On automorphisms and the cone conjecture for Enriques surfaces in odd characteristic

TL;DR

This paper proves that automorphism groups of Enriques surfaces in odd characteristic are finitely generated and act with a rational polyhedral fundamental domain, and constructs a related surface with a non-finitely generated automorphism group.

Contribution

It establishes the finite generation of automorphism groups for Enriques surfaces in odd characteristic and explores their action on the nef cone, also providing a counterexample with a non-finitely generated automorphism group.

Findings

Automorphism group of Enriques surfaces in odd characteristic is finitely generated.

Automorphism group acts on the nef cone with a rational polyhedral fundamental domain.

Existence of a surface birational to an Enriques surface with a non-finitely generated automorphism group.

Abstract

We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated.