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

## TL;DR

This paper investigates the FFRT property of two-dimensional normal graded rings in positive characteristic by linking it to orbifold curves and Frobenius push-forwards, revealing conditions under which FFRT occurs.

## Contribution

It introduces a novel approach connecting FFRT of graded rings to orbifold curves and Frobenius push-forwards, providing new criteria for FFRT in terms of singularity types and characteristic divisibility.

## Key findings

- FFRT relates to the singularity being log terminal.
- FFRT occurs only when the characteristic divides a weight of the orbifold curve.
- The approach uses algebraic stacks and Frobenius push-forwards to analyze FFRT.

## Abstract

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 $F^e_*(L^i)$ 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 $\mathfrak C$.