The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$

TL;DR

This paper investigates the second fundamental form of the Siegel metric on the moduli space of cubic threefolds, revealing its image lies in the kernel of a specific multiplication map using advanced geometric tools.

Contribution

It establishes a new relation between the second fundamental form and a multiplication map in the context of cubic threefolds and Prym varieties.

Findings

The second fundamental form's image is contained in the kernel of a multiplication map.

Utilizes conic bundle structures, Prym theory, and Gaussian maps to analyze the geometry.

Provides new insights into the geometry of the moduli space of cubic threefolds.

Abstract

We study the second fundamental form of the Siegel metric in $\mathcal A_5$ restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals.