A PARAMETRIZATION OF THE THETA DIVISOR OF THE QUARTIC DOUBLE SOLID

D.G MARKUSHEVICH AND A.S. TIKHOMIROV

Abstract. Let M_{X}(2; 0, 3) be the moduli space of rank-2 stable vector bundles with Chern classes c_{1} = 0, c_{2} = 3 on the Fano threefold X, the double solid ℙ^{3} of index two. We prove that the vector bundles obtained by Serre’s construction from smooth elliptic quintic curves on X form an open part of an irreducible component M of M_{X}(2; 0, 3) and that the Abel-Jacobi map ϕ : M → J(X) into the intermediate Jacobian J(X) of X defined by the second Chern class is generically finite of degree 84 onto a translate Θ + const of the theta-divisor. We also prove that the family of elliptic quintics on a general X is irreducible and of dimension 10.

0. Introduction

After the famous paper of Clemens and Griffiths [CG], in which they determined the intermediate Jacobian and its theta divisor for the cubic threefold, the next Fano threefold to study was naturally the quartic double solid, that is the double cover π : X_{2} ⟶^{2:1} ℙ^{3} ramified in a quartic surface W ⊂ ℙ^{3}. In 1981, Welters [We] found a family of curves parametrizing the intermediate Jacobian J(X_{2}), the septics of genus 4, and this is the simplest known family of generically irreducible curves with this property. He developed also techniques that allow one to decide whether the Abel-Jacobi images of certain families of curves lie in a translate Θ+const of the theta divisor, but failed to find one parametrizing the whole of Θ + const. Such a family was found in 1986 (with a use of the techniques of Welters) by Tikhomirov [T-2]: these are the sextics of genus 3, or the so called Reye sextics. In 1991, Clemens [C-2] re-proved Tikhomirov’s result in degenerating the double solid into a pair of ℙ^{3}’s meeting each other transversely along a quadric, the technique he developed earlier in [C-1]. Several other papers have contributed, since 1981, into the study of the quartic double solids: [T-1], [SV], [De], [V].

In 2001, Tikhomirov [T-3] found a new and simpler parametrization of Θ, the one by the family of elliptic quintics C_{5}^{1}(X). Its advantage is in that the elliptic quintics define stable vector bundles on X via Serre’s construction, and the Abel-Jacobi map factors through the moduli space of vector bundles.

The study of moduli spaces of stable vector bundles on Fano threefolds of indices 1 and 2 is quite a recent topic. Remind that the index of a Fano threefold X is the maximal integer ν dividing K_{X} in the Picard group of X. The cubic X_{3} and the quartic double solid X_{2} are Fano of index 2. The nature of the results obtained depends strongly on the index. In the index-2 case, the answers are given in terms of the Abel–Jacobi map defined by the second Chern class c_{2} modulo rational equivalence. In the index-1 case, the Abel–Jacobi map does not bring much new information about the moduli spaces.

1991 Mathematics Subject Classification. 14J30.

Partially supported by the grant INTAS-OPEN-2000-269.

The results which are known so far include the desciption of one component of a moduli space M_{X}(2; c_{1}, c_{2}) of rank-2 vector bundles with c_{1} = 0 or 1 and small c_{2} on each one of the following five Fano threefolds: the cubic X_{3} [MT-1], [IM-1], [Dr], [B], the quartic [IM-2], the Fano threefolds X_{2g−2} of the main series of genus 7 [IM-3] and 9 [IR], and the double solid X_{2}, treated in the present paper and in [T-3]. It is noteworthy that in most cases the interesting components of moduli are those parametrizing the vector bundles obtained by Serre’s construction from elliptic quintics. For the cubic X_{3}, the relevant moduli space is M_{X_{3}}(2; 0, 2), and the Abel-Jacobi map identifies it with an open set in the intermediate Jacobian J(X_{3}). For the double solid, the corresponding moduli space M_{X}(2; 0, 3) is reducible: it has at least two components, the one, say, M, associated to elliptic quintics, and the other to disjoint unions of two elliptic curves in X whose images in ℙ^{3} are a plane cubic and a line. Leaving apart the question whether there are any other components, we restrict ourselves to the study of M.

The main result of the paper states that the Abel-Jacobi map Φ sends the family of elliptic quintics C_{5}^{1}(X) onto an open subset of Θ + const, and that the Serre construction defines a factorization C_{5}^{1}(X) ⟶^{Serre} M ⟶^{g} Θ + const with g generically finite of degree 84 (Theorems 2.7 and 5.1). We also prove the irreducibility of C_{5}^{1}(X) (Theorem 1.4), so that M is really an irreducible component of the moduli space, rather than a union of components. The facts that the image of Φ is dense in Θ + const and that g is generically finite were stated in [T-3], so the main goal of our paper is to determine the degree of g. Nonetheless, we provide a complete proof of these facts too, for their proof was only sketched in loc. cit.

We use Welters’ criterion (Proposition 2.4) which gives sufficient conditions for a certain subfamily {C_{t}}_{t∈T_{l}} of a given family {C_{t}}_{t∈T} of curves of degree k(k +1)−3 to be mapped by the Abel-Jacobi map into a translate of the theta divisor. Here l is a line in X and

T_{l} = {t ∈ T | C_{t} ∩ l = ∅ and C_{t} ∪ l lies on a surface from |O_{X}(k)|},

where O_{X}(k) := π^{∗}O_{ℙ^{3}}(k). We construct such a family T with k = 4 as a partial compactification of the one of reducible curves C = C_{1} ∪ C_{2}, where C_{1} is a genus-4 septic, C_{2} a pullback of a genus-2 quintic \underline{C}_{2} ⊂ ℙ^{3}, and C_{1}, C_{2} meet each other quasitransversely in 13 points. We show that T_{l} contains a subfamily of curves of the form C′ ∪ C′′ ∪ C_{2} such that C′ ∈ C_{5}^{1}(X) and C′′ = π^{−1}(m), where m ⊂ ℙ^{3} is a trisecant of π(C′), whose Abel-Jacobi images differ from those of C′ by a constant translation. Hence Φ_{C_{5}^{1}(X)}(C_{5}^{1}(X)) ⊂ Φ_{T_{l}}(T_{l}) + const ⊂ Θ + const. The opposite inclusion follows from the computation of the differential of the Abel-Jacobi map. The generic fiber of Φ_{C_{5}^{1}(X)} is a union of ν copies of ℙ^{1}, which can be interpreted either as pencils of elliptic quintics {C_{λ}}_{λ∈ℙ^{1}} in the K3 surfaces S(C_{λ}) from the linear system |O_{X}(2)|, or as pencils of the zero loci of sections s ∈ H^{0}(X, E), where E is the vector bundle obtained by Serre’s construction from C_{λ}.

To compute the number ν of copies of ℙ^{1} in the generic fiber of Φ_{C_{5}^{1}(X)}, we use the degeneration {X_{t}}_{t∈Δ} of X into the union X_{0} = ℙ^{3′} ∪ℙ^{3′′} of two copies of ℙ^{3} meeting each other in a smooth quadric Q = {G = 0}, where Δ is a disc in ℂ. The fiber X_{t} is defined as the double cover π_{t} : X_{t} ⟶^{2:1} ℙ^{3} ramified in W_{t} = {tF + G^{2} = 0}, where W = {F = 0} is the branch locus of X⟶ℙ^{3}. The Hodge theory and the Neron model of the relative intermediate Jacobian of such a degeneration were studied by Clemens in [C-1]. He made also the following important observation: let C_{0} ∈ X_{0} be an irreducible curve of degree d which is the limit of a family of curves C_{s} ⊂ X_{t}, t = s^{e}. Assume that C_{s} for s ≠ 0 is projected birationally onto its image \underline{C}_{s} ⊂ ℙ^{3}; this means that \underline{C}_{s} is tangent to W_{t} in 2d points. Then C_{0} is a 2d-secant to the octic curve B = W ∩ Q. Thus the limits of elliptic quintics from C_{5}^{1}(X_{t}) are elliptic quintics in ℙ^{3′} or ℙ^{3′′} which are 10-secant to B. The work of Clemens allows us to extend the Abel-Jacobi map to such curves in X_{0} and the question on the number of copies of ℙ^{1} in the generic fiber of the Abel-Jacobi map Φ_{0} at t = 0 is reduced to that on the number of elliptic quintics in ℙ^{3} which are 10-secant to a given octic B.

By a result of Getzler [Ge], there are 42 elliptic quintics passing through 10 generic points in ℙ^{3}. We show that the 10-uples of points lying in the smooth complete intersections of a quartic and a quadric in ℙ^{3} form a divisor in the main component H of Hilb^{10}(ℙ^{3}) and that this divisor is not contained in the branch locus of the 42-sheeted covering defined by the elliptic quintics, hence there are 2 · 42 = 84 copies of ℙ^{1} in the generic fiber of Φ_{0} (42 copies in each one of the two ℙ^{3}’s constituting X_{0}). We show also that the elliptic quintics, 10-secant to B, are acquired with multiplicity 1 in the relative Hilbert scheme of the family {X_{t}}_{t∈Δ}, hence the number of copies of ℙ^{1} in the fiber of Φ_{t} is invariant under the deformation and ν = 84.

To this end, we show that the family of 10-secant elliptic quintics to a given generic B is irreducible and construct a family {C_{t}}_{t∈Δ} of cycles of 5 lines, or pentagons in X_{t}, such that C_{0} is 10-secant to B and is strongly smoothable into an elliptic quintic in X_{0}, 10-secant to B. This implies that a generic 10-secant elliptic quintic in X_{0} admits a local cross-section passing through it in the relative Hilbert scheme of {X_{t}}_{t∈Δ}, hence the fiber over t = 0 is acquired with multiplicity 1.

Describe now the contents of the paper by sections.

The main result of Section 1 is the proof of the irreducibility of the family of elliptic quintics C_{5}^{1}(X). We start by proving the irreducibility of the families C_{2}^{0}(X) of conics and C_{3}^{0}(X) of twisted cubics. The proofs are based on the splitting of the curves under consideration into two components of smaller degrees. As concerns elliptic quintics, we split them into a twisted cubic and an ”elliptic conic”, that is, π^{−1}(m) for a line m ⊂ ℙ^{3}. As soon as the irreducibility of the family of split curves is established, we get a distinguished component of C_{5}^{1}(X), namely the one containing the smoothings of the split curves. Then we use the irreducibility of the monodromy action on the set of all the components of C_{5}^{1}(X) to deduce that the number of components is 1.

In Section 2 we prove the irreducubility of the family C_{7}^{4}(X) of genus-4 septics in X, which dominate J(X) according to Welters, and construct the above mentioned family T of curves of degree k(k + 1) − 3 = 17 for k = 4, to which the criterion of Welters is applied. We conclude by the proof of Theorem 2.7, stating that the Abel-Jacobi image of the elliptic quintics is a translate of Θ.

In Section 3 we explain the Clemens’ degeneration technique and describe the limiting Abel-Jacobi map Φ_{0} on the degenerate double solid, which consists of two copies of ℙ^{3}. We also show that the central fiber in the domain of the Abel-Jacobi map is reduced by studying the deformations of 10-secant pentagons. This implies that the number of connected components in the generic fiber of Φ_{t} is the same as for Φ_{0}.

In Section 4 we prove that the number of elliptic quintics in ℙ^{3} passing through 10 generic points remains unchanged if we specialize the 10-uple of points to a one lying on a generic complete intersection of a quadric and a quartic. This implies that the number of connected components in the generic fiber of Φ_{0} is 84.

In Section 5 we introduce the Serre construction and state the main theorem in terms of moduli of vector bundles.

1. Conics, twisted cubics and elliptic quintics on X

1.1. Notations.

• X, a general quartic double solid.

• π : X⟶ℙ^{3}, the natural double covering map.

• G(1, 3), the Grassmannian of lines in ℙ^{3}.

• C_{2}^{0} = C_{2}^{0}(X) := {C ∈ Hilb(X)| C is a smooth irreducible conic in X}.

• C_{2}^{1}(X) := {C ∈ Hilb(X) | C = π^{−1}(l), l ∈ G(1, 3)}; we call the curves from C_{2}^{1}(X) elliptic conics on X.

• C_{3}^{0} = C_{3}^{0}(X) := {C ∈ Hilb(X)| C is a smooth irreducible twisted cubic in X}.

• C_{5}^{1} = C_{5}^{1}(X) := {C ∈ Hilb(X)| C is a smooth irreducible elliptic quintic in X}.

• F = F(X), the Fano surface of X, that is the base of the family of lines on X.

The main result of this section is the irreducibility of the family of elliptic quintics on X (Theorem 1.4), but before we need to prove that the families C_{2}^{0}(X) and C_{3}^{0}(X) of conics and, respectively, twisted cubics on X are irreducible. 1.2. Conics. First recall the well known facts about lines on a general X (see, e.g., [T-1], [We]). Lemma 1.1. Let X be a general quartic double solid. Then the following assertions hold:

(i) The Fano surface F = F(X) of lines on X is an irreducible surface having at most isolated quadratic points as singularities, and the incidence divisor D_{F} = {(l_{1}, l_{2}) ∈ F × F | l_{1} ∩ l_{2} ≠ ∅} on F × F is irreducible.

(ii) For a general line l ∈ F(X), the normal bundle N_{l/X} is isomorphic to 2O_{ℙ^{1}}.

Next we will prove the following statement: Lemma 1.2. Let X be a general quartic double solid. Then we have:

(i) C_{2}^{0}(X) is irreducible of dimension 4.

(ii) For a general conic C ∈ C_{2}^{0}(X), the normal bundle N_{C/X} is isomorphic either to 2O_{ℙ^{1}}(1) or to O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(2).

Proof. Let (l_{1}, l_{2}) ∈ D_{F} be generic. Then l_{1} meet each other l_{2} quasi-transversely (that is, with different tangents) at some point x ∈ X and, by Lemma 1.1, N_{l_{i}/X} ≃ 2O_{ℙ^{1}}. Let C = l_{1} ∪ l_{2}. Then we have the exact sequences

0 → N_{l_{i}/X} → N_{C/X}|l_{i} ⟶^{ε_{i}} C_{x} → 0, i = 1, 2.

So

(1)  N_{C/X}|l_{i} ≃ O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(1), i = 1, 2,

and the exact triple

0 → N_{C/X} → ⨁_{i=1}^{2} N_{C/X}|l_{i} → N_{C/X}|_{x} → 0, N_{C/X}|_{x} ≃ C_{x}^{2},

implies that

(2)  h^{1}(N_{C/X}) = 0.

Also, (1) and the exact triple

0 → N_{C/X}|l_{2}(−x) → N_{C/X} ⟶^{R} N_{C/X}|l_{1} → 0,

in which R is the usual restriction map, imply the surjectivity of the map H^{0}R : H^{0}(N_{C/X}) → H^{0}(N_{C/X}|l_{1}). Hence the composition of the natural maps

τ_{x} : H^{0}(N_{C/X}) ⟶^{H^{0}R} H^{0}(N_{C/X}|l_{1}) ⟶^{H^{0}ε_{1}} H^{0}(C_{x})

is surjective. Hence (2) and [HH, Prop. 1.1] imply that the conic C = l_{1} ∪ l_{2} is strongly smoothable in X, that is there exists an analytic family π : S ↪^{i} X × Δ ⟶^{pr_{2}} Δ over the disk Δ such that S is a smooth complex surface, π is a flat morphism, π^{−1}(0) = C and C_{t} = i(π^{−1}(t)) ≃ π^{−1}(t) is a smooth conic in X for any t ∈ Δ ∖ {0}. Hence we can take a section s : Δ ↪ S, t ↦ y_{t}, of π such that y_{0} ≠ x, y_{0} ∈ l_{1}. It follows now from (1) and the exact triple

0 → N_{C/X}|l_{2}(−x) → N_{C/X}(−y_{0}) → N_{C/X}|l_{1}(−y_{0}) → 0

that h^{1}(N_{C/X})(−y_{0}) = 0, so that by semicontinuity (possibly after shrinking Δ) we obtain h^{1}(N_{C_{t}/X})(−y_{t}) = 0 for t ∈ Δ ∖ {0}. Combining this with the observation that det(N_{C_{t}/X}) ≃ O_{ℙ^{1}}(2) for t ∈ Δ ∖ {0}, we see that the statement (ii) of the lemma holds for a general conic from the irreducible component C_{2}^{0∗}(X) of C_{2}^{0}(X) containing smoothings of reducible conics. It follows that

(3)  dim C_{2}^{0∗}(X) = 4.

To prove (i), recall that there is a natural embedding (see, e.g., [T-1], [We])

(4)  X ↪ Y = Proj(O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))

of X as a divisor of the linear series |L^{⊗2}|:

(5)  X ∈ |L^{⊗2}| ≃ ℙ^{45},

where L = O_{Y/ℙ^{3}}(1) is the Grothendieck sheaf. Recall that

(6)  p_{Y ∗}L = O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2), p_{Y ∗}(L^{⊗2}) = O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2) ⊕ O_{ℙ^{3}}(4),

where p_{Y} : Y → ℙ^{3} is the natural projection. The divisors of the linear series (5) constitute the family X ↪ Y × ℙ^{45} with natural projections Y ⟵^{p} X ⟶^{q} ℙ^{45}. Consider the following open subset in ℙ^{45}:

(7)  U = {t ∈ ℙ^{45} | X_{t} = p(q^{−1}(t)) is smooth}.

Denote by ρ : Y → ℙ^{3} the natural projection. Then for any t ∈ U, any conic C ∈ C_{2}^{0}(X_{t}) is projected by ρ to a conic C_{0} ⊂ ℙ^{3}, and ρ|C factors into the composition ρ|C : C ↪^{i_{C}} F_{C} ⟶ C_{0}, where

(8)  F_{C} := ρ^{−1}(C_{0}) = Proj((O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}}), i_{C}(C) ∈ |L|F_{C}| .

Consider the varieties

C_{2}^{0}(ℙ^{3}) = {C_{0} ∈ Hilb(ℙ^{3}) | C_{0} is a smooth conic in ℙ^{3}}

and

C_{2}^{0}(Y) = {C ∈ Hilb(Y) | ρ|_{C} : C → ρ(C) is an isomorphism onto a conic ρ(C) ∈ C_{2}^{0}(ℙ^{3})}.

Clearly

(9)  dim C_{2}^{0}(ℙ^{3}) = 8,

and for any C_{0} ∈ C_{2}^{0}(ℙ^{3}) we have (O_{ℙ^{3}}⊕O_{ℙ^{3}}(2))|_{C_{0}} ≃ O_{ℙ^{1}}⊕O_{ℙ^{1}}(4). This together with (8) implies that there exists a rank-6 vector bundle E over C_{2}^{0}(ℙ^{3}) with fiber E_{{C_{0}}} = H^{0}(O_{ℙ^{1}}⊕ O_{ℙ^{1}}(4)) over an arbitrary point C_{0} ∈ C_{2}^{0}(ℙ^{3}) (we identify C_{0} with ℙ^{1}), and C_{2}^{0}(Y) is a dense open subset of Proj(E) such that the natural projection C_{2}^{0}(Y) → C_{2}^{0}(ℙ^{3}), C ↦ ρ(C), coincides with the composition C_{2}^{0}(Y) ↪ Proj(E) ⟶^{pr} C_{2}^{0}(ℙ^{3}), where pr is the structure morphism. It is obvious that C_{2}^{0}(ℙ^{3}) is irreducible, hence so is C_{2}^{0}(Y), and from (9) we obtain

(10)  dim C_{2}^{0}(Y) = 13.

Consider the incidence variety Γ ↪ U × C_{2}^{0}(Y):

Γ = {(t, C) ∈ U × C_{2}^{0}(Y) | C ⊂ X_{t}}, U ⟵^{p_{1}} Γ ⟶^{p_{2}} C_{2}^{0}(Y).

It follows from (5) that

(11)  p_{2}^{−1}(C) = U ∩ |I_{C/Y} ⊗ L^{⊗2}| ↪^{open} |I_{C/Y} ⊗ L^{⊗2}|.

Any conic C ∈ C_{2}^{0}(Y) is a zero-set of a regular section of the sheaf G = p_{Y}^{∗} (O_{ℙ^{3}}(1) ⊕ O_{ℙ^{3}}(2)) ⊕ L. Hence the ideal sheaf I_{C/Y} has the standard Koszul resolution

(12)  0 → ∧^{3}G^{∨} → ∧^{2}G^{∨} → G^{∨} → I_{C/Y} → 0.

Now a simple computation using (6) and the projection formula for p_{Y} proves that the only nonzero numbers h^{i}(∧^{j}G^{∨} ⊗ L^{⊗2}) are h^{0}(G^{∨} ⊗ L^{⊗2}) = 46 and h^{0}(∧^{2}G^{∨} ⊗ L^{⊗2}) = 9. Twisting (12) by L^{⊗2}, we deduce that h^{0}(I_{C/Y} ⊗ L^{⊗2}) = 37, that is |I_{C/Y} ⊗ L^{⊗2}| = ℙ^{36}. Now (10), (11) imply that Γ is an irreducible variety of dimension 49. Next, the fibre p_{1}^{−1}(t) = C_{2}^{0}(X_{t}), t ∈ U, contains a distinguished irreducible component C_{2}^{0∗}(X_{t}) of dimension 4 (see (3)), in accordance with the surjectivity of the morphism p_{1} : Γ → U and the equalites dim U = 45, dim Γ = 49. Since both Γ and U are irreducible, it follows that C_{2}^{0}(X_{t}) = C_{2}^{0∗}(X_{t}). □

1.3. Twisted cubics. Now we will prove the irreducibility of the family of rational cubics on X. Lemma 1.3. Let X be a general quartic double solid. Then the following assertions hold:

(i) C_{3}^{0}(X) is irreducible of dimension 6.

(ii) For a general smooth cubic C ∈ C_{3}^{0}(X), the normal bundle N_{C/X} is isomorphic to O_{ℙ^{1}}(k) ⊕ O_{ℙ^{1}}(4 − k), 0 ≤ k ≤ 4.

Proof. The proof is similar to that of Lemma 1.2. Let us start by proving (ii). Using Lemmas 1.1 and 1.2 pick a general line l ∈ F and a conic C′ ∈ C_{2}^{0}(X) meeting each other quasi-transversely in a unique point x and such that

N_{l/X} ≃ 2O_{ℙ^{1}}, N_{C′/X} ≃ O_{ℙ^{1}}(k) ⊕ O_{ℙ^{1}}(2 − k), 0 ≤ k ≤ 1.

Let C = l ∪ C′. Then we have the exact sequences

0 → N_{l/X} → N_{C/X}|l ⟶^{ε} C_{x} → 0, 0 → N_{C′/X} → N_{C/X}|C′ → C_{x} → 0.

Hence

(13)  N_{C/X}|C′ ≃ O_{ℙ^{1}}(a) ⊕ O_{ℙ^{1}}(b), 0 ≤ a ≤ b, a + b = 3,

and the exact triple

0 → N_{C/X} → N_{C/X}|l ⊕ N_{C/X}|C′ → N_{C/X}|_{x} → 0, N_{C/X}|_{x} ≃ C_{x}^{2},

implies that

(14)  h^{1}(N_{C/X}) = 0.

Further, (13) and the restriction triple

0 → N_{C/X}|C′(−x) → N_{C/X} ⟶^{R_{l}} N_{C/X}|l → 0

imply the surjectivity of the map H^{0}R_{l} : H^{0}(N_{C/X}) → H^{0}(N_{C/X}|l). Hence the composition τ_{x} : H^{0}(N_{C/X}) ⟶^{H^{0}R_{l}} H^{0}(N_{C/X}|l) ⟶ H^{0}(C_{x}) is surjective. Similarly to the case of conics, we deduce from the surjectivity of τ_{x} and (14) the existence of a flat family of cubic curves C_{t} ⊂ X over a disc Δ together with a section t ∈ Δ ↦ y_{t} ∈ C_{t} such that C_{0} = C, C_{t} is smooth for t ≠ 0 and h^{1}(N_{C_{t}/X}(−y_{t})) = 0 for all t. The fact that det(N_{C_{t}/X}) ≃ O_{ℙ^{1}}(4) for t ∈ Δ ∖ {0} together with (i) of Lemma 1.2 imply the wanted assertion for a generic cubic from any irreducible component of C_{3}^{0}(X) which contains smoothings of a generic decomposable curve. Let us show that there is only one such irreducible component.

By [We], Corollary (5.4), the curve D_{l} ⊂ F of lines in X meeting a given generic line l is irreducible and is a 6-sheeted covering of l, because there are 6 lines passing through the generic point of X. So, if we take a generic reducible conic C = l_{1} + l_{2} ⊂ X, then the curve D_{C} of lines meeting C is a 6-sheeted covering of C, which is irreducble as a covering, that is there is no subcurve D′ ⊂ D_{C} which is a d-sheeted covering of C for some d < 6. The irreducibility of a covering is stable under small deformations, so we can conclude that D_{C} stays irreducible when we deform C into a smooth conic. The family of conics in X being irreducible, the incidence variety D of pairs (l, C′) ∈ F × C_{2}^{0} such that l ∩ C′ ≠ ∅ is irreducible. By the above, l + C′ is a smooth point of Hilb(X) for generic (l, C′) ∈ D, so there is only one component C_{3}^{0∗}(X) of C_{3}^{0}(X) containing the smoothings of the curves l + C′ for generic (l, C′) ∈ D.

Now we can prove the assertion (i); it implies, in particular, that C_{3}^{0∗}(X) is the whole of C_{3}^{0}(X) and hence (ii) is verified for a generic cubic C from C_{3}^{0}(X).

We have dim C_{3}^{0∗}(X) = 6, and any irreducible component C_{3}^{0′}(X) of C_{3}^{0}(X) has dimension at least χ(N_{C/X}) = 6. This together with an easy dimension count implies that

Č_{3}^{0}(X) = {C ∈ C_{3}^{0}(X) | ρ|C : C → ρ(C) is an isomorphism}

is a dense open subset of C_{3}^{0}(X).

In the notations of (5) – (7), any cubic C ∈ Č_{3}^{0}(X_{t}) for t ∈ U is projected by ρ to a twisted cubic C_{0} ⊂ ℙ^{3}. Consider the varieties

C_{3}^{0}(ℙ^{3}) = {C_{0} ∈ Hilb(ℙ^{3}) | C_{0} is a smooth cubic in ℙ^{3}}

and

Č_{3}^{0}(Y) = {C ⊂ Y | ρ|_{C} : C → ρ(C) is an isomorphism onto a cubic ρ(C) ∈ C_{3}^{0}(ℙ^{3})}.

We have

(15)  dim C_{3}^{0}(ℙ^{3}) = 12.

As in the proof of Lemma 1.2, we obtain (O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}} ≃ O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(6) for any C_{0} ∈ C_{3}^{0}(ℙ^{3}). Hence there exists a rank-8 vector bundle E over C_{3}^{0}(ℙ^{3}) with fiber E_{{C_{0}}} = H^{0}(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(6)) over C_{0} ∈ C_{3}^{0}(ℙ^{3}), and Č_{3}^{0}(Y) is a dense open subset of Proj(E) such that the natural projection C_{3}^{0}(Y) → C_{3}^{0}(ℙ^{3}), C ↦ ρ(C), coincides with the composition Č_{3}^{0}(Y) ↪ Proj(E) ⟶^{pr} C_{3}^{0}(ℙ^{3}), where pr is the structure morphism. C_{3}^{0}(ℙ^{3}) being irreducible, C_{3}^{0}(Y) is irreducible too and by (15)

(16)  dim Č_{3}^{0}(Y) = 19.

Consider the incidence variety Γ ↪ U × Č_{3}^{0}(Y):

(17)  Γ = {(t, C) ∈ U × Č_{3}^{0}(Y) | C ⊂ X_{t}},  U ⟵^{p_{1}} Γ ⟶^{p_{2}} Č_{3}^{0}(Y).

From (5) it follows that

(18)  p_{2}^{−1}(C) = U ∩ |I_{C/Y} ⊗ L^{⊗2}| ↪^{open} |I_{C/Y} ⊗ L^{⊗2}|.

Now any twisted cubic C_{0} ∈ C_{3}^{0}(ℙ^{3}) has the following well known resolution of its structure sheaf: 0 → 2O_{ℙ^{3}}(−3) → 3O_{ℙ^{3}}(−2) → O_{ℙ^{3}} → O_{C_{0}} → 0. Hence the twisted ideal sheaf I_{C/Y} × L^{⊗2} of any C ∈ Č_{3}^{0}(Y) has a resolution of the form

0 → 2L(−3h) → 3L(−2h) ⊕ 2L^{⊗2}(−3h) → L ⊕ 3L^{⊗2}(−2h) → I_{C/Y} × L^{⊗2} → 0.

An easy computation with this resolution gives |I_{C/Y} ⊗L^{⊗2}| = ℙ^{32}. An argument similar to that used in the end of the proof of Lemma 1.2 shows then the irreducubility of C_{3}^{0}(X). □

1.4. Elliptic quintics. Now we proceed to the study of the family C_{5}^{1}(X) of elliptic quintics on X. The following theorem is the main result of Section 1. Theorem 1.4. Let X be a general quartic double solid. Then C_{5}^{1}(X) is irreducible and is of dimension 10.

Proof. The proof goes along the lines of those of Lemmas 1.2 and 1.3.

Using Lemma 1.3, pick a general cubic C_{1} ∈ C_{3}^{0} and a smooth elliptic conic C_{2} ∈ C_{2}^{1}(X) meeting each other quasi-transversely at a unique point x and such that

N_{C_{1}/X} ≃ O_{ℙ^{1}}(k) ⊕ O_{ℙ^{1}}(4 − k), 0 ≤ k ≤ 2, N_{C_{2}/X} ≃ 2O_{C_{2}}(x + y), {x, y} = π^{−1}(π(x)).

Let C = C_{1} ∪ C_{2}. Then we have the exact sequences

0 → N_{C_{i}/X} → N_{C/X}|C_{i} ⟶^{ε_{i}} C_{x} → 0, i = 1, 2.

Exactly as in the proofs of Lemmas 1.2 and 1.3, one shows that there exists an analytic family π : S ↪^{i} X × Δ ⟶^{pr_{2}} Δ over a disk Δ such that S is a smooth complex surface, π is a flat morphism, π^{−1}(0) = C and C_{t} = i(π^{−1}(t)) ≃ π^{−1}(t) is a smooth elliptic quintic in X for any t ∈ Δ ∖ {0}. Let C_{5}^{1∗}(X) denote the unique irreducible component of Hilb(X) which contains the of generic reducible curve C = C_{1} ∪ C_{2} as above. The uniqueness is proved exactly as in the proof of Lemma 1.3. From the values of h^{i}(N_{C/X}), we see that dim C_{5}^{1∗}(X) = 10. For any other irreducible component C_{5}^{1∗}(X) of C_{5}^{1}(X) we have dim C_{5}^{1∗}(X) ≥ 10. An easy dimension count shows that

(19)  Č_{5}^{1}(X) = {C ∈ C_{5}^{1}(X) | ρ|C : C → ρ(C) is an isomorphism}

is a dense open subset of C_{5}^{1}(X).

Now for t ∈ U any quintic C ∈ Č_{5}^{1}(X_{t}) is projected by ρ to an elliptic quintic C_{0} ⊂ ℙ^{3}. Consider the varieties

C_{5}^{1}(ℙ^{3}) = {C_{0} ∈ Hilb(ℙ^{3}) | C_{0} is a smooth elliptic quintic in ℙ^{3}}

and

Č_{5}^{1}(Y) = {C ∈ Hilb(Y) | ρ|_{C} : C → ρ(C) is an isomorphism onto its image ρ(C) ∈ C_{5}^{1}(ℙ^{3})}.

Here C_{5}^{1}(ℙ^{3}) is an irreducible variety of dimension 20 and, arguing as in the proof of Lemma 1.2, for any C_{0} ∈ C_{5}^{1}(ℙ^{3}) we have (O_{ℙ^{3}}⊕O_{ℙ^{3}}(2))|_{C_{0}} ≃ O_{C_{0}} ⊕O_{C_{0}}(D), deg D = 10, so that h^{0}((O_{ℙ^{3}} ⊕O_{ℙ^{3}}(2))|_{C_{0}}) = 11. This implies that there exists a rank-11 vector bundle E over C_{5}^{1}(ℙ^{3}) with fiber E_{{C_{0}}} = H^{0}((O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}}) over C_{0} ∈ C_{5}^{1}(ℙ^{3}), and Č_{5}^{1}(Y) is a dense open subset of Proj(E) such that the natural projection Č_{5}^{1}(Y) → C_{5}^{1}(ℙ^{3}), C ↦ ρ(C), coincides with the composition Č_{5}^{1}(Y) ↪ Proj(E) ⟶^{pr} C_{5}^{1}(ℙ^{3}), where pr is the structure morphism. Hence Č_{5}^{1}(Y) is irreducible and dim Č_{5}^{1}(Y) = 30.

Consider the incidence variety Γ ↪ U × Č_{5}^{1}(Y):

Γ = {(t, C) ∈ U × Č_{5}^{1}(Y) | C ⊂ X_{t}}, U ⟵^{p_{1}} Γ ⟶^{p_{2}} Č_{5}^{1}(Y).

From (5) it follows that

p_{2}^{−1}(C) = U ∩ |I_{C/Y} ⊗ L^{⊗2}| ↪^{open} |I_{C/Y} ⊗ L^{⊗2}|.

Now take any quintic C ∈ Č_{5}^{1}(Y). Let ρ_{C} = ρ|_{C} : C → C_{0} be the natural isomorphic projection of C onto a quintic C_{0} ∈ C_{5}^{1}(ℙ^{3}). We have p_{Y ∗}(L^{⊗2} ⊗ O_{C}) ≃ O_{ℙ^{3}}(4)|C_{0}. Consider the standard exact triple

(20)  0 → I_{C/Y} ⊗ L^{⊗2} → L^{⊗2} ⟶^{ε} L^{⊗2} ⊗ O_{C} → 0.

Applying ρ_{C∗}, we obtain the following one:

(21)  0 → p_{Y ∗}I_{C/Y} ⊗ L^{⊗2} → O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2) ⊕ O_{ℙ^{3}}(4) ⟶^{ε_{0}} O_{ℙ^{3}}(4)|C_{0} → 0.

The map ε_{0}|O_{ℙ^{3}}(4) : O_{ℙ^{3}}(4) → O_{ℙ^{3}}(4)|C_{0} coincides with the restriction one in the triple

0 → I_{C_{0},ℙ^{3}}(4) → O_{ℙ^{3}}(4) ⟶^{res} O_{ℙ^{3}}(4)|C_{0} → 0.

