A similar argument can be made for the types (3, 3, 1), (4, 3, 1), (5, 3, 1), (6, 3, 1) and (7, 3, 1). Here the identity 3e_{3} < g − 4 gives that the third directrix curve (a line) consists of base points for the linear system. The identity 3e_{2} ≥ g − 4 gives that there are no other base points. The identity e_{1} + 2e_{3} = g − 4 gives that in each fiber the curve that arises as the intersection of that fiber and the zero scheme of a section of 3H − (g − 4)F is smooth at (0, 0, 1), provided we choose the section with a non-zero cZ_{1}Z_{3}^{2}-term. The total zero scheme is also smooth then.

The remaining possible smooth scroll types for g ≤ 13 on the list above are different, in the sense that for a general section of 3H − (g − 4)F, the zero scheme of the section is singular at finitely many points. It turns out that for these types, which are (4, 2, 1), (5, 2, 1), (5, 3, 2) and (6, 3, 2), the general zero schemes are singular at exactly one point each.

The reason is the following: Since 3e_{3} < g − 4, the third directrix curve consists of base points for the linear system. Since 3e_{2} ≥ g − 4 for these types, there are no other base points. Since e_{2} +2e_{3} < g −4, there is no Z_{2}Z_{3}^{2}-term for any section. Since e_{1} +2e_{3} > g −4, in fact e_{1} + 2e_{3} = (g − 4) + 1 for all these types, there is no cZ_{1}Z_{3}^{2}-term with c a constant, but there is an L(t, u)Z_{1}Z_{3}^{2}-term with L(t, u) a linear expression in t and u. If the section is chosen general, L(t, u) is not identically equal to zero. For all fixed (t, u) where L(t, u) ≠ 0, the zero scheme of the section of 3H − (g − 4)F is then smooth. For the single zero of L(t, u), the zero scheme is however singular.

A comment about the types not appearing on the list above: The smooth scroll types (4, 4, 1), (5, 4, 1), (6, 4, 1), (5, 5, 1) are eliminated the following way: A section of 3H−(g−4)F can have no term containing a Z_{3}^{2}Z_{i}-term for these scroll types. Hence all plane cubics in the pencil are singular where they meet the linear directrix. But this is a contradiction, since the general element in the pencil |D| is smooth. Here we have used the identity e_{1} + 2e_{3} < g − 4 for these types. The other types are eliminated because Z_{1} must be a factor in each relevant section, a contradiction. These are the types with 3e_{2} < g − 4.

The necessary and sufficient condition “e_{1} + 2e_{3} < g − 4 or 3e_{2} < g − 4” for eliminating scroll types is given in [Re], as quoted in [St, Lemma 1.8.]. In [St, p.9] one also describes on which scroll types a general section of 3H − (g − 4)F is singular in a finite number of points.

In Section 10 we will show that all the types listed above for g ≤ 10 actually occur.

Remark 8.9. One does not have to use the resolution from Proposition 6.2 to see that a projective model S′ of a K3 surface of Clifford index one in a smooth scroll T as above must be of divisor type 3H−(g−4)F in T . Define the vector space W = H^{0}(J_{S′}(3))/H^{0}(J_{T} (3)). In a natural way W represents the space of cubic functions on T that vanish on S′.

Study the exact sequences:

0 ⟶ H^{0}(J_{S′}(3)) ⟶ H^{0}(O_{P^{g}}(3)) ⟶ H^{0}(O_{S′}(3)) ⟶ H^{1}(J_{S′}(3)) ⟶ 0

and

0 ⟶ H^{0}(J_{T} (3)) ⟶ H^{0}(O_{P^{g}}(3)) ⟶ H^{0}(O_{T} (3)) ⟶ H^{1}(J_{T} (3)) ⟶ 0.

One obtains dim W = h^{0}(J_{S′}(3))−h^{0}(J_{T} (3)) = g −3, since h^{0}(O_{T} (3))−h^{0}(O_{S′}(3)) = g −3 and h^{1}(J_{S′}(3)) = h^{1}(J_{T} (3)) = 0 (see [SD, Prop 1.5(i)]). Take g −3 arbitrary fibers F of the ruling on T , that is g − 3 planes. For each plane it is one linear condition on the elements in W to contain it (since this is equivalent to contain an point in the plane outside S′). Hence containing all the g − 3 planes imposes g − 3 conditions. These conditions must be independent, since otherwise there would be a cubic in P^{g}, not containing T , and containing the union of S′ and g − 3 planes. This union has degree (2g − 2) + (g − 3) = 3g − 5. But by Bezout’s theorem the cubic and T intersect in a surface of degree 3g − 6. Hence, in particular, any choice of g − 4 planes gives independent conditions, and there is one, and only one, hypercubic (modulo the ideal of T ), which contains S′ and g − 4 planes in the pencil. By Bezout’s theorem, it does not contain more. Hence S′ in a natural way is a section of 3H − (g − 4)F.

8.2. Projective models with c = 2. We have g ≥ 7. Let the scroll T formed from S′ as usual have type (e_{1}, e_{2}, e_{3}, e_{4}). Proposition 6.2(a), gives that S′ is a complete intersection in T of two divisors of type 2H − b_{1}F and 2H − b_{2}F. By convention, we set b_{1} ≥ b_{2}. Part (d) of the same proposition gives the well-known fact that b_{1} + b_{2} = g − 5. Such a situation has been thoroughly investigated in [Br]. As already mentioned in the proof of Corollary 7.27, it follows from [Br, Thm. 5.1] that S′ is singular along a curve if b_{1} > \frac{2(e_{1}+e_{2}+e_{3})}{3}, or equivalently b_{2} < \frac{g−9+2e_{4}}{3} . Hence b_{2} ≥ 0 for g ≥ 7, for complete intersections with only finitely many singularities. Since in particular b_{2} ≥ −1, it is clear that part (b) and (c) of Proposition 6.2 can be used to give a resolution of S′ in P^{g}. This resolution is minimal because of the comment in [Sc, Example 3.6]. The fact that b_{2} ≥ 0 means that S′ can be viewed geometrically as a complete intersection of one hyperquadric containing b_{1} threeplanes in the pencil, and another containing b_{2} three-planes (throwing away the three-planes and taking the closure of what remains).

Let us study projective models in smooth scrolls for c = 2 and D^{2} = 0 in general. We see from Proposition 8.1 that the set parametrizing the scrolls having the same type as T has dimension (g + 1)^{2} − 19 − δ_{1} = dim(Aut(P^{g})) − 18 − δ_{1}, where δ_{1} = ∑_{i,j} max(0, e_{i} − e_{j} − 1). Therefore one expects the set of projective models of smooth K3 surfaces in each scroll to have dimension 36 if the scroll type is reasonably well balanced, to get a set of total dimension dim(Aut(P^{g})) + 18. This “expectation” is based on the natural assumption that a set of total dimension dim(Aut(P^{g}))+18 arises from S′ that sit inside maximally balanced scrolls. In this case there are two different sources of imbalance; that of the e_{i}, and that of the b_{k}. We will look more closely at this.

Set δ_{2} = max(0, b_{1} − b_{2} − 1). Assume first b_{1} > b_{2}. We calculate

dim |O_{T} (2H − b_{1}F)| = 5g − 6 − 10b_{1} + δ_{3},

where δ_{3} = h^{1}(Sym^{2} E ⊗ O_{P^{1}}(−b_{1})) = 0 if and only if e_{4} ≥ \frac{b_{1}−1}{2}.

By the same sort of calculation we of course get

dim |O_{T} (2H − b_{2}F)| = 5g − 6 − 10b_{2} + δ_{4},

where δ_{4} = h^{1}(Sym^{2} E ⊗ O_{P^{1}}(−b_{2})) = 0 if and only if e_{4} ≥ \frac{b_{2}−1}{2}.

Now fix a zero scheme Y of a section s of 2H − b_{1}F, and study the exact sequence

0 ⟶ O_{T} ((b_{1} − b_{2})F) ⟶ O_{T} (2H − b_{2}F) ⟶ O_{Y} (2H − b_{2}F) ⟶ 0.

This induces a sequence

0 ⟶ H^{0}(O_{T} ((b_{1} − b_{2})F)) ⟶ H^{0}(O_{T} (2H − b_{2}F))
⟶ H^{0}(O_{Y} (2H − b_{2}F)) ⟶ H^{1}(O_{T} ((b_{1} − b_{2})F)) ⟶ 0.

Since h^{0}(O_{T} ((b_{1} − b_{2})F)) = b_{1} − b_{2} + 1 and h^{1}(O_{T} ((b_{1} − b_{2})F)) = 0, we obtain

dim |O_{Y} (2H − b_{2}F)| = dim |O_{T} (2H − b_{2}F)| − (b_{1} − b_{2} + 1)
= 5g − 6 − 10b_{2} + δ_{4} − (b_{1} − b_{2} + 1)
= 5g − 8 − 10b_{2} + δ_{4} − δ_{2}.

Summing up, we obtain

dim O_{T} (2H − b_{1}F) + dim O_{Y} (2H − b_{2}F)
= 5g − 6 − 10b_{1} + δ_{3} + 5g − 8 − 10b_{2} + δ_{4} − δ_{2}
= 10g − 14 − 10(b_{1} + b_{2}) − δ_{2} + δ_{3} + δ_{4}
= 36 − δ_{2} + δ_{3} + δ_{4}.

In particular, if g is even, b_{1} − b_{2} = 1 and e_{4} ≥ \frac{b_{1}−1}{2} = \frac{g−6}{4}, we get 36. We remark that \frac{g−6}{4} is just \frac{3}{4} less than the average values of the e_{i}. In general there is thus a (36 − δ_{2} + δ_{3} + δ_{4})- dimensional set of complete intersections of type (2H−b_{1}F, 2H−b_{2}F), provided the section s is uniquely determined. The latter fact follows, for example by the same kind of argument as in [Sc], section (6.2), where scrolls from tetragonal curves are treated (see also the proof of Proposition 8.12 below).

We now assume b_{1} = b_{2}(= b = \frac{g−5}{2}), which can only occur if g is odd. Then h^{0}(2H−bF) = h^{0}(P^{1}, Sym^{2}(E) ⊗ O_{P^{1}}(b)). This number is of the form 20 + δ_{3} = 20 + δ_{4}, where δ_{3} and δ_{4} are defined as in the case above. We see that δ_{3} = δ_{4} = 0 if and only if 2e_{4} − b ≥ −1, or equivalently e_{4} ≥ \frac{g−7}{4}. The average value of the e_{i} is \frac{g−3}{4}, which is just one more. The set of complete intersections corresponds to an open set in the Grassmannian G(2, h^{0}(2H − bF)), which has dimension 36 + 2δ_{3}. Hence we get the expected number if b_{1} = b_{2}, and the e_{i} are well balanced. Whether b_{1} = b_{2} or not, we have now proved:

Proposition 8.10. The set of complete intersection schemes of type (2H − b_{1}F, 2H − (g − 5 − b_{1})F) on a smooth rational normal scroll of dimension 4 in P^{4} of a fixed scroll type, is either empty or of dimension 36 − δ_{2} + δ_{3} + δ_{4}.

Corollary 8.11. The set of complete intersection schemes of type (2H−b_{1}F, 2H−(g −5− b_{1})F) with no or finitely many singularities on a smooth rational normal scroll of dimension 4 in P^{4} of a fixed scroll type, is either empty or of dimension at least 36 if δ_{1} ≥ 1.

Proof. Set s = b_{1} − \frac{g−5}{2}. If s = 0, then δ_{2} = 0, and there is nothing to prove. If s ≥ \frac{1}{2}, then δ_{2} = 2s − 1. We split into 4 subcases: e_{1} + e_{2} + e_{3} + e_{4} = 4e + h, where h = 0, 1, 2, 3. If h = 0, 1, we have if δ_{1} ≥ 1: 2e_{4} ≤ 2(⌊\frac{g−3}{4}⌋ − 1) ≤ \frac{g−7}{2}, and e_{3} + e_{4} ≤ \frac{g−3}{2} − 1 = \frac{g−5}{2}. This gives b_{1} −2e_{4} −1 ≥ \frac{g−5}{2} +s− \frac{g−7}{2} −1 = s, and b_{1} −(e_{3} +e_{4})−1 ≥ \frac{g−5}{2} +s− \frac{g−1}{2} −1 = s−1. Hence

δ_{3} = h^{1}(P^{1}, Sym^{2}(E) ⊗ O_{P^{1}}(b_{1})) ≥ s + (s − 1) = 2s − 1 = δ_{2},

and then the result follows from Proposition 8.10.

If h = 3, essentially the same method works (look at the three terms b_{1} − 2e_{3} − 1, b_{1} − e_{3} − e_{4} − 1 and b_{1} − 2e_{4} − 1).

If h = 2, then essentially the same method works (look at the six terms of the form b_{1} −e_{i} −e_{j} −1, for i, j = 2, 3, 4), except in the case s = 1, and (e_{1}, e_{2}, e_{3}, e_{4}) = (e+2, e, e, e). But in that case b_{1} = \frac{g−1}{2} + 1 = 2e + 1. Using Lemma 7.26 we see that that Z_{1} is a factor in every section of 2H − b_{1}F, so this case simply does not occur. See also the appendix of [Br].

If we add the assumption that there exists at least one smooth model S′ of a given intersection type, giving rise to a scroll of a given type, then we can conclude that there is a set of dimension (36 − δ_{2} + δ_{3} + δ_{4}) parametrizing smooth projective models S′ in a scroll of the given type. We see that if the scroll type or (b_{1}, b_{2})-type is unbalanced, the dimension of the set of complete intersections (smooth or singular) can a priori go up, or it can go down from its “expected” value 36. From the last corollary, however, we see that the dimension goes down only if the scroll type is maximally balanced, and the intersection type is not.

If we just start with an arbitrary scroll type (e_{1}, e_{2}, e_{3}, e_{4}), with e_{4} ≥ 1, ∑_{i−1}^{c+2} e_{i} = g − 3 and “intersection type” b_{1} ≥ b_{2} ≥ 0, with b_{1} + b_{2} = g − 5, it is an intricate question to decide whether there are any that are smooth projective models of K3 surface, or any that are not singular along a curve. This problem is studied in detail in [Br], and we will study the cases for low g in Subsection 8.4.

The question whether a projective model S′ can be included in several scrolls of the same type simultaneously is not as simple to answer as in the case c = 1. In the case c ≥ 2 the scroll T is no longer the intersection of the hyperquadrics containing S′; in fact S′ itself is that intersection [SD]. The question is essentially how many divisor classes of elliptic curves E with E.L = c + 2 there are on S.

On the other hand it is clear that a projective model cannot be contained in a positive dimensional set of scrolls of the type in question. There is a discrete family of divisor classes on S, so the dimension of the family of smooth projective models of K3 surfaces on scrolls of the type in question is now, as for c = 1, equal to the sum of the dimension of the set of scrolls of a given type and the dimension of the set of smooth projective models of that type. This sum is

(g + 1)^{2} − 19 − δ_{1}) + (36 − δ_{2} + δ_{3} + δ_{4})
= dim(Aut(P^{g})) + 18 − δ_{1} − δ_{2} + δ_{3} + δ_{4}.

To obtain the dimension of the set of projective equivalence classes, since only a finite number of automorphisms of P^{g} fixes a K3 surface, we subtract the number dim(Aut(P^{g})) and get

18 − δ_{1} − δ_{2} + δ_{3} + δ_{4}.

