NEW COMPONENT OF THE MODULI SPACE M(2;0,3) OF STABLE VECTOR BUNDLES ON THE DOUBLE SPACE P^{3} OF INDEX TWO

A. S. TIKHOMIROV

Introduction

In 1998 D.Markushevich and the author found a first relation between the moduli spaces of stable vector bundles on Fano threefolds and their intermediate Jacobians (see [MT-1]). They investigated the case of a cubic hypersurfuce, and their results were developed in further works [MT-2],[IM],[D],[B]. In this paper we are interested in the case of the quartic double solid X, the next after cubic in the series of Fano varieties of index 2 with nontrivial intermediate Jacobian. We consider stable rank-2 vector bundles on X. Let W be a smooth quartic surface in P^{3} and π : X → P^{3} the double covering ramified in W. (It is clear that X is defined by the embedding W ↪ P^{3} uniquely up to an isomorphism.) We study the moduli scheme M(2; 0, 3) of rank-2 stable vector bundles with Chern classes c_{1} = 0, c_{2} = 3 on X. We show that there is a distinguished component M in M_{X}(2; 0, 3), of the prescribed dimension 9, obtained via the Serre construction from the family H of smooth elliptic quintics on X. It is proved that the Stein factorization of the Abel-Jacobi map of H into the intermediate Jacobian J(X) of X gives a quasi-finite (probably birational) morphism of (an open part of) M to (a translate of) the Poincare theta divisor Θ of J(X). As a byproduct we obtain a new parametrisation of Θ via Abel-Jacobi image of the family of elliptic quintics on X (the known parametrisation of Θ via sextics of genus 3 was found in [T-1]).

Acknowledgements. The author acknowledges with pleasure the hospitality of the Max Planck Institute of Mathematics at Bonn (December 2000 – February 2001), where a part of this work was carried out.

1. Construction of the theta divisor of the double space X via the family of elliptic quintics on X

1.1. Notations. We adopt the standard notation for the degree deg(C) of a curve C in X as the degree deg(C) := deg(O_{X}(1)|C), where O_{X}(1) is the positive generator of P ic(X).

Let

• H^{∗} := {C ∈ Hilb(X)| C is a smooth irreducible elliptic quintic curve in X},

• H closure of H^{∗} in Hilb(X).

By a standard parameter counting one proves the following

Lemma 1.2. H is nonempty and equidimensional of dimension 10.

For any irreducible component H_{0} of H a general curve C ∈ H_{0} lies in a unique K3-surface S(C) of the linear series |O_{X}(2)| and satisfies the condition that π : C → π(C) is an isomorphism.

We also can formulate here

Conjecture 1.3. H is irreducible.

1991 Mathematics Subject Classification. 14J30.

The work is partially supported by the grant INTAS-OPEN-2000-269.

Now let H_{0} be any fixed irreducible component of H and denote

• H_{6,4}(P^{3}) := {C ∈ Hilb(P^{3})| C is a smooth complete intersection curve of a cubic and a quadric surfaces in P^{3}},

• H_{12,19} := {C ∈ Hilb(X)| C = π^{−1}(C_{0}), C_{0} ∈ H_{6,4}(P^{3})},

• H := {C ∈ Hilb(X)| = _{1} ∪ C_{2}, where C_{1} ∈ H_{0}, C_{2} ∈ H_{12,19}, π : C_{1} → π(C_{1}) is an isomorphism, and C_{1} and C_{2} intersect transversally at 10 points}.

Remark 1.4. Clearly, H_{12,19} ≃ H_{6,4} is a rational irreducible variety, and any curve C ∈ H_{12,19} has degree 12 and arithmetic genus 19. Respectively, any curve C ∈ H has degree 17 and arithmetic genus 28.

We have a natural embedding H ↪ H_{0} × H_{12,19} : C = _{1} ∪ C_{2} ↦ (C_{1}, C_{2}), and lemma 1.2 together with an easy dimension count implies

\overline{pr_{1}(H)} = \overline{H_{0}}.  (1)

Convention: for any subvariety B of the Hilbert scheme of curves on X we denote by Φ_{B} : B → J(X) the Abel-Jacobi map (defined uniquely up to a choice of the reference point in B).

The remark 1.4 above shows immediately that

Φ_{H_{12,19}}(H_{12,19}) = {point}.  (2)

Let F be the Fano surface of X, i.e. the base of the family of lines on X. (This is a smooth irreducible surface, see, e.g., [T-2] or [We].)

