A combinatorial criterion decides finite generation of valuation semigroups on smooth toric surfaces for non-toric maximal-rank valuations, plus a lattice polytope where none from one-parameter subgroups at a non-toric point are finitely generated.
Brunn- M inkowski inequality for multiplicities
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On the finite generation of valuation semigroups on toric surfaces
A combinatorial criterion decides finite generation of valuation semigroups on smooth toric surfaces for non-toric maximal-rank valuations, plus a lattice polytope where none from one-parameter subgroups at a non-toric point are finitely generated.