By Theorem 3.8 this number is equal to 18 for at least one scroll type, where there exists smooth complete intersections of that type, and where the fibers of the complete intersection represent a free Clifford divisor on S with c = 2. It is clear that if we choose the most balanced scroll type for a fixed g, then δ_{1} = 0. If in addition we choose the most balanced (b_{1}, b_{2})-type, then δ_{2} = δ_{3} = δ_{4} = 0. Using Lemma 7.26 it is also easy to prove that for all g the most balanced scroll and intersection type (the unique combination for fixed g with δ_{1} = δ_{2} = 0) then a general complete intersection will be a smooth projective model of a K3 surface.

8.3. An interpretation of b_{1} and b_{2}. We will briefly study the case c = 2 in an analogous manner as the case c = 1 was studied in Remark 8.9 when we showed that a projective model of a K3 surface of Clifford index one with smooth asssociated scroll T must be of divisor type 3H + (g − 4)F in T without using the resolution from Proposition 6.2.

Define the vector space W = H^{0}(J_{S′}(2))/H^{0}(J_{T} (2)). In a natural way W represents the space of quadric functions on T that vanish on S′.

As in Remark 8.9 one obtains dim W = h^{0}(J_{S′}(2))−h^{0}(J_{T} (2)) = g−3, since h^{0}(O_{T} (2))− h^{0}(O_{S′}(2)) = g − 3 and h^{1}(J_{S′}(2)) = h^{1}(J_{T} (2)) = 0 (see [SD, Prop. 1.5(i) and Theorem 6.1(ii)]. Assume g is odd. Take b = \frac{g−5}{2} arbitrary fibers F of the ruling on T , that is threeplanes. For each three-plane we have two independent linear conditions on the quadric hypersurfaces to contain it. (First, take one point in the three-plane, not on S′. There is only one quadric surface in the threespace containing this point and the intersection with S′. Then take another point outside this quadric surface. To contain these two points and S′ is equivalent to containing the threespace and S′.) So, one naively expects there to be 2b = g − 5 conditions to contain all the b three-planes. Hence there should be a pencil, and only a pencil, of elements of W doing so. Intersecting the elements of the pencil, one would expect to get the projective model of the K3 surface. If it really were so simple, however, all intersection types (b_{1}, b_{2}) would be completely balanced. This is not always true, and one reason is that two different elements of W may intersect T in a common threedimensional component dominating P^{1} in the fibration on T .

We are therefore not able to imitate the reasoning of Remark 8.9, and thereby establish the fact that the ideal of S′ in T is generated as it is, without using Proposition 6.2. On the other hand we may use the knowledge that we have from Proposition 6.2, that S′ is indeed of intersection type (2H − b_{1}F, 2H − b_{2}F) in its scroll. Make no assumption on the parity of g.

Proposition 8.12. The invariant b_{1} is equal to the largest number k, such that there exists a non-zero element Q of W (a hyperquadric in P^{g} containing S′, but not T ) containing k three-planes in the pencil.

Moreover, b_{2} is the largest number m, such that there exists a non-zero element of W containing m three-planes in the pencil, and intersecting T in a different 3-dimensional dominant component than Q does.

Proof. Define two elements in W to be congruent if they have the same dominating threedimensional component (but possibly differ in which three-planes they contain). Define the index of an element of W as the number of three-planes it contains (if necessary, counted with multiplicity). It is clear that if two elements of W are congruent, then they have the same index (they correspond to a well defined divisor class 2H − iF, where i is the index). It follows from a Bezout argument that if the sum of the indices of two elements is larger than 2b = g − 5, then they must be congruent. This shows both assertions. It also shows that the element in 2H−b_{1}F which any element in W with index larger than b gives rise to, is the same. (This element is nothing but the congruence class of the element in W.)

8.4. Possible scroll types for c = 2. Almost all the information in this subsection can also be found in [Br] and [St], taken together, but we include it for completeness, and present it in our own way, for the sake of the reader.

Also in the case c = 2 it is possible to use Lemma 7.26 to obtain useful conclusions for many concrete scroll types. First we will look at possible smooth scroll types for g ≤ 10.

For g = 7 and g = 8, the only possible smooth scroll types are (1, 1, 1, 1) and (2, 1, 1, 1), respectively. For g = 7, Lemma 7.26 immediately gives b_{1} ≤ 2. Here b_{1} + b_{2} = 2, so b_{2} ≥ 0.

For g = 8 we see that Z_{1} is a factor in every section of 2H − b_{1}F, for b_{1} ≥ 3. Since a reducible section which is the sum a section H − 2F and a section H − (b − 2)F would intersect another section of type 2H − b_{2}F in something reducible, alternatively since S′ is non-degenerate, we must have b_{1} = 2 and b_{2} = 1.

For g = 9 we have b_{1} + b_{2} = 4 and two smooth scroll types (2, 2, 1, 1) and (3, 1, 1, 1). For the latter type Lemma 7.26 gives b_{1} = b_{2} = 2, since any section of 2H − bF, with b ≥ 3 must be a sum of a section H − 3F and H. For the type (2, 2, 1, 1) any section of 2H − 3F has a zero scheme containing the subscroll generated by the two linear directrices (a quadric surface Q). If b_{1} = 3, then b_{2} = 1, and the section 2H − b_{2}F intersects Q as a curve of type (1, 2) on Q. This is a rational twisted cubic Γ. This corresponds to the fact that δ_{1} = δ_{3} = δ_{4} = 0 and δ_{2} = 1, so that the set of complete intersections inside T has dimension at most 36 − δ_{2} + δ_{3} + δ_{4} = 35, and taking the union over all T of the same type we get dimension at most dim(Aut(P^{g})) + 17.

Any section of 2H − 4F has a zero scheme, which restricts to two lines in each fiber of T . This is impossible if this scheme shall contain a (necessarily non-degenerate) model S′. We also have h^{0}(2H − bF) = 0 for b ≥ 5. Hence b_{1} is 2 or 3.

For g = 10, we have b_{1} + b_{2} = 5 and a priori three possible scroll types (4, 1, 1, 1), (3, 2, 1, 1) and (2, 2, 2, 1). But the first cannot occur, since any section of 2H − b_{1}F must have total weight −b_{1} ≤ −3, and then Z_{1} must be a factor, using Lemma 7.26. Since S′ is non-degenerate, this is impossible. Likewise, if T has type (3, 2, 1, 1) and b_{1} ≥ 5, we conclude that Z_{1} must be a factor, again impossible. The cases b_{1} = 4 and b_{1} = 3 are however possible.

If T has type (2, 2, 2, 1), then b_{1} ≤ 4, since h^{0}(2H − bF) = 0 if b ≥ 5. If b_{1} = 4, then the zero scheme of any section of 2H − b_{1}F = 2H − 4F contains the linear directrix of T twice (its equation is a homogeneous quadric in Z_{1}, Z_{2}, Z_{3} involving neither Z_{4}, t nor u). If we intersect with a section of 2H − b_{2}F = 2H − F, and interpret it as the intersection with a quadric containing a fiber, and throw away the fiber, the residual intersection with T must contain one point of its linear directrix. This must then be a singular point of S′. Hence only b_{1} = 3, b_{2} = 2 gives a smooth S′ for this scroll type. For these invariants δ_{i} = 0, for i = 1, 2, 3, 4, so we get a family of total dimension dim(Aut(P^{g})) + 18.

So far we have studied smooth scroll types for 7 ≤ g ≤ 10. At this point we could either proceed with smooth scroll types for g ≥ 11, or look at singular scroll types for g ≥ 7. These topics are closely related. Assume we have a singular scroll T (c, D, {D_{λ}}) for g ≥ 7, D^{2} = 0, c = 2. Then the associated smooth scroll T_{0} is contained in P^{g+4}, with a resolution as in Proposition 7.14:

0 ⟶ O_{T_{0}}(−4H_{0} + (g − 1)F) ⟶ ⊕_{k=1}^{2}O_{T_{0}}(−2H_{0} + b_{k}F) ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.

So, at this point we can use the method of rolling factors to check what scroll types in P^{g+4} that may contain a surface like S′′. Scroll types (e_{1} + 1, …, e_{4} + 1) for T_{0} correspond to types (e_{1}, …, e_{4}) for T .

Let us study complete intersection surfaces in smooth scroll types for g = 11 with this dual viewpoint. Now b_{1} + b_{2} = 6 and deg T = 8 and there are a priori 5 different possible scroll types:

(5, 1, 1, 1), (4, 2, 1, 1), (3, 2, 2, 1), (3, 3, 1, 1) and (2, 2, 2, 2).

The type (5, 1, 1, 1) cannot occur, for the same reason that (4, 1, 1, 1) cannot occur for g = 10. For (4, 2, 1, 1) and (3, 3, 1, 1) we can conclude that Z_{1} or Z_{2} is a factor in every term of every section of 2H − 3F. This gives that the subscroll formed by the two linear directrices is contained in S′ (or S′′). This is clearly impossible. Hence b_{1} ≥ 4 for these types.

If b_{1} ≥ 5, then Z_{1} is a factor in every section of 2H − b_{1}F, for each of the types (4, 2, 1, 1), (3, 2, 2, 1) and (2, 2, 2, 2), which gives a contradiction. For the type (3, 3, 1, 1) we argue as follows: If b_{1} ≥ 5, then no term of the form Z_{i}Z_{3} or Z_{i}Z_{4} can occur as factor in a monomial of a section of 2H − b_{1}F, so for each fixed value of (t, u) we get a quadric in Z_{1}, Z_{2} only. This defines a union of two planes in each P^{3} which is a fiber of T (or T_{0}). Hence each fiber of S′ (or S′′) is degenerate, a contradiction. So b_{1} = 4.

We make the same kind of considerations for all g ≤ 14. We end up with the following a priori possible combinations of smooth scroll type and b_{1}, for 7 ≤ g ≤ 14 (of course b_{2} = g − 5). For each scroll type and intersection type (b_{1}, b_{2}) we indicate whether the general zero scheme of a complete intersection of type (2H − b_{1}F, 2H − b_{2}F) is smooth or singular. See also Remark 8.13. In the column with headline “ # mod.” we give the value of 18 − δ_{1} − δ_{2} + δ_{3} + δ_{4}. This table contains information that can also be found in [Br] and [St], taken together. In [Br] all possible scroll and intersection types for smooth scrolls for all g ≥ 7 are listed, and in [St] the information on the moduli of the corresponding families are given. We include the list for g ≤ 14, since it will be useful in the study of projective models on singular scrolls, and for the lattice-theoretical considerations in Section 10.

g scroll type b_{1} comp. int. # mod. g scroll type b_{1} comp. int. # mod.
7 (1, 1, 1, 1) 1 Smooth  18  12 (4, 3, 1, 1) 5 Smooth  16
7 (1, 1, 1, 1) 2 Smooth  17  12 (4, 2, 2, 1) 4 Singular  15
8 (2, 1, 1, 1) 2 Smooth  18  13 (3, 3, 2, 2) 4 Smooth  18
9 (2, 2, 1, 1) 2 Smooth  18  13 (3, 3, 2, 2) 5 Smooth  17
9 (2, 2, 1, 1) 3 Smooth  17  13 (3, 3, 3, 1) 4 Smooth  17
9 (3, 1, 1, 1) 2 Smooth  15  13 (3, 3, 3, 1) 6 Smooth  18
10 (2, 2, 2, 1) 3 Smooth  18  13 (4, 2, 2, 2) 4 Smooth  15
10 (2, 2, 2, 1) 4 Singular  17  13 (4, 3, 2, 1) 4 Singular  16
10 (3, 2, 1, 1) 3 Smooth  16  13 (4, 3, 2, 1) 5 Smooth  16
10 (3, 2, 1, 1) 4 Singular  17  13 (4, 3, 2, 1) 6 Smooth  18
11 (2, 2, 2, 2) 3 Smooth  18  13 (5, 3, 1, 1) 6 Smooth  17
11 (2, 2, 2, 2) 4 Smooth  17  14 (3, 3, 3, 2) 5 Smooth  18
11 (3, 2, 2, 1) 3 Smooth  17  14 (3, 3, 3, 2) 6 Singular  17
11 (3, 2, 2, 1) 4 Smooth  17  14 (4, 3, 2, 2) 5 Smooth  16
11 (4, 2, 1, 1) 4 Singular  15  14 (4, 3, 2, 2) 6 Singular  17
11 (3, 3, 1, 1) 4 Smooth  16  14 (4, 3, 3, 1) 5 Smooth  17
12 (3, 2, 2, 2) 4 Smooth  18  14 (4, 4, 2, 1) 5 Singular  16
12 (3, 3, 2, 1) 4 Smooth  17  14 (5, 3, 2, 1) 5 Singular  15
12 (3, 3, 2, 1) 5 Smooth  17  14 (5, 3, 2, 1) 6 Singular  16

For a perfect Clifford divisor D we get the following possibilities for the corresponding singular T in P^{g}, for 7 ≤ g ≤ 10 (subtracting 2 from all values of b_{i} for T_{0}, for i = 1, 2):

g sing. scroll type b_{1} (S′′)_{virt} g sing. scroll type b_{1}  (S′′)_{virt}
7  (2, 1, 1, 0)  1, 2 Smooth 9  (4, 2, 0, 0)  4  Smooth
7  (2, 2, 0, 0)  2 Smooth 9  (3, 2, 1, 0)  2, 3 Smooth only for b_{1} = 3
7  (3, 1, 0, 0)  2 Singular 9  (2, 2, 2, 0)  2  Smooth
8  (3, 2, 0, 0)  3 Smooth 10  (4, 2, 1, 0)  3, 4  Singular
8  (3, 1, 1, 0)  2 Singular 10  (3, 3, 1, 0)  3  Singular
8  (2, 2, 1, 0)  2 Smooth 10  (3, 2, 2, 0)  3  Smooth

Remark 8.13. For each smooth scroll T and intersection type where the general element S′ is smooth (on the upper list of types for 7 ≤ g ≤ 14) it is clear that we have a smooth projective model of a K3 surface. For the remaining cases (on that list) it is natural to interpret them as projective models S′ of K3 surfaces by non-ample linear systems. The types on the upper list are the only ones for g ≤ 14 where a general complete intersection (2H − b_{1}F, 2H − b_{2}F), with b_{1} + b_{2} = g − 5, is either smooth, or singular in a finite number of points. In Section 10, moreover, we describe all projective models for low g, including those with c = 2. All scroll types listed above (smooth or singular) for g ≤ 10 reappear in the list in Section 10.

To decide which intersections that are in general smooth, which intersections that are in general singular in finitely many points, and which intersections that are in general singular along a curve (or even reducible) one uses rolling factors (Lemma 7.26) as in [Br] and [St]. In particular we have checked with the Appendix in [Br], which gives a list of smooth complete intersection K3 surfaces in 4-dimensional smooth rational normal scrolls and also a list of relevant intersections with only finitely many singularities.

On the lower list, concerning singular scrolls for ≤ 10, we have listed all scroll and intersection types which might a priori appear as “images” of scrolls T_{0} and surfaces S′′ on the upper list.

In the columns with heading (S′′)_{virt} we have indicated whether a general complete intersection of type in question on T_{0}, which contains the exceptional divisor of the map i from T_{0} to T , is smooth or singular. We call such a complete intersection (S′′)_{virt}, since we do not a priori know that it is an S′′. If D is perfect and the scroll T is singular, each occurring projective model S′ on T is of course also singular, but S′′ smooth means that all singularities of S′ are due to contractions across the fibers; there are no contractions in the individual fibers.

