On the finite generation of valuation semigroups on toric surfaces

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.

22 pages, 13 figures

References in corpus (2)

• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces