Automorphisms of cubic surfaces in positive characteristic

TL;DR

This paper classifies automorphism groups of smooth cubic surfaces over algebraically closed fields of any characteristic, explores their moduli space, and examines liftability to characteristic zero.

Contribution

It provides a complete classification of automorphism groups, normal forms, and conditions for lifting cubic surfaces from positive characteristic to zero.

Findings

Automorphism groups are classified for all characteristics.

The moduli space of smooth cubic surfaces is rational in every characteristic.

Explicit normal forms are provided for each automorphism group class.

Abstract

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.