The moduli of smooth hypersurfaces with level structure

TL;DR

This paper constructs the moduli space of smooth hypersurfaces with level structure over certain rings, demonstrating uniformizability and establishing a Torelli theorem for cubic threefolds, with implications for automorphism actions.

Contribution

It introduces a new construction of the moduli space with level structure and proves its uniformizability, along with a global Torelli theorem for cubic threefolds in odd characteristic.

Findings

Automorphisms act faithfully on cohomology of hypersurfaces.

The moduli stack is uniformisable by a smooth affine scheme for large N.

A global Torelli theorem is established for cubic threefolds over fields of odd characteristic.

Abstract

We construct the moduli space of smooth hypersurfaces with level $N$ structure over $\mathbb{Z}[1/N]$. As an application we show that, for $N$ large enough, the stack of smooth hypersurfaces over $\mathbb{Z}[1/N]$ is uniformisable by a smooth affine scheme. To prove our results, we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic threefolds over fields of odd characteristic.