Complete moduli of cubic threefolds and their intermediate Jacobians

TL;DR

This paper extends the intermediate Jacobian map for cubic threefolds to a modular compactification, enabling detailed study of degenerations and geometric properties of these threefolds.

Contribution

It constructs a morphism from the wonderful compactification of cubic threefolds to the second Voronoi compactification, refining the understanding of degenerations of intermediate Jacobians.

Findings

Extended the Jacobian map to a modular compactification.

Classified degenerations of Jacobians of torus rank 1 and 2.

Provided a framework for studying cubic threefold degenerations.

Abstract

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 the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.