Combinatorial characterization of curves governing finite generation

Characterize how the existence of the curves on the blowup of the toric variety X_P at the identity point e=(1,\dots,1), whose existence determines Cutkosky's finite-generation criterion for the section ring R_P, depends on the combinatorics of a lattice triangle P.

Background

The paper identifies finite generation of the harmonic algebra H_P with finite generation of the section ring R_P associated with the blowup \operatorname{Bl}_e X_P of the toric variety X_P at the identity point of its dense torus.

For lattice triangles, Cutkosky's geometric criterion for finite generation depends on the existence of certain curves on this blowup. The paper explains that examples of both finite and non-finite generation are known, but the relationship between the relevant curves and the combinatorial structure of the triangle is not understood. This leaves a concrete unresolved question about translating the geometric criterion into polyhedral or lattice-combinatorial terms.