Hence, by (20) and (21), the equality h^{1}(I_{C/Y} ⊗ L^{⊗2}) = 0 is a consequence of h^{1}(I_{C_{0},ℙ^{3}}(4)) = 0. To prove the last one, take two general cubic surfaces S_{1}, S_{2} through C_{0} such that the residual curve of their intersection is a smooth rational quartic C_{1} meeting C_{0} in a scheme Z of length 10. Then we have two exact triples

0 → I_{C_{0}∪C_{1},ℙ^{3}}(4) → I_{C_{0},ℙ^{3}}(4) → O_{C_{1}}(4)(−Z) → 0,

where O_{C_{1}}(4)(−Z) = O_{ℙ^{1}}(6), and

0 → O_{ℙ^{3}}(−2) → 2O_{ℙ^{3}}(1) → I_{C_{0}∪C_{1},ℙ^{3}}(4) → 0.

Their long cohomology exact sequences imply the vanishing of h^{1}(I_{C_{0},ℙ^{3}}(4)) = 0. Hence h^{0}(I_{C/Y} ⊗L^{⊗2}) = 26 and |I_{C/Y} ⊗L^{⊗2}| = ℙ^{25}. The remaining part of the proof is the same as in Lemma 1.2. □

In Section 2 we will use the following auxiliary result about quintics on X.

Lemma 1.5. Let X be general so that C_{5}^{1}(X) is irreducible and let \overline{C_{5}^{1}(X)} be the closure of C_{5}^{1}(X) in Hilb(X). Then there exists a strongly smoothable reduced reducible quintic C_{0} ∈ \overline{C_{5}^{1}(X)} of the form

(22)  C_{0} = l_{1} ∪ ... ∪ l_{5}, l_{i} ∈ F, i = 1, ..., 5,

such that

(23)  a_{1} = l_{1} ∩ l_{2}, a_{2} = l_{2} ∩ l_{3}, a_{3} = l_{3} ∩ l_{4}, a_{4} = l_{4} ∩ l_{1}, a_{5} = l_{1} ∩ l_{5},

are distinct points satisfying the following conditions:

(24)  N_{l_{i}/X} = 2O_{l_{i}}, i = 1, ..., 5,

(25) N_{C_{0}/X}|l_{1} ≃ O_{l_{1}}(1) ⊕ O_{l_{1}}(2), N_{C_{0}/X}|l_{i} ≃ 2O_{l_{i}}(1), i = 2, 3, 4, N_{C_{0}/X}|l_{5} ≃ O_{l_{1}} ⊕ O_{l_{1}}(1).

Proof. For a general line l ∈ F on X, we have N_{l/X} ≃ 2O_{l}. Hence, for a general plane ℙ^{2} in ℙ^{3} the del Pezzo surface S = π^{−1}(ℙ^{2}) is smooth and all the lines in it have trivial normal bundle in X. One easily finds a configuration C_{0} of five lines in S satisfying (22) and (23). Let C_{1} = l_{1} ∪ ... ∪ l_{4}. Since each line l_{i} ∈ C_{0} is a (−1)-curve on S, it follows that N_{C_{1}/S}|l_{i} ≃ O_{l_{i}}(1), i = 1, ..., 4. Hence the exact triple

0 → N_{C_{1}/S}|l_{i} → N_{C_{1}/X}|l_{i} → N_{S/X}|l_{i} → 0

and the obvious equality det(N_{C_{1}/X}|l_{i}) ≃ O_{l_{i}}(2) imply N_{C_{1}/X}|l_{i} ≃ 2O_{l_{i}}(1), i = 1, ..., 4. Adding l_{5} to C_{1} yields C_{0}, and one immediately obtains (25).

To show that C_{0} is strongly smoothable, denote

(26)  Z_{a} = {a_{1}, ..., a_{5}}

and consider the natural exact triples

(27) 0 → N_{a} → N_{C_{0}/X} → O_{Z_{a}} → 0, 0 → N′_{a} → N_{a} → N_{l_{1}/X} ⊕ N_{l_{3}/X} → 0,

where N′_{a} := (N_{C_{0}/X}|l_{2} ∪ l_{4} ∪ l_{5})(−Z_{a}); then the strong smoothability of C_{0} follows from the equality h^{1}(N_{a}) = 0 (see [HH]). To prove the last equality, remark that, by (24), one has N′_{a} ≃ 2O_{l_{2}}(−1) ⊕ 2O_{l_{4}}(−1) ⊕ O_{l_{5}} ⊕ O_{l_{5}}(−1), hence h^{1}(N′_{a}) = 0. Also, (25) implies h^{1}(N_{l_{1}/X} ⊕ N_{l_{3}/X}) = 0, so the second triple (27) implies h^{1}(N_{a}) = 0. □

2. The Abel-Jacobi map of the family of elliptic quintics

In this section we will study the Abel-Jacobi maps of three families of curves on a general double solid X: genus-4 septics C_{7}^{4}(X), elliptic quintics C_{5}^{1}(X) and a certain family T of reducible curves of degree 17 and arithmetic genus 29 which will set a connection between the two other families. 2.1. Notations. We will use the following symbols, in addition to those introduced in 1.1:

• C_{7}^{4}(X) := {C ∈ Hilb(X)| C is a smooth irreducible septic of genus 4 in X}.

• \overline{C_{7}^{4}(X)}, the closure of C_{7}^{4}(X) in Hilb(X).

• \overline{C_{7}^{4}(X)}^{∗} := \{C ∈ \overline{C_{7}^{4}(X)} | C is a reduced curve with at worst ordinary double points\}, a dense open subset of \overline{C_{7}^{4}(X)} containing C_{7}^{4}(X).

• C_{7}^{4}(ℙ^{3}) := {C ∈ Hilb(ℙ^{3}) | C is a smooth irreducible septic of genus 4 in ℙ^{3}}.

• C_{5}^{2}(ℙ^{3}) := {C ∈ Hilb(ℙ^{3})| C is a reduced quintic curve of arithmetic genus 2 on a smooth quadric surface Q ⊂ ℙ^{3} of bidegree (3,2) }.

• π^{∗}C_{5}^{2}(ℙ^{3}) := \{C ∈ Hilb(X)| C = π^{−1}(\underline{C}), \underline{C} ∈ C_{5}^{2}(ℙ^{3})\}.

• C_{(3,3)}(ℙ^{3}) := {C ∈ Hilb(ℙ^{3})| C is a curve of arithmetic genus 4 on a smooth quadric surface Q ⊂ ℙ^{3} of bidegree (3,3) (that is C is a complete intersection of Q and a cubic surface)}.

• π^{∗}C_{(3,3)}(ℙ^{3}) := \{C ∈ Hilb(X)| C = π^{−1}(\underline{C}), \underline{C} ∈ C_{(3,3)}(ℙ^{3})\}.

• \overline{C_{5}^{1}(X)}, the closure of C_{5}^{1}(X) in Hilb(X).

• \overline{C_{5}^{1}(X)}^{∗} = \{C ∈ \overline{C_{5}^{1}(X)} | C is a reduced curve with at most ordinary double points\}, an open subset of \overline{C_{5}^{1}(X)}.

• C_{5+2} := {C ∈ Hilb(X) | C is a reduced curve of the form C = C′ ∪ C′′, where C′ ∈ \overline{C_{5}^{1}(X)}^{∗}, C′′ ∈ C_{2}^{1}(X) and C′ meets C′′ quasi-transversely in 3 points\}.

• Č_{5+2} := {C′ ∪ C′′ ∈ C_{5+2} | C′ ∈ Č_{5}^{1}(X)}, where Č_{5}^{1}(X) is defined by (19).

• Φ_{B} : B → J(X), the Abel-Jacobi map of a given subvariety B of the Hilbert scheme of curves on X (defined uniquely up to a choice of a reference point in B).

2.2. Septics of genus 4. We are interested in the family of septics C_{7}^{4}(X) because of the following result of Welters [We, 6.18]: Proposition 2.1. For a general quartic double solid X, the Abel-Jacobi map Φ_{C_{7}^{4}(X)} : C_{7}^{4}(X) → J(X) is dominant.

We begin with the following lemma. Lemma 2.2. Let X be a general quartic double solid. Then:

(i) Č_{5+2} is irreducible of dimension 21 and the natural projection p′′ : Č_{5+2} → Č_{5}^{1}(X), C′ ∪ C′′ ↦ C′, is surjective.

(ii) Let \overline{Č_{5+2}} be the closure of Č_{5+2} in C_{5+2}. There exists a strongly smoothable curve C′_{0} ∪ C′′_{0} ∈ \overline{Č_{5+2}} ∖ Č_{5+2} such that

(28)  C′_{0} = l_{1} ∪ ... ∪ l_{5} ∈ \overline{C_{5}^{1}(X)} ∖ C_{5}^{1}(X)

is the curve (22) from Lemma 1.5 and C′′_{0} ∈ C_{2}^{1}(X) is a smooth elliptic conic such that

(29)  Z_{b} := C′_{0} ∩ C′′_{0} = {b_{1}, b_{2}, b_{3}}, b_{1} ∈ l_{2}, b_{2} ∈ l_{4}, b_{3} ∈ l_{5}, Z_{b} ∩ Z_{a} = ∅,

where Z_{a} = SingC′_{0} was introduced in (26).

(iii) The set

(30)  C_{5+2}^{∗} := \{C ∈ Č_{5+2} | C is strongly smoothable\} ⊂ C_{5+2} ∩ \overline{C_{7}^{4}(X)}^{∗}

is an irreducible dense open subset of C_{5+2} and the projection p′′ : C_{5+2}^{∗} → Č_{5}^{1}(X), C′ ∪ C′′ ↦ C′, is dominant.

(iv) Let C_{7}^{4∗}(X) be an irreducible component of \overline{C_{7}^{4}(X)} containing C_{5+2}^{∗}. Then dim C_{7}^{4∗}(X) = 14 and C_{7}^{4∗}(X)^{0} = {C ∈ C_{7}^{4∗}(X) | C is smooth and π|C : C → π(C) is an isomorphism} is a nonempty open subset of C_{7}^{4∗}(X).

Proof. (i) The curve \underline{C′} = π(C′) is an elliptic quintic in ℙ^{3}. For any elliptic quintic B ⊂ ℙ^{3} and for any x ∈ B, there are at most two trisecant lines of B passing through x, and the family Z of trisecant lines of B is an irreducible curve. To prove this, consider the linear projection π_{x} of B with center x to a plane ℙ^{2}. The image of B is a plane quartic of geometric genus 1, so we have the following two possibilities for its singular locus: 1) two singular points, each one of which is either a node, or a cusp, 2) one tacnode. The inverse image in B of a singular point of π_{x}(B) is a pair of points, say, z_{1}, z_{2}, and the line \overline{z_{1}z_{2}} is a trisecant of B, which meets B in three distinct points x, z_{1}, z_{2}. Hence the family Z of trisecant lines of B is a curve. To prove its irreducibility, consider the incidence curve D = {(z, l) ∈ B × Z | z ∈ l} in B × Z with its natural projections B ⟵^{p} D ⟶^{q} Z. The above argument shows that p is a degree-two map. It suffices to verify that it has at least one simple ramification point to see that both D and Z are irreducible.

The points x with only one inverse image l in D are those for which π_{x}(B) has a tacnode. Equivalently, the tangent lines T_{z_{1}}B and T_{z_{2}}B are coplanar. In this case l is either a point of intersection of two local branches of D, or a cusp, or a simple ramification point of p. To eliminate the first two possibilities, it suffices to see that l is a nonsingular point of D. Assume that it is singular, then all the three curves π_{z_{1}}(B), π_{z_{2}}(B), π_{x}(B) are also tacnodal. This implies that the tangent lines T_{x}B, T_{z_{1}}B and T_{z_{2}}B of B are coplanar. This is absurd, because the intersection index of B with a plane in ℙ^{3} is equal to 5.

So, p has a simple ramification point if and only if it possesses a trisecant l such that the tangent lines T_{z_{1}}B and T_{z_{2}}B are coplanar for a pair z_{1}, z_{2} of points of l ∩ B; let us call such a trisecant special. One can deduce the existence of special trisecants from the Chasles-Cayley-Brill formula (see [SR], Chapter XII, Corollary 3.52) which gives the number N of fixed points of a correspondence G ⊂ C × C on a smooth curve C of genus g(C) > 0:

N = n_{1} + n_{2} + 2g(C)γ(G),

where n_{i} is the degree of the projection p_{i} : G⟶C to the i-th factor, and γ(G) is the valence of the correpondence, defined as the integer m such that the linear equivalence class of mu + p_{1}^{−1}(u) does not depend on u ∈ C.

We will now describe a correspondence G in B × B for our elliptic curve B, whose fixed points correspond to the special trisecants of B. We will specify a divisor G_{u} = p_{1}^{−1}(u) on B for every u ∈ B. So, let u ∈ B be any point, and m_{1}, m_{2} the two trisecant lines of B passing through u. Let m_{i} ∩ B = {u, a_{i}, b_{i}}, i = 1, 2. Let h_{1} be the section of B by the plane containing m_{1} and the tangent line T_{a_{1}}B. We have

h_{1} = u + 2a_{1} + b_{1} + c_{1}.

Similarly, define the hyperplane sections h_{2}, h_{3}, h_{4} of B such that

h_{2} = u + a_{1} + 2b_{1} + d_{1}, h_{3} = u + 2a_{2} + b_{2} + c_{2}, h_{4} = u + a_{2} + 2b_{2} + d_{2}.

Define G_{u} = c_{1} + c_{2} + d_{1} + d_{2}. For such G, we have n_{1} = 4, n_{2} ≥ 1. Moreover, since h_{0} = x + a_{1} + a_{2} + b_{1} + b_{2} is also a hyperplane section of B, we have

h_{1} + h_{2} + h_{3} + h_{4} − 3h_{0} = u + G_{u} ∼ h_{0},

and hence G is of valence γ(G) = 1. Then the Chasles-Cayley-Brill formula gives the estimate N ≥ 7. Now, it is easy to see that all the fixed points of G provide special trisecants of B. Indeed, by definition, u is a fixed point of G if and only if u coincides with one of the points of G_{u}. Let, for example, u = c_{1}. Then h_{1} = 2u + 2a_{1} + b_{1}, and m_{1} is a special trisecant of B.

Thus, we have proved that B has at least one special trisecant, hence D and Z are irreducible. Apply this to B = \underline{C′}. The fiber p′′^{−1}(C′) is identified with a nonempty open subset of the family of trisecant lines of \underline{C′}, so it is 1-dimensional and irreducible, which implies the assertion (i).

(ii) Remark that, in Lemma 1.5, one can choose the curve C′_{0} = l_{1} ∪ ... ∪ l_{5} in such a way that the lines m_{i} = π(l_{i}), i = 2, 4, 5 in ℙ^{3} are disjoint. Then it is obvious that a small (in the classical topology over ℂ) smooth deformation C′_{t} of C′_{0} can be lifted to a small deformation C′_{t} ∪ C′′_{t} ∈ Č_{5+2} of C′_{0} ∪ C′′_{0}. By the strong smoothability of C′_{0} proved in Lemma 1.5, C := C′_{0} ∪ C′′_{0} ∈ \overline{Č_{5+2}}. To see that C is strongly smoothable, we consider the natural exact triples similar to (27):

(31)  0 → N_{a} → N_{C/X} → O_{Z_{a}} → 0, 0 → N′_{a} → N_{a} → N_{C_{1}/X} → 0,
(32)  0 → N_{b} → N_{C_{0}/X} → O_{Z_{b}} → 0, 0 → N′_{b} → N_{b} → N_{C′′_{0}/X} → 0,

where C_{1} := l_{1} ∪ l_{2} ∪ l_{4} ∪ C′′, N′_{a} := (N_{C/X}|l_{3} ∪ l_{5})(−Z_{a}), N′_{b} := (N_{C/X}|C′_{0})(−Z_{b}). Then the strong smoothability of C will follow from the equalities

(33)  h^{1}(N_{a}) = 0,
(34)  h^{1}(N_{b}) = 0.

To prove the first of them, one deduces from Lemma 1.5 that N′_{a} = 2O_{l_{3}}(−1) ⊕ 2O_{l_{5}}(−1). Then the exact triples

0 → 2O_{l_{2}}(−1) ⊕ 2O_{l_{4}}(−1) → N_{C_{1}/X} → N_{C_{1}/X}|C′′_{0} ⊕ 2O_{l_{1}}(1) → 0,
0 → N_{C′′_{0}/X} → N_{C_{1}/X}|C′′_{0} → ℂ_{b_{1}} ⊕ ℂ_{b_{2}} → 0

together with the second triple (31) give (33).

To prove (34), we include N′_{b} into the series of exact triples following from the construction of C:

0 → N_{1} → N′_{b} → O_{l_{5}} ⊕ O_{l_{5}}(−1) → 0, 0 → 2O_{l_{3}}(−1) → N_{1} → N_{2} → 0,

0 → N_{3} → N_{2} → O_{l_{2}} ⊕ O_{l_{2}}(1) → 0, 0 → O_{l_{4}} ⊕ O_{l_{4}}(−1) → N_{3} → O_{l_{4}} ⊕ O_{l_{4}}(−1) → 0. As C′′_{0} = π^{−1}(m), where m denotes a line in ℙ^{3}, one immediately verifies that h^{1}(N_{C′′_{0}/X}) = 0. This together with the second triple (32) gives (34).

(iii) follows from (i) and (ii) since the strong smoothability is an open condition in \overline{Č_{5+2}}.