Lemma 1.5. (i) For any curve C ∈ H, h^{0}(J_{C,X}(4)) = 4.

(ii) For a general point (l, C_{1}) ∈ F × H_{0} there exists a curve C = C_{1} ∪ C_{2} ∈ H such that C ∩ l = ∅ and there exists a surface S ∈ |O_{X}(4)| containing the curve C ∪ l.

Proof. (i) We have C = C_{1} ∪ C_{2}, where Z := C_{1} ∩ C_{2} = {x_{1}, ..., x_{10}}, C_{1} ∈ H_{0}, π : C_{1} → C_{10} := π(C_{0}) is an isomorphism, and C_{2} = π^{−1}(C_{20}), where C_{20} ∈ H_{6,4}. This means, in particular, that C_{20} lies in a uniquely defined quadric, say, Q. Let Z_{0} := π(Z), deg(Z_{0}) = deg(Z) = 10. Since by construction C_{10}∩Q ⊃ Z_{0} (as a scheme-theoretic inclusion) and deg(C_{10}∩Q) = 10 = deg(Z_{0}), it follows that

Z_{0} = C_{10} ∩ Q.  (3)

Using (3) and the fact that O_{Q}(C_{20}) = O_{Q}(3) one derives the exact triple:

0 → J_{Q∪C_{10},P^{3}}(4) → J_{C_{10}∪C_{20},P^{3}}(4) → O_{Q}(1) → 0.  (4)

Next, take any smooth cubic surface S ⊂ P^{3} through the smooth quintic curve C_{10}. We have the exact triple 0 → J_{Q∪S,P^{3}} → J_{Q∪C_{20},P^{3}} → J_{(S∩Q)∪C_{10},S} → 0. Here J_{(S∩Q)∪C_{10},S} ≃ O_{S}(−2)(−C_{10}) since S is smooth, hence twisting the above triple by O_{P^{3}}(4) we obtain:

0 → O_{P^{3}}(−1) → J_{Q∪C_{20},P^{3}}(4) → O_{S}(2)(−C_{10}) → 0.  (5)

One quickly checks that the natural map H^{0}(O_{S}(2)) → H^{0}(O_{C_{10}}(2)) is surjective, so that h^{1}(O_{S}(2)(−C_{10})) = 0. Besides, h^{1}(O_{P^{3}}(−1) = 0. Hence (5) implies h^{1}(J_{Q∪C_{10},P^{3}}(4)) = 0. This together with (4) and the equality h^{1}(O_{Q}(1)) = 0 gives

h^{1}(J_{C_{10}∪C_{20},P^{3}}(4)) = 0.  (6)

Hence the exact triple 0 → J_{C_{10}∪C_{20},P^{3}}(4) → J_{C_{20},P^{3}}(4) → J_{Z_{0},C_{10}}(4) → 0 implies the surjective map

H^{0}(J_{C_{20},P^{3}}(4)) ↠ H^{0}(J_{Z_{0},C_{10}}(4)).  (7)

Now since π : C_{1} → C_{10} is an izomorphism, we have π_{∗}J_{Z,C_{1}}(4) ≃ J_{Z_{0},C_{10}}(4), this giving the exact triple

0 → J_{Z_{0},C_{10}}(4) → π_{∗}O_{C}(4) → π_{∗}O_{C_{2}}(4) → 0.  (8)

Whence one quickly obtains:

h^{0}(π_{∗}O_{C}(4)) = χ(O_{C}(4)) = 40.  (9)

Next, consider the composition of natural (restriction) maps: 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 natural (restriction) map of sections

f : H^{0}(π_{∗}O_{X}(4)) → H^{0}(π_{∗}O_{C}(4)),  (10)

we obtain:

coker(f) = coker(H^{0}(π_{∗}J_{C_{2},X}(4)) →^{res} H^{0}(J_{Z_{0},C_{10}}(4))).  (11)

Next, by the projection formula π_{∗}J_{C_{2},X}(4) = π_{∗}π^{∗}J_{C_{20},P^{3}}(4) = π_{∗}π^{∗}J_{C_{20},P^{3}}(4)⊕π_{∗}π^{∗}J_{C_{20},P^{3}}(2), and the composition H^{0}(π_{∗}π^{∗}J_{C_{20},P^{3}}(4)) ↪ H^{0}(π_{∗}J_{C_{2},X}(4)) →^{res} H^{0}(J_{Z_{0},C_{10}}(4)) coincides with the surjective map (6), so (11) implies coker(f) = 0. This together with (9) and the equality h^{0}(O_{X}(4)) = h^{0}(O_{P^{3}}(4)) ⊕ h^{0}(O_{P^{3}}(2)) = 45 gives:

H^{0}(J_{C,X}(4)) = H^{0}(π_{∗}J_{C,X}(4)) = ker(f) = k^{5}.  (12)

(ii) This is a standard dimension count. For simplicity one could look for the surface S with the prescribed properties, e.g., in the linear subseries π^{∗}|O_{P^{3}}(4)| of |O_{X}(4)|. Lemma is proved.

Now we invoke the following important technical statement due to G.Welters [We, Prop. 6.17].

Proposition 1.6. 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 dominating;

(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 don’t contain all lines of X;

(iii) the incidence divisor between T and the family of lines F (the Fano surface) of X is reduced.

Then for any line l ∈ F the subvariety of T defined as T_{l} = {t ∈ T |C_{t} ∩ l = ∅} is mapped into (a translate of) the theta divisor Θ of J(X).

Now take the closure T̃ of H in Hilb(X) with a reduced structure, i.e. as an irreducible variety, and let T = T̃ \ Sing(T̃) be its smooth open part. In view of the statement (i) of lemma 1.5 we can apply the above proposition to T with k = 4, d = 17. (Here we possibly remove from T a proper subset so that the condition (iii) on the incidence divisor is satisfied.) We thus obtain Φ_{T} (T) ⊂ Θ + const, hence also

ΦT̃(T̃) ⊂ Θ + const.  (13)

As a corollary we have

Φ_{H}(H) ⊂ Θ + const.  (14)

Now remark that, since by construction H ⊂ H_{0} ×H_{12,19}, hence for any point C = (C_{1}, C_{2}) ∈ H we have Φ_{H}(C) = Φ_{H_{0}}(C_{1}) + Φ_{H_{12,19}}(C_{2}). Thus,

Φ_{H}(H) ⊂ Φ_{H_{0}}(H_{0}) + Φ_{H_{12,19}}(H_{12,19}).  (15)

Now (2) together with (14) and (15) gives the inclusion:

Φ_{H_{0}}(H_{0}) ⊂ Θ + const.  (16)

Next, the description of the differential of the composition H_{0} →^{Φ_{H_{0}}} Φ_{H_{0}}(H_{0}) →^{γ} P^{9} = P (T_{J(X)}(0)), (where γ is the Gauss map) given in [We], we quickly obtain in view of lemma 1.2 that Φ_{H_{0}}(H_{0}) is a divisor in J(X). Hence, since Θ is irreducible, (16) implies the following

Theorem 1.7. Φ_{H_{0}}(H_{0}) = Θ + const.

2. H.Clemens’ approach to the description of the Abel-Jacobi map of the family of elliptic quintics on X

Let {F = 0} be the equation of a smooth branch quartic W of the double covering π : X → P^{3} and let Q be a genericaly chosen quadric in P^{3} with equation {G = 0}, so that D = W ∩ Q is a smooth curve. We consider a family π_{t} : X_{t} → P^{3} of double solids defined by the branch quartics of the pencil W_{t} = {F + tG^{2} = 0}. Thus W_{t} degenerate at t = 0 to 2Q, and the double solids X_{t} degenerate to the reducible threefold X_{0} which is the union of two copies, say P′ and P′′, of P^{3} intersecting along the quadric Q. Moreover, according to H.Clemens [C-1],[C-2] the corresponding family of intermediate Jacobians X_{t} has Neron model J with fibre over t = 0 given by J(X_{0}), where J(X_{0}) is made up of two copies of a C^{∗}-extension J_{0} of the Jacobian J(D). The extension data that characterizes J_{0} is given by (L · D) − (M · D), where L and M are the two rulings on Q. Now one makes J proper over the t-line A^{1} by pasting in two copies of J(D) in the fiber over t = 0. Denote by J̄ this compacified Neron model.