We will illustrate that the issues whether a smooth scroll T (2, D) and an associated intersection type for a model S′ appears on the upper list, is different from the issue whether the scroll and intersection type appears for a T_{0} and a S′′. If D is perfect, it is a priori possible that all complete intersections, or a general one, represents a model S′, but not an S′′. As an example, look at the type (4, 3, 2, 1), with (b_{1}, b_{2}) = (6, 2). Then S′ consists of the common zeroes of two sections of the form

P_{2}(t, u)Z_{1}^{2} + P_{1}(t, u)Z_{1}Z_{2} + c_{1}Z_{2}^{2} + c_{2}Z_{1}Z_{3}, and
f(t, u, Z_{1}, Z_{2}, Z_{3}, Z_{4}) + c_{4}Z_{4}^{2},

where f(t, u, Z_{1}, Z_{2}, Z_{3}, Z_{4}) is contained in the ideal generated by Z_{1}, Z_{2}, Z_{3}. The general such intersection is smooth. But in order to be a surface of the form S′′, the intersection must contain the last directrix. This forces c_{4} to be zero. In that case the intersection is no longer smooth, in fact it contains the directrix in its singular locus, and hence it cannot be an S′′. Hence scroll type (4, 3, 2, 1) with b_{1} = 6 appears on the upper list, but the corresponding “pushed down” type (3, 2, 1, 0) does not appear in combination with (the revised) b_{1} = 4.

A similar, but slightly different case, is scroll type (3, 2, 2, 1) and intersection type (4, 2). Then S′ consists of the common zeroes of two sections of the form:

P_{2,1}(t, u)Z_{1}^{2} + P_{1,1}(t, u)Z_{1}Z_{2} + P_{1,2}(t, u)Z_{1}Z_{3} + c_{1}Z_{2}^{2} + c_{2}Z_{2}Z_{3} + c_{3}Z_{3}^{2} + c_{4}Z_{1}Z_{4}

f(t, u, Z_{1}, Z_{2}, Z_{3}) + P_{2,2}(t, u)Z_{1}Z_{4} + P_{1,3}(t, u)Z_{2}Z_{4} + P_{1,4}(t, u)Z_{3}Z_{4} + c_{5}Z_{4}^{2}.

If this is an S′′, then it contains the last directrix, which means c_{5} = 0. Even if c_{5} = 0, the intersection will in general be smooth if c_{4} ≠ 0, and P_{1,3} and P_{1,4} have no common roots. In this example only a subfamily of positive codimension of the (dim Aut(P^{g}) + 18 − δ_{1} − δ_{2} + δ_{3} + δ_{4})-dimensional family of all complete intersections of that type are of the form S′′.

Remark 8.14. If we do not assume that D is perfect, then we are not able to eliminate cases that appear in the upper list, from the lower list.

For instance we would also get the possibility of scroll type (2, 2, 2, 0) with b_{1} = 4 if Sing T ∩ S′ = ∅. This concludes the proof of Proposition 7.31.

8.5. Projective models with c = 3. From Proposition 6.2(a) we get that O_{S′} has a resolution as a O_{T} -module of the following form:

0 → O_{T} (−5H + (g − 6)F) → ⊕_{k=1}^{5}O_{T} (−3H + b_{k}F) →
⊕_{k=1}^{5}O_{T} (−2H + a_{k}F) → O_{T} → O_{S′} → 0.

We see from [Sc] that we are in a situation very similar to that of a pentagonal canonical curve, which is natural, since a general hyperplane section of S′ is such a curve. We do not intend to say as much about this situation as about the cases c = 1 and 2. Study the skew-symmetrical map Φ in the resolution above, already introduced in Corollary 7.27:

Φ : ⊕_{k=1}^{5}O_{T} (−3H + b_{k}F) → ⊕_{k=1}^{5}O_{T} (−2H + a_{k}F).

Recall that the Pfaffians of this map generate the ideal of S′ in T . Assume only that S′ is contained in a rational normal scroll T of degree g − 4 in P^{g} and of type (e_{1}, …, e_{5}), where e_{5} ≥ 1.

From Proposition 8.1 the dimension of the set of scrolls of type e in P^{g} is equal to (g + 1)^{2} − 28 − δ_{1} = dim(Aut(P^{g}) − 27 − δ_{1}, where δ_{1} = ∑_{i,j} max(0, e_{i} − e_{j} − 1). To obtain the number 18 + dim(Aut(P^{g}) for the dimension of the set of projective models of K3 surfaces in scrolls of some type, one expects a 45-dimensional set of such models in a given scroll, provided the scroll type is reasonably well balanced. We will look into this issue, but we will not give a rigorous proof that we can find such a 45-dimensional set.

A given projective model S′ is characterized by the ten above-diagonal entries of a fiveby-five matrix description of the map Φ. These entries are sections of:

H − (b_{2} − a_{1})F, H − (b_{3} − a_{1})F, H − (b_{4} − a_{1})F, H − (b_{5} − a_{1})F, H − (b_{3} − a_{2})F, H − (b_{4} − a_{2})F, H − (b_{5} − a_{2})F, H − (b_{4} − a_{3})F, H − (b_{5} − a_{3})F, H − (b_{5} − a_{4})F.

We have h^{0}(T , H−(b_{i}−a_{j})F) = g+1−5(b_{i}−a_{j})+δ_{2,i,j}, where δ_{2,i,j} = h^{1}(P^{1}, E⊗O_{P^{1}}(a_{j}−b_{i})) and is zero if and only if e_{5} − (b_{i} − a_{j}) ≥ −1. In all, there is a set of choices of the ten linear terms, such that this set has dimension

10(g + 1) − ∑_{i>j}(5(b_{i} − a_{j}) + δ_{2,i,j}).

Moreover, we have

∑_{i>j}(b_{i} − a_{j}) = b_{2} + 2b_{3} + 3b_{4} + 4b_{5} − 4a_{1} − 3a_{2} − 2a_{3} − a_{4}
= b_{2} + 2b_{3} + 3b_{4} + 4b_{5} − 4(g − 6 − b_{1}) − 3(g − 6 − b_{2})
−2(g − 6 − b_{3}) − (g − 6 − b_{4})
= 4(b_{1} + ⋯ + b_{5}) − 10(g − 6)
= 2g − 12,

where we have used the self-duality of the resolution (Proposition 6.2(b)) which gives a_{i} = g − 6 − b_{i}, for i = 1, …, 5) and Proposition 6.2(d) (which gives b_{1} + ⋯ + b_{5} = 3g − 18). Inserting this in the expression above we obtain the number

10(g + 1) − 5(2g − 12) + δ_{2} = 70 + δ_{2},

where δ_{2} = ∑_{i>j} δ_{2,i,j} for the dimension of the set of choices of entries in the matrix determining the map Φ.

We see from a_{i} = g − 6 − b_{i}, for i = 1, …, 5, that ∑_{i=1}^{5} a_{i} = 2g − 12, so the average value of the b_{i} − a_{j} is \frac{g−6}{5}. The average value of the e_{i} is \frac{g−4}{5}, so if both the e_{i} and the b_{j} (and therefore the a_{j}) are maximally balanced, we will in fact have e_{5} − (b_{i} − a_{j}) ≥ −1, so δ_{2,i,j} = 0, for each i > j.

To obtain the desired value 45 in the maximally balanced situation, one needs to argue that it is correct to subtract 25, in the sence that there is typically a 25-dimensional family of matrix decriptions giving rise to each projective model of a K3 surface as described. We do not know how to do this in a rigorous way, but the problem is related to the one mentioned in [B-E, p. 457], where one treats matrix descriptions of maps between two free modules of rank 5 over a ring (see also [Be]). Translating the discussion in [B-E] into our situation, the issue is: Do two matrices A′ and A have the same Pfaffian ideal if and only if there is a matrix B, such that A′ = BAB^{t}? In an extremely simple case, take g = 11 and a_{i} = 2 for all i (and consequently b_{j} = 3 for all j), so that all entries in the matrix representation A of Φ are sections of the same line bundle on T (in this case H − F). One can imagine the set of five-by-five matrices acting on the matrix A representing Φ as A → BAB^{t}, for all B in GL(5). In a situation where the a_{i} are less balanced, one can imagine an analogous matrix B with entries in suitably manufactured line bundles, so that the “shape” of A is preserved under a similar action. By this we mean that if entry A_{i,j} of A is a section of a line bundle L_{i,j}, then entry A′_{i,j} of BAB^{t} is also a section of L_{i,j}. One must then count the sections in the entries of B, control the stabilizers of the action, and show that all A with the same Pfaffian ideal are in the same orbit by the action.

A natural candidate for such a matrix B is one where the entry B_{ij} is chosen as a general section of O_{T} ((a_{j} − a_{i})F) = O_{T} ((b_{i} − b_{j})F), for all (i, j). Since h^{0}(O_{T} ((a_{j} − a_{i})F)) + O_{T} ((a_{i} − a_{j})F) = 2 + max(0, |a_{i} − a_{j}| − 1), we see that ∑_{i,j} h^{0}(O_{T} ((a_{j} − a_{i})F) = 25 if and only if the a_{i} are chosen in a maximally balanced way. Set δ_{3} = ∑_{i>j} max(0, |a_{i} − a_{j}| − 1). Then the dimension of the set of choices of matrix B as described is 25 + δ_{3} (We see that det B is a constant, and we look at the closed subset of those B with non-zero determinant). One checks that BAB^{t} is antisymmetric, and has entries that are sections in the same line bundles as the corresponding ones for A. This leads to the following:

Conjecture 8.15. The dimension of the set M of projective models of K3 surfaces of Clifford index 3, contained in a maximally balanced smooth rational normal scroll T (3, D) of dimension 5 is equal to 45. For an arbitrary scroll type, and given combination (a_{1}, …, a_{5}) the corresponding set M is empty, or it has dimension 45 + δ_{2} − δ_{3}. We have δ_{2} ≥ δ_{3} if δ_{1} ≥ 1.

Remark 8.16. The last statement of the conjecture does not follow directly from purely numerical considerations. As an example, take the case g = 11, scroll type (3, 1, 1, 1, 1) and (a_{1}, …, a_{5}) = (1, 2, 2, 2, 3), which gives (b_{1}, …, b_{5}) = (4, 3, 3, 3, 2). Here δ_{1} = 4, and δ_{3} = 1. For all (i, j) with i > j, we have e_{5} − (b_{i} − a_{j}) ≥ −1, so δ_{2} = 0.

On the other hand the entries outside the diagonal in the first row of a matrix description of Φ are sections of H − (b_{2} − a_{1})F, H − (b_{3} − a_{1})F, H − (b_{4} − a_{1})F and H − (b_{5} − a_{1})F, which here are H − 2F, H − 2F, H − 2F and H − F. Taking the submaximal minor where we disregard the term H − F, we see that Z_{1} is a factor, since Z_{1} is a factor in every section of H − 2F. This is a contradiction, and hence the case does not occur.

8.6. Speculations for higher values of c. From Proposition 8.1 we see that the dimension of the set of scrolls of a given type in P^{g} is (g + 1)^{2} − 3 − (c + 2)^{2} − δ_{1}, where δ_{1} is a non-negative number, which is zero if and only if the scroll type is maximally balanced. Since we know that for all c in the range in question there exists a set of dimension dim(Aut(P^{g})) + 18 = (g + 1)^{2} + 17 parametrizing projective model of K3 surfaces in P^{g} with Clifford-index c fibered by elliptic curves on a scroll of some type, we know that for this type, the set of projective models of K3 surfaces of Clifford index c, with smooth associated scrolls T , has dimension at least

(g + 1)^{2} + 17 − ((g + 1)^{2} − 3 − (c + 2)^{2}) = (c + 2)^{2} + 20.

A scroll type with δ_{1} = 0 is then a natural candidate. It is therefore a challenge to try to prove the following conjecture:

Conjecture 8.17. Let g ≥ 5 and 1 ≤ c < ⌊\frac{g−1}{2}⌋. Then the dimension of the set M of projective models of K3 surfaces of Clifford index c, contained in a maximally balanced smooth rational normal scroll T (c, D, {D_{λ}}) of dimension c + 2 is equal to (c + 2)^{2} + 20. For the scroll types with δ_{1} > 0, the set M is empty, or

(c + 2)^{2} + 20 ≤ dim M ≤ (c + 2)^{2} + 20 + δ_{1}.

Remark 8.18. The inequality (c + 2)^{2} + 20 ≤ dim M is inspired from Proposition 8.2, Corollary 8.11 and Remark 8.16. (Many examples above reveal that a strict inequality is impossible here.)

The cases c = 1, scroll type (7, 3, 1), and c = 2, scroll types (3, 3, 3, 1) with b_{1} = 6, and (4, 3, 2, 1) with b_{1} = 6, reveal that a strict inequality dim M < (c + 2)^{2} + 20 + δ_{1} is impossible. In Section 10 one sees that these cases indeed occur with the fiber D a perfect Clifford divisor. In these cases both the most balanced scroll/intersection type and the mentioned non-balanced types give families of dimension dim(Aut(P^{g})) + 18.

9. BN generality and Clifford generality

It is shown in [Mu1] that a general projective model of a K3 surface in P^{g}, for g = 6, 7, 8, 9 and 10, is a complete intersection in a homogeneous spaces described below.

We recall the following definition of Mukai:

Definition 9.1. [Mu2] A polarized K3 surface (S, L) of genus g is said to be Brill-Noether (BN) general if the inequality h^{0}(M)h^{0}(N) < h^{0}(L) = g + 1 holds for any pair (M, N) of non-trivial line bundles such that M ⊗ N ≃ L.

Clearly the polarized K3 surfaces which are BN general form a 19-dimensional Zariski open subset in the moduli space of polarized K3 surfaces of a fixed genus g.

The following theorem is due to Mukai. We use the following convention: For a vector space V^{i} of dimension i, we write G(r, V^{i}) (resp. G(V^{i}, r)) for the Grassmann variety of r-dimensional subspaces (resp. quotient spaces) of V.

The variety Σ_{12}^{10} ⊆ P^{15} is a 10-dimensional spinor variety of degree 12. Let V^{10} be a 10-dimensional vector space with a nondegenerate second symmetric tensor λ. Then Σ_{12}^{10} is one of the two components of the subset of G(V^{10}, 5) consisting of 5-dimensional totally isotropic quotient spaces ^{1}.

The variety Σ_{16}^{6} ⊆ P^{13} is the Grassmann variety of 3-dimensional totally isotropic quotient spaces of a 6-dimensional vector space V^{6} with a nondegenerate second skew-symmetric tensor σ. It has dimension 6 and degree 16.

Also, Σ_{18}^{5} = G/P ⊆ P^{13}, where G is the automorphism group of the Cayley algebra over C and P is a maximal parabolic subgroup. The variety has dimension 5 and degree 18.

Finally, in the case g = 12, let V^{7} be a 7-dimensional vector space and N ⊆ ∧^{2}V^{∨} a 3-dimensional vector space of skew-symmetric bilinear forms, with basis {m_{1}, m_{2}, m_{3}}. We denote by Grass(3, V^{7}, m_{i}) the subset of Grass(3, V^{7}) consisting of 3-dimensional subspaces u of V such that the restriction of m_{i} to U × U is zero. Then Σ_{12}^{3} = Grass(3, V^{7}, N) := ∩ Grass(3, V^{7}, m_{i}). It has dimension 3 and degree 12.

Theorem 9.2. [Mu2] The projective models of BN general polarized K3 surfaces of small genus are as follows:

genus projective model of BN general polarized K3 surface
2  S_{2} ⟶ P^{2} double covering with branch sextic
3  (4) ⊆ P^{3}
4  (2, 3) ⊆ P^{4}
5  (2, 2, 2) ⊆ P^{5}
6  (1, 1, 1, 2) ∩ G(2, V^{5}) ⊆ P^{6}
7  (1^{8}) ∩ Σ_{12}^{10} ⊆ P^{7}
8  (1^{6}) ∩ G(V^{6}, 2) ⊆ P^{8}
9  (1^{4}) ∩ Σ_{16}^{6} ⊆ P^{9}
10  (1^{3}) ∩ Σ_{18}^{5} ⊆ P^{10}
12  S_{12} = (1) ⊆ Σ_{12}^{3}

We will in this section compare the notions of BN generality with our notion of Clifford generality. For reasons of space, we will leave out the case g = 12.

We have

Proposition 9.3. Let (S, L) be a polarized K3 surface of genus g = 2, 3 …, 7 or 9. Then (S, L) is BN general if and only if it is Clifford general.

If g = 8 resp. 10, then (S, L) is Clifford general but not BN general if and only if there is an effective divisor D satisfying D^{2} = 2 and D.L = 7 resp. 8, and there are no divisors satisfying the conditions (∗) for c < 3 resp. 4.

Proof. We must investigate the condition that there exists an effective decomposition L ∼ D + F such that h^{0}(F)h^{0}(D) ≥ g + 1, but Cliff O_{C}(D) ≥ ⌊\frac{g−1}{2}⌋ for any smooth curve C ∈ |L|.

1A quotient f : V → V′ is totally isotropic with respect to λ if (f ⊗ f)(λ) is zero on V′ ⊗ V′.

Since Cliff O_{C}(D) ≤ g + 1 − h^{0}(D) − h^{0}(F), we easily see that we must be in one of the two cases above.

Since the divisor D in the proposition satisfies Cliff L = Cliff O_{C}(D), we have D ∈ A^{0}(L), so we get the following from Propositions 2.4 and 2.5:

Corollary 9.4. Any divisor D as in Proposition 9.3 must satisfy h^{1}(D) = h^{1}(L − D) = 0, and among all such divisors we can find one satisfying the conditions (C1)-(C5).

By arguing as in the proof of Proposition 3.9, with the lattice ZL ⊕ ZD, with

[ \begin{matrix}L^{2} & L.D \\ D.L & D^{2} \end{matrix} ] = [ \begin{matrix}2(g − 1) & c + 4 \\ c + 4 & 2 \end{matrix} ]

for g = 8 and 10 and c = ⌊\frac{g−1}{2}⌋ = 3 and 4 respectively, we see that there exists an 18dimensional family of isomorphism classes of polarized K3 surfaces that are Clifford general but not BN general for both g = 8 and g = 10.

We will now investigate these two cases. A choice of a subpencil {D_{λ}} of |D| gives as before a rational normal scroll T within which ϕ_{L}(S) = S′ is contained. Unfortunately, as we will see, we no longer have such a nice result about V = Sing T as Theorem 4.7, since the Clifford index c is now the general one. We will however be able to describe these particular cases in a similar manner, too.

Let us first consider the case g = 8, where c = 3. We have D.L = 7, h^{0}(L) = 9 and h^{0}(L − D) = 3. Since (L − 2D)^{2} = −6 and (L − 2D).L = 0, we have h^{0}(L − 2D) = 0 or 1 and h^{0}(L − 3D) = 0.

Recall that the type (e_{1}, …, e_{d}) of the scroll T , with d = d_{0} = dim T , is given by

e_{i} = #{j | d_{j} ≥ i} − 1, (40)

where

d_{i} := h^{0}(L − iD) − h^{0}(L − (i + 1)D).

We have d_{≥3} = 0 and (d_{0}, d_{1}, d_{2}) = (6, 3 − h^{0}(L − 2D), h^{0}(L − 2D)) and the two possible scroll types

(e_{1}, …, e_{6}) = \{ \begin{matrix}(1, 1, 1, 0, 0, 0) & if h^{0}(L − 2D) = 0 \\ (2, 1, 0, 0, 0, 0) & if h^{0}(L − 2D) = 1 \end{matrix}

We first study the case h^{0}(L − 2D) = 0, that is h^{1}(L − 2D) = 1. We have V = Sing T = P^{2}. Here we already see that Theorem 4.7 will not apply, since it is clear by the examples given by the lattice above that there are such cases with no contractions across the fibers. Denote the two base points of the pencil {D_{λ}} by p_{1} and p_{2} and their images under ϕ_{L} by x_{1} and x_{2}. We have the following result:

Lemma 9.5. Either (i) R_{L,D} = ∅, or (ii) R_{L,D} = {Γ} and V intersects S′ in x_{1}, x_{2} and y := ϕ_{L}(Γ), and V =< x_{1}, x_{2}, y >.

Proof. We first show that R_{L,D} is either empty or contains at most one curve.

Choose any smooth D_{0} ∈ |D|. Set F := L − D as usual. Since deg F_{D_{0}} = c + 2 = 5 = 2p_{a}(D) + 1, one has that F_{D_{0}} is very ample, and by arguing as in Lemma 5.1, we get that D.Δ = 0 or 1. This shows the assertion.

By arguing as in the proof of Theorem 4.7, we get that V intersects S′ in at most three points (two of which must of course be x_{1} and x_{2}), and that these three points are then independent.

By this lemma, there are only two cases occurring for h^{0}(L − 2D) = 0, which we denote by (CG1) and (CG2), since they are Clifford general:

(CG1) R_{L,D} = ∅, (CG2) R_{L,D} = {Γ}.

If h^{0}(L − 2D) = 1, then since F^{2} = 2, we have

L ∼ 2D + Δ,

where Δ is the base divisor of |F | and satisfies Δ^{2} = −6, Δ.L = 0 and Δ.D = 3. By arguing as in the proof of Proposition 3.5, we find that L is as in one of the three following cases (where all the Γ and Γ_{i} are smooth rational curves):

(CG3) L ∼ 2D + Γ_{1} + Γ_{2} + Γ_{3}, with the following configuration:

[figure: D; Γ_{1}; Γ_{2}; Γ_{3}]

and R_{L,D} = {Γ_{1}, Γ_{2}, Γ_{3}},

(CG4) L ∼ 2D + Γ + 2Γ_{0} + 2Γ_{1} + ⋯ 2Γ_{N} + Γ_{N+1} + Γ_{N+2}, with the following configuration:

[figure: D; Γ_{0}; Γ_{N}; Γ_{N+2}; Γ; Γ_{N+1}]

and R_{L,D} = {Γ, Γ_{0}},

(CG5) L ∼ 2D + 3Γ_{1} + 2Γ_{2} + 2Γ_{3} + Γ_{4} + Γ_{5}, with the following configuration:

[figure: D; Γ_{1}; Γ_{2}; Γ_{4}; Γ_{3}; Γ_{5}]

and R_{L,D} = {Γ_{1}}.

Defining Z_{λ} as in (13)-(16), we have by arguing as in the proof of Theorem 4.7 in these three cases that for any D ∈ D:

V =< Z_{λ} >= P^{3},

any subscheme of length 4 spans a P^{3} and V ∩ S′ has support only on this scheme.

For the cases (CG1)-(CG5) we can now argue as in Section 7. In particular, we get a commutative diagram as on page 36, and Propositions 7.6, 7.10, Remark 7.12, Proposition 7.17, and Corollary 7.19 still apply.

In the cases (CG1) and (CG2) the type of T_{0} is (2, 2, 2, 1, 1, 1). We leave it to the reader to use Lemma 7.26 to show that the only possible combinations of the b_{i}’s are

(b_{1}, …, b_{8}) = (4, 3, 3, 3, 2, 2, 2, 1), (4, 3, 3, 2, 2, 2, 2, 2) and (3, 3, 3, 3, 2, 2, 2, 2).

In the cases (CG3), (CG4) and (CG5) the type of T_{0} is (3, 2, 1, 1, 1, 1). We again leave it to the reader to show that (b_{1}, …., b_{8}) = (4, 3, 3, 2, 2, 2, 2, 2) is the only possibility.

The case g = 10 is very similar. We have c = 4, D.L = 8, h^{0}(L) = 11 and h^{0}(L−D) = 4. Since (L−2D)^{2} = −6 and (L−2D).L = 2, we have h^{0}(L−2D) = 0 or 1 and h^{0}(L−3D) = 0.

This gives as before d_{≥3} = 0 and (d_{0}, d_{1}, d_{2}) = (7, 4 − h^{0}(L − 2D), h^{0}(L − 2D)) and the two possible scroll types

(e_{1}, …, e_{6}) = \{ \begin{matrix}(1, 1, 1, 1, 0, 0, 0) & if h^{0}(L − 2D) = 0 \\ (2, 1, 1, 0, 0, 0, 0) & if h^{0}(L − 2D) = 1 \end{matrix}

We now get exactly analogous cases (CG1)’ and (CG2)’ as for g = 8, corresponding to the scroll type (1, 1, 1, 1, 0, 0, 0). If h^{0}(L − 2D) = 1, write as usual F := L − D and denote by Δ the base divisor of |F |, so that we have

L ∼ 2D + A + Δ.

for some A > 0 satisfying A.L = (L − 2D).L = 2 and A.Δ = 0. We can now show that 2 = h^{1}(R) = Δ.D, so that A^{2} = −2 and A.D = 2. By arguing as in the proof of Proposition 3.5 again, we find that L is as in one of the two following cases (where all the Γ_{i} are smooth rational curves such that Γ_{i}.A = 0):

(CG3)’ L ∼ 2D + A + Γ_{1} + Γ_{2}, with Γ_{1}.D = Γ_{2}.D = 1, Γ_{1}.Γ_{2}. = 0 and R_{L,D} = {Γ_{1}, Γ_{2}}, (CG4)’ L ∼ 2D + A + 2Γ_{0} + 2Γ_{1} + ⋯ + 2Γ_{N} + Γ_{N+1} + Γ_{N+2}, with all the Γ_{i} having a

configuration as in (E2), Γ_{i}.A = 0, R_{L,D} = {Γ_{0}}, Defining Z_{λ} as in (13)-(16), we have by arguing as in the proof of Theorem 4.7 in these two cases that for any D ∈ D:

V =< Z_{λ} >= P^{3}

and V ∩ S′ has support only on this scheme.

For the cases (CG1)’ and (CG2)’ the type of T_{0} is (2, 2, 2, 2, 1, 1, 1). Again one can use Lemma 7.26 to show that the only possible combinations of the b_{i}’s are

(4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2) and (3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2)

for the case (CG1)’, and

(4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2), (3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2) and
(4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 1)

for the case (CG2)’.

The type of T_{0} for the cases (CG3)’ and (CG4)’ is (3, 2, 2, 1, 1, 1, 1). The only possible cases for the b_{i}’s are found to be

(4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2) and (4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2),

We conclude this section by showing that all the cases (CG1)-(CG5) and (CG1)’-(CG4)’ actually exist, by arguing with the help of Propositions 1.4 and 1.5.

We first consider the case g = 8.

The case (CG1) can be realized by the lattice just below Corollary 9.4 and therefore has number of moduli 18.

We now show that the case (CG2) can be realized by the lattice ZD ⊕ ZF ⊕ ZΓ, with intersection matrix:

[ \begin{matrix}D^{2} & D.F & D.Γ \\ F.D & F^{2} & F.Γ \\ Γ.D & Γ.F & Γ^{2} \end{matrix} ] = [ \begin{matrix}2 & 5 & 1 \\ 5 & 2 & −1 \\ 1 & −1 & −2 \end{matrix} ]

One easily checks that this matrix has signature (1, 2), so by Proposition 1.4 there is an algebraic K3 surface with this lattice as its Picard lattice.

Set L := D + F. By Proposition 1.5 we can assume that L is nef, whence by Riemann-Roch D, F > 0.

We first show that L is base point free and that Cliff L = 3. Since D.L − D^{2} − 2 = 3, we only need to show that there is no effective divisor B on S satisfying either

B^{2} = 0, B.L = 1, 2, 3, 4, or
B^{2} = 2,  B.L = 6.

Setting B ∼ xD + yF + zΓ, one finds

B.L = 7(x + y),

which is not equal to any of the values above. Furthermore, D forces (S, L) to be non-BN general. Since one easily sees that we cannot be in any of the cases (CG1), (CG3)-(CG5), we must be in case (CG2).

We can argue in the same way for the cases (CG3)-(CG5), with the obvious lattices. The number of moduli of these cases are 16, 15 and 14, respectively. We leave these cases to the reader.

In the same way, we can show the existence of each of the types (CG1)’-(CG4)’ for g = 10 and show that their number of moduli 18, 17, 16 and 15 respectively.

These results will all be summarized in the next section, together will all non-Clifford general projective models for g ≤ 10.

10. Projective models of K3 surfaces of low genus

In this section we will use the results obtained to classify all projective models of K3 surfaces of low genus. Together with the description of the general models, we are able to give a complete classification and characterization.

We first need to describe the possible perfect Clifford divisors for c = 1, 2 and 3, and also in some more detail the cases where h^{1}(L − 2D) > 0, since this last number determines the singular locus of the scroll T by equation (10).

The reason why we concentrate on perfect Clifford divisors is purely to make the classification simpler. If we did not restrict to perfect Clifford divisors, we would get more projective models, but the extra projective models would also have been possible to describe with a perfect Clifford divisor, whence they would belong to our list.

In the list below, A and R are defined as in Section 5. Furthermore, we have:

• Γ is a smooth rational curve such that Γ.D = 1 and Γ.A = 0.

• Γ_{1} and Γ_{2} are smooth rational curves such that Γ_{1}.D = Γ_{2}.D = 1 and Γ_{1}.A = Γ_{2}.A = Γ_{1}.Γ_{2} = 0.

• Δ_{0} := 2Γ_{0}′ + 2Γ_{1}′ + ⋯ + 2Γ_{N}′ + Γ_{N+1}′ + Γ_{N+2}′, for N ≥ 0, with a configuration with respect to D as in (E2) and such that A.Γ_{i}′ = 0 for i = 0, …, N + 2.

Also we denote the different cases by {c, D^{2}}.

Here is the list of all possible perfect Clifford divisors for c = 1, 2 and 3, and the cases where h^{1}(R) > 0:

c = 1, L^{2} ≥ 8

{1, 0} D^{2} = 0, D.L = 3, dim T = 3. {1, 2} D^{2} = 2, L^{2} = 10, L ∼ 2D + Γ as in (E0), dim T = 4.

Moreover, h^{1}(R) ≠ 0 if and only if L is as in the following case: {1, 0}^{a} L ∼ 2D + A + Γ, A^{2} ≥ −2, D.A = 2, L^{2} = A^{2} + 10 ≤ 18 with equality if and only if

L ∼ 6D + 3Γ, h^{1}(R) = 1, R_{L,D} = {Γ}.

c = 2, L^{2} ≥ 12

{2, 0} D^{2} = 0, D.L = 4, dim T = 4. {2, 2} D^{2} = 2, D.L = 6, L^{2} ≤ 18 with equality if and only if L ∼ 3D, dim T = 5. {2, 4} D^{2} = 4, L^{2} = 16, L ∼ 2D as in (Q), dim T = 6.

