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.

• 14C20 - Divisors, linear systems, invertible sheaves
• 14M25 - Toric varieties, Newton polyhedra, Okounkov bodies
• 52B20 - Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

• 1 zbMATH Open