The moduli space of cubic fourfolds via the period map

TL;DR

This paper describes the structure of the moduli space of cubic fourfolds using the period map, identifying its GIT compactification with a specific Looijenga compactification, and extends previous work on this topic.

Contribution

It characterizes the image of the period map for cubic fourfolds with simple singularities and links the GIT compactification to a Looijenga compactification.

Findings

The image of the period map is the complement of a hyperplane arrangement.

The GIT compactification is isomorphic to Looijenga's compactification.

The work extends previous results on the moduli space of cubic fourfolds.

Abstract

We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga's compactification associated to this arrangement. This work builds on and is a natural continuation of our previous paper on the GIT compactification of the moduli space of cubic fourfolds.