Moreover, h^{1}(R) ≠ 0 if and only if L is as in one of the following cases: {2, 0}^{a} L ∼ 2D + A + Γ, A^{2} ≥ −2, D.A = 3, L^{2} = A^{2} + 14 ≤ 32 with equality if and only if

L ∼ 8D + 4Γ, h^{1}(R) = 1, R_{L,D} = {Γ}. {2, 0}^{b} L ∼ 2D + A + Γ_{1} + Γ_{2}, A^{2} ≥ 0, D.A = 2, L^{2} = A^{2} + 12 ≤ 16 with equality if and only

if L ∼ 4D + 2Γ_{1} + 2Γ_{2}, h^{1}(R) = 2, R_{L,D} = {Γ_{1}, Γ_{2}} ^{2}. {2, 0}^{c} L ∼ 2D + A + Δ_{0}, A^{2} ≥ 0, D.A = 2, L^{2} = A^{2} + 12 ≤ 16 with equality if and only if

L ∼ 4D + 2Δ_{0}, h^{1}(R) = 2, R_{L,D} = {Γ_{0}} ^{3}. {2, 2}^{a} L ∼ 2D + Γ_{1} + Γ_{2} as in (E1), L^{2} = 12, h^{1}(R) = 1, R_{L,D} = {Γ_{1}, Γ_{2}}. {2, 2}^{b} L ∼ 2D + Δ_{0} as in (E2), L^{2} = 12, h^{1}(R) = 1, R_{L,D} = {Γ_{0}}.

c = 3, L^{2} ≥ 16

{3, 0} D^{2} = 0, D.L = 5, dim T = 5. {3, 2} D^{2} = 2, D.L = 7, L^{2} ≤ 22, dim T = 6. {3, 4} D^{2} = 4, L^{2} = 18, L ∼ 2D + Γ as in (E0), dim T = 7.

Moreover, h^{1}(R) ≠ 0 if and only if L is as in one of the following cases: {3, 0}^{a} L ∼ 2D + A + Γ, A^{2} ≥ −2, D.A = 4, L^{2} = A^{2} + 18 ≤ 50 with equality if and only if

L ∼ 10D + 5Γ, h^{1}(R) = 1, R_{L,D} = {Γ}. {3, 0}^{b} L ∼ 2D + A + Γ_{1} + Γ_{2}, A^{2} ≥ 0,D.A = 3, L^{2} = A^{2} + 16 ≤ 24, h^{1}(R) = 2, R_{L,D} =

{Γ_{1}, Γ_{2}}. {3, 0}^{c} L ∼ 2D + A + Δ_{0}, A^{2} ≥ 0, D.A = 3, L^{2} = A^{2} + 16 ≤ 24, h^{1}(R) = 2, R_{L,D} = {Γ_{0}}. {3, 2}^{a} L ∼ 2D + A + Γ, A^{2} = −2, D.A = 2, L^{2} = 16, h^{1}(R) = 1, R_{L,D} = {Γ}.

This list is obtained by using the relations (∗) and (3) in Section 3 together with Propositions 4.3, 4.5 and 4.6. We now show how it works for c = 3.

The three cases {3, 0}, {3, 2} and {3, 4} follow directly from the relations (∗). If D^{2} > 0, then by (3) we must have L^{2} ≤ 24. Assume L^{2} = 24 and consider the divisor E := L − 3D. This satisfies E^{2} = 0 and E.L = 3, thus inducing a Clifford index 1 on L, a contradiction. So L^{2} ≤ 22.

Now assume we are in case {3, 0} and h^{1}(R) > 0. By Propositions 4.5 and 4.6 we have 1 ≤ D.Δ ≤ 2. Since

5 = D.L = A.D + Δ.D,

we have the two possibilities:

(a) Δ.D = 1 and D.A = 4,

(b) Δ.D = 2 and D.A = 3.

In case (a), there has to exist a smooth rational curve Γ in the support of Δ such that Γ.D = 1 and Γ.A = 0 (the last equality follows since Γ.(D + A) ≤ 1 by [SD, (2.7.3)]). Write

L ∼ 2D + A + Γ + Δ′.

2If L^{2} = 14, then the moving part of A is a perfect Clifford divisor of type {2, 2} containing D, and if L^{2} = 16, then A is a perfect Clifford divisor of type {2, 4} containing D.

3Same comment as above.

Clearly D.Δ′ = A.Δ′ = 0, and by 0 = Γ.L = 2 − 2 + Γ.Δ′, we also get Γ.Δ′ = 0, whence

(2D + A + Γ).Δ′ = 0,

and we must have Δ′ = 0 since L is numerically 2-connected. This establishes case {3, 0}^{a}. From the Hodge index theorem on L and A it follows that L^{2} ≤ 50 with equality if and only if 4L ∼ 5A.

In case (b), there either exist two (and only two) disjoint smooth rational curves Γ_{1} and Γ_{2} in the support of Δ such that Γ_{1}.D = Γ_{2}.D = 1 and Γ_{1}.A = Γ_{2}.A = 0, or there exists one and only one smooth rational curve Γ_{0} in the support of Δ (necessarily with multiplicity 2) such that Γ_{0}.D = 1 and Γ_{0}.A = 0. Arguing as in the proof of Proposition 3.5, these two cases give the cases {3, 0}^{b} and {3, 0}^{c} respectively. Again it follows from the Hodge index theorem on L and A that L^{2} ≤ 24.

Assume we are in case {3, 2} and h^{1}(R) > 0. By Propositions 4.5 and Proposition 4.6 we have L^{2} = 16, D.Δ = 1 and Δ^{2} = −2. There has to exist a smooth rational curve Γ in the support of Δ such that Γ.D = 1 and Γ.A = 0. Arguing as above, we easily find that L ∼ 2D + A + Γ. Since 7 = D.L = 2D^{2} + A.D + Δ.D, we have A.D = 2, and since 16 = L^{2} = 18 + A^{2}, we must have A^{2} = −2. This is case {3, 2}^{a}.

We leave the easier cases c = 1 and 2 to the reader, but make a comment on the cases {2, 0}^{b} and {2, 0}^{c}.

From the Hodge index theorem on L and A we get that L^{2} ≤ 16 with equality if and only if L ∼ 2A. If A^{2} = 2 or 4, one calculates

A.L − A^{2} − 2 = 2,

(A−D)^{2} ≥ −2 and (A−D).D = 2, whence by Riemann-Roch A ≥ D, so D does not satisfy the condition (C6). However, since A computes the Clifford index of L, we have h^{1}(A) = 0, whence D is perfect by Lemma 5.10. If L ∼ 2A, one easily sees that A is base point free, whence perfect.

These cases are particularly interesting, since S′ is contained in two scrolls of different types.

Note that for g ≤ 10 (equivalently L^{2} ≤ 18) a polarized K3 surface of non-general Clifford index must have c ≤ 3, so the above cases are sufficient to consider these surfaces. We know that the general K3 surface has general Clifford index. The following proposition considers the dimension of the families in the list above.

Proposition 10.1. The number of moduli of polarized K3 surfaces of genus g, with 5 ≤ g ≤ 10, and non-general Clifford index c > 0 of each of the types {1, 0}, {1, 2}, {2, 0}, {2, 2} with g ≤ 9, {3, 0}, {3, 2} and {3, 4} is 18, and of each of the types {2, 2} with g = 10 and {2, 4} is 19.

Furthermore the general projective model of each of these types satisfies h^{1}(L − 2D) = 0, and the general projective model of each of these types except for the types {1, 2} and {3, 4} is smooth.

The number of moduli of each of these types with h^{1}(L − 2D) > 0 is ≤ 17, except for the type {1, 0}^{a} for g = 10, whose number is 18.

Proof. In the cases {2, 2} with g = 10 and {2, 4} we have L ∼ 3D and L ∼ 2D respectively, so it is clear that those cases can be realized with a Picard group of rank 1 and hence live in 19-dimensional families.

In the other cases, one easily sees that L and D are linearly independent, and we will show that these cases can all be realized with a Picard group of rank 2. Arguing as in the proof of Proposition 3.9 we easily see that there is a K3 surface S such that Pic S ≃ ZL ⊕ ZD such that L^{2}, L.D and D^{2} have the values corresponding to the different cases in question and such that D is a perfect Clifford divisor for L. This has already been done for D^{2} = 0 in the proof of Proposition 3.9, and a case by case study establishes the proof in the other cases.

Recall now that h^{1}(L−2D) > 0 if and only if there exists a smooth rational curve Γ such that Γ.L = 0 and Γ.D = 1, and (since c > 0) ϕ_{L}(S) is singular if and only if there exists a smooth rational curve Γ such that Γ.L = 0 and Γ.D = 0 or 1.

Assuming that the rank of the Picard group is two, we can write Γ = aL + bD, for a, b ∈ Q. The conditions Γ^{2} = −2, Γ.L = 0 and Γ.D = 0 or 1 give the equations:

a^{2}(g − 1) + ab(c + 2 + D^{2}) + \frac{b^{2}D^{2}}{2} = −1,
a(2g − 2) + b(c + 2 + D^{2}) = 0 and
a(c + 2 + D^{2}) + bD^{2} = 0 or 1.

A case by case check reveals that we have a solution only when Γ.D = 1 and then in the following cases:

(a) {1, 2}, with Γ ∼ L − 2D,

(b) {3, 4}, with Γ ∼ L − 2D,

(c) {1, 0} for g = 10, with L ∼ 6D + 3Γ. One can easily show that case (c) can be realized with a lattice of the form ZD ⊕ ZΓ, with D^{2} = 0, D.Γ = 1 and Γ^{2} = −2.

This concludes the proof of the Proposition.

We now would like to study which scroll types are possible for each value of (g, c, D^{2}) with 5 ≤ g ≤ 10 and 1 ≤ c ≤ 3. Recall that the type (e_{1}, …, e_{d}) of the scroll T , with d = dim T , is given by

e_{i} = #{j | d_{j} ≥ i} − 1, (41)

where

d = d_{0} := h^{0}(L) − h^{0}(L − D) = c + 2 + \frac{1}{2}D^{2},
d_{1} := h^{0}(L − D) − h^{0}(L − 2D) = d_{0} − r,
⋮
d_{i} := h^{0}(L − iD) − h^{0}(L − (i + 1)D),
⋮

with

(42) r = \{ \begin{matrix}D^{2} + h^{1}(L − 2D) & if L ≁ 2D (equiv. D^{2} ≠ c + 2), \\ D^{2} − 1 & if L ∼ 2D (equiv. D^{2} = c + 2) \end{matrix}

In the cases {1, 2} and {3, 4}, which are both of type (E0), and the case {2, 4}, which is of type (Q), we have h^{0}(L − 2D) = 1 and h^{0}(L − iD) = 0 for all i ≥ 3, so the scroll types are immediately given.

In the case {2, 2} with g = 10, we have L ∼ 3D, so h^{0}(L−2D) = h^{0}(D) = 2, h^{0}(L−3D) = 1 and h^{0}(L − iD) = 0 for all i ≥ 3.

We will now consider one by one the remaining cases, and gather the result in the tables below.

If c = 1 or 2 and D^{2} = 0 the possible scroll types are given in Section 8. We now briefly review these cases.

Let us first consider the case c = 1 and D^{2} = 0 (case {1, 0}).

For g = 5 the two possible scroll types are (1, 1, 1) and (2, 1, 0). One easily sees that the first case corresponds to (d_{0}, d_{1}, d_{2}) = (3, 3, 0), whence h^{0}(L − 2D) = h^{1}(L − 2D) = 0 and the second corresponds to (d_{0}, d_{1}, d_{2}) = (3, 2, 1), whence h^{1}(L − 2D) = 1 and we are in case {1, 0}^{a}.

For g = 6 we have three possible scroll types: (2, 1, 1), (2, 2, 0) and (3, 1, 0). Comparing with the possible values of the d_{i}, one finds that the first case corresponds to h^{0}(L−2D) = 1 (and h^{1}(L − 2D) = 0). Moreover, the two last cases corresponds to the case {1, 0}^{a} with A ≯ D and A > D respectively.

For g = 7 there are four possible scroll types: (2, 2, 1), (3, 1, 1), (3, 2, 0) and (4, 1, 0). We see that the two first cases correspond to h^{1}(L − 2D) = 0, with h^{0}(L − 3D) = 0 and 1 respectively. The two last cases have h^{1}(L − 2D) = 1 and therefore correspond to {1, 0}^{a} with A ≯ 2D and A > 2D respectively.

We now leave the cases g = 8, 9 and 10 to the reader.

If c = 2 and D^{2} = 0 (case {2, 0}), then 12 ≤ L^{2} ≤ 18.

We leave the easiest case g = 7 to the reader.

If g = 8, we have seen that the four possible scroll types are (2, 1, 1, 1), (2, 2, 1, 0), (3, 1, 1, 0) and (3, 2, 0, 0). The scroll (2, 1, 1, 1) corresponds to h^{1}(R) = 0, whereas the scrolls (2, 2, 1, 0) and (3, 1, 1, 0) correspond to the case {1, 0}^{a} with A ≯ D and A > D respectively. The type (3, 2, 0, 0) corresponds to a polarized surface that also has a different perfect Clifford divisor, and is hence contained in another scroll as well, by the footnote on page 71.

If g = 9, we have seen that the five possible scroll types are (2, 2, 1, 1), (3, 1, 1, 1), (2, 2, 2, 0), (3, 2, 1, 0) and (4, 2, 0, 0). The types (2, 2, 1, 1) and (3, 1, 1, 1) correspond to h^{1}(R) = 0 with h^{0}(L − 3D) = 0 and 1 respectively. (One easily sees that the scroll type (3, 1, 1, 1) can be realized by a K3 surface S with Picard group Pic S ≃ ZD ⊕ ZΓ, for a smooth rational curve Γ satisfying Γ.D = 2, and with L ∼ 3D + Γ. Therefore, it has number of moduli 18). The scroll types (2, 2, 2, 0) and (3, 2, 1, 0) correspond to the case {1, 0}^{a} with A ≯ D and A > D respectively. The type (4, 2, 0, 0) corresponds to a polarized surface that also has a different perfect Clifford divisor, and is hence contained in another scroll as well, by the footnote on page 71.

If g = 10, there are again five possible scroll types: (2, 2, 2, 1), (3, 2, 1, 1), (3, 2, 2, 0), (3, 3, 1, 0) and (4, 2, 1, 0). Again the two first correspond to h^{1}(R) = 0 with h^{0}(L − 3D) = 0 and 1 respectively. The three last cases correspond to the case {1, 0}^{a} with h^{0}(A − D) = 1 and 2 respectively, but A ≯ 2D for the two first cases, and A > 2D for the last case.

If c = 2 and D^{2} = 2 (case {2, 2}), then 12 ≤ L^{2} ≤ 16 (the case L^{2} = 18 being already treated). We have

(L − 3D).L = L^{2} − 18 < 0.

By the nefness of L we must have h^{0}(L − 3D) = 0. Since (L − 2D)^{2} = L^{2} − 16, we get by Riemann-Roch h^{0}(L − 2D) = \frac{1}{2}L^{2} − 6 + h^{1}(R). This gives d_{≥3} = 0 and the two possibilities (d_{0}, d_{1}, d_{2}) = (5, 3, \frac{1}{2}L^{2} − 6) or (5, 2, 1), the latter occurring if and only if L^{2} = 12 and L is of type (E1) or (E2) (the special cases {2, 2}^{a} and {2, 2}^{b}). The corresponding scroll types in the first situation are then (1, 1, 1, 0, 0) for g = 7, (2, 1, 1, 0, 0) for g = 8 and (2, 2, 1, 0, 0) for g = 9. For g = 7 and L of type (E1) or (E2), the scroll type is (2, 1, 0, 0, 0).