Following the arguments of H.Clemens [C-1],[C-2], we consider the family H_{t} of smooth elliptic quintics on X_{t}. Every smooth elliptic quintic that lies in X_{0} must lie in one of the two copies P′ and P′′ of P^{3}. Since curves that are not locally complete intersection of X_{0} with a surface (inside the total space of the family {X_{t}}) cannot deform, the image in P^{3} of any curve from Z_{0} must meet D in 10 points. Next, any continuous choice of linear equivalence class {C_{t}} in the ”carrier” K3-surface S(C_{t}) (see notations of lemma 1.2) gives a (possibly multivalued) normal function ν(t) near t = 0 with values in J(X_{t}).

We are going to reprove theorem 1.7 and at the same time obtain some additional piece of information about the fibers of the Abel-Jacobi map Φ_{H_{t}}. For this, we follow the divisor Φ_{H_{t}}(H_{t}) as t approaches zero, and we show that there is a divizor D in J̄ such that

(i) D gives Φ_{H_{t}}(H_{t}) for t ≠ 0,

(ii) D cuts the two components of the compactification of J(X_{0}) in divisors Σ′ and Σ′′ respectively, such that exactly one of the two, say Σ′, is reduced and projects birationally onto J(D), while the other projects into a proper subvariety of J(D). (Here inside the fiber J(X_{t}) we have a real-dimensional two-torus, which represents the homology class given by the Pontryagin product of of the vanishing cycle and a transverse one-cycle and degenerates to a curve made up of the closure of the union of a C^{∗}-fiber from each component of J(X_{0}). The condition (ii) ensures that the intersection number of D with this torus is exactly one.)

Now copying the argument of H.Clemens from [C-2, pp.96-97] (applied to the families H_{t} of elliptic quintics on X_{t}’s instead of the families of sextics of genus 3 treated therein) we see that, to check (i)–(ii), we need to show the following for the family H_{t} = {C_{t}} of elliptic quintics:

(iii) Given a generic divisor Δ of degree 10 on the curve D, there is a unique elliptic quintic C_{0} that meets D at the points of Δ.

We are going to check the condition (iii). For this we need to do some preparation. First introduce some more notation. Let Hilb^{n}P^{3} be the Hilbert scheme of 0-dimensional subschemes of length n of P^{3}, W_{n} its irreducible open subscheme consisting of reduced subschemes (i.e. of n-ples of distinct points in P^{3}) and W̄_{n} its closure in Hilb^{n}P^{3} . Consider the closed subset B = {Δ ∈ W_{10}|there exists a quadric in P^{3} containing Δ} of W_{10}. Clearly,

(a) the closure B̄ of B in W̄_{10} is an irreducible divisor in W̄_{10}, and

(b) B̄ ⊄ Sing W̄_{10}.

Next let Y be the open subset of HilbP^{3} consisting of smooth irreducible elliptic quintics in P^{3}. This is an irreducible 20-dimensional variety (see, e.g., [H-H, Prop. 6.1]). Let Ȳ be the closure of Y in HilbP^{3}. The following result is apparently well known, but we give here its proof because of the lack of an appropriate reference.

Lemma 2.1. Through a general set of points Δ ∈ W_{10} there pass a finite number of quintics from Ȳ.

Proof. Take a reducible elliptic quintic curve C = C_{0} ∪ l consisting of a line l and s a smooth elliptic quartic C_{0}:

C_{0} = Q_{1} ∩ Q_{2},  (17)

where Q_{1}, Q_{2} are smooth quadrics in P^{3} and where l and C_{0} intersect transversally at a point, say, x_{0}. Clearly, C ∈ Ȳ; besides, by (32)

N_{C_{0}/P^{3}} ≃ O_{C_{0}}(2) ⊕ O_{C_{0}}(2), N_{l/P^{3}} ≃ O_{l}(1) ⊕ O_{l}(1).  (18)

As it is known (see, e.g., [H-H, Prop.3.1]), the restricted normal bundle N_{C/P^{3}}|C_{0} (respectively, N_{C/P^{3}}|l) is an elementary transform of N_{C_{0}/P^{3}} (respectively, of N_{l/P^{3}}) by means of the skyscraper sheaf k(x_{0}), so that (18) imply:

N_{C/P^{3}}|C_{0} ≃ O_{C_{0}}(2)(x_{0}) ⊕ O_{C_{0}}(2), N_{C/P^{3}}|l ≃ O_{l}(1) ⊕ O_{l}(2).  (19)

