Prym varieties and cubic threefolds over $\mathbb{Z}$

TL;DR

This paper extends the theory of Prym varieties and cubic threefolds to characteristic 2 fields, proving non-rationality and Torelli theorems over arbitrary fields, and resolving Deligne's conjecture on arithmetic Torelli maps.

Contribution

It develops a new theory of Prym varieties and cubic threefolds in characteristic 2, proving non-rationality and Torelli theorems over any field, and confirms Deligne's conjecture.

Findings

Proved smooth cubic threefolds are non-rational over any field.

Established Torelli theorem for cubic threefolds over arbitrary fields.

Solved Deligne's conjecture on arithmetic Torelli maps.

Abstract

We develop a theory of Prym varieties and cubic threefolds over fields of characteristic $2$. As an application, we prove that smooth cubic threefolds are non-rational over an arbitrary field and solve a conjecture of Deligne regarding arithmetic Torelli maps. We also prove the Torelli theorem for cubic threefolds over arbitrary fields.