The FFRT property of two-dimensional normal graded rings and orbifold curves

We study the finite F-representation type (abbr. FFRT) property of a two-dimensional normal graded ring R in characteristic p>0, using notions from the theory of algebraic stacks. Given a graded ring R, we consider an orbifold curve C, which is a root stack over the smooth curve C=ProjR, such that R is the section ring associated with a line bundle L on C. The FFRT property of R is then rephrased with respect to the Frobenius push-forwards F⁎e(Li) on the orbifold curve C. As a result, we see that if the singularity of R is not log terminal, then R has FFRT only in exceptional cases where the characteristic p divides a weight of C.