Now the result of gluing of these locally free sheaves on components of C into the sheaf N_{C/P^{3}} leads evidently to one of the possibilities:

(I) N_{C/P^{3}} ≃ E_{1} ⊕ E_{2}, where E_{1}|C_{0} = O_{C_{0}}(2)(x_{0}), E_{1}|l ≃ O_{l}(1), E_{2}|l ≃ O_{C_{0}}(2), E_{2}|l ≃ O_{l}(2), or

(II) N_{C/P^{3}} ≃ E_{1} ⊕ E_{2}, where E_{1}|C_{0} = O_{C_{0}}(2)(x_{0}), E_{1}|l ≃ O_{l}(2), E_{2}|l ≃ O_{C_{0}}(2), E_{2}|l ≃ O_{l}(1).

Let Q be the quadric of the pencil spanned by Q_{1} and Q_{2} that touches the line l at the point x_{0}, and let P_{x_{0}}^{2} be the projective plane in P^{3} touching Q at x_{0}. Take a simple effective divisor D′ of degree 8 on C_{0} (i.e. a set of 8 distinct points on C_{0}), respectively, a pair D′′ of distinct points on l such that

O_{C_{0}}(D′) ≠ O_{C_{0}}(2), x_{0} ∉ D′′,  (20)

and consider Δ = D′ ∪ D′′ ∈ W_{10} as a divisor on C. We show now that the case (II) above is impossible. In fact, this case would mean that there exists a section 0 ≠ s ∈ H^{0}(E_{1}(−D)) ⊂ H^{0}(N_{C/P^{3}}(−D)) such that the corresponding infinitesimal deformation C_{s} of the curve C would consist of two components C′_{s} and C′′_{s} corresponding to the restrictions of the section s onto C_{0} and l respectively, where by construction C′_{s} ⊂ Q, C′′_{s} ⊂ P_{x_{0}}^{2}, and in view of (20) x_{0} ∉ C′_{s} ∪ C′′_{s}. Since clearly l ⊄ Q (for, otherwise, C wouldn’t belong to Ȳ), it follows that C′_{s}∩C′′_{s} = ∅, contrary to the fact that C′_{s} ∪ C′′_{s} connected as a deformation of C. Thus the case (II) is excluded. Now in view of (20) an easy computation shows that in case (I) h^{0}(E_{1}(−D)) = h^{0}(E_{2}(−D)) = 0, hence also h^{0}(N_{C/P^{3}}(−D)) = 0. This precisely means that through Δ there passes a finite set of curves from Ȳ (including the curve C itself). This property is open by an easy dimension count, hence the same is true for a general point Δ ∈ W_{10}.

Since Ȳ is irreducible, from this lemma and the above properties (a) and (b) of the variety B follows

Corollary 2.2. Through a general set of points Δ ∈ B there pass a finite number of quintics from Ȳ, and these quintics are smooth.

Now consider in W_{5} × W_{5} the open subset U = {(P, P′) ∈ W_{5} × W_{5}|P ∩ P′ = ∅} and let ϕ : U → W_{10} be the natural map (P, P′) ↦ P ∪P′. Denote B = ϕ^{−1}(B). Since B is irreducible, B is irreducible as well. Let B^{∗} be the open subset of B consisting of points z = (P, P′) such that:

(1) Δ = P ∪P′ lies on a unique smooth quadric Q and the set of curves from Ȳ passing through Δ is finite and consists of smooth elliptic quintics;

(2) take any such elliptic quintic C through Δ and consider the linear series |J_{C,P^{3}}(3)| ≃ P^{4} and its restriction res_{Q}(P^{4}) onto Q. (By construction, res_{Q}(P^{4}) ⊂ |J_{Δ,Q}(3, 3)| ≃ P^{5}; next, for each point x ∈ P consider the linear subseries P_{x}^{2} = {X ∈ res_{Q}(P^{4})|x ∈ Sing X}, take the Grassmanian G = Gr(1, res_{Q}(P^{4})) and let M_{x} = {P^{1} ∈ G|P^{1} ∩ P_{x}^{2} ≠ ∅}; here clearly M_{z} := ∩_{x∈P}M_{x} ≠ ∅; now for any pencil of curves P^{1} ∈ M_{z} on Q let B(P^{1}) be its base-point set

(by construction this set contains Δ as a subscheme); the open condition that we impose on z is:

length_{x}(B(P^{1})) = 2, x ∈ P,  (21)

for a general point P^{1} of any irreducible component of M_{z}. (Here by the length of a 0dimensional subscheme of Q at a given point x we denote the length of its subscheme with a support at x.)

