The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

TL;DR

This paper explores how degenerations of intermediate Jacobians relate to the compactification of the moduli space of cubic threefolds, connecting classical results with modern compactification techniques.

Contribution

It provides a detailed description of the compactification of the moduli space of cubic threefolds via degenerations of their intermediate Jacobians and compares it with existing compactifications.

Findings

Description of degenerations of intermediate Jacobians

Relation between different compactifications of moduli space

Insights into the structure of the moduli space of cubic threefolds

Abstract

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.