If c = 3 and D^{2} = 0 (case {3, 0}), then L^{2} = 16 or 18. We have

(L − 3D).L = L^{2} − 15 ≤ 3

and

(L − 4D).L = L^{2} − 20 < 0.

This gives immediately h^{0}(L − iD) = 0 for all i ≥ 3, whence d_{≥4} = 0. Also, since c = 3, we must have h^{0}(L − 3D) ≤ 1. We also have by Riemann-Roch h^{0}(L − 2D) = \frac{1}{2}L^{2} − 8 + h^{1}(R).

Let us first consider the case g = 9. Then we have (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 5−h^{1}(R), h^{1}(R)− h^{0}(L − 3D), h^{0}(L − 3D)). If h^{1}(R) = 0, then h^{0}(L − 2D) = h^{0}(L − 3D) = 0 and (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 5, 0, 0). The corresponding scroll type is (1, 1, 1, 1, 1). The cases with h^{0}(R) = h^{1}(R) > 0 are {3, 0}^{a}, {3, 0}^{b} and {3, 0}^{c}. In the first we have h^{0}(R) = h^{1}(R) = 1, whence h^{0}(L − 3D) = 0 and (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 4, 1, 0). The corresponding scroll type is (2, 1, 1, 1, 0). In the cases {3, 0}^{b} and {3, 0}^{c}, we have h^{0}(R) = h^{1}(R) = 2. If h^{0}(L−3D) = 0 (eqv. A ≯ D), we get (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 3, 2, 0) and the scroll type is (2, 2, 1, 0, 0). If h^{0}(L − 3D) = 1 (eqv. A > D), we get (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 3, 1, 1) and the scroll type is (3, 1, 1, 0, 0).

If g = 10, we have (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 5 − h^{1}(R), 1 + h^{1}(R) − h^{0}(L − 3D), h^{0}(L − 3D)). If h^{1}(R) = 0, then h^{0}(L − 2D) = 1 and h^{0}(L − 3D) = 0 and (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 5, 1, 0). The corresponding scroll type is (2, 1, 1, 1, 1). The cases with h^{1}(R) > 0 are {3, 0}^{a}, {3, 0}^{b} and {3, 0}^{c} as in the case g = 9. Arguing as in that case, we get (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 4, 2, 0) and scroll type (2, 2, 1, 1, 0) in the case {3, 0}^{a} (where h^{1}(R) = 1), and we get (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 3, 3, 0) and scroll type (2, 2, 2, 0, 0) if h^{0}(L−3D) = 0 (eqv. A ≯ D), and (d_{0}, d_{1}, d_{2}, d_{3}) = (5, 3, 2, 1) and scroll type (3, 2, 1, 0, 0) if h^{0}(L − 3D) = 1 (eqv. A > D) in the two latter cases (where h^{1}(R) = 2).

If c = 3 and D^{2} = 2 (case {3, 2}), then L^{2} = 16 or 18. We have

(L − 3D).L = L^{2} − 21 < 0,

whence h^{0}(L − iD) = 0 for all i ≥ 3, whence d_{≥3} = 0. By Riemann-Roch, h^{0}(L − 2D) = \frac{1}{2}L^{2} − 8 + h^{1}(R) and we have (d_{0}, d_{1}, d_{2}) = (6, 4 − h^{1}(R), \frac{1}{2}L^{2} − 8 + h^{1}(R), 0).

If g = 9 and h^{1}(R) = 0, then (d_{0}, d_{1}, d_{2}) = (6, 4, 0) and the corresponding scroll type is (1, 1, 1, 1, 0, 0). The case with h^{1}(R) > 0 is given by {3, 2}^{a}. In this case we have (d_{0}, d_{1}, d_{2}) = (6, 3, 1) and the corresponding scroll type is (2, 1, 1, 0, 0, 0).

If g = 10, then we automatically have h^{1}(R) = 0, whence (d_{0}, d_{1}, d_{2}) = (6, 4, 1) and the corresponding scroll type is (2, 1, 1, 1, 0, 0).

We will summarize these results below.

Furthermore, we can prove, by arguing with lattices that all the cases mentioned above exist, and calculate the number of their moduli. In many cases, we can also explicitly find an expression for L in terms of D and some smooth rational curves on the surface. Also, by studyng the Picard lattices, we can find the curves that are contracted by L, and hence find the mildest singularities of the surfaces in the families.

All these informations are also summarized below.

We will first focus on some concrete examples, to give the reader an idea of the proofs, and leave all the other cases to the reader.

Example 10.2. We start with an easy case: g = 6, c = 1, D^{2} = 0 and the scroll type (3, 1, 0). This occurs if L is of type {1, 0}^{a} with A > D (and also R_{L,D} = {Γ}). By arguing as in the proof of Proposition 3.5 we find that L ∼ 3D + 2Γ + Γ_{0} + Γ_{1}, where Γ, Γ_{0} and Γ_{1} are smooth rational curves, with the following configuration:

D  Γ  Γ_{1} Γ_{0}.

By Propositions 1.4 and 1.5 there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ ⊕ ZΓ_{0} ⊕ ZΓ_{1} and intersection matrix

[ \begin{matrix}D^{2} & D.Γ & D.Γ_{0} & D.Γ_{1} \\ Γ.D & Γ^{2} & Γ.Γ_{0} & Γ.Γ_{1} \\ Γ_{0}.D & Γ_{0}.Γ & Γ_{0}^{2} & Γ_{0}.Γ_{1} \\ Γ_{1}.D & Γ_{1}.Γ & Γ_{1}.Γ_{0} & Γ_{1}^{2} \end{matrix} ] = [ \begin{matrix}0 & 1 & 1 & 0 \\ 1 & −2 & 0 & 1 \\ 1 & 0 & −2 & 0 \\ 0 & 1 & 0 & −2 \end{matrix} ] ,

and such that L := 3D + 2Γ + Γ_{0} + Γ_{1} is nef (whence by Riemann-Roch D and Γ_{0} > 0).

We have D.L − D^{2} − 2 = 1. To show that L is base point free and of Clifford index 1, it suffices to show that there is no effective divisor E such that E^{2} = 0 and E.L = 1 or 2.

Set E ∼ xD + yΓ + zΓ_{0} + wΓ_{1}. Since we can assume E ∈ A^{0}(L), and E base point free, we easily see that

E.Γ_{0} = x − 2z = 0 or 1,

whence

E.L = 3x + z = 7z or 7z + 3,

which can never be equal to 1 or 2.

By Riemann-Roch either Γ > 0 or −Γ > 0. If the latter is the case, write Γ = −γ, and we then have D = D_{0} + γ with D_{0} > 0, since D.γ = −1. Therefore, we can write

L ∼ 3(D_{0} + γ) − 2γ + Γ_{0} + Γ_{1} = 3D_{0} + 2γ + Γ_{0} + Γ_{1}.

We can use the same argument if −Γ_{1} > 0, so possibly after a change of basis, we can assume D, Γ, Γ_{0} and Γ_{1} > 0. It is then easy to check that D is nef, whence a perfect Clifford divisor.

Example 10.3. Let us consider the case g = 9, c = 2, D^{2} = 0 and the scroll type (3, 2, 1, 0). This occurs if L is of type {2, 0}^{a} with A > D (and also R_{L,D} = {Γ}). An analysis as in the proof of Proposition 3.5 shows that L is one of the following three types:

(a) L ∼ 3D + 2Γ + Γ_{1} + Γ_{2}, with the following configuration:

[figure: D; Γ; Γ_{2}; Γ_{1}]

(b) L ∼ 3D + 2Γ + Γ_{1} + Γ_{2}, with the following configuration:

D  Γ  Γ_{1} Γ_{2}

(c) L ∼ 3D + 2Γ + Γ_{1} + Γ_{2} + ⋯ + Γ_{n+3}, for n ≥ 0 (in general n = 0) with the following configuration:

[figure: D; Γ_{2}; Γ_{1}; Γ; Γ_{n+3}.]

(Actually case (b) can be looked at as a special case of case (c), with “n = −1”.)

One can easily show that both cases (a) and (b) do not occur with a Picard group of rank < 4, and case (c) does not occur with a Picard group of rank < 5. We now show that both case (a) and (b) occur with a Picard group of rank 4.

We first consider case (a).

By Propositions 1.4 and 1.5 there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ ⊕ ZΓ_{1} ⊕ ZΓ_{2} and intersection matrix corresponding to the configuration above, and such that L := 3D + 2Γ + Γ_{1} + Γ_{2} is nef (whence D, Γ_{1}, Γ_{2} > 0 by Riemann-Roch).

We calculate D.L − D^{2} − 2 = 2. To show that L is base point free and that Cliff L = 2 with D as a perfect Clifford divisor, it will suffice to show that there are no divisor B on S satisfying B^{2} = 0, B.L = 1, 2, 3 or B^{2} = 2, B.L = 6, and that D is nef.

Write B ∼ xD + yΓ + zΓ_{1} + wΓ_{2}. Since we can assume B ∈ A^{0}(L), and B base point free, we easily see that

B.Γ_{2} = x − 2w = 0 or 1,

and

B.Γ_{1} = x + y − 2z = 0, 1 or 2.

By the Hodge index theorem one also finds

B.D = y + z + w = \{ \begin{matrix}1 & if B^{2} = 0, \\ 2 & if B^{2} = 2 \end{matrix}

Also, we have

B.L = 4x + 3z + w = \{ \begin{matrix}1, 2, 3 & if B^{2} = 0, \\ 6 & if B^{2} = 2 \end{matrix}

One checks by inspection that these four equations have no integer solutions.

By Riemann-Roch, either Γ > 0 or −Γ > 0. As in the previous example, possibly after a change of basis one can assume that Γ > 0 and that D is nef, whence perfect.

We now consider case (b).

Again by Propositions 1.4 and 1.5 there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ ⊕ ZΓ_{1} ⊕ ZΓ_{2} and intersection marix corresponding to the configuration for (b) above, and such that L := 3D + 2Γ + Γ_{1} + Γ_{2} is nef (whence D and Γ_{2} > 0 by Riemann-Roch).

We calculate D.L − D^{2} − 2 = 2. To show that L is base point free and that Cliff L = 2 with D as a perfect Clifford divisor, it will again suffice to show that there are no divisor B on S satisfying B^{2} = 0, B.L = 1, 2, 3 or B^{2} = 2, B.L = 6, and that D is nef.

Write B ∼ xD + yΓ + zΓ_{1} + wΓ_{2} as before. Again by the Hodge index theorem and since we can assume B ∈ A^{0}(L), and B base point free, we get

B.D = y + 2w = \{ \begin{matrix}1 & if B^{2} = 0, \\ 2 & if B^{2} = 2, \end{matrix}

B.Γ_{2} = 2(x − w) = 0 or 2,

and

B.Γ_{1} = y − 2z = −1, 0, or 1

(since we do not know whether it is Γ_{1} or −Γ_{1} which is effective). Combining these equations with

B.L = 2(2x + y + 2w),

we find no integer solutions. Again, possibly after a change of basis, we get that D is perfect and that all D, Γ, Γ_{1} and Γ_{2} > 0.

We can also check which curves are contracted by L.

In case (a), the only contracted curve is in general Γ, so all surfaces in that family has an A_{1} singularity, and the general surface has only such a singularity. Furthermore S′′ is then in general smooth.

In case (b), the only contracted curves are in general Γ and Γ_{1} , so all surfaces in that family has an A_{2} singularity, and the general surface has only such a singularity. Furthermore S′′ is then necessarily singular.

By comparing with the table on page 62, we then find that case (a) has b_{1} = 3 and case (b) has b_{1} = 2.

Example 10.4. As an easy example we consider the case g = 10, c = 1, D^{2} = 0 and the scroll type (5, 2, 1). This occurs if R_{L,D} = ∅ and h^{0}(L−5D) = 1. An analysis as in the proof of Proposition 3.5 shows that L ∼ 5D + 3Γ_{1} + 2Γ_{2} + Γ_{3}, with the following configuration:

D  Γ_{1}  Γ_{2}  Γ_{3}.

One can easily show that this cannot be achieved with a Picard group of rank < 4.

By Propositions 1.4 and 1.5 again there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ_{1} ⊕ ZΓ_{2} ⊕ ZΓ_{3} and intersection marix corresponding to the configuration above, and such that L := 5D + 3Γ_{1} + 2Γ_{2} + Γ_{3} is nef (whence D and Γ_{1} > 0 by Riemann-Roch).

We calculate D.L − D^{2} − 2 = 1. To show that L is base point free and that Cliff L = 1 with D as a perfect Clifford divisor, it will suffice to show that there is no divisor E on S satisfying E^{2} = 0, E.L = 1, 2 and that D is nef.

By the Hodge index theorem 36E.D ≤ (E + D)^{2}L^{2} ≤ ((E + D).L)^{2} ≤ 25, whence E.D = 0. Writing E ∼ xD + yΓ_{1} + zΓ_{2} + wΓ_{3}, we get

E.D = y = 0,

whence

E.L = 3x + y = 3x ≠ 1 or 2.

Possibly after a change of basis, we get that D is perfect and that all D, Γ_{1}, Γ_{2} and Γ_{3} > 0.

One finds that the only contracted curves with this Picard group are Γ_{2} and Γ_{3}, so the general surface in this family has an A_{2} singularity.

Example 10.5. We give a more involved example: g = 10, c = 2, D^{2} = 0 and the scroll type (3, 2, 1, 1). This occurs if R_{L,D} = ∅ and h^{0}(L − 3D) = 1. By the table on page 62, we must have b_{1} = 3 or 4, and we will now show that both these cases exist (with the number of moduli 17 and 16 respectively).

One easily sees that there is no way to achieve this situation with a Picard group of rank < 3. We will now show that it is possible with a Picard group of rank 3.

By Propositions 1.4 and 1.5 there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ_{1} ⊕ ZΓ_{2} and intersection matrix

[ \begin{matrix}D^{2} & D.Γ_{1} & D.Γ_{2} \\ Γ_{1}.D & Γ_{1}^{2} & Γ_{1}.Γ_{2} \\ Γ_{2}.D & Γ_{2}.Γ_{1} & Γ_{2}^{2} \end{matrix} ] = [ \begin{matrix}0 & 2 & 0 \\ 2 & −2 & 1 \\ 0 & 1 & −2 \end{matrix} ] ,

and such that L := 3D + 2Γ_{1} + Γ_{2} is nef (whence by Riemann-Roch D and Γ_{1} > 0).

We have D.L − D^{2} − 2 = 2. To show that L is base point free and of Clifford index 2 with D as a perfect Clifford divisor, it suffices to show that there is no effective divisor B such that B^{2} = 0, B.L = 1, 2, 3, or B^{2} = 2, B.L = 6.

By the Hodge index theorem one has

18(B^{2} + 2B.D) = L^{2}(B + D)^{2} ≤ ((B + D).L)^{2} = (B.L + 4)^{2},

which gives B.D ≤ 1.

Writing B ∼ xD + yΓ_{1} + zΓ_{2}, we have B.D = 2y, whence y = 0.

Since either Γ_{2} > 0 or −Γ_{2} > 0 and we can assume B ∈ A^{0}(L), we must have B.Γ_{2} = y − 2z = −2z = −1, 0, 1. We therefore get z = 0.

So B is a multiple of D, a contradiction.

Possibly after a change of basis, we get that D is perfect and that also Γ_{2} > 0.

One finds that the only contracted curve with this Picard group is Γ_{2}, so that all surfaces in this family have at least an A_{1} singularity, and the general such surface has such a singularity. By comparing with the table on page 62, we see that we must have b_{1} = 4.

But there is also another family of surfaces. Again we find that there is an algebraic K3 surface S with Picard group Pic S = ZD ⊕ ZΓ_{1} ⊕ ZΓ_{2} ⊕ ZΓ_{3} and intersection matrix

[ \begin{matrix}D^{2} & D.Γ_{1} & D.Γ_{2} & D.Γ_{3} \\ Γ_{1}.D & Γ_{1}^{2} & Γ_{1}.Γ_{2} & Γ_{1}.Γ_{3} \\ Γ_{2}.D & Γ_{2}.Γ_{1} & Γ_{2}^{2} & Γ_{2}.Γ_{3} \\ Γ_{3}.D & Γ_{3}.Γ_{1} & Γ_{3}.Γ_{2} & Γ_{3}^{2} \end{matrix} ] = [ \begin{matrix}0 & 2 & 1 & 1 \\ 2 & −2 & 0 & 0 \\ 1 & 0 & −2 & 0 \\ 1 & 0 & 0 & −2 \end{matrix} ] ,

and such that L := 3D + Γ_{1} + Γ_{2} + Γ_{3} is nef.

One can show that Cliff L = 2 with D as a perfect Clifford divisor (again after possibly changing the basis). Furthermore, one finds that with this lattice, there are no contracted curves, whence S′ is smooth. By comparing with the table on page 62, we see that we must have b_{1} = 3.

We will now summarize all our knowledge of birational projective models S′ of K3 surfaces of genera 5 ≤ g ≤ 10. In some cases we are able to give a resolution of S′ in its scroll T . When we are not able to do this, we give the vector bundle a section of which cuts out S′′ in T_{0} ≃ P(E) (which is the dual of the vector bundle F_{1} in the resolution

⋯ ⟶ F_{2} ⟶ F_{1} ⟶ O_{T_{0}} ⟶ O_{S′′} ⟶ 0.)

This vector bundle is a direct sum of line bundles, which we write as a linear combination of the line bundles H and F on P(E), where H = i^{∗}O_{P^{g}}(1) and F = π^{∗}O_{P^{1}}(1), with

\xymatrix{P(E) \ar[r]^{i} \ar[d]^{π} & T ⊆ P^{g} \\ P^{1}.}

Also note that we in all cases have J_{S′/T} = i_{∗}S′′/T_{0} by Proposition 7.15, and that we in most cases, by Remark 7.28, the sections of F_{1}^{∨} are constant on the fibers of i, whence they also give “equations” cutting out S′ in T set-theoretically.

g = 5

The general projectice model is a complete intersection of three hyperquadrics. The others are as follows:

c D^{2} scroll type # mod.  type of L  sing.
1 0 (1, 1, 1)  18  h^{0}(L − 2D) = 0 sm.
1 0 (2, 1, 0)  17  {1, 0}^{a}, A^{2} = −2 A_{1}

In these cases O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

g = 6

The general projective model is a hyperquadric section of a Fano 3-fold of index 2 and degree 5. The others are as follows:

c D^{2} scroll type # mod.  type of L  sing.
1 0 (2, 1, 1)  18  h^{0}(L − 2D) = 0  sm.
1 0 (2, 2, 0)  17  {1, 0}^{a}, A^{2} = 0, A ≯ D A_{1}
1 0 (3, 1, 0)  16  {1, 0}^{a}, A^{2} = 0, A > D ^{(i)} A_{2}
1 2 (2, 1, 0, 0)  18  (E0)  A_{1}

In the three first cases O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + 2F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

In the last case, S′ has a resolution:

0 ⟶ O_{T} (−4H + 2F) ⊕ O_{T} (−4H + F)
⟶ O_{T} (−2H + 2F) ⊕ O_{T} (−3H + F) ⊕ O_{T} (−3H)
⟶ O_{T} ⟶ O_{S′} ⟶ 0

Comments on the types of L:

(i) L ∼ 3D + 2Γ + Γ_{0} + Γ_{1}, with the following configuration:

D  Γ  Γ_{1} Γ_{0}

g = 7

The general projective model is a complete intersection of 8 hyperplanes in Σ_{12}^{10}, as described in the beginning of Section 9.

The other projective models are as follows:

c D^{2} scroll type # mod.  type of L  sing.
1 0  (2, 2, 1)  18  h^{0}(L − 2D) = 2, h^{0}(L − 3D) = 0 sm.
1 0  (3, 1, 1)  16  h^{0}(L − 2D) = 2, h^{0}(L − 3D) = 1 sm.
1 0  (3, 2, 0)  17  {1, 0}^{a}, A^{2} = 2, A > D, A ≯ 2D A_{1}
1 0  (4, 1, 0)  16  {1, 0}^{a}, A^{2} = 2, A > 2D ^{(i)}  A_{3}
2 0 (1, 1, 1, 1)  18  h^{0}(L − 2D) = 0  sm.
2 0 (2, 1, 1, 0)  17  {2, 0}^{a}, A^{2} = −2  A_{1}
2 0 (2, 2, 0, 0)  16  {2, 0}^{b} or {2, 0}^{c}, A ≯ D ^{(ii)}  2A_{1}
2 0 (3, 1, 0, 0)  15  {2, 0}^{b} or {2, 0}^{c}, A > D ^{(iii)}  2A_{2}
2 2 (1, 1, 1, 0, 0)  18  h^{0}(L − 2D) = 0  sm.
2 2 (2, 1, 0, 0, 0)  17  (E1) or (E2) ^{(iv)}  2A_{1}

In the cases (c, D^{2}) = (1, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + 3F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

.

In the cases (c, D^{2}) = (2, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + (g − 1)F) ⟶
O_{T} (−2H + b_{1}F) ⊕ O_{T} (−2H + b_{2}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

with (b_{1}, b_{2}) = (1, 1) or (2, 0) for the scroll types (1, 1, 1, 1) and (2, 1, 1, 0) and (b_{1}, b_{2}) = (2, 0) for the scroll types (2, 2, 0, 0) and (3, 1, 0, 0).

In the cases (c, D^{2}) = (2, 2) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

⊕_{i=1}^{4}O_{T_{0}}(2H − b_{i}F),

where (b_{1}, b_{2}, b_{3}, b_{4}) = (1, 1, 1, 0) or (2, 1, 0, 0) for the type (1, 1, 1, 0, 0), and (2, 1, 0, 0) for the type (2, 1, 0, 0, 0).

Comments on the types of L:

(i) L ∼ 4D + 3Γ + 2Γ_{1} + Γ_{2}, with the following configuration:

D  Γ  Γ_{1}  Γ_{2}

(ii) The number of moduli of the case {2, 0}^{c} is 15, with mildest singularity A_{3}. (iii) In the case {2, 0}^{b} we have L ∼ 3D + 2Γ_{1} + 2Γ_{2} + Γ′_{1} + Γ′_{2}, with the following config-

uration:

[figure: D; Γ_{1}; Γ′_{1}; Γ_{2}; Γ′_{2},]

and in the case {2, 0}^{c} we have L ∼ 3D+4Γ_{0} +3Γ_{1} +2Γ_{2} +2Γ_{3} +Γ_{4}, with the following configuration:

D  Γ_{0}  Γ_{1}  Γ_{3} Γ_{2}  Γ_{4}.

The mildest singularity of this latter case is A_{5}.

(iv) The number of moduli of the case (E2) is 16, with mildest singularity A_{3}.

g = 8

The general projective model is a complete intersection of 5 hyperplanes in Grass(V^{6}, 2) ⊆ P^{14}.

The others are as follows:

c D^{2} scroll type # mod.  type of L  sing.
1 0  (2, 2, 2)  18  h^{0}(L − 2D) = 3, h^{0}(L − 3D) = 0  sm.
1 0  (3, 2, 1)  17  h^{0}(L − 2D) = 3, h^{0}(L − 3D) = 1  sm.
1 0  (4, 2, 0)  17  {1, 0}^{a}, A^{2} = 4  A_{1}
2 0  (2, 1, 1, 1)  18  h^{0}(L − 2D) = 1  sm.
2 0  (2, 2, 1, 0)  17  {2, 0}^{a}, A^{2} = 0, A ≯ D  A_{1}
2 0  (3, 1, 1, 0)  15  {2, 0}^{a}, A^{2} = 0, A > D ^{(i)}  A_{2}
2 0  (3, 1, 1, 0)  14  {2, 0}^{a}, A^{2} = 0, A > D ^{(ii)}  A_{2} + 2A_{1}
2 0  (3, 2, 0, 0)  16  {2, 0}^{b} or {2, 0}^{c} , A^{2} = 2, A > D ^{(iii)}  2A_{1}
2 2 (2, 1, 1, 0, 0)  18  h^{0}(L − 2D) = 1  sm.
3 2 (1, 1, 1, 0, 0, 0)  18  (CG1) or (CG2) ^{(iv)}  sm.
3 2 (2, 1, 0, 0, 0, 0)  16  (CG3),(CG4) or (CG5) ^{(v)}  3A_{1}

In the cases (c, D^{2}) = (1, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + 4F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

.

In the cases (c, D^{2}) = (2, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + (g − 1)F) ⟶
O_{T} (−2H + b_{1}F) ⊕ O_{T} (−2H + b_{2}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

with (b_{1}, b_{2}) = (2, 1), except for the type (3, 2, 0, 0), where (b_{1}, b_{2}) = (3, 0). In this latter case, S also contains a different perfect Clifford divisor (by the footnote on page 71), so S′ can also be described as for the case (c, D^{2}) = (2, 2).

In the cases (c, D^{2}) = (2, 2) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

⊕_{i=1}^{4}O_{T_{0}}(2H − b_{i}F),

where (b_{1}, b_{2}, b_{3}, b_{4}) = (3, 2, 0, 0), (3, 1, 1, 0) or (2, 2, 1, 0).

In the cases (CG1) and (CG2) then S′′ is cut out in T_{0} by a section of

O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H − F)^{2} ⊕ O_{T_{0}}(2H)^{5} or
O_{T_{0}}(2H − F)^{4} ⊕ O_{T_{0}}(2H)^{4}  or
O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H − F)^{3} ⊕ O_{T_{0}}(2H)^{3} ⊕ O_{T_{0}}(2H + F)

(which is constant on the fibers of i in the first four cases). In the cases (CG3),(CG4), (CG5) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H − F)^{2} ⊕ O_{T_{0}}(2H)^{5}

Comments on the types of L:

(i) L ∼ 3D + 2Γ + Γ′ + Γ_{1} + Γ_{2}, with the following configuration:

[figure: D; Γ; Γ′_{1}; Γ_{1}; Γ_{2}]

(ii) L ∼ 3D + 2Γ + Γ′ + 2Γ_{0} + 2Γ_{1} + ⋯ + 2Γ_{N} + Γ_{N+1} + Γ_{N+2}, for N ≥ 0 (in general N = 0) with the following configuration:

[figure: D; Γ; Γ′; Γ_{0}; Γ_{1}; Γ_{N}; Γ_{N+2}; Γ_{N+1}]

(iii) The number of moduli of the case {2, 0}^{c} is 15, with mildest singularity A_{3}. (iv) The number of moduli of the case (CG2) is 17, with mildest singularity A_{1}.

(v) The number of moduli of the cases (CG4) and (CG5) are 15 ands 14 respectively, with mildest singularities A_{1} + A_{3} and A_{5} respectively.

g = 9

The general projective model is a complete intersection of 4 hyperplanes in Σ_{16}^{6}, as described in the beginning of Section 9.

The others are as follows:

c D^{2} scroll type # mod.  type of L  sing.
1 0  (3, 2, 2)  18  h^{0}(L − 2D) = 4, h^{0}(L − 3D) = 1  sm.
1 0  (3, 3, 1)  17  h^{0}(L − 2D) = 4, h^{0}(L − 3D) = 2, h^{0}(L − 4D) = 0  sm.
1 0  (4, 2, 1)  16  h^{0}(L − 2D) = 4, h^{0}(L − 3D) = 2, h^{0}(L − 4D) = 1 ^{(i)}  A_{1}
1 0  (5, 2, 0)  17  {1, 0}^{a}, A^{2} = 6  A_{1}
2 0  (2, 2, 1, 1)  18  h^{0}(L − 2D) = 2, h^{0}(L − 3D) = 0  sm.
2 0  (3, 1, 1, 1)  15  h^{0}(L − 2D) = 2, h^{0}(L − 3D) = 1  sm.
2 0  (2, 2, 2, 0)  17  {2, 0}^{a}, A^{2} = 2, A ≯ D  A_{1}
2 0  (3, 2, 1, 0)  16  {2, 0}^{a}, A^{2} = 2, A > D ^{(ii)}  A_{1}
2 0  (3, 2, 1, 0)  16  {2, 0}^{a}, A^{2} = 2, A > D ^{(iii)}  A_{2}
2 0  (4, 2, 0, 0)  17  {2, 0}^{b} (L ∼ 4D + 2Γ_{1} + 2Γ_{2}) or {2, 0}^{c} (L ∼ 4D + 2Δ_{0} ^{(iv)} 2A_{1}
2 2 (2, 2, 1, 0, 0)  18  h^{0}(L − 2D) = 2, h^{0}(L − 3D) = 0  sm.
2 4 (2, 1, 1, 0, 0, 0)  19  L ∼ 2D  sm.
3 0 (1, 1, 1, 1, 1)  18  h^{0}(L − 2D) = 0  sm.
3 0 (2, 1, 1, 1, 0)  17  {3, 0}^{a}, A^{2} = −2  A_{1}
3 0 (2, 2, 1, 0, 0)  16  {3, 0}^{b} or {3, 0}^{c}, A^{2} = 0, A ≯ D ^{(v)}  2A_{1}
3 0 (3, 1, 1, 0, 0)  14  {3, 0}^{b} or {3, 0}^{c}, A^{2} = 0, A > D ^{(iv)} ^{(vi)}  2A_{2}
3 2 (1, 1, 1, 1, 0, 0)  18  h^{0}(L − 2D) = 0  sm.
3 2 (2, 1, 1, 0, 0, 0)  17  {3, 2}^{a}  A_{1}

In the cases (c, D^{2}) = (1, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + 5F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

.

In the cases (c, D^{2}) = (2, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + (g − 1)F) ⟶
O_{T} (−2H + b_{1}F) ⊕ O_{T} (−2H + b_{2}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

with (b_{1}, b_{2}) = (2, 2) or (3, 1) for the scroll type (2, 2, 1, 1); (b_{1}, b_{2}) = (2, 2) for the scroll types (3, 1, 1, 1), (2, 2, 2, 0) and (3, 2, 1, 0); (b_{1}, b_{2}) = (3, 1) for the scroll type (3, 2, 1, 0) and (b_{1}, b_{2}) = (4, 0) for the scroll type (4, 2, 0, 0). In this latter case S also contains a different perfect Clifford divisor (by the footnote on page 71, so S′ can also be described as in the case (c, D^{2}) = (2, 2) (with perfect Clifford divisor 2D + Γ_{1} + Γ_{2} or D + Δ_{0}).