(iv) Take the curve C = C′_{0} ∪ C′′_{0} ∈ \overline{Č_{5+2}} ∖ Č_{5+2} from (ii) above. From (31)-(34) one immediately obtains h^{1}(N_{C/X}) = 0, h^{0}(N_{C/X}) = 14, hence dim C_{7}^{4∗}(X) = 14. Next, consider C as the union C = C_{3} ∪ C_{4}, where C_{3} := l_{5} ∪ C′′_{0} and C_{4} := l_{1} ∪ l_{2} ∪ l_{3} ∪ l_{4}. One easily verifies that the reducible elliptic cubic C_{3} is strongly smoothable and

(35)  dim_{{C_{3}}} Hilb(X) = 6

and thus there exists a unique 6-dimensional irreducible component C_{3}^{1∗}(X) of Hilb(X) containing C_{3}. Denote R = {C̃ ∈ C_{3}^{1∗}(X) | C̃ is reducible of the form C̃ = l ∪ π^{−1}(m), where l is a line on X and m is a line on ℙ^{3}}. By construction, C_{3} ∈ R. It follows easily that: a) for any C̃ ∈ C_{3}^{1∗}(X) ∖ R the map π|C̃ : C̃ → π(C̃) is an isomorphism of C̃ onto a plane cubic π(C̃); b) dim R = 5, and dim_{{C_{3}}} R(l_{1}, l_{2}, l_{4}) = 2, where R(l_{1}, l_{2}, l_{4}) = {C̃ ∈ R | C̃ intersects each one of the lines l_{1}, l_{2}, l_{4} in one point}. Thus, by (35), there exists C̃ ∈ C_{3}^{1∗}(X) ∖ R intersecting each of the lines l_{1}, l_{2}, l_{4} in one point, so that π maps C̃ ∪ C_{4} isomorphically onto its image. By construction, C̃ ∪C_{4} ∈ C_{7}^{4∗}(X). The property of a curve in X to be mapped isomorphically onto its image in ℙ^{3} is open and a general curve in C_{7}^{4∗}(X) is smooth as a deformation of the strongly smoothable curve C_{3} ∪ C_{4}, so C_{7}^{4∗}(X)^{0} is open in C_{7}^{4∗}(X), and this ends the proof. □

Lemma 2.3. Let X be a general quartic double solid. Then C_{7}^{4}(X) is irreducible of dimension 14.

Proof. This is proved in the same way as Lemmas 1.2, 1.3 and 1.4. First, remark that C_{7}^{4}(ℙ^{3}) is an irreducible variety of dimension 28 (see, e.g., [H, Ch.4]), and for any C_{0} ∈ C_{7}^{4}(ℙ^{3}) we have (O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}} ≃ O_{C_{0}} ⊕ O_{C_{0}}(D), deg D = 14, so that h^{0}((O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}}) = 9. This implies that there exists a rank-9 vector bundle E over C_{7}^{4}(ℙ^{3}) with fiber E_{{C_{0}}} = H^{0}((O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2))|_{C_{0}}) over C_{0} ∈ C_{7}^{4}(ℙ^{3}). Thus, denoting Č_{7}^{4}(Y) = {C ⊂ Y | ρ|_{C} : C → ρ(C) is an isomorphism onto its image ρ(C) ∈ C_{7}^{4}(ℙ^{3})}, we see that Č_{7}^{4}(Y) is a dense open subset of Proj(E) such that the natural projection Č_{7}^{4}(Y) → C_{7}^{4}(ℙ^{3}), C ↦ ρ(C), coincides with the composition Č_{7}^{4}(Y) ↪ Proj(E) ⟶^{pr} C_{7}^{4}(ℙ^{3}), where pr is the structure morphism. Hence C_{7}^{4}(Y) is irreducible and

(36)  dim Č_{7}^{4}(Y) = 36.

Consider the incidence variety Γ ↪ U × Č_{7}^{4}(Y):

Γ = {(t, C) ∈ U × Č_{7}^{4}(Y) | C ⊂ X_{t}}, U ⟵^{p_{1}} Γ ⟶^{p_{2}} Č_{7}^{4}(Y).

From (5) it follows that

(37)  p_{2}^{−1}(C) = U ∩ |I_{C/Y} ⊗ L^{⊗2}| ↪^{open} |I_{C/Y} ⊗ L^{⊗2}|.

Now take any curve C ∈ Č_{7}^{4}(Y). Let ρ_{C} = ρ|_{C} : C → C_{0} be the natural isomorphic projection of C onto a quintic C_{0} ∈ C_{7}^{4}(ℙ^{3}). We have p_{Y ∗}(L^{⊗2} ⊗ O_{C}) ≃ O_{ℙ^{3}}(4)|C_{0}. Consider the standard exact triple

(38)  0 → I_{C/Y} ⊗ L^{⊗2} → L^{⊗2} ⟶^{ε} L^{⊗2} ⊗ O_{C} → 0.

Apply ρ_{C∗}:

(39)  0 → p_{Y ∗}I_{C/Y} ⊗ L^{⊗2} → O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2) ⊕ O_{ℙ^{3}}(4) ⟶^{ε_{0}} O_{ℙ^{3}}(4)|C_{0} → 0.

The map ε_{0}|O_{ℙ^{3}}(4) : O_{ℙ^{3}}(4) → O_{ℙ^{3}}(4)|C_{0} is iedntified with the restriction one in the triple

0 → I_{C_{0},ℙ^{3}}(4) → O_{ℙ^{3}}(4) ⟶^{res} O_{ℙ^{3}}(4)|C_{0} → 0.

Hence, by (38), (39), the vanishing of h^{1}(I_{C/Y} ⊗ L^{⊗2}) follows from that of h^{1}(I_{C_{0}/ℙ^{3}}(4)). To prove the last one, take two general cubic surfaces S_{1}, S_{2} through C_{0} (such cubics exist since, by Riemann-Roch, h^{0}(O_{C_{0}}(3)) = 18.) There are two possibilities for S_{1} and S_{2}:

1) both S_{1} and S_{2} are irreducible, and 2) at least one of S_{1}, S_{2} is reducible. Consider them separately.

1) In this case S_{1} ∩S_{2} = C_{0} ∪C′, where C′ is a curve of degree 2. We have the following two subcases: 1.a) C′ a plane conic, or 1.b) C′ is a pair of two skew lines, possibly coincident, that is a double line on a smooth quadric. Consider the scheme Z = C_{0} ∩ C′. One easily verifies that ξ(O_{Z}) = 7 in case 1.a) and ξ(O_{Z}) = 8 in case 1.b), and that ξ(O_{Z∩l}) = 4 for any line l ⊂ C′. Thus, in both cases one obtains h^{1}(I_{Z,C′}(4)) = 0. Then we have two exact triples 0 → I_{C_{0}∪C_{1}/ℙ^{3}}(4) → I_{C_{0}/ℙ^{3}}(4) → I_{Z/C′}(4) → 0, and 0 → O_{ℙ^{3}}(−2) → 2O_{ℙ^{3}}(1) → I_{C_{0}∪C′/ℙ^{3}}(4) → 0. Their long cohomology exact sequences imply h^{1}(I_{C_{0}/ℙ^{3}}(4)) = 0.

2) In this case a smooth curve C_{0} lies on an irreducible quadric, say, Q, since C_{0} is not a plane curve. Then Q is either a smooth quadric, or a cone. Consider the case of a smooth quadric Q, the case of a cone being treated similarly. Then C_{0} is a curve of bedegree (5, 2) on Q, so that the sheaf I_{C_{0}/ℙ^{3}}(4) is included into the exact triple 0 → I_{Q/ℙ^{3}}(4)⟶I_{C_{0}/ℙ^{3}}(4)⟶I_{C_{0}/Q}(4) → 0, where clearly I_{Q/ℙ^{3}}(4) = O_{ℙ^{3}(2)} and I_{C_{0}/Q}(4) = O_{Q}(−1, 3), so that h^{1}(I_{Q/ℙ^{3}}(4)) = h^{1}(I_{C_{0}/Q}(4)) = 0 and, as a corollary, h^{1}(I_{C_{0}/ℙ^{3}}(4)) = 0.

Hence h^{0}(I_{C/Y} ⊗ L^{⊗2}) = 24 and |I_{C/Y} ⊗ L^{⊗2}| = ℙ^{23}, so that by (36)-(37) we deduce that Γ is irreducible of dimension 59. The remaining part of the proof goes along the same lines as the end of the proofs of Lemmas 1.2, 1.3 and 1.4. Namely, by Lemma 2.2 (iv), for a general double solid X ∈ U, there is a distinguished component C_{7}^{4∗}(X) of the fibre p_{1}^{−1}(X) = C_{7}^{4}(X) which plays the role of C_{2}^{0∗}(X), C_{3}^{0∗}(X), C_{5}^{1∗}(X) in the proofs of the above mentioned lemmas. (Remark, that, since dim Γ = 59 = 14 + 45 = dim C_{7}^{4∗}(X) + dim U, it follows that p_{1} : Γ → U is dominant.) □

2.3. Curves of degree 17 and arithmetic genus 29.

Let

M := {(Q, l, Z, Z_{l}) | Q is a smooth quadric in ℙ^{3}, l a line on Q, Z = {y_{4}, ..., y_{10}} a 7-uple of distinct points on Q ∖ l, Z_{l} = {y_{1}, y_{2}, y_{3}} a triple of distinct points on l}

and

M^{∗} = {(Q, l, Z, Z_{l}) ∈ M | dim |J_{Z∪Z_{l}/Q}(3, 2)| = 1},

where we assume that l is of bidegree (0,1) on Q. Clearly, M is irreducible and M^{∗} is nonempty, hence dense in M. Next, let

Π = \{(C, (Q, l, Z, Z_{l})) ∈ Č_{5}^{1}(X) × M | \underline{C} ∩ Q = Z ∪ Z_{l}, where \underline{C} = π(C)\}.

It has two natural projections: Č_{5}^{1}(X) ⟵^{p_{Π}} Π ⟶^{q_{Π}} M. It is easy to see that Π is nonempty. Indeed, for any C ∈ Č_{5}^{1}(X), consider the smooth quintic \underline{C} = π(C), take any trisecant line l of \underline{C} and any smooth quadric Q through l and let Z = C ∩ (Q ∖ l) and Z_{l} = \underline{C} ∩ l; then, by construction, (C, (Q, l, Z, Z_{l})) ∈ Π. We can factor p_{Π} as

(40)  \begin{matrix}p_{Π} : Π ⟶^{p′} C_{5+2} ⟶^{p′′} Č_{5}^{1}(X), \\ p′ : (C′, (Q, l, Z, Z_{l})) ↦ C′ ∪ π^{−1}(l), p′′ : C′ ∪ C′′ ↦ C′, \end{matrix}

and our construction shows that p′ is surjective with an irreducible and rational fiber of dimension 7. Besides, as it was already mentioned in the proof of Lemma 2.2, for C′ ∈ Č_{5}^{1}(X), the fiber p′^{−1}(C′) is isomorphic to a dense open subset of the 1-dimensional family of trisecant lines of the smooth quintic π(C′) in ℙ^{3}. This description of the fibres of p′ and p′′ together with (40) implies, in particular, that Π is irreducible for general X.

Let us show now that

(41)  Π^{∗} := p′^{−1}(C_{5+2}^{∗}) ∩ q_{Π}^{−1}(M^{∗}) ≠ ∅.

Choose any C ∈ Č_{5}^{1}(X) and a trisecant line l of \underline{C} such that C ∪ π^{−1}(l) ∈ C_{5+2}^{∗}, where \underline{C} = π(C). Then Z_{l} = \underline{C} ∩ l = \{x_{1}^{0}, x_{2}^{0}, x_{3}^{0}\} is a triple of distinct points. Through each x_{i}^{0} there is, in addition to l, one more trisecant l_{0}^{i} of \underline{C}, and we denote \{x_{2i+2}^{0}, x_{2i+3}^{0}\} = l_{i}^{0} ∩ \underline{C} \ \{x_{i}^{0}\}, i = 1, 2, 3. Since \underline{C} has degree 5, it follows immediately that the lines l_{1}^{0}, l_{2}^{0}, l_{3}^{0} are disjoint, hence there exists a unique smooth quadric Q_{0} containing l, l_{1}^{0}, l_{2}^{0} and l_{3}^{0}. Let x_{10}^{0} be the residual point of intersection of \underline{C} and Q^{0}: \underline{C} ∩ Q_{0} ∖ \{x_{1}^{0}, ..., x_{10}^{0}\}. One can easily verify that for sufficiently general l, the 10 points x_{1}^{0}, ..., x_{10}^{0} are distinct. Denote Z^{0} = {x_{4}^{0}, ..., x_{10}^{0}} and let l_{4}^{0} be a line on Q^{0} through x_{10}^{0} in the same ruling as l. Let it be the ruling |O_{Q_{0}}(0, 1)|. Then

|J_{Z^{0}∪Z_{l}/Q_{0}}(3, 2)| = ⋃_{i=1}^{4} l_{i}^{0} + |O_{Q_{0}}(0, 1)| ≃ ℙ^{1}.

Now consider the triple x_{0} = (x_{1}^{0}, x_{2}^{0}, x_{3}^{0}) as a point of Sym^{3}(l) ≃ ℙ^{3} and take a sufficiently small neighborhood U of x_{0} in the classical topology of Sym^{3}(l) so that for any x = (x_{1}, x_{2}, x_{3}) ∈ U the following properties are satisfied: (i) for each i = 1, 2, 3, there is a unique trisecant line l_{i} of \underline{C} passing through x_{i} and close to l_{i}^{0} in the sense that \{x_{2i+2}, x_{2i+3}\} = l_{i} ∩\underline{C} \\{x_{i}^{0}\} is close to the pair \{x_{2i+2}^{0}, x_{2i+3}^{0}\} in Sym^{2}(C); (ii) there exists a unique quadric Q containing l, l_{1}, l_{2}, l_{3} and close to Q_{0} in the space of quadrics; for this quadric, the residual point x_{10} = \underline{C} ∩ Q ∖ \{x_{1}, ..., x_{9}\} is close to x_{10}^{0}.

Denote Z_{l} = {x_{1}, x_{2}, x_{3}}, Z = {x_{4}, ..., x_{10}} and let l_{4} be the line on Q through x_{10} belonging to the same ruling as l. These data define a point m = (Q, l, Z, Z_{l}) ∈ M. Now we will determine the exact sense of the condition that U is sufficiently small:

(42)  |J_{Z∪Z_{l}/Q}(3, 2)| ≃ ℙ^{1}.

Remark that, by construction, ⋃_{i=1}^{4} l_{i} + |O_{Q}(0, 1)| ⊂ |J_{Z∪Z_{l}/Q}(3, 2)|, so that

(43)  |J_{Z∪Z_{l}/Q}(3, 2)| = ⋃_{i=1}^{4} l_{i} + |O_{Q}(0, 1)|.

The condition (42) is open and clearly nonempty. So, for any x_{0} ∈ Sym^{3}(l), we obtain a point m = m(x_{0}) = (Q, l, Z, Z_{l}) ∈ M^{∗}. Then (C, m) ∈ Π^{∗}, and (41) follows. As Π is irreducible and M^{∗} is open in M, this implies that Π^{∗} is dense and open in Π and, consequently,

(44)  p′ : Π^{∗} → C_{5+2}^{∗} is dominant.

Hence (40) and Lemma 2.2(i) show that

(45)  p_{Π} : Π^{∗} → Č_{5}^{1}(X), (C′, (Q, l, Z, Z_{l})) ↦ C′, is dominant.

Remark that for any C_{2} ∈ π^{∗}|J_{Z∪Z_{l}/Q}(3, 2)|, we have :

(46)  C′′ ∪ C_{2} ∈ π^{∗}C_{(3,3)}(ℙ^{3}).

Define, for any given z = (C′, (Q, l, Z, Z_{l})) ∈ Π^{∗},

(47)

C_{1} = p′(z) = C′ ∪ C′′, C′′ = π^{−1}(l),
f_{z} = \{C_{2} ∈ π^{∗}C_{5}^{2}(ℙ^{3}) | C_{2} = π^{−1}(\underline{C}_{2}), \underline{C}_{2} ∈ |J_{Z∪Z_{l}/Q}(3, 2)|\}, f_{z} ≃ ℙ^{1},
f_{z}^{∗} = {C_{2} ∈ f_{z} | C_{2} is a reduced curve},

Clearly, f_{z}^{∗} is a nonempty open subset of f_{z}. By construction, for any C_{2} ∈ f_{z}^{∗}, the curves C_{1} and C_{2} are reduced and meet each other quasi-transversely in 13 points:

(48)  C_{1} ∩ C_{2} = {y_{1}, y_{2}, y_{3}, y′_{1}, y′_{2}, y′_{3}, y_{4}, ..., y_{10}},

Z̃_{l} = π^{−1}(Z_{l}) = {y_{1}, y_{2}, y_{3}, y′_{1}, y′_{2}, y′_{3}} = C′′ ∩ C_{2}, Z̃ = (π|C)^{−1}(Z) = {y_{4}, ..., y_{10}} = C′ ∩ C_{2}.

Hence

(49)  C_{0} = C′ ∪ C′′ ∪ C_{2}

is a curve on X of degree 17 and arithmetic genus 29.

Remark also that π^{∗}C_{5}^{2}(ℙ^{3}) ≃ C_{5}^{2}(ℙ^{3}) is clearly a rational irreducible variety of dimension 20:

(50)  dim π^{∗}C_{5}^{2}(ℙ^{3}) = 20.