One can quickly find a point z ∈ U satisfying the condition (2) above; respectively, the condition (1) above is also satisfied by the general point of U because of the corollary 2.2. Hence B^{∗} is dense open in B.

Now consider the set T of smooth irreducible rational quartic curves in P^{3}. This is an irreducible 16-dimensional open subset of HilbP^{3} (see, e.g. [H-H, Prop. 6.1]). Let T̄ be its closure in HilbP^{3}. Fix a point z = (P, P′) ∈ B^{∗}, where Δ = P ∪ P′ lies on a smooth quadric Q, and let C_{Δ} = {C_{1}, ..., C_{N}} be the set of all quintics from Ȳ passing through Δ (these quintics all being smooth by the definition of B^{∗}). Next, consider the set T_{P} = {C ∈ T |C ⊃ P} and its open subset T_{P}^{∗} = {C ∈ T_{P}|mult_{x}(C ∩ Q) = 1 for each x ∈ P}, respectively, its closure T̄_{P} in T̄. Since each smooth rational quartic C ∈ T lies on a unique quadric which is smooth, it follows immediately that T̄_{P} is an irreducible 6-dimensional variety and T_{P}^{∗} is dense open in T_{P}.

Consider the two linear series P_{Δ}^{5} = |J_{Δ,Q}(3, 3)|, P_{Δ}^{9} = |J_{Δ,P^{3}}(3)| and the corresponding Grassmanians Gr = Gr(1, P_{Δ}^{5}), Gr′ = Gr(1, P_{Δ}^{9}). The natural rational restriction map res_{Q} : P_{Δ}^{9} −−→ P_{Δ}^{5} : S ↦ S ∩ Q induces the corresponding restriction map res_{Q} : Gr′ −−→ Gr and thus the rational composition map

f : T_{P} ⟶^{ψ} Gr′ −−→^{res_{Q}} Gr : C ↦ P^{1}(C) := |J_{C∪P′,P^{3}}(3)| ↦^{res_{Q}} P^{1}(Q).  (22)

(here ψ is clearly an embedding). Next, for each x ∈ P we consider the linear subseries P_{x}^{3} = {C ∈ P_{Δ}^{5}|x ∈ Sing C}, and let L_{x} = {P^{1} ∈ Gr|P^{1} ∩ P_{x}^{3} ≠ ∅} be the corresponding hyperplane section of Gr. Denote C_{P} = f(T ) ∩ ( ⋂_{x∈P}L_{x}), C_{P}^{∗} = f(T^{∗}) ∩ ( ⋂_{x∈P}L_{x}) and let X_{P} be

the closure in Gr′ of the set f^{−1}(C_{P}). Since res_{Q} : Gr′ −−→ Gr is a linear rational projection, X_{P} is an intersection of 5 hyperplane sections of Gr′, hence it is connected. Besides, since ψ is an embedding, we can identify X_{P} with its preimage in T̄_{P} under ψ. Consider the map g : X_{P} → Ȳ : C ↦ P^{1}(C) ↦ F ∩ F′ ∖ C for any two cubics F, F′ ∈ P^{1}(C). By construction, for a general point C of any irreducible component of X_{P} we have g(C) ∈ C_{Δ}; hence, since X_{P} is connected, g(X_{P}) ⊂ C_{Δ}, i.e. it is a finite set. Again, since X_{P} is connected, g(X_{P}) is a point:

g(X_{P}) = {C′}, C′ ∈ C_{Δ} ⊂ Y.  (23)

Now show that, moreover,

g(X_{P}) = C_{Δ}.  (24)

In fact, take any smooth elliptic quintic C′ ∈ C_{Δ} and, in notations of (21) pick a general point P^{1} of some irreducible component of M_{z} satisfying (21), where z is our point (P, P′) fixed above. Let C = F ∩ F′ ∖ C′. By construction, C′ = g(C), and we are done. Now (23) and (24) imply that C_{Δ} = {C′}. We thus have proved

Proposition 2.3. For a general quadric Q in P^{3} and a general 10-ple of points Δ ⊂ Q there is a unique smooth elliptic quintic curve passing through Δ.

Clearly one can take a general quartic surface W through Δ, so that Δ ⊂ D for D = W ∩ Q; thus the proposition 2.3 is just the condition (iii) we were looking for.

Moreover, from proposition 2.3 we obtain certain information about the Abel-Jacobi map Φ_{H_{t}}. Namely, by construction, for t = 0,

a) the preimage Φ_{H_{0}}^{−1}(Σ′) of the component Σ′ of Φ_{H_{0}}(H_{0}), where Σ′ is birational to J(D), coincides with the closure H̄′ in Ȳ of the set H′ = {C ∈ Y|C intersects D in 10 points} (clearly, H′ is irreducible, as Ȳ and D are irreducible);

b) the composition map Φ_{0} : H′ ⟶^{Φ_{H_{0}}} Σ′ ⟶^{bir} J(D) coincides with the map

C ↦ cl(C ∩ Q − Δ_{0}),  (25)

where Δ_{0} is some fixed divisor of degree 10 on D. Now, by proposition 2.3, for a general 10-ple of points Δ on C through a general divisor of the linear series |Δ| there passes a unique elliptic quintic from H′, and |Δ| ≃ P^{1}. Thus it follows quickly by dimension reasons (since H′ is irreducible) that through any divisor of the linear series |Δ| there passes a unique quintic from H′. By (25), this precisely means that, for a general point y = cl(Δ − Δ_{0}) ∈ J(D), the fiber Φ^{−1}(y) equals |Δ| ≃ P^{1}. Thus the fiber of the Abel-Jacobi map Φ_{H_{0}} over a general point of Σ′ equals P^{1}. One can expect that the similar property of the Abel-Jacobi map is true also for t ≠ 0. In other words, we may formulate the following

Conjecture 2.4. In conditions and notations of theorem 1.7 the general fiber of the Abel-Jacobi map Φ_{H_{0}} : H_{0} → Θ + const is isomorphic to P^{1}.

3. The component of the moduli space M(2; 0, 3) of rank-2 stable vector bundles on X associated to the family of elliptic quintics on X

Consider the irreducible component H_{0} of Hilb(X) from theorem 1.7 and, using notations of subsection 1.1, let

• H_{0}^{∗} := {C ∈ H^{∗} ∩ H_{0}| h^{0}(J_{C,X}(2)) = 1 and π : C → π(C) is an isomorphism}.

According to lemma 1.2, H_{0}^{∗} is a dense open subset of H_{0}. Now take any curve C ∈ H_{0}^{∗} and consider the so called Serre construction defining the O_{X}-sheaf E_{C} as an extension of O_{X}-sheaves

0 → O_{X}(−1) → E_{C} → J_{C,X}(1) → 0.  (26)

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 (here we use the well known equalities h^{i}O_{X}(−2) = 0, i < 3, and the equality ω_{C} ≃ O_{C} for the smooth elliptic curve C). Thus E_{C} is defined uniquely up to an isomorphism by the curve C. Besides, since for the elliptic curve C a nonzero section of ω_{C} vanishes nowhere on C, it follows that E_{C} is locally free (this is the well known property of the Serre construction). Now from (26) one quickly computes the Chern classes of the vector bundle E_{C}:

c_{1}(E_{C}) = 0, c_{2}(E_{C}) = 3  (27)

(here we understand, as usual, the second Chern class c_{2} of a given O_{X}-shef as an element of the group B_{1} of classes of algebraic equivalence of 1-dimensional cycles on X, where B_{1} = ℤ · l ≃ ℤ and l is the class of a line on X). Next, since π : C → π(C) is an isomorphism onto a quintic π(C) not lying in a plane and, as it is easy to see, H^{0}(O_{X}(1)) = π^{∗}H^{0}(O_{P^{3}}(1)), it follows quickly that H^{0}(J_{C,X}(1)) ≃ H^{0}(J_{π(C),P^{3}}(1)) = 0. Hence the triple (26) implies that h^{0}(E_{C}) = 0. This clearly means that E_{C} is stable in the sense of Gieseker (cf. [MT-1, Prop. 2.6]). Thus, denoting

• M(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 see from (27) that

[E_{C}] ∈ M(2; 0, 3)  (28)

(here we use the standard notation [E] for the isomorphism class of a given O_{X}-sheaf 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 (26) also

h^{i}(E_{C}) = 0, i > 0.  (29)