In the case (c, D^{2}) = (2, 2) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

O_{T_{0}}(2H − 3F) ⊕ O_{T_{0}}(2H − 2F)^{2} ⊕ O_{T_{0}}(2H).

In the case (c, D^{2}) = (2, 4) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of:

O_{T_{0}}(2H − 2F)^{3} ⊕ O_{T_{0}}(2H − F)^{2} ⊕ O_{T} (2H)^{2}.

We also have that S′ is the 2-uple embedding of the quartic ϕ_{D}(S) if and only if D is not hyperelliptic. If D is hyperelliptic, then S′ has a O_{T′}-resolution as follows:

0 ⟶ O_{T′}(−4H + (g − 1)F) ⟶
O_{T′}(−2H + b_{1}F) ⊕ O_{T′}(−2H + b_{2}F) ⟶ O_{T′} ⟶ O_{S′} ⟶ 0,

for a suitable scroll T′ in P^{9}. If D is linearly equivalent to a sum of two smooth elliptic curves intersecting twice, then (b_{1}, b_{2}) = (2, 2) or (4, 0), and the type of T′ is (2, 2, 2, 0). Otherwise (b_{1}, b_{2}) = (4, 0) and the type of T′ is (4, 2, 0, 0) (see Proposition 7.31).

In the cases (c, D^{2}) = (3, 0) we have an O_{T} -resolution of O_{S′} of the following type:

\begin{aligned}0 & ⟶ & O_{T} (−5H + 8F) & ⟶ & ⊕_{i=1}^{5}O_{T} (−3H + a_{i}F) \\ & ⟶ & ⊕_{i=1}^{5}O_{T} (−2H + b_{i}F) & ⟶ & O_{T} ⟶ O_{S′} ⟶ 0, \end{aligned}

with a_{i} = 3 − b_{i}, for all i. For the smooth scroll type (1, 1, 1, 1, 1) we have (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 1, 1, 1, 1) or (2, 2, 2, 0, 0). For the scroll type (2, 1, 1, 1, 0) we have (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 1, 1, 1, 1), (2, 2, 1, 1, 0) or (2, 2, 2, 0, 0). For the remaining two singular scroll types we have (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 2, 1, 1, 0) or (2, 2, 2, 0, 0).

In the case (c, D^{2}) = (3, 2) with scroll type (1, 1, 1, 1, 0, 0) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of:

\begin{matrix}O_{T_{0}}(2H − 2F) & ⊕ & O_{T_{0}}(2H − F)^{5} & ⊕ & O_{T_{0}}(2H)^{2} & or \\ O_{T_{0}}(2H − F)^{7} & ⊕ & O_{T_{0}}(2H) \end{matrix}

In the case (c, D^{2}) = (3, 2) with scroll type (2, 1, 1, 0, 0, 0) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of:

\begin{matrix}O_{T_{0}}(2H − 2F)^{3} & ⊕ & O_{T_{0}}(2H − F) & ⊕ & O_{T_{0}}(2H)^{4} & or \\ O_{T_{0}}(2H − 2F)^{2} & ⊕ & O_{T_{0}}(2H − F)^{3} & ⊕ & O_{T_{0}}(2H)^{3}. \end{matrix}

Comments on the types of L:

(i) L ∼ 4D + 2Γ_{1} + Γ_{2} + Γ_{3}, with the following configuration:

D  Γ_{1}  Γ_{2} Γ_{3},

or L ∼ 4D + 3Γ_{1} + ⋯ + 3Γ_{N} + 2Γ_{N+1} + Γ_{N+2} + Γ_{N+3}, for N ≥ 1 (in general N = 0) with the following configuration:

[figure: D; Γ_{1}; Γ_{N}; Γ_{N+1}; Γ_{N+2}; Γ_{N+3}]

(ii) L ∼ 3D + 2Γ + Γ_{1} + Γ_{2}, with the following configuration:

[figure: D; Γ; Γ_{2}; Γ_{1}]

(iii) L ∼ 3D + 2Γ + Γ_{1} + Γ_{2}, with the following configuration:

D  Γ  Γ_{1} Γ_{2}

or L ∼ 3D + 2Γ + Γ_{1} + Γ_{2} + ⋯ + Γ_{N+3}, for N ≥ 0 (in general N = 0) with the following configuration:

[figure: D; Γ_{2}; Γ_{1}; Γ; Γ_{N+3}]

(iv) The number of moduli of the case {2, 0}^{c} is 16, with mildest singularity A_{3}.

(v) The number of moduli of the case {3, 0}^{c} is 15, with mildest singularity A_{3}. (vi) In the case {3, 0}^{b} we have L ∼ 3D + 2Γ_{1} + 2Γ_{2} + Γ_{3} + Γ_{4} + Γ_{5}, with the following

configuration:

[figure: D; Γ_{1}; Γ_{3}; Γ_{5}; Γ_{2}; Γ_{4}.]

or L ∼ 3D + 2Γ_{1} + Γ′_{1} + 3Γ_{2} + ⋯ + 3Γ_{N} + 2Γ_{N+1} + Γ_{N+2} + Γ_{N+3}, with the following configuration:

[figure: D; Γ_{2}; Γ_{N}; Γ_{N+1}; Γ_{N+2}; Γ_{1}; Γ′_{1}; Γ_{N+3}.]

In the case {3, 0}^{c} we have L ∼ 3D + Γ + 4Γ_{0} + 3Γ_{1} + 2Γ_{2} + 2Γ_{3} + Γ_{4}, with the following configuration:

D  Γ_{0}  Γ_{1}  Γ_{3} Γ  Γ_{2}  Γ_{4}.

The mildest singularity of this case is A_{5}.

g = 10

The general projective model is a complete intersection of 2 hyperplanes in the homogeneous variety Σ_{18}^{5}, as described in Section 9.

The others are as follows:

c D^{2}  scroll type  # mod.  type of L  sing.
1 0  (3, 3, 2)  18  h^{0}(L − 2D) = 5, h^{0}(L − 3D) = 2, h^{0}(L − 4D) = 0 sm.
1 0  (4, 2, 2)  16  h^{0}(L − 2D) = 5, h^{0}(L − 3D) = 2, h^{0}(L − 4D) = 1 sm.
1 0  (4, 3, 1)  17  h^{0}(L − 2D) = 5, h^{0}(L − 3D) = 3, h^{0}(L − 4D) = 1 sm.
1 0  (5, 2, 1)  16  h^{0}(L − 5D) = 1 ^{(i)}  A_{2}
1 0  (6, 2, 0)  18  {1, 0}^{a}, L ∼ 6D + 3Γ  A_{1}
2 0  (2, 2, 2, 1)  18  h^{0}(L − 2D) = 3, h^{0}(L − 3D) = 0  sm.
2 0  (3, 2, 1, 1)  17  h^{0}(L − 2D) = 3, h^{0}(L − 3D) = 1 ^{(ii)}  A_{1}
2 0  (3, 2, 1, 1)  16  h^{0}(L − 2D) = 3, h^{0}(L − 3D) = 1 ^{(iii)}  sm.
2 0  (3, 2, 2, 0)  17  {2, 0}^{a}, A^{2} = 4, h^{0}(A − D) = 1, A ≯ 2D ^{(iv)}  A_{1}
2 0  (3, 3, 1, 0)  16  {2, 0}^{a}, A^{2} = 4, h^{0}(A − D) = 2, A ≯ 2D ^{(v)}  A_{2}
2 0  (4, 2, 1, 0)  16  {2, 0}^{a}, A^{2} = 4, A > 2D ^{(vi)}  2A_{1}
2 2  (3, 2, 1, 0, 0)  19  L ∼ 3D  sm.
3 0  (2, 1, 1, 1, 1)  18  h^{0}(L − 2D) = 0  sm.
3 0  (2, 2, 1, 1, 0)  17  {3, 0}^{a}, A^{2} = 0  A_{1}
3 0  (2, 2, 2, 0, 0)  16  {3, 0}^{b}, A^{2} = 2, A ≯ D  2A_{1}
3 0  (2, 2, 2, 0, 0)  15  {3, 0}^{c}, A^{2} = 2, A ≯ D  A_{3}
3 0  (3, 2, 1, 0, 0)  15  {3, 0}^{b}, A^{2} = 2, A > D ^{(vii)}  2A_{1}
3 0  (3, 2, 1, 0, 0)  15  {3, 0}^{b}, A^{2} = 2, A > D ^{(viii)}  A_{1} + A_{2}
3 0  (3, 2, 1, 0, 0)  14  {3, 0}^{c}, A^{2} = 2, A > D ^{(ix)}  A_{3}
3 0  (3, 2, 1, 0, 0)  14  {3, 0}^{c}, A^{2} = 2, A > D ^{(x)}  A_{4}
3 2 (2, 1, 1, 1, 0, 0)  18  h^{0}(L − 2D) = 1  sm.
3 4 (2, 1, 1, 0, 0, 0, 0)  18  (E0)  A_{1}
4 2 (1, 1, 1, 1, 0, 0, 0)  18  (CG1)’ or (CG2)’ ^{(xi)}  sm.
4 2 (2, 1, 1, 0, 0, 0, 0)  16  (CG3)’ or (CG4)’ ^{(xii)}  3A_{1}

In the cases (c, D^{2}) = (1, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−3H + 6F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0.

.

In the cases (c, D^{2}) = (2, 0) O_{S′} has the following O_{T} -resolution:

0 ⟶ O_{T} (−4H + (g − 1)F) ⟶
O_{T} (−2H + b_{1}F) ⊕ O_{T} (−2H + b_{2}F) ⟶ O_{T} ⟶ O_{S′} ⟶ 0,

with (b_{1}, b_{2}) = (3, 2) or (4, 1) for the scroll type (4, 2, 1, 0), (b_{1}, b_{2}) = (3, 2) for the scroll types (2, 2, 2, 1), (3, 2, 1, 1) (smooth), (3, 2, 2, 0) and (3, 3, 1, 0), and (b_{1}, b_{2}) = (4, 1) for the scroll type (3, 2, 1, 1) (A_{1}-sing.).

In the case (c, D^{2}) = (2, 2) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

O_{T_{0}}(2H − 4F) ⊕ O_{T_{0}}(2H − 3F) ⊕ O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H).

or of

O_{T_{0}}(2H − 3F)^{3} ⊕ O_{T_{0}}(2H).

In the cases (c, D^{2}) = (3, 0) we have an O_{T} -resolution of O_{S′} of the following type:

\begin{aligned}0 & ⟶ & O_{T} (−5H + 9F) & ⟶ & ⊕_{i=1}^{5}O_{T} (−3H + a_{i}F) \\ & ⟶ & ⊕_{i=1}^{5}O_{T} (−2H + b_{i}F) & ⟶ & O_{T} ⟶ O_{S′} ⟶ 0, \end{aligned}

with a_{i} = 4 − b_{i} for all i.

For the smooth scroll type (2, 1, 1, 1, 1) we have (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 2, 2, 1, 1). For the scroll type (2, 2, 2, 0, 0) we have (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 2, 2, 1, 1) or (2, 2, 2, 2, 0). For the remaining two singular scroll types we have (b_{1}, b_{2}, b_{3}, b_{4}) = (b_{1}, ..., b_{4}) are (3, 3, 2, 0, 0), (3, 3, 1, 1, 0), (3, 2, 2, 1, 0), (2, 2, 2, 1, 1) or (2, 2, 2, 2, 0).

In the case (c, D^{2}) = (3, 2) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

O_{T_{0}}(2H − 2F)^{3} ⊕ O_{T_{0}}(2H − F)^{4} ⊕ 2O_{T_{0}}(H).

In the case (c, D^{2}) = (3, 4) then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

O_{T_{0}}(2H − 2F)^{3} ⊕ O_{T_{0}}(2H − F)^{4} ⊕ O_{T} (2H)^{5}.

In the cases (CG1)’ and (CG2)’ then S′′ is cut out in T_{0} by a section of

O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H − F)^{7} ⊕ O_{T_{0}}(2H)^{5} or
O_{T_{0}}(2H − F)^{9} ⊕ O_{T_{0}}(2H)^{4}  or
O_{T_{0}}(2H − 2F) ⊕ O_{T_{0}}(2H − F)^{8} ⊕ O_{T_{0}}(2H)^{3} ⊕ O_{T_{0}}(2H + F),

where the last option occurs only for the case (CG2)’ (the section is constant on the fibers of i in the first four cases).

In the cases (CG3)’, (CG4)’ then S′′ is cut out in T_{0} by a section (which is constant on the fibers of i) of

\begin{matrix}O_{T_{0}}(2H − 2F)^{2} & ⊕ & O_{T_{0}}(2H − F)^{5} & ⊕ & O_{T_{0}}(2H)^{6} & or \\ O_{T_{0}}(2H − 2F)^{3} & ⊕ & O_{T_{0}}(2H − F)^{3} & ⊕ & O_{T_{0}}(2H)^{7} \end{matrix}

Comments on the types of L:

(i) L ∼ 5D + 3Γ_{1} + 2Γ_{2} + Γ_{3}, with the following configuration:

D  Γ_{1}  Γ_{2}  Γ_{3}

(ii) L is in general of the form L ∼ 3D + 2Γ_{1} + Γ_{2}, with the following configuration:

D  Γ_{1}  Γ_{2}

(iii) L is in general of the form L ∼ 3D + Γ_{1} + Γ_{2} + Γ_{3}, with the following configuration:

[figure: D; Γ_{1}; Γ_{2}; Γ_{3}]

(iv) L is in general of the form L ∼ 3D + 2Γ + Γ_{1}, with the following configuration:

[figure: D; Γ; Γ_{1}]

(v) L ∼ 3D + E + 2Γ + Γ_{1}, where E is a smooth elliptic curve, with the following configuration:

D  Γ  Γ_{1} E

(vi) L is in general of the form L ∼ 4D + 2Γ + 2Γ_{1} + Γ_{2}, with the following configuration:

D  Γ_{1}  Γ_{2} Γ

(vii) L is in general of the form L ∼ 3D+2Γ_{1}+2Γ_{2}+Γ_{3}+Γ_{4}, with the following configuration:

[figure: D; Γ_{1}; Γ_{3}; Γ_{2}; Γ_{4}]

(viii) L is in general of the form L ∼ 3D+2Γ_{1}+2Γ_{2}+Γ_{3}+Γ_{4}, with the following configuration:

[figure: D; Γ_{1}; Γ_{4}; Γ_{3}; Γ_{2}]

(ix) L is in general of the form L ∼ 3D + 4Γ_{0} + 2Γ_{1} + 2Γ_{2} + Γ_{3} + Γ_{4}, with the following configuration:

[figure: D; Γ_{0}; Γ_{3}; Γ_{4}; Γ_{1}; Γ_{2}]

(x) L is in general of the form L ∼ 3D + 4Γ_{0} + 3Γ_{1} + 2Γ_{2} + 2Γ_{3} + Γ_{4}, with the following configuration:

[figure: D; Γ_{0}; Γ_{2}; Γ_{4}; Γ_{3}; Γ_{1}]

(xi) The number of moduli of the case (CG2)’ is 17, with mildest singularity A_{1}. (xii) The number of moduli of the case (CG4)’ is 15, with mildest singularity A_{1} + A_{3}.

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Mathematics Subject Classification: 14J28 (14H45). E-mail: johnsen@mi.uib.no, andreask@mi.uib.no.

Dept. of Mathematics, University of Bergen, Johs. Brunsgt 12, N-5008 Bergen, Norway