Now consider the incidence variety

Γ = \{(C_{1}, C_{2}) ∈ \overline{C_{7}^{4}(X)}^{∗} × π^{∗}C_{5}^{2}(ℙ^{3}) | C_{1} and C_{2} are reduced and meet each other quasi-transversely in 13 points\}

with projections \overline{C_{7}^{4}(X)}^{∗} ⟵^{p_{1}} Γ ⟶^{p_{2}} π^{∗}C_{5}^{2}(ℙ^{3}) and define its locally closed subset Σ^{∗} = p_{1}^{−1}(p′(Π^{∗})). From (40), (44) and (45) it follows immediately that the natural projection

(51)  p_{Σ^{∗}} := p′′ ∘ p_{1} : Σ^{∗} → Č_{5}^{1}(X), (C′ ∪ C′′, C_{2}) ↦ C′, is dominant.

Moreover, for any z ∈ Π^{∗}, by (47) and by the description of p′ in (40), we obtain for C_{1} = p′(z):

(52)  p_{1}^{−1}(C_{1}) is a rational 7-dimensional variety.

This implies that, for general X, Σ^{∗} is an irreducible variety of dimension 17, and there is an isomorphism

(53)  ψ : Σ^{∗} ⟶^{∼} Σ′, (C′ ∪ C′′, C_{2}) ↦ C′ ∪ C′′ ∪ C_{2},

where

Σ′ = {C_{0} ∈ Hilb(X) | C_{0} is defined by (48)-(49) for some z ∈ Π^{∗}}.

In view of (53), we rewrite (51) as follows:

p_{Σ′} : Σ′ → C_{5}^{1}(X), C_{0} = (C′ ∪ C′′ ∪ C_{2}) ↦ C′, is dominant.

Also, by (46), Im(q_{Σ′}) ⊂ π^{∗}C_{(3,3)}(ℙ^{3}), where q_{Σ′} is the projection C_{0} = (C′ ∪ C′′ ∪ C_{2}) ↦ C′′ ∪ C_{2}.

Next, looking at the local equations of Γ in W = \overline{C_{7}^{4}(X)}^{∗} × π^{∗}C_{5}^{2}(ℙ^{3}) at a point w = (C_{1}, C_{2}) ∈ Σ^{∗}, we see that the transversal intersection of the curves C_{1} and C_{2} in each one of the 13 points (48) gives one local equation for Γ in Y; hence by (50), the dimension of the fiber of p_{1} at w is at least 20-13=7. Since by (52) this dimension is precisely 7, it follows that there exists an irreducible component, say, Γ_{0} of Γ, which contains w and dominates \overline{C_{7}^{4}(X)}^{∗} via the projecion p_{1}:

(54)  \overline{p_{1}(Γ_{0})} = \overline{C_{7}^{4}(X)}^{∗},

where the closure is taken in \overline{C_{7}^{4}(X)}^{∗} and we use the irreducibility of \overline{C_{7}^{4}(X)}^{∗} by Lemma 2.3. Hence, taking into account the irreducibility of Σ^{∗}, (52) and (53), it follows that Σ_{0} = ψ(Γ_{0} ∩ Σ^{∗}) is a dense open subset of Σ; also, by (51), the natural projection

(55)  p_{Σ_{0}} : Σ_{0} → C_{5}^{1}(X), C_{0} = (C′ ∪ C′′ ∪ C_{2}) ↦ C′, is dominant.

Now consider in Hilb(X) the following subset:

Γ̃_{0} = {C ∈ Hilb(X) | C is a reducible curve of the form C = C_{1} ∪ C_{2}, where (C_{1}, C_{2}) ∈ Γ_{0}}.

There is an isomorphism ψ : Γ_{0} ⟶^{∼} Γ̃_{0}, (C_{1}, C_{2}) ↦ C_{1} ∪ C_{2}, similar to (53). 2.4. Criterion of Welters.

Now we invoke the following important technical result of G.Welters [We, Prop. 6.17], giving a sufficient condition for a family of curves to parametrize, via the Abel-Jacobi map, a translate of the theta divisor of J(X). Proposition 2.4. Let T be a smooth irreducible (not necessarily complete) variety parametrizing a family {C_{t}|t ∈ T } of reduced curves of degree d = k(k + 1) − 3 on X with the following properties:

(i) The Abel-Jacobi map Φ_{T} : T → J(X) is dominant;

(ii) for all t ∈ T, the linear system of surfaces |J_{C_{t}/X}(k)| has dimension k and the surfaces of this linear system do not contain all the lines of X;

(iii) the incidence divisor in T × F is reduced.

Then for any line l ∈ F, the subvariety of T defined by

T_{l} = {t ∈ T | C_{t} ∩ l = ∅ and C_{t} ∪ l lies on a surface from |O_{X}(k)|}

is sent by the Abel-Jacobi map into a translate of the theta divisor Θ ⊂ J(X).

Let us construct such a T from the curves introduced in the previous subsection. Consider the Abel-Jacobi map ΦΓ̃_{0} : Γ̃_{0} → J(X) and the induced map

(56)  Φ_{0} : Γ_{0}⟶^{ψ} ; ∼ Γ̃_{0} ⟶^{ΦΓ̃_{0}} J(X), (C_{1}, C_{2}) ↦ ΦΓ̃_{0}(C_{1} ∪ C_{2}),

where

(57) ΦΓ̃_{0}(C_{1} ∪ C_{2}) = Φ\overline{C_{7}^{4}(X)}^{∗}(C_{1}) + Φ_{π^{∗}C_{5}^{2}(ℙ^{3})}(C_{2}), C_{1} ∈ \overline{C_{7}^{4}(X)}^{∗}, C_{2} ∈ π^{∗}C_{5}^{2}(ℙ^{3}).

As π^{∗}C_{5}^{2}(ℙ^{3}) ≃ C_{5}^{2}(ℙ^{3}) is rational,

(58)  Φ_{π^{∗}C_{5}^{2}(ℙ^{3})} = const, Im(Φ_{π^{∗}C_{5}^{2}(ℙ^{3})}) = {c}.

Hence, taking the closure of the image of the Abel-Jacobi map in J(X) and using (54), (56)-(58) and Proposition (2.1), we obtain:

(59) \overline{Im(ΦΓ̃_{0})} = \overline{Im(Φ_{0})} = \overline{c + Im(Φ\overline{C_{7}^{4}(X)}^{∗}(\overline{p_{1}(Γ_{0})} ))} = \overline{c + Im(Φ\overline{C_{7}^{4}(X)}^{∗})}

= c + \overline{Im(Φ_{C_{7}^{4}(X)})} = c + J(X) = J(X),

that is the Abel-Jacobi map ΦΓ̃_{0} : Γ̃_{0} → J(X) is dominant. Now, consider the Abel-Jacobi map Φ_{Σ_{0}} : Σ_{0} → J(X). As π^{∗}C_{(3,3)}(ℙ^{3}) is rational, Im(Φ_{π^{∗}C_{(3,3)}(ℙ^{3})}) = {c′} is a point, and we obtain:

(60)  \overline{Im(Φ_{Σ_{0}})} = \overline{Im(Φ_{C_{5}^{1}(X)})} + c′.

Now let σ : T ⟶^{bir} Γ̃_{0} be any desingularization of Γ̃_{0}. Define

(61)  Σ ⊂ σ^{−1}(Σ_{0}), an irreducible component of σ^{−1}(Σ_{0}) such that σ : Σ ↠ Σ_{0}.

Consider the Abel-Jacobi maps Φ_{T} = ΦΓ̃_{0} ∘ σ : T → J(X) and Φ_{Σ} = Φ_{Σ_{0}} ∘ σ : Σ → J(X). Then (60) gives

(62)  \overline{Im(Φ_{Σ})} = \overline{Im(Φ_{C_{5}^{1}(X)})} + c′.

We obtain the following proposition: Proposition 2.5. Let X be a general quartic double solid. Then there exists a smooth irreducible variety T parametrizing some family of curves of degree 17 and arithmetic genus 29 such that:

(i) the Abel-Jacobi map Φ_{T} : T → J(X) is dominant;

(ii) T contains the subset Σ, defined by (61) and satisfying (62), such that the curves C_{t} for t ∈ Σ split into three components as in (48)-(49);

(iii) the incidence divisor D = {(C, l) ∈ T × F | C ∩ l ≠ ∅}, endowed with its natural scheme-theoretic structure, is reduced.

Proof. (i) and (ii) folow from (59), (55) and (62). Let us prove (iii). There are 6 distinct lines, counted with multiplicity 1 in the Fano scheme F, which pass through a general point of X. So, it is sufficient to show that there exists a curve C from T which pass through a generam point of X: in this case, the curve D_{C} is a reduced 6-sheeted covering of C, and hence D is reduced. But this is clear, since by the construction of Σ, the components of the reduced curves C_{0} ∈ Σ sweep all of X as C_{0} runs through Σ. □

We proceed now to the proof of the remaining hypotheses of the Welters criterion for our family T. Lemma 2.6. Let X be a general quartic double solid. Then the following assertions are verified:

(i) For any curve C ∈ Σ, h^{0}(J_{C/X}(4)) = 5;

(ii) for a general (hence any) line l ∈ F such that C ∩ l = ∅, there exists a surface S ∈ |O_{X}(4)| containing C ∪ l;

(iii) for general C ∈ Σ, the surfaces of the linear system |J_{C/X}(4)| do not contain all the lines of X.

Proof. (i) First, take a curve C ∈ Σ. By the definition of Σ we have C = C_{1} ∪ C_{2}, where C_{1} ∈ Č_{5}^{1}(X), π : C_{1} → \underline{C}_{1} := π(C_{1}) is an isomorphism, C_{2} = π^{−1}(\underline{C}_{2}), where \underline{C}_{2} ∈ C_{(3,3)}(ℙ^{3}), and Z := C_{1} ∩ C_{2} = \{y_{1}, ..., y_{10}\} is a subset of the 13-uple of points (48). This means, in particular, that \underline{C}_{2} lies in a uniquely defined quadric, say, Q. Let Z_{0} := π(Z), deg(Z_{0}) = deg(Z) = 10. Since by construction \underline{C}_{1} ∩ Q ⊃ Z_{0} (as a schemetheoretic inclusion) and deg(\underline{C}_{1}∩Q) = 10 = deg(Z_{0}), it follows that, scheme-theoretically, Z_{0} = \underline{C}_{1} ∩ Q. Using the fact that O_{Q}(\underline{C}_{2}) = O_{Q}(3), we get the following exact triple:

(63)  0 → J_{Q∪\underline{C}_{1}/ℙ^{3}}(4) → J_{\underline{C}_{1}∪\underline{C}_{2}/ℙ^{3}}(4) → O_{Q}(1) → 0.

Let S ⊂ ℙ^{3} be any smooth cubic surface passing through the smooth quintic curve \underline{C}_{1}. We have the exact triple

0 → J_{Q∪S/ℙ^{3}} → J_{Q∪\underline{C}_{2}/ℙ^{3}} → J_{(S∩Q)∪\underline{C}_{1}/S} → 0.

Here J_{(S∩Q)∪\underline{C}_{1}/S} ≃ O_{S}(−2)(−\underline{C}_{1}), because S is smooth, hence twisting by O_{ℙ^{3}}(4), we obtain:

(64)  0 → O_{ℙ^{3}}(−1) → J_{Q∪\underline{C}_{2}/ℙ^{3}}(4) → O_{S}(2)(−\underline{C}_{1}) → 0.

One easily verifies that the natural map H^{0}(O_{S}(2)) → H^{0}(O_{\underline{C}_{1}}(2)) is surjective, so that h^{1}(O_{S}(2)(−\underline{C}_{1})) = 0. Besides, h^{1}(O_{ℙ^{3}}(−1) = 0. Hence (64) implies h^{1}(J_{Q∪\underline{C}_{1}/ℙ^{3}}(4)) = 0. This together with (63) and the equality h^{1}(O_{Q}(1)) = 0 gives

(65)  h^{1}(J_{\underline{C}_{1}∪\underline{C}_{2}/ℙ^{3}}(4)) = 0.

Hence the exact triple 0 → J_{\underline{C}_{1}∪\underline{C}_{2}/ℙ^{3}}(4) → J_{\underline{C}_{2}/ℙ^{3}}(4) → J_{Z_{0}/\underline{C}_{1}}(4) → 0 yields the surjectivity H^{0}(J_{\underline{C}_{2}/ℙ^{3}}(4)) ↠ H^{0}(J_{Z_{0}/\underline{C}_{1}}(4)).

Now, since π : C_{1} → \underline{C}_{1} is an isomorphism, we have π_{∗}J_{Z/C_{1}}(4) ≃ J_{Z_{0}/\underline{C}_{1}}(4), which provides the exact triple

0 → J_{Z_{0}/\underline{C}_{1}}(4) → π_{∗}O_{C}(4) → π_{∗}O_{C_{2}}(4) → 0.

From here, one immediately obtains:

(66)  h^{0}(π_{∗}O_{C}(4)) = χ(O_{C}(4)) = 40.

Next, consider the natural restriction map g : π_{∗}O_{X}(4) ↠ π_{∗}O_{C}(4) ↠ π_{∗}O_{C_{2}}(4). Here ker(g) = π_{∗}J_{C_{2}/X}(4), so passing to the map on global sections f : H^{0}(π_{∗}O_{X}(4)) → H^{0}(π_{∗}O_{C}(4)), we obtain:

(67)  coker(f) = coker(H^{0}(π_{∗}J_{C_{2}/X}(4)) ⟶^{res} H^{0}(J_{Z_{0}/\underline{C}_{1}}(4))).

By the projection formula for π, we have π_{∗}J_{C_{2}/X}(4) = π_{∗}π^{∗}J_{\underline{C}_{2}/ℙ^{3}}(4) = J_{\underline{C}_{2}/ℙ^{3}}(4) ⊕ J_{\underline{C}_{2}/ℙ^{3}}(2), and the composition H^{0}(J_{\underline{C}_{2}/ℙ^{3}}(4)) ↪ H^{0}(π_{∗}J_{C_{2}/X}(4)) ⟶^{res} H^{0}(J_{Z_{0}/\underline{C}_{1}}(4)) coincides with the surjective map (65), so (67) implies coker(f) = 0. Hence (66) and the equality h^{0}(O_{X}(4)) = h^{0}(O_{ℙ^{3}}(4)) ⊕ h^{0}(O_{ℙ^{3}}(2)) = 45 imply that

H^{0}(J_{C/X}(4)) = H^{0}(π_{∗}J_{C/X}(4)) = ker(f) = ℂ^{5}.

(ii) This is a standard dimension count. Take a cubic surface S containing the above smooth quintic \underline{C}_{1} and take for the above curve \underline{C}_{2} the complete intersection \underline{C}_{2} = F ∩Q, where Q is a general quadric. Then C_{1} ∪ C_{2} ⊂ π^{−1}(S). Now take a general plane ℙ^{2} in ℙ^{3} and take a line l in the del Pezzo surface π^{−1}(ℙ^{2}) which does not intersect C_{1} ∪ C_{2}. Clearly such a line exists for a general choice of ℙ^{2}. Then S = π^{−1}(F ∪ ℙ^{2}) ∈ |O_{X}(4)| is the desired surface.

(iii) Again, this is a standard dimension count. One should embed the curve C into Y = Proj(O_{ℙ^{3}} ⊕ O_{ℙ^{3}}(2)) (see (4)) and consider the linear system L = |J_{C/Y} ⊗ O_{Y/ℙ^{3}}(2)| on Y. From (i), we deduce that dim L = 5. Take now a line l ⊂ Y mapped isomorphically onto a line in ℙ^{3} by the natural projection p_{Y} : Y → ℙ^{3}. If it is sufficiently general, then there exists a unique smooth X ∈ L containing l; this X is the wanted double solid, and the lemma is proved. □

We are ready now to prove the main result of this section: Theorem 2.7. \overline{Im(Φ_{C_{5}^{1}(X)})} = Θ + const.

Proof. The family of curves T constructed in Proposition 2.5 satisfies the hypotheses of Proposition 2.4 with k = 4. According to Lemma 2.6, for general l ∈ F, we have T_{l} ⊃ Σ, where we replace Σ by a nonempty open subset, if necessary. Hence Proposition 2.4 and (62) imply:

(68)  \overline{Im(Φ_{C_{5}^{1}(X)})} ⊂ Θ + const.

To prove the opposite inclusion, let us remind a well known description of the image of the differential of the Abel-Jacobi map (see, e.g., [We, (2.11.0) and (6.19)]), in applying it specifically to the map Φ := Φ_{C_{5}^{1}(X)} : C_{5}^{1}(X) → J(X). Take any unobstructed quintic C ∈ Č_{5}^{1}(X), so that T_{C}C_{5}^{1}(X) = H^{0}(N_{C/X}) = ℂ^{10}. Assume, moreover, that C does not lie on the ramification quartic W of X and denote C_{0} = π(C), Z_{0} = π(C ∩ W). Consider the composition

ϕ : ℂ^{10} = T_{C} Č_{5}^{1}(X) ⟶^{dΦ} T_{Φ(C)}J(X) ⟶^{∼} T_{0}J(X) ≃ (H^{0}(O_{ℙ^{3}}(2)))^{∨}.

