Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves

TL;DR

This paper provides examples over various fields where rings of invariants are not finitely generated, linking this failure to elliptic fibrations, and proposes a generalization of the Morrison-Kawamata cone conjecture, proving it for certain surfaces.

Contribution

It constructs new examples of non-finitely generated invariant rings over arbitrary fields and extends the cone conjecture to klt Calabi-Yau pairs, with a proof in dimension 2.

Findings

Examples of non-finitely generated invariant rings over arbitrary fields.

Failure of finite generation linked to elliptic and abelian surface fibrations.

Proof of the cone conjecture for minimal rational elliptic surfaces in dimension 2.

Abstract

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces.