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

This study examines 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 $\mathfrak{C}$, which is a root stack over the smooth curve $C=\text{Proj}R$, such that $R$ is the section ring associated with a line bundle $L$ on $\mathfrak{C}$. The FFRT property of $R$ is then rephrased with respect to the Frobenius push-forwards $F_{\ast }^e(L^i)$ on the orbifold curve $\mathfrak{C}$. 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 $\mathfrak{C}$.