Then the kernel of the dual map ϕ^{∨} is described as follows:

(69) Ker(ϕ^{∨}) = Ker{ℂ^{10} = H^{0}(O_{ℙ^{3}}(2)) ⟶^{H^{0}(r_{C_{0}})} H^{0}(O_{C_{0}}(2)) ⟶^{H^{0}(r_{Z_{0}})} H^{0}(O_{Z_{0}}(2))},

where r_{C_{0}} : O_{ℙ^{3}}(2) → O_{C_{0}}(2), r_{Z_{0}} : O_{C_{0}}(2) → O_{Z_{0}}(2) are the restriction maps, and Im ϕ is the orthogonal complement of Ker(ϕ^{∨}). By definition, C_{0} is a smooth elliptic quintic in ℙ^{3}, hence it does not lie on a quadric, so H^{0}(r_{Z_{0}}) is an isomorphism, and Ker(H^{0}(r_{Z_{0}})) ≃ H^{0}(O_{C_{0}}) = ℂ, as follows from the exact triple 0 → O_{C_{0}} ⟶^{·Z_{0}} O_{C_{0}}(2) ⟶^{r_{Z_{0}}} O_{Z_{0}}(2) → 0. Hence dim Ker(ϕ^{∨}) = 1 and dim Im Φ = 9. This together with (68) and the irreducibility of Θ (see [We]) implies the wanted assertion. □

3. Pentagons on a degenerate double solid

3.1. Degeneration into a pair of P^{3}’s.

In this section, we will use the technique of degeneration of the double solid into a reducible variety, developed by Clemens in [C-1], [C-2]. Let π : X⟶Δ be a family of double covers π_{t} : X_{t}⟶ℙ^{3} branched in the quartics W_{t} = {tF + G^{2} = 0} ⊂ ℙ^{3} (t ∈ Δ), where Δ is an open disc in ℂ, the equation F = 0 defines a smooth quartic W and G = 0 a smooth quadric Q such that W ∩ Q = B is a smooth octic curve. For t ∈ Δ^{∗} = Δ \ {0}, the fiber X_{t} is a smooth double solid, and for t = 0 a union of two copies of ℙ^{3}, say, ℙ′, ℙ′′, meeting each other transversely along Q.

One can associate to π the compactified family F⟶Δ of Fano surfaces, whose fiber F_{t} over t ≠ 0 parametrizes lines in X_{t}, and the Neron model J⟶Δ of the family of the intermediate Jacobians J_{t} of the threefolds X_{t}. The compactifications of both families are described by Clemens. Namely, F_{0} is the union of two components F′ ∪ F′′, where F′ (resp. F′′) parametrizes the bisecant lines to C in ℙ′ (resp. ℙ′′). As follows from a remark of Clemens on p. 211 in [C-1], the fiber F_{0} is acquired in the family F/Δ with multiplicity 1. The lines parametrized by F_{0}, that is, the bisecants to C in either one of the two copies of ℙ^{3}, will be called lines in X_{0}. The Neron model J is a family of abelian Lie groups such that the fiber J_{0} over t = 0 is the union of two components J^{+}, J^{−}, both isomorphic to a ℂ^{∗}-extension of the Jacobian J(B) of B. The Neron model has a natural compactification \overline{J}, obtained by pasting in two copies of J(B), so that the central fiber \overline{J}_{0} is the union of two components \overline{J}^{±} which are both ℙ^{1}-bundles over J(B) and which meet each other transversely along two disjoint sections of the ℙ^{1}-bundles.

In [C-2], Clemens studies the family of sextics of genus 3 in the varieties X_{t}. We will treat in the same way the one of elliptic quintics.

Let H⟶Δ be the subscheme of the relative Hilbert scheme Hilb(X/Δ), such that the fiber H_{t} over t ≠ 0 is the component of Hilb(X_{t}) whose generic point represents an elliptic quintic in X_{t}, and the points of H_{0}, the fiber of H over t = 0, represent the curves which are limits of elliptic quintics in X_{t} as t → 0, that is, H is the closure of ⋃_{t∈Δ^{∗}} H_{t} in Hilb(X/Δ). Let Δ̃⟶Δ, s ↦ t = s^{e}, be some base change and C_{s} ∈ H_{s^{e}} a family of curves in X_{s^{e}} which are smooth elliptic quintics for s ≠ 0. Their images \underline{C}_{s} = π_{t}(C_{s}) ⊂ ℙ^{3} are elliptic quintics, possibly with self-intersections, which are totally tangent to W_{t}. Each C_{s} for s ≠ 0 defines a unique K3 surface S(C_{s}) ∈ |O_{X_{t}}(2)| such that C_{s} ⊂ S(C_{s}), where we denote by O_{X_{t}}(k) the sheaf π_{t}^{∗}(O_{ℙ^{3}}(k)). The image π_{t}(S(C_{s})) is a quartic \underline{S(C_{s})} in ℙ^{3}, containing \underline{C}_{s} and tangent to W_{t} along the curve \underline{S(C_{s})} ∩ W_{t}, so that π_{t}^{−1}(\underline{S(C_{s})}) is the union of two components: S(C_{s}) and ιS(C_{s}), where ι denotes the Galois involution of π_{t}. For s = 0, there is a unique limit quartic in ℙ^{3}, possibly singular, which we denote by \underline{S(C_{0})}. So the family S(C_{s}) extends over s = 0 with fiber S(C_{0}) ≃ \underline{S(C_{0})} lying in one of the two components ℙ′, ℙ′′ of X_{0}, and the limiting curve C_{0} lies entirely in ℙ′ or in ℙ′′. As remarks Clemens, a component of C_{0} of degree d which does not lie in Q = ℙ′ ∩ ℙ′′ has to meet B = W ∩ Q in 2d points counting multiplicities, because otherwise the deformed curve \underline{C}_{s} for small s will not be totally tangent to W_{t}, which is absurd. We will call such components 2d-secants to B.

Let H′, resp. H′′ be the subset of H_{0} consisting of the curves C ⊂ X_{0} such that:

(1) C is smoothable in X_{0}, (2) C_{red} has no components contained in Q, and (3) all the components of C lie in ℙ′, resp. in ℙ′′. Denote H^{0} = H′ ∪ H′′.

We want now to define a relative Abel-Jacobi map α = α_{H} : H⟶\overline{J}. To do this, we need a local analytic family of algebraic 1-cycles {Z_{t}}_{t∈Δ} in X_{t} (we can shrink Δ if necessary) such that Z_{t} is homologous in X_{t} to the curves C_{t} ∈ H_{t} for any t ∈ Δ^{∗}. As H^{2}(X_{t}, ℤ) ≃ ℤ, we can choose for Z_{t} a multiple of a line in X_{t}. We have mentioned the fact that F_{0} is a reduced fiber of F, so there exists a local analytic family of lines l_{t} ∈ F_{t} near t = 0, and we can use Z_{t} = 5l_{t} as the section of reference points for the definition of α. According to [C-1], Sect. 8, α extends as a regular map to all of H^{0} ⊂ H_{0}, and the image of H′ (resp. H′′) lies in one component J′ (resp. J′′) of the smooth locus J_{0} = J^{+} ∪ J^{−} of \overline{J}_{0} (thus, J′ = J^{+} or J^{−}, and J′′ may a priori coincide either with J′, or with the other component of J_{0}). If we consider the composition ϕ of α|_{H^{0}} with the natural projection η : J_{0}⟶J(B), we obtain the following description modulo a constant translation by a fixed divisor on B depending on the choice of the reference point: ϕ(C) is the class of B · C considered as a degree-10 divisor on B. In particular, H′ dominates J(B) via ϕ if and anly if H′′ does.

We will use these considerations to prove the following proposition. Proposition 3.1. Let π : X⟶Δ be a family of degenerating quartic double solids as above which is generic, that is, F and G are generic. Then there exists a morphism

α = α_{H} : H \ A⟶\overline{J}

over a possibly shrinked disc Δ, where A = H_{0} \ H^{0} is a proper closed subset of H_{0}, such that the following properties are verified:

(i) The restriction α_{t} : H_{t}⟶J_{t} of α to any fiber H_{t} with t ∈ Δ^{∗} is, modulo a constant translation in J_{t}, the Abel–Jacobi map of H_{t}.

(ii) The generic fiber of α_{t} for any t ∈ Δ is the union of finitely many copies of ℙ^{1} corresponding to elliptic pencils in the K3 surfaces S(C_{t}).

(iii) The closure D of α(H \ A) is a relative divisor in \overline{J} over Δ, that is D_{t} = D ∩ \overline{J}_{t} is a divisor in \overline{J}_{t} for every t ∈ Δ, and D_{t} is a translate of the theta divisor of J_{t} if t ∈ Δ^{∗}.

(iv) The smooth locus J_{0} of the central fiber \overline{J}_{0} of \overline{J} consists of two components J^{±}, and only one of them, say, J′ contains α_{0}(H^{0}). Moreover, α_{0}(H^{0}) = α_{0}(H′) = α_{0}(H′′) ⊂ J′.

(v) α_{0}(H^{0}) is a section of the natural projection J′⟶J(C) over an open subset of J(C).

Proof. We have already proved (i) and (ii); (iii) is a consequence of Theorem 2.7. To prove (iv) and (v), we will use the characterization of the divisors in \overline{J}_{0} which can be limits of a family of principal polarizations in the neighboring fibers J_{t}, given by Clemens in [C-2], (4.2.1)-(4.2.2). It states that: 1) only one of the irreducible components of D_{0}, say, D′, dominates J(B), and 2) if \overline{J}^{ϵ} (ϵ = + or −) is the component of \overline{J}_{0} containing D′, then D′ is a section of the natural projection \overline{J}^{ϵ}⟶J(B) over an open subset of J(B).

As we have remarked, H′ dominates J(B) via ϕ if and only if H′′ does. Hence, if we assume that H′ dominates J(B), then α_{0}(H′) and α_{0}(H′′) both dominate J(B), hence J′ = J′′ and α_{0}(H′) = α_{0}(H′′) is one and the same rational section of J′⟶J(B).

It remains to see that H′ dominates J(B). This follows from the fact that for generic W, Q and for a generic 10-uple of points Z in B = W ∩ Q, there is at least one elliptic quintic in ℙ^{3} passing through Z (see Corollary 4.6). Alternatively, one can prove, by an argument similar to that of Clemens on p. 97 of [C-2], that the adherence values of α along A project to a proper closed subset of J(B), and thus at least one of the components of H_{0} \ A = H′ ∪ H′′ has to project to a rational section of \overline{J}^{ϵ}⟶J(B). □

3.2. Deforming pentagons from X_{0} to X_{t}.

We want now to prove that the fiber H_{0} is reduced at a generic point of H′. To this end, we will construct a local cross-section of the projection H⟶Δ in the neighborhood of 0 taking its values in the subschemes of H_{t} parametrizing cycles of five lines, or pentagons in X_{t}. Definition 3.2. Let X be a quartic double solid. A pentagon in X is a reducible curve Γ = ⋃_{i=1}^{5} ℓ_{i}, where ℓ_{i} are lines in X such that ℓ_{i} ∩ℓ_{i+1} ≠ ∅ (addition of subscripts modulo 5). If ℓ_{i} meets ℓ_{i+1} quasi-transversely at one point and these are the only intersection points of ℓ_{i}, ℓ_{j} (i ≠ j), we will call Γ a good pentagon.

Consider the 5-uple fibered product of F with itself over Δ:

Y = F_{Δ}^{5} = F ×_{Δ} F ×_{Δ} F ×_{Δ} F ×_{Δ} F

Let Y_{t} denote the fiber of Y over t ∈ Δ and D_{i} the subvariety of Y parametrizing the 5-uples (ℓ_{1}, …, ℓ_{5}) of lines in the X_{t} defined by the incidence relation ℓ_{i} ∩ ℓ_{i+1} ≠ ∅ (i = 1, …, 5, 5 + 1 = 1). Then the variety parametrizing all the pentagons in X_{t}, t ∈ Δ can be represented as the intersection Σ := ⋂_{i=1}^{5} D_{i}. Proposition 3.3. Let the quartic W and the quadric Q defining the family X⟶Δ be generic. Let Σ_{0} be the fiber of Σ over t = 0. Then there exists a component of Σ_{0}, such that its generic point y represents a good pentagon in X_{0} whose five vertices are not in Q, the 6 subvarieties Y_{0}, D_{1}, …, D_{5} are nonsingular at y and their intersection is transversal at y.

Proof. It suffices to prove the assertions of the proposition for special W, Q with a degenerate octic C = W ∩ Q. Choose them in such a way that C decomposes into four smooth conics C_{k}, k = 1, …, 4, so that the planes Π_{k} = ⟨C_{k}⟩ of these conics are in general position in ℙ^{3}. Consider the component Σ′ of Σ_{0} parametrizing the pentagons ⋃_{i=1}^{5} ℓ_{i} in ℙ′ such that ℓ_{k} is a bisecant of C_{k} for k = 1, 2, 3, 4 and ℓ_{5} meets C_{2} and C_{3}. Choose affine coordinates (u_{i}, v_{i}), i = 1, …, 5, on an affine open set of Y_{0} as follows. For i = 1, …, 4, (u_{i}, v_{i}) are affine coordinates in the plane ℙ_{i}^{2} parametrizing lines in Π_{i}, and u_{5}, v_{5} are the parameters of some birational parametrizations of the conics C_{2}, C_{3} respectively. As F_{0} is acquired in F with multiplicity 1, these ten rational functions extend to local parameters on an affine open set of the neighboring fibers F_{t}.

We will show that the equations defining D_{i} with respect to (u_{i}, v_{i}), i = 1, …, 5 form a system of maximal rank at some point y_{0} ∈ Σ′. The transversality of the D_{i} with Y_{0} follows automatically from the fact that the 11 functions t, u_{1}, v_{1}, …, u_{5}, v_{5} form a local (analytic) coordinate system in Y near y_{0}, and from the observation that the equations of D_{i} do not depend on t. In fact, for our purposes it suffices to see that the linearizations of the equations at y_{0} are linearly independent.

In order to assure that Σ′ is nonempty and to choose a point y_{0} in such a way that the lines of the corresponding pentagon have particularly simple coordinates, let us fix a pentagon in ℙ^{3} first: ℓ_{i}^{0} = \overline{P_{i}P_{i+1}} (addition of subscripts modulo 5), where P_{1} = (1 : 0 : 0 : 0), P_{2} = (0 : 1 : 0 : 0), P_{3} = (0 : 0 : 1 : 0), P_{4} = (0 : 0 : 0 : 1), P_{5} = (1 : 1 : 1 : 1). Then choose W as the union of 4 generic planes Π_{k} subject to the conditions ℓ_{k}^{0} ⊂ Π_{k}, k = 1, …, 4, and afterwards choose a generic quadric Q passing through the two points A_{1}^{0} = Π_{2} ∩ ℓ_{5}^{0} and A_{2}^{0} = Π_{3} ∩ ℓ_{5}^{0}. Choose for y_{0} the point of Y_{0} representing this pentagon and assign to it the values of parameters u_{i} = v_{i} = 0 for all i.

Now, let the plane of the first conic Π_{1} be {x_{3} − λx_{4} = 0}. Then we can define the line ℓ_{1} corresponding to parameters u_{1}, v_{1} in the neighborhood of ℓ_{1}^{0} as the one passing by the two points (1 : 0 : λu_{1} : u_{1}), (0 : 1 : λv_{1} : v_{1}) ∈ Π_{1}. Its Plücker coordinates are

p_{12} = 1, p_{13} = λv_{1}, p_{14} = v_{1}, p_{23} = −λu_{1}, p_{24} = −u_{1}, p_{34} = O(2),

where O(2) stands for the terms of second order in (u_{1}, v_{1}, …, u_{5}, v_{5}) or smaller. Similarly, Π_{2} can be given by \{x_{1} − μx_{4} = 0\} and ℓ_{2} = \overline{(μu_{2} : 1 : 0 : u_{2})(μv_{2} : 0 : 1 : v_{2})} has Plücker coordinates

q_{12} = −μv_{2}, q_{13} = μu_{2}, q_{14} = O(2), q_{23} = 1, q_{24} = v_{2}, q_{34} = −u_{2}.

The incidence relation between two lines in ℙ^{3} represented by their Plücker coordinates (p_{ij}), (q_{ij}) is given by a single equation:

p_{12}q_{34} + p_{34}q_{12} − p_{13}q_{24} − p_{24}q_{13} + p_{14}q_{23} + p_{23}q_{14} = 0.

Thus the equation of D_{1}, linearized at y_{0}, has the form

D_{1} : −u_{2} + v_{1} = 0.

Continuing the calculation in the same manner, we put:

Π_{3} = \{x_{1} − νx_{2} = 0\} , ℓ_{3} = \overline{(νu_{3} : u_{3} : 1 : 0)(νv_{3} : v_{3} : 0 : 1)} ,

Π_{4} = \{x_{1} − x_{3} − η(x_{2} − x_{3}) = 0\} , ℓ_{4} = \overline{(ηu_{4} : u_{4} : 0 : 1)(1 + ηv_{4} : 1 + v_{4} : 1 : 1)} .

To determine the dependence of ℓ_{5} on u_{5}, v_{5}, remark first that ℓ_{5}^{0} = \overline{A_{1}^{0}A_{2}^{0}}, where A_{1}^{0} = (μ : 1 : 1 : 1), A_{2}^{0} = (ν : 1 : 1 : 1). So, to deform ℓ_{5}^{0}, we will deform A_{i}^{0} to A_{i} along the conic C_{1+i} and represent the deformed line ℓ_{5} as the one joining A_{1} and A_{2}. We have:

ℓ_{5} = \overline{A_{1}A_{2}} , A_{1} = (μ : 1 + (1 + α)u_{5} + O(2) : 1 + u_{5} + O(2) : 1) , A_{2} = (ν : 1 : 1 + (1 − β)v_{5} + O(2) : 1 + v_{5} + O(2)) ,

where α, β are the parameters corresponding to tangent directions of C_{1}, resp. C_{2} at A_{1}, resp. A_{2}, and we have replaced the conics by their tangent lines, which explains the appearance of the terms O(2) in the formulas for A_{i}.

