The class of the locus of intermediate Jacobians of cubic threefolds

Abstract

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 expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat

Stable cohomology of the perfect cone toroidal compactification of 𝒜 g

“…The second construction is by going to a level cover A Perf (2) (such as the simplicial locus) which descend to A Perf g . To avoid unnecessary multiplicities in our notation we normalize the pushforward by dividing by the order of the deck group, as was also done in [GH12,Section 4]. This construction provides us with well defined cohomology classes on the simplicial locus A Perf g,simp .…”

“…This construction provides us with well defined cohomology classes on the simplicial locus A Perf g,simp . It was used in [GH12], where especially in Sections 8 and 9 similar constructions were performed, and we freely use the notation and results from there. We recall that the intersection of two different boundary divisors D m 1 ∩ D m 2 ⊂ A Perf g (2) is non-empty if and only if m 1 and m 2 span an isotropic subspace, i.e.…”

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić)

“…With the work in [ABH02], translating from our results to the language of stable semiabelic pairs is straightforward ( §2.4, §9). In addition, one of our original motivations for this work was investigating the extension of the period map for cubic threefolds to a morphism from a suitable GIT compactification of the moduli space of threefolds to a suitable compactification of A 5 , stemming from our work [CML09] and [CML13], and using some of the results of our work [GH12]. The methods we use in this paper apply in that setting also, and we will return to the study of the period map for cubic threefolds in subsequent work.…”

The locus of plane quartics with a hyperflex

“…The preimage p −1 D is reducible, and its irreducible components are indexed by non-zero characteristics: p −1 D = ∪ n∈(Z/2Z) 6 −0 D n . This enumeration of the components of ∆ 0 is also discussed in [GH12].…”

“…Extension to the Boundary. Extension of theta constants and theta gradients to the boundary component D n is done in [GH12]. The vanishing orders are computed using the Fourier-Jacobi expansion of the theta function (this expansion is convenient for the computation that we will later do on ∆ 1 ):…”