The moduli space of cubic threefolds with a non-Eckardt type involution via intermediate Jacobians

TL;DR

This paper investigates cubic threefolds with a specific type of involution, establishing a Torelli-type theorem by linking their intermediate Jacobians to Prym varieties and analyzing the period map.

Contribution

It proves the global Torelli Theorem for cubic threefolds with non-Eckardt involutions using Prym varieties and the bigonal construction, extending previous work on involutions.

Findings

The period map is injective for these threefolds.

Invariant intermediate Jacobians can be described as Prym varieties.

The two descriptions of the Jacobian are related by the bigonal construction.

Abstract

There are two types of involutions on a cubic threefold: the Eckardt type (which has been studied by the first named and the third named authors) and the non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type involution, whose fixed locus consists of a line and a cubic curve. Specifically, we consider the period map sending a cubic threefold with a non-Eckardt type involution to the invariant part of the intermediate Jacobian. The main result is that the global Torelli Theorem holds for the period map. To prove the theorem, we project the cubic threefold from the pointwise fixed line and exhibit the invariant part of the intermediate Jacobian as a Prym variety of a (pseudo-)double cover of stable curves. The proof relies on a result of Ikeda and Naranjo-Ortega on the injectivity of the related Prym map. We also describe the invariant part of the intermediate Jacobian via the projection from a general invariant line and show that the two descriptions are related by the bigonal construction.