The class of the locus of intermediate Jacobians of cubic threefolds

TL;DR

This paper computes the classes of loci of intermediate Jacobians of cubic threefolds and related abelian varieties within moduli spaces, assuming expected codimension, and provides explicit formulas for genera up to 5.

Contribution

It introduces a method to compute the classes of these loci in Chow, homology, and tautological rings, extending to compactifications and specific cases up to genus 5.

Findings

Computed classes of loci in moduli spaces for genus up to 5.

Explicit formulas for loci of products of elliptic curves and hyperelliptic curves.

Analyzed intersections of boundary divisors in toroidal compactifications.

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.