Finally, the linearized system of equations of D_{1}, …, D_{5} with respect to variables u_{1}, v_{1}, …, u_{5}, v_{5} has the following matrix:

[ \begin{matrix}0 & 1 & −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −μ & ν & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 − ν & η − 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ν − μ & α(1 − ν) & (1 − μ)(1 − β) \\ (1 − ν)(ν − μ) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & −β \end{matrix} ]

One immediately sees that its rank is 5 as soon as μ ≠ 0, ν − μ ≠ 0, ν − 1 ≠ 0. Hence the wanted transversality is proved. □

3.3. Smoothability of good pentagons.

Now we will check that the good pentagons can be deformed to elliptic quintics. Proposition 3.4. Under the hypotheses and in the notation of Proposition 3.3, there exists a local cross-section Δ⟶H, t ↦ y(t), over a possibly shrinked disc Δ, such that y_{0} ∈ H′ and the corresponding curves Γ_{y(t)} ⊂ X(t) are good pentagons, strongly smoothable into elliptic quintics in X_{t} for all t ∈ Δ.

Remind that Γ_{0} = Γ is called smoothable if there exists a flat family of curves Γ_{s} such that its generic member is smooth. If there exists such a deformation whose base and the total space are both smooth, then Γ_{0} is strongly smoothable.

Proof of Proposition 3.4. As above, we specialize our family X⟶Δ to the case when C is the union of four conics C_{i}. Let Γ = ⋃_{i=1}^{5} ℓ_{i}^{0} be a good pentagon constructed in the proof of Proposition 3.3. Lemma 3.5. The Hilbert scheme of X_{0} = ℙ′ ∪ ℙ′′ is smooth of local dimension 20 at Γ.

Proof. We will denote by N_{Z} the normal sheaf of a curve Z in ℙ′ ≃ ℙ^{3}. The normal sheaf of a line in ℙ^{3} is 2O(1). Using the relation between the restriction of the normal sheaf of a reducible curve to its component with the normal sheaf of this component (see [HH]), we see that N_{Γ}|_{ℓ_{i}} is obtained by two positive elementary tranformations from N_{ℓ_{i}} ≃ 2O(1). Hence it is either 2O(2), or O(1) ⊕ O(3). The second case occurs only if the proper transforms of ℓ_{i−1}, ℓ_{i+1} in the blowup of ℙ′ with center ℓ_{i} meet the exceptional divisor E = ℙ(N_{ℓ_{i}}) ≃ ℙ^{1} × ℙ^{1} in two points lying on one cross-section of the natural projection E⟶ℓ_{i}. This may happen only if ℓ_{i−1}, ℓ_{i}, ℓ_{i+1} lie in one plane, which is impossible in a good pentagon. So, N_{Γ}|_{ℓ_{i}} ≃ 2O(2) for any i.

Now, denote N_{Γ} by N_{1} and consider the following chain of restriction exact sequences:

0⟶N_{i+1}⟶N_{i}⟶N_{i}|_{ℓ_{i}}⟶0 , i = 1, …, 4,

in which SuppN_{i} = ⋃_{j=i}^{5} ℓ_{j}. We have already determined N_{1}|_{ℓ_{1}} ≃ 2O_{ℓ_{1}}(2). For i = 2, …, 5 one finds successively:

N_{i}|_{ℓ_{i}} ≃ 2O_{ℓ_{i}}(1) (i = 2, 3, 4),

whilst

N_{5} = N_{5}|_{ℓ_{5}} ≃ 2O .

Now, reading the same chain of exact sequences backwards, one easily calculates h^{k}(N_{i}) for all i, k and concludes that h^{0}(N_{Γ}) = 20, h^{1}(N_{Γ}) = 0. This ends the proof of the lemma. □

Now we will impose on the curves parametrized by Hilb(X_{0}) the incidence conditions with the four conics C_{i}. Lemma 3.6. In the situation of the preceding lemma, let U_{C} denote the locally closed subset of Hilb(X_{0}) representing the curves meeting transversely C at 10 points. Then the point y_{0} representing Γ is a smooth point of U_{C}, dim_{y_{0}} U_{C} = 10, and the generic point of the component of U_{C} contaning y_{0} represents a nonsingular curve in ℙ′.

Proof. By Lemma 3.5, y_{0} is a smooth point of Hilb(X_{0}). It has a neighborhood U consisting only of smooth points and contained entirely in Hilb(ℙ′). Let F ⊂ ℙ′ be any reduced curve meeting Γ transversely at r points x_{1}, …, x_{r} which are nonsingular both on F and on Γ, and let ξ_{i} for i = 1, …, r denote the point of ℙ(N_{Γ}) corresponding to the direction of the branch of F passing through x_{i}. Denote by elm_{ξ_{1},...,ξ_{r}}^{−}N_{Γ} the negative elementary transformation of the normal bundle N_{Γ} w.r.t. the ξ_{i} (see [HH]). Let U_{F} ⊂ U be the locus of curves meeting F transversely at r distinct points and represented by elements of U. Then U_{F} is smooth at y_{0} if h^{1}(Γ, elm_{ξ_{1},...,ξ_{r}}^{−}N_{Γ}) = 0 and in this case we have for the tangent space T_{y_{0}}U_{F} = H^{0}(Γ, elm_{ξ_{1},...,ξ_{r}}^{−}N_{Γ}) (see [Ran], Remark 5.1).

Let us apply this observation to F = C = ⋃_{i=1}^{4} C_{i}, r = 10, C_{i} ∩ ℓ_{i} = {x_{2i−1}, x_{2i}}, i = 1, …, 5. Let N = elm_{ξ_{1},...,ξ_{10}}^{−}N_{Γ}. As we saw in the proof of Lemma 3.5, N_{Γ}|_{ℓ_{i}} ≃ 2O(2) for any i. As the branches of C passing through x_{2i−1}, x_{2i} lie in one plane for each i = 1, …, 4 and they do not for i = 5, we have N |_{ℓ_{i}} ≃ O ⊕ O(2) for i = 1, …, 4 and N |_{ℓ_{5}} ≃ 2O(1). Similarly to the proof of Lemma 3.5, one can easily show that h^{0}(N) = 10, h^{1}(N) = 0. It remains to prove that Γ is smoothed out by a generic deformation inside U_{C}. It suffices to show that the natural map T_{y_{0}}U_{C}⟶T_{p_{i}}^{1} to the Schlesinger’s space T^{1} is surjective for any i = 1, …, 5, where p_{i} = ℓ_{i} ∩ ℓ_{i+1} (the addition of subscripts modulo 5). We do it in the same way as the authors of [HH] in the proof of Theorem 4.1.

Let F = ℓ_{i}, G = ⋃_{j≠i} ℓ_{j}. We have the following natural exact sequences:

0⟶N ⟶N |_{F} ⊕ N |_{G}⟶N |_{{p_{i−1},p_{i}}}⟶0 ,
0⟶elm_{{p′_{i−1}}}^{+}elm_{{ξ_{2i−1},ξ_{2i}}}^{−}N_{F} ⟶N |_{F} ⟶T_{p_{i}}^{1}⟶0 ,

where p′_{i} is the point of ℙ(elm_{{ξ_{2i−1},ξ_{2i}}}^{−}N_{F}) corresponding to the tangent direction of l_{i+1} at p_{i}. It follows that if the natural maps H^{0}(N |_{G})⟶N |_{{p_{i}}} and H^{0}(N |_{F})⟶T_{p_{i}}^{1} are both surjective, then H^{0}(N)⟶T_{p_{i}}^{1} is surjective as well. We have N |_{F} ≃ O ⊕O(2) or 2O(1), so the surjectivity of the second map is obvious. Next, G is a chain of four lines ℓ_{i+1}, …, ℓ_{i+4} and

H^{0}(N |_{G}) = {(s_{i+1}, s_{i+2}, s_{i+2}, s_{i+4}) ∈ ∑_{k=i+1}^{i+4} H^{0}(N |_{ℓ_{k}}) | s_{k}(p_{k}) = s_{k+1}(p_{k})

for k = i + 1, i + 2, i + 3}

Each restriction N |_{ℓ_{k}} is generated by global sections, so for arbitrary s_{i+1}, we can choose a s_{i+2} such that s_{i+2}(p_{i+1}) = s_{i+1}(p_{i+1}), then choose a s_{i+3} such that s_{i+3}(p_{i+2}) = s_{i+2}(p_{i+2}), and finally, choose a s_{i+4} such that s_{i+4}(p_{i+3}) = s_{i+3}(p_{i+3}). Thus any s_{i+1} ∈ H^{0}(N |_{ℓ_{i+1}}) extends to a section of H^{0}(N |_{G}), hence the map H^{0}(N |_{G})⟶N |_{{p_{i}}} is surjective and we are done. □

Now we are coming back to the proof of Proposition 3.4. By Proposition 3.3, we can find a local cross-section of Σ representing a flat family of good pentagons in the varieties X_{t}, whose fiber over 0 is a good pentagon in ℙ′. By the universal property of the Hilbert scheme, this family is induced via a cross-section Δ⟶ Hilb(X/Δ) over Δ, which we will denote by t ↦ y(t). It suffices to show that y_{0} ∈ H′. We have verified in Lemma 3.6 that a good pentagon Γ in ℙ′ becomes strongly smoothable when C is specialized into a union of four conics. By the semi-continuity of h^{i}(N), and because the surjectivity of the maps H^{0}(N)⟶T_{p_{i}}^{1} is an open condition, a generic good pentagon in some component of good pentagons in ℙ′ is strongly smoothable for generic X⟶Δ. In fact, we do not check that the family of good pentagons in ℙ′ is irreducible, so we are speaking here about one of its components.

As Γ is a connected curve of degree 5 and arithmetic genus 0, so is its smoothing, hence it is an elliptic quintic in ℙ′. By construction, it meets transversely C at 10 points and is an element of H′. This ends the proof of Proposition 3.4.

This immediately implies Corollary 3.7. With the same assumptions as above, there is a component of H′ which is acquired with multiplicity 1 in the fiber H_{0} of H⟶Δ.

We will prove in Section 4 that H′ is irreducible, hence H′ and H′′ are irreducible components of H_{0} acquired with multiplicity 1.

4. Elliptic quintics through ten points

4.1. Elliptic quintics, 10-secant to B = W ∩ Q.

In this section we will prove the irreducibility of the subscheme H′ of Hilb(X_{0}) introduced in Sect. 3.1. Lemma 4.1. Let Q (resp. S) be a generic quadric (resp. quartic) in ℙ^{3}, and B = Q ∩ S. Let C_{5}^{1}[B]_{10} ⊂ C_{5}^{1}(ℙ^{3}) be the base of the family of elliptic quintics meeting B in a subscheme of length 10. Then C_{5}^{1}[B]_{10} is irreducible.

Proof. Let ℙ^{34} be the space of quartic surfaces in ℙ^{3} and consider the incidence variety Γ = {(S, C) ∈ C_{5}^{1}(ℙ^{3}) × ℙ^{34} | C ⊂ S} with projections ℙ^{34} ⟵^{p_{1}} Γ ⟶^{p_{2}} C_{5}^{1}(ℙ^{3}). First remark that for any quintic C ∈ C_{5}^{1}(ℙ^{3}) one has h^{0}(I_{C/ℙ^{3}}(4)) = 15. In fact, since C does not lie on a quadric, it follows that the restriction map H^{0}(O_{ℙ^{3}}(2)) → H^{0}(O_{C}(2)) is surjective, hence the exact triple 0 → I_{C/ℙ^{3}}(2) → O_{ℙ^{3}}(2) → O_{C}(2) → 0 gives h^{1}(I_{C/ℙ^{3}}(2)) = 0 and a fortiori h^{1}(I_{C/ℙ^{3}}(4)) = 0, so that we obtain the desired equality from the exact triple 0 → H^{0}(I_{C/ℙ^{3}}(4)) → H^{0}(O_{ℙ^{3}}(4)) → H^{0}(O_{C}(4)) → 0. Hence p_{2}^{−1}(C) ≃ ℙ^{14}, and as C_{5}^{1}(ℙ^{3}) is irreducible of dimension 20, Γ is irreducible as well and dim Γ = 34. Moreover, denoting C_{5}^{1}(ℙ^{3})^{∗} := C_{5}^{1}(ℙ^{3}) ∖ Sing(C_{5}^{1}(ℙ^{3})), Γ^{∗} = p_{2}^{−1}(C_{5}^{1}(ℙ^{3})^{∗}), we see that Γ^{∗} is smooth and is a projective bundle over C_{5}^{1}(ℙ^{3})^{∗}.

Next, clearly, a general quartic S in the fibre p_{2}^{−1}(C) is smooth for general C ∈ C_{5}^{1}(ℙ^{3}). Hence, denoting D = p_{1}(Γ), we obtain that D is irreducible. Besides, for any smooth quartic S = p_{1}(S, C) ∈ D the fibre p_{1}^{−1}(S) is a dense open subset of the linear series |O_{S}(C)| ≃ ℙ^{1}; hence dim D = 33, i.e. D is an irreducible divisor on ℙ^{34}. Moreover, p_{1} : Γ → D is a smooth morphism of relative dimension 1.

Next, consider the closure \overline{D} of D in ℙ^{34} and prove that

(70)  codim\overline{D} Sing \overline{D} ≥ 2.

In fact, if codim\overline{D} Sing \overline{D} = 1, i.e. dim Sing \overline{D} = 32, then, taking any C ∈ C_{5}^{1}(ℙ^{3})^{∗}, we see that p_{1}(p_{2}^{−1}(C)) ∩ Sing \overline{D} = ℙ^{14} ∩ Sing \overline{D} ≠ ∅; hence, since p_{1}|Γ^{∗} is a smooth morphism, p_{2}^{−1}(C) ∩ Sing Γ^{∗} ⊂ p_{2}^{−1}(C) ∩ p_{1}^{−1}(Sing \overline{D}) ≠ ∅, which contradicts the smoothness of Γ^{∗}.

Next let C_{2·4}(ℙ^{3}) := {C ∈ Hilb(X) | C is a smooth complete intersection curve of a smooth quartic surface and a smooth quadric surface in ℙ^{3}}. Clearly, C_{2·4}(ℙ^{3}) is a smooth irreducible variety of dimension 33. Let U be the dense open subset of smooth quartics in ℙ^{34} and consider the incidence variety Π = {(C, F) ∈ C_{2·4}(ℙ^{3}) × ℙ^{34} | C ⊂ F } with projections C_{2·4}(ℙ^{3}) ⟵^{q_{1}} Π ⟶^{q_{2}} ℙ^{34}. One immediately verifies, using the resolution 0 → O_{ℙ^{3}}(−2) → O_{ℙ^{3}}(2) ⊕ O_{ℙ^{3}} → I_{C/ℙ^{3}}(4) → 0 for C ∈ C_{2·4}(ℙ^{3}), that h^{0}(I_{C/ℙ^{3}}(4)) = 11. Besides, by the definition of C_{2·4}(ℙ^{3}), a general quartic in q_{1}^{−1}(C) is smooth, hence q_{1}^{−1}(C) is a dense open part of the projective space ℙ^{10} = P (H^{0}(I_{C/ℙ^{3}}(4))). Thus, Π is irreducible of dimension 43. Moreover, for any point (C, S) ∈ Π, we have (C, S) ∈ q_{2}^{−1}(S′) with S′ ∈ U, and q_{2}^{−1}(S′) is an open subset of the linear series |O_{S′}(C)| = |O_{S′}(2)| ≃ ℙ^{9} consisting of smooth curves. Clearly, D′ := U ∩ D ≠ ∅, hence D′ is an open subset of D, so that D_{Π} := q_{2}^{−1}(D′) is an irreducible divisor in Π.

Let us look now at the general fiber of the projection q_{D} := q_{1}|D_{Π} : D_{Π} → C_{2·4}(ℙ^{3}). Let \overline{Π} be the closure of Π in \overline{C} × ℙ^{34} with natural projections C ⟵^{\overline{q}_{1}} \overline{Π} ⟶^{\overline{q}_{2}} ℙ^{34}, where \overline{C} is the closure of C_{2·4}(ℙ^{3}) in Hilb(ℙ^{3}), and \overline{D}_{Π} = \overline{q}_{2}^{−1}(\overline{D}). Let Z ⊂ \overline{D} be the subset of \overline{D} which consists of reducible quartics, codim\overline{D}Z ≥ 11, D^{∗} = \overline{D} ∖ Z, D_{Π}^{∗} := \overline{q}_{2}^{−1}(D^{∗}). Clearly, for any S ∈ D^{∗}, the linear series |O_{S}(2)| ≃ ℙ^{9} lies in \overline{C}. Hence \overline{q}_{2}|_{D_{Π}^{∗}} : D_{Π}^{∗} → D^{∗} is a projective bundle with fiber ℙ^{9}, in particular, it is a smooth morphism of relative dimension 9. Now by construction the fiber of the projection \overline{q}_{D} = \overline{q}_{1}|\overline{D} : \overline{D}_{Π} → \overline{C} over any point C ∈ C_{2·4}(ℙ^{3}) is a divisor in \overline{q}_{1}^{−1}(C) = ℙ^{10} which contains q_{D}^{−1}(C) as an open subset. Assume that q_{D}^{−1}(C) is reducible. Then its closure \overline{q_{D}^{−1}(C)} is also a reducible divisor in ℙ^{10}, hence it has singularities in codimension 1. Since C ∈ C_{2·4}(ℙ^{3}) is smooth, it follows that codim_{\overline{D}_{Π}} Sing \overline{D}_{Π} = 1. As codim\overline{D}Z ≥ 11, this implies that codim_{D_{Π}^{∗}} Sing D_{Π}^{∗} = 1. Hence, since \overline{q}_{2}|_{D_{Π}^{∗}} : D_{Π}^{∗} → D^{∗} is a smooth morphism, it follows that codim_{D^{∗}} Sing D^{∗} = 1 and a fortiori codim\overline{D} Sing \overline{D} ≥ 2, which contradicts (70).

Thus, for general C ∈ C_{2·4}(ℙ^{3}), both q_{D}^{−1}(C) and \overline{q_{D}^{−1}(C)} are irreducible. The generic quartic S from \overline{q_{D}^{−1}(C)} is smooth and contains a unique pencil |E| of elliptic curves, and these curves belong to C_{5}^{1}[B]_{10}, where B = Q ∩ S, Q being the unique quadric containing C. Conversely, taking B as in the hypothesis of the lemma, any smooth elliptic quintic E meeting B in 10 points lies in a generically smooth quartic surface S through B, i.e. such that (B, S) ∈ \overline{q}_{D}^{−1}(B), and varies on S in a pencil. Hence we have a fibration H′ → \overline{q}_{D}^{−1}(C) with an open subset of ℙ^{1} as a fiber. This implies the irreducibility of C_{5}^{1}[B]_{10}. □

Corollary 4.2. Let Q ⊂ ℙ^{3} be a smooth quadric surface, W ⊂ ℙ^{3} a generic quartic one, B = Q ∩ W. Then the subscheme H′ (resp. H′′) of Hilb(X_{0}) introduced in Sect. 3.1 and C_{5}^{1}[B]_{10} have a common dense open subset, hence H′ (resp. H′′) is irreducible. This implies, by Corollary 3.7, that the fiber H_{0} of H over t = 0 is reduced at the generic points of H′ and H′′. 4.2. 42 elliptic quintics through 10 points

Consider the following open subset of Hilb ℙ^{3}:

U = {C | C is a smooth irreducible elliptic quintic such that {C} is a smooth point of Hilb ℙ^{3}}.

There is a universal family of quintics C ⊂ U × ℙ^{3} over U with projection p_{C} : C → U. Consider the 10-th relative symmetric power S := Sym^{10}(C/U) of C over U with projection p : S → U and fiber

(71)  p^{−1}({C}) = Sym^{10} C, {C} ∈ U.

Thus S is a smooth irreducible variety of dimension 30. We can identify it with a subvariety of U × Hilb^{10} ℙ^{3}:

S = {({C}, {Z}) | {C} ∈ U, {Z} ∈ Hilb^{10} ℙ^{3} and Z ⊂ C as a scheme},

and we have a natural projection

π : S → Hilb^{10} ℙ^{3}, ({C}, {Z}) ↦ {Z}.

There exists at least one quintic from the closure of C_{7}^{4}(ℙ^{3}) passing through any 10 points in ℙ^{3}, and for a general 10-uple of points in ℙ^{3}, the number of such quintics is finite and they are smooth. In fact, we have:

Lemma 4.3 (E. Getzler, [Ge], Table 2). For a general 10-uple Z of points in ℙ^{3}, there are 42 elliptic quintics passing through Z.

Denote by H^{10} the component of Hilb^{10} ℙ^{3} contating a 10-uple of distinct points. It follows that H := π(S) is an open subset of H^{10} and that π : S → H is generically finite of degree 42. We need to replace in the above lemma “a general 10-uple Z of points in ℙ^{3}” by “a general 10-uple Z of points in a smooth quadric Q ⊂ ℙ^{3}”, that is, we should prove that the 10-uples of points in a given smooth quadric form a divisor in H which is not contained in the branch locus of π.

Let

D = {({C}, {Z}) ∈ S | Z = C ∩ Q for some quadric Q in ℙ^{3}}.

Clearly, D is a divisor in S, and by (71), for any {C} ∈ U,

(72) D ∩ p^{−1}({C}) = {{Z} ∈ Sym^{10} C | O_{C}(Z) = O_{ℙ^{3}}(2)|C} = |O_{ℙ^{3}}(2)|C|} ≃ ℙ^{9}.

As U is irreducible, it follows from (72) that D is irreducible too, hence so is B := π(D). It is easy to see that B is a divisor. Indeed, for any smooth elliptic quintic C in ℙ^{3} and for any Z ∈ |O_{C}(2)|, there is a unique quadric Q passing through Z, so the dimension of B is at least that of the family of quadrics Q in ℙ^{3} plus that of the the family of the 10-uples of points in Q, which gives 29.

Further, let R ⊂ S be the ramification divisor of π, so that

(73)  K_{S} = p^{∗}K_{U} ⊗ K_{S/U} = π^{∗}K_{H} ⊗ O_{S}(R).

Our aim is to prove the following Proposition 4.4. (i) D is not contained in R, hence π : S → H is étale in a general point of D.

(ii) π is an étale covering over an open subset V of H such that B ∩ V ≠ ∅.

Proof. We start by (ii), in assuming (i) already proved.

For a general 10-uple of points Z on a general smooth quadric Q, there exists a quartic surface passing through Z and intersecting Q in a smooth curve, say, C_{0}. Further, by Corollary 4.2, the set of all the smooth elliptic quintics meeting C_{0} in 10-uples of points is irrreducible. Hence (π^{−1}(Sym^{10}(C_{0})))_{red} is irreducible. Hence π has the same ramification indices in all the points of π^{−1}(Z), where {Z} ∈ B is the above general point of B. But according to (i), π is unramified in at least one point of π^{−1}(Z). Hence π is unramified, that is étale, in all the points of π^{−1}(Z).

Prove now (i). To see that D is not contained in R, it suffices to show, by virtue of (72), that

(74)  R = p^{−1}(R_{0})

for some divisor R_{0} on U. Take any point {C} ∈ U. Then π embeds S_{C} := Sym^{10} C into H. Moreover, H is the union of the subvarieties π(S_{C}) ≃ S_{C} when {C} runs through U. Since S_{C} is a fiber of p, (74) follows from the relation

(75)  R ∩ S_{C} ∼^{alg} 0 on S_{C},

which we are going to prove now.

First, clearly

(76)  K_{S/U}|S_{C} = K_{S_{C}}.

Next, consider S_{C} as a subvariety of H, embedded by π|_{S_{C}}, and compute the normal sheaf N_{S_{C}/H}. Let Γ ⊂ S × S_{C} be the incidence divisor and C ⟵^{p_{1}} S × S_{C} ⟶^{p_{2}} the natural projections.

Lemma 4.5.

(77)  N_{S_{C}/H} = p_{2∗}p_{1}^{∗}(N_{C/ℙ^{3}}|Γ), R^{1}p_{2∗}p_{1}^{∗}(N_{C/ℙ^{3}}|Γ) = 0.

Proof. Take any {Z} ∈ S_{C}. Since Z is a curvilinear 0-dimensional subscheme in ℙ^{3}, hence a locally complete intersection, we have:

(78) Ext^{1}(I_{Z/C}, O_{Z}) = 0, Hom(I_{Z/C}, O_{Z}) = T_{{Z}}S_{C} ≃ ℂ^{10}, Hom(I_{Z/ℙ^{3}}, O_{Z}) = T_{{Z}}H ≃ ℂ^{30}, Hom(I_{C/ℙ^{3}}, O_{Z}) = H^{0}(N_{C/ℙ^{3}} ⊗ O_{Z}).

Thus, applying the functor Hom(·, O_{Z}) to the exact triple 0 → I_{C/ℙ^{3}} → I_{Z/ℙ^{3}} → I_{Z/C} → 0, we obtain the exact triple:

(79)  0 → T_{{Z}}S_{C} → T_{{Z}}H → H^{0}(N_{C/ℙ^{3}} ⊗ O_{Z}) → 0.

Hence

(80)  N_{S_{C}/H}|_{Z} = H^{0}(N_{C/ℙ^{3}} ⊗ O_{Z}).

From here, taking into account (78), we deduce:

(81)  H^{0}(N_{C/ℙ^{3}} ⊗ O_{Z}) = p_{2∗}p_{1}^{∗}(N_{C/ℙ^{3}}|Γ)|_{Z}, H^{1}(N_{C/ℙ^{3}} ⊗ O_{Z}) = 0.

so that, globalizing (80) over S_{C}, we get (77) by base change. □

Coming back to the proof of Proposition 4.4, take any cubic surface F in ℙ^{3} through C, smooth along C; we can shrink U, if necessary, so that such a cubic exists for all C ∈ U. Then the exact sequence of normal bundles of the embeddings C ↪ F ↪ ℙ^{3}, in view of the equality det(N_{C/ℙ^{3}}) = O_{C}(4), can be written in the form 0 → O_{C}(1) → N_{C/ℙ^{3}} → O_{C}(3) → 0. Denote E_{O_{C}(1)} = p_{2∗}(O_{C}(1) ⊠ O_{S_{C}}|Γ, E_{O_{C}(3)} = p_{2∗}(O_{C}(3) ⊠ O_{S_{C}}|Γ and apply R^{i}p_{2∗}(N_{C/ℙ^{3}} ⊠ O_{S_{C}} ⊗ ·) to the last exact triple. Using (77), we get

(82)  c_{1}(N_{S_{C}/H}) = c_{1}(E_{O_{C}(1)}) + c_{1}(E_{O_{C}(3)}).

By [F, Ex. 14.4.17(b)] we obtain: c_{1}(E_{O_{C}(1)}) ∼^{alg} −5x, c_{1}(E_{O_{C}(3)}) ∼^{alg} 5x, where x is the algebraic equivalence class of the image of Sym^{9}C under the embedding Sym^{9}C ↪ Sym^{10}C : D ↦ D + P_{0}, and P_{0} ∈ C is a fixed point. Hence

(83)  c_{1}(N_{S_{C},H}) ∼^{alg} 0.

so that, identifying S_{C} with π(S_{C}), we get K_{H}|π(S_{C}) ∼^{alg} K_{π(S_{C})}, hence also

(84)  π^{∗}K_{H}^{−1}|S_{C} ∼^{alg} (π|S_{C})^{∗}(K_{π(S_{C})}^{−1}) = K_{S_{C}}^{−1}.

Substituting (76) and (84) into (73)|S_{C}, we obtain (75), and this ends the proof. □

The following statements are obvious corollaries of what we have proved:

Corollary 4.6. Let Q ⊂ ℙ^{3} be a smooth quadric surface. Then there are 42 elliptic quintics in ℙ^{3} passing through 10 generic points of Q. There are also 42 elliptic quintics in ℙ^{3} passing through 10 generic points on the curve B = Q ∩ W for a generic quartic surface W ⊂ ℙ^{3}.

In fact, the number 42 given by the result of Getzler is that of stable pointed genus-1 curves in ℙ^{3} with incidence relation in 10 prescribed generic points. The moduli space of these curves is projective, so the result implies that there are no singular curves in \overline{C_{5}^{1}(ℙ^{3})} which pass through the 10 generic points, because all the 42 solutions to the problem are in the interior part of the moduli space, but not on its boundary, corresponding to singular curves. We thus obtain: Corollary 4.7. Let α_{0} : H′ ∪ H′′⟶J^{+} be the map introdiced in Proposition 3.1, (ii). Then the number of copies of ℙ^{1} in the generic fiber of α_{0} is equal to 84.

As H′ ∪H′′ contains all the elliptic quintics which are the limits of elliptic quintics from the neighboring fibers, we deduce from Corollaries 4.7 and 4.2 that the number of ℙ^{1}’s in the generic fiber of α_{t} is 84 for all sufficiently small t ∈ Δ. This implies the following result. Theorem 4.8. Let X be a general quartic double solid. Then the generic fiber of the Abel Jacobi map Φ_{C_{5}^{1}(X)} : C_{5}^{1}(X)⟶Θ + const is a dense open subset in the union of 84 copies of ℙ^{1}.

5. Vector bundles

Let X be general, so that C_{5}^{1}(X) is irreducible, and consider the dense open subset Č_{5}^{1}(X) of C_{5}^{1}(X) defined in (19). Take any curve C ∈ Č_{5}^{1}(X) and use the Serre construction to define the O_{X}-sheaf E_{C} as an extension of O_{X}-sheaves:

(85)  0 → O_{X}(−1) → E_{C} → J_{C/X}(1) → 0.

Any such extension is determined uniquely up to an isomorphism by a 1-dimensional subspace of the group Ext^{1}(J_{C/X}(1), O_{X}(−1)) ≃ H^{0}(Ext^{1}(J_{C/X}(1), O_{X}(−1))) ≃ H^{0}(Ext^{2}(O_{C}(1), O_{X}(−1))) ≃ H^{0}(Ext^{2}(O_{C}, ω_{X})) ≃ H^{0}(ω_{C}) ≃ ℂ (we are using h^{i}O_{X}(−2) = 0, i < 3, and ω_{C} ≃ O_{C} C). Thus E_{C} is determined uniquely up to an isomorphism by C. Besides, by a well known property of the Serre construction, the local triviality of Ext^{1}(J_{C/X}(1), O_{X}(−1)) ≃ ω_{C} implies that E_{C} is locally free.

From (85), one easily computes the Chern classes:

c_{1}(E_{C}) = 0, c_{2}(E_{C}) = 3

(here we understand c_{2} as an element of the group B_{1} of classes of algebraic equivalence of 1-dimensional cycles on X, generated by the class of a line on X). As π : C → π(C) is an isomorphism onto a quintic π(C) that does not lie in a plane and H^{0}(O_{X}(1)) = π^{∗}H^{0}(O_{ℙ^{3}}(1)), it follows that H^{0}(J_{C/X}(1)) ≃ H^{0}(J_{π(C)/ℙ^{3}}(1)) = 0. Hence the triple (85) gives h^{0}(E_{C}) = 0. This means that E_{C} is Gieseker stable (see [MT-1, Prop. 2.6]). Thus, denoting by M_{X}(2; 0, 3) the Gieseker-Maruyama moduli scheme of stable rank-2 vector bundles on X with Chern classes c_{1} = 0, c_{2} = 3, we have [E_{C}] ∈ M(2; 0, 3), where [E] stands for the isomorphism class of E. Remark that the equality h^{0}(J_{C/X}(2)) = 1 immediately implies h^{i}(J_{C/X}(2)) = 0, i > 0; hence by (85) also

h^{i}(E_{C}) = 0, i > 0.

Let M̃ = {[E] ∈ M_{X}(2; 0, 3)| h^{i}(E) = 0, i > 0}. By semicontinuity, this is an open subset of M_{X}(2; 0, 3) containing [E_{C}]. Let M be the irreducible component of M̃ containing [E_{C}]. We have a natural map

f : Č_{5}^{1}(X) → M, C′ ↦ [E_{C′}].

It is a standard matter to show that f is a dominant morphism (see [MT-1, Lemmas 5.2 and 5.3]), whose fibre f^{−1}(f(C′)) is open in P (H^{0}(E_{C′}(1)) ≃ ℙ^{1}:

\overline{f^{−1}(f(C′))} = ℙ^{1}, C′ ∈ Č_{5}^{1}(X).

The closure is taken in \overline{C_{5}^{1}(X)}, and we keep in mind that, by Riemann-Roch, h^{0}(E_{C′}(1)) = 2, since by the definition of M̃, h^{i}(E_{C′}(1)) = 0, i > 0. The equality h^{0}(J_{C′/X}(2)) = 1 means that the linear series |J_{C′/X}(2))| consists of a unique K3 surface S(C′), and by Serre construction, the fibre \overline{f^{−1}(f(C′))} = ℙ^{1} is just the pencil of elliptic curves on S(C′), which are nothing else than the zero loci of sections oh E_{C′}(1) :

(86)  \overline{f^{−1}(f(C′))} = |O_{S(C′)}(C′)| = ℙ^{1}.

Now consider the Stein factorization of the Abel-Jacobi map Φ = Φ_{C_{5}^{1}(X)}|Č_{5}^{1}(X) : Č_{5}^{1}(X) → Θ + const. Using the above description (86) of fibres of f and the well-known interpretation of the differential of the Abel-Jacobi map in terms of the branch quartic W of π : X → ℙ^{3} (see the end of Section 2, or [T-1, Corollary 1], or [We, Prop.2.13]), we obtain the following Theorem 5.1. Let X be a general quartic double solid. Then there exists a quasi-finite dominant morphism g : M_{0} → Θ + const of degree 84 from an open part M_{0} of the above component M of the moduli space M_{X}(2; 0, 3) to a translate of the theta divisor of J(X) which gives the Stein factorization of the Abel-Jacobi map Φ of Č_{5}^{1}(X):

(87)  \xymatrix{Č_{5}^{1}(X) \ar@{-->}[dr]_{f} \ar[rr]^{Φ} &  & Θ + const \\  & M_{0} \ar[ur]_{g}}

The fibers of f : Č_{5}^{1}(X) → M_{0} are open subsets of ℙ^{1}.

The proof is similar to that of Theorem 5.6 in [MT-1].

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D. M.: Mathématiques - bât. M2, Université Lille 1, F-59655 Villeneuve d’Ascq Cedex, France

  E-mail address: markushe@agat.univ-lille1.fr

Department of Mathematics, State Pedagogical University, Respublikanskaya Str. 108 150 000 Yaroslavl, Russia

  E-mail address: tikhomir@yaroslavl.ru