Now let M̃ = {[E] ∈ M(2; 0, 3)| h^{i}(E) = 0, i > 0}. By semicontinuity, this is clearly an open subset of M(2; 0, 3) containing a point E_{C} in view of (29). Let M be an irreducible component of M̃ containing this point E_{C}. We then have a natural map

f : H_{0}^{∗} → M : C′ ↦ E_{C′}.  (30)

It is a standard matter to show that f is a dominating morphism (cf. [MT-1, lemmas 5.2 and 5.3]), with the fibre f^{−1}(f(C′)) open in P (H^{0}(E_{C′}(1)) ≃ P^{1}:

\overline{f^{−1}(f(C′))} = P^{1}, C′ ∈ H_{0}^{∗}.  (31)

(Here the closure is taken in H_{0}; besides, we keep in mind that, by Riemann-Roch, h^{0}(E_{C′}(1)) = 2, since by the property of M, h^{i}(E_{C′}(1)) = 0, i > 0.) Remark that 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 the Serre construction, the fibre \overline{f^{−1}(f(C′))} = P^{1} is just the pencil of elliptic curves on S(C′), these curves being the zero loci of sections from H^{0}(E_{C′}(1)) :

\overline{f^{−1}(f(C′))} = |O_{S(C′)}(C′)| = P^{1}.  (32)

Now consider the Stein decomposition of the Abel-Jacobi map Φ = Φ_{H_{0}}|H_{0}^{∗} : H_{0}^{∗} → Θ + const. Using the above description (32) of fibres of f and the well-known interpretation of the differential of the Abel-Jacobi map Φ in terms of the branch quartic W of the double covering π : X → P^{3} (see, e.g., [T-1, corollary 1] or [We, Prop.2.13]), we can apply the argument similar to that used in the proof of Theorem 5.6 of [MT-1] and obtain the following

Theorem 3.1. There exists a quasi-finite dominating morphism of (an open part of) the component M of the moduli space M(2; 0, 3) to (a translate of) the theta divisor:

g : M → Θ + const  (33)

which gives the Stein decomposition Φ = g ∘ f of the Abel-Jacobi map Φ of H_{0}^{∗}:

\begin{aligned}H_{0}^{∗} & ⟶^{Φ} & Θ + const \\ f ↘ & & ↗ g \\ & M & , \end{aligned}  (34)

where the the fibers of f : H_{0}^{∗} → M are open subsets in P^{1}.

Now theorem 3.1 and conjecture 2.4 give a hint to the following

Conjecture 3.2. The morphism g : M → Θ + const in (34) is birational.

References

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89–101.

[D] Druel, S.: Espace des modules des faisceaux semi-stables de rang 2 et de classes de Chern c_{1} = 0, c_{2} = 2 et c_{3} = 0 sur une hypersurface lisse de P^{4}. E-print Math. AG/0002058 (2000).

[H-H] Hartshorne, R.; Hirschowitz, A.: Smoothing algebraic space curves, In: Algebraic geometry, Sitges

(Barcelona), 1983, Lecture Notes in Math., 1124. Springer, Berlin-New York, 1985, 98–131.

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[MT-1] Markushevich, D., Tikhomirov, A.S.: The Abel-Jacobi map of a moduli component of vector bundles

on the cubic threefold. J. of Alg. Geometry 10 (2001), 37-62. (E-print Math. AG/9809140 (1998).)

[MT-2] Markushevich, D., Tikhomirov, A.S.: Symplectic structure on a moduli space of sheaves on the

cubic fourfold. Preprint MPIM 99-114 (1999), Bonn (E-print Math.AG/9910063), 1-21.

[T-1] Tikhomirov, A.S.: The Abel-Jacobi map of sextics of genus 3 on double spaces of P^{3} of index two.

Soviet Math. Dokl. 33 (1986), 204-206.

[T-2] Tikhomirov, A.S.: The geometry of the Fano surface of the double cover of P^{3} branched in a

quartic, Math. USSR, Izv. 16 (1981), 373–397.

[We] Welters, G.E.: Abel–Jacobi isogenies for certain types of Fano threefolds. Mathematical Centre Tracts 141, Amsterdam, 1981.

A. T.: Department of Mathematics, State Pedagogical University of Yaroslavl, Respublikanskaya 108, Yaroslavl 150000, Russia

E-mail address: tikhomir@yaroslavl.ru