The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds
Sebastian Casalaina-Martin, Zheng Zhang
备注：
33 pages, AMS LaTeX, final version, minor update of the published version: fixed several typos, added reference to Saccà for Proposition 3.10, clarified statement of Corollary 5.17
We study the moduli space of pairs consisting of a smooth cubic surface and a smooth hyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second named author. The construction associates to such a pair a so-called Eckardt cubic threefold, admitting an involution, and the period map sends the pair to the anti-invariant part of the intermediate Jacobian of this cubic threefold, with respect to this involution. Our main result is that the global Torelli theorem holds for this period map; i.e., the period map is injective. To prove the result, we describe the anti-invariant part of the intermediate Jacobian as a Prym variety of a branched cover. Our proof uses results of Naranjo-Ortega, Bardelli-Ciliberto-Verra, and Nagaraj-Ramanan, on related Prym maps. In fact, we are able to recover the degree of one of these Prym maps by describing positive dimensional fibers, in the same spirit as a result of Donagi-Smith on the degree of the Prym map for connected étale double covers of genus 6 curves.
Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a Jacobian over k'. The aim of this note is to give a geometric construction of this extension.
Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrising lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper we show that if X is generic, the Fano surface is the only surface of minimal class in J.
We define, basing upon semiorthogonal decompositions of $\Db(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.
The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to…
Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.
A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.
A new proof of the non-rationality of a generic cubic threefold is given as follows: If a generic cubic threefold were rational then the associated intermediate Jacobian would be a product of Jacobians of curves. We degenerate a generic cubic threefold to the Segre Cubic Threefold and so there is a degeneration of intermediate Jacobians as well. Associated to the degenerating family of Pryms is a unimodular system of vectors. Rationality of the generic cubic threefold would imply that the unimodular system would be cographic dicing. However, we show that the unimodular system obtained is a wel…
It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6 x 6 skew-symmetric matrix of linear forms in P^4. To compactify the space of such Pfaffian representations of a given cubic and to study the construction in families as well as for singular or reducible cubics, it is thus natural to consider the incidence correspondence of Pfaffian representations inside the product of the space of semistable skew-symmetric 6 x 6 matrices of linear forms in P^4 and the space of cubics. Here we describe concretely the irreducible component of this incidence correspondenc…
An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of $\mathbb F_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over $\mathbb F_q$ which attains Perret's and Weil's upper boun…
We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressi…
The Abel-Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic fiber is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich-Tikhomirov, who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers $c_1=0, c_2=2$ obtained by Serre's construction from elliptic quintics, and…
In this paper we study principally polarized complex abelian varieties that admit an automorphism of order 3. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves. As an application, we get another proof of the already known (thanks to Clemens and Griffiths) fact that intermediate jacobians of certain cubic threefolds are not jacobians of curves.
We exhibit the isogeny classes of supersingular abelian threefolds over F_{2^n} containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curve over F_{2^n}. All the curves can be obtained explicitly.
This paper was written in 1982. Ideas and methods of "Clemens C.H., Griffiths Ph. The intermediate Jacobian of a cubic threefold" are applied to a Fano threefold X of genus 6 -- intersection of Grassmann sixfold with two hyperplanes and a quadric. We prove: 1. The Fano surface F(X) of X is smooth and irreducible. Hodge numbers and some other invariants of F(X) are calculated. 2. Tangent bundle theorem for X, and its geometric interpretation. It is shown that F(X) defines X uniquely. 3. The Abel - Jacobi map from the Albanese of F(X) to the middle Jacobian of X is an isogeny.
We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstraß point defined over the same field. We also construct homogeneous quartic polynomials on the Kummer variety and show that they represent the duplication map.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
This work makes a parallel construction for curves on threefolds to a ``current-theoretic'' proof of Abel's theorem giving the rational equivalence of divisors P and Q on a Riemann surface when Q - P is (equivalent to) zero in the Jacobian variety of the Riemann surface. The parallel construction is made for homologous ''sub-canonical'' curves P and Q on a general class of threefolds. If P and Q are algebraically equivalent and Q - P is zero in the (intermediate) Jacobian of a threefold, the construction ''almost'' gives rational equivalence.
Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that the intermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an elliptic curve $E$ and the self-product of an abelian surface $B$ with real multiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of some algebraic curves related to the construction of $J(X)$ as a Prym variety. This…
We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K_X. We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in P5, relati…
Let $X$ be a smooth cubic threefold and $J(X)$ be its intermediate Jacobian. We show that there exists a codimension 2 cycle $Z$ on $J(X)\times X$ with $Z_{t}$ homologically trivial for each $t\in J(X)$, such that the morphism $ϕ_{Z}: J(X)\rightarrow J(X)$ induced by the Abel-Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.
In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold $X$. We use the standard realization of $X$ as a conic bundle and a $4-$dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle structure. The additional data of an even theta characteristic on the curves in the family gives us a model for the theta divisor.
Let X be a smooth cubic threefold, M the moduli space of stable rank 2 vector bundles on X with trivial determinant and c_2=2 (the smallest value for which this space is non-empty). Recent results of Druel, Iliev, Markushevich and Tikhomirov lead to a precise description of M and its Maruyama compactification M': the latter is isomorphic to the intermediate Jacobian of X blown-up along the Fano surface. In this survey paper we explain these results and discuss some applications.
We show that the only summands of theta divisors on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the obvious powers of the curve and the Fano surface of lines on the threefold. The proof uses the decomposition theorem for perverse sheaves, a bit of representation theory and a computation of characteristic cycles for Brill-Noether sheaves.
Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form χ_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one ca…
The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.
We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
We show that on the Jacobian $(JC,θ)$ of a smooth curve $C$ of genus $g$, any effective cycle in $JC$ with cohomology class $θ^d/d!$ is a translate of $W_{g-d}(C)$ or $-W_{g-d}(C)$. We then use this result to prove that for $1<d<g$, the Jacobian locus (\resp the locus of intermediate Jacobians of cubic threefolds) is an irreducible component of the set of principally polarized abelian varieties of dimension $g$ for which $θ^d/d!$ (\resp $θ^3/3!$) is the class of an effective algebraic cycle. Moreover, on the intermediate Jacobian of a generic cubic threefold, $θ^2/2!$ is not the class of an ef…
This paper proves the following converse to a theorem of Mumford: Let $A$ be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then $A$ is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of $A$, and eventually to show that $A$ is the Prym variety of a possibly singular plane quintic.
Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$ which are preimages of lines bitangent to $Y$, gives an isogeny between the Albanese variety of $\tilde F(X)$ and the intermediate Jacobian of $X$.
Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.
For a general K3 surface of genus g = 2,3,...,10, we prove that the intermediate jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable system.
This paper extends joint work with R. Friedman to show that the closure of the locus of intermediate Jacobians of smooth cubic threefolds, in the moduli space of principally polarized abelian varieties (ppav's) of dimension five, is an irreducible component of the locus of ppav's whose theta divisor has a point of multiplicity three or more. This paper also gives a sharp bound on the multiplicity of a point on the theta divisor of an irreducible ppav of dimension less than or equal to five; for dimensions four and five, this improves the bound due to J. Kollár, R. Smith-R. Varley, and L. Ein-R…
The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibil…
Let $Σ$ be a smooth projective complex curve and $\mathfrak{g}$ a simple Lie algebra of type ${\sf ADE}$ with associated adjoint group $G$. For a fixed pair $(Σ, \mathfrak{g})$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(Σ,\mathfrak{g})$. Our main result establishes an isomorphism between the Calabi-Yau integrable system, whose fibers are the intermediate Jacobians of this family of Calabi-Yau threefolds, and the Hitchin system for $G$, whose fibers are Prym varieties of the corresponding spectral c…
We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…
The paper is joined with arXiv:0911.5428 and improved. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactificatio…
Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism t of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of X to the product of the intermediate algebraic Jacobians of Y. When the orthogonal complement of T in D^b(X) has a trivial Jacobian (e.g. wh…
For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym varieties. Applications of this theorem to cubic threefolds, and Prym varieties of dimension five, are also considered.
In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds. The base of this system is a moduli space of gauged Calabi-Yaus in the family, and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the threefolds. This system has several interesting properties: the multivalued sections obtained as Abel-Jacobi images, or ``normal functions'', of a family of curves on the generic variety of the family, are always Lagrangian; the natural affine coordinates on the base, which are used in…
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve G, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of G. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c_1=-K_X and deg c_2=6 and that the latter is birational to the singular locus of the theta divisor of J(X).
