A note on the cone conjecture for K3 surfaces in positive characteristic

TL;DR

This paper proves that for K3 surfaces in characteristic p > 2, the automorphism group acts with a rational polyhedral fundamental domain on the nef cone and has finitely many orbits on nodal classes, leading to finiteness results for linear systems.

Contribution

It establishes the cone conjecture for K3 surfaces in positive characteristic p > 2, showing automorphism group actions have rational polyhedral fundamental domains.

Findings

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

Finitely many orbits of nodal classes under automorphism group.

Finiteness of linear systems of fixed genus up to automorphisms.

Abstract

We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group.