On the finite generation of valuation semigroups on toric surfaces

TL;DR

This paper establishes a combinatorial criterion for when valuation semigroups on smooth toric surfaces are finitely generated and provides a counterexample showing some are not.

Contribution

It introduces a new criterion for finite generation of valuation semigroups and constructs a specific example where these semigroups are not finitely generated.

Findings

A combinatorial criterion for finite generation is provided.

A lattice polytope example shows non-finite generation in certain cases.

Valuation semigroups can be non-finitely generated even on polarized toric varieties.

Abstract

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.