$F$-rationality of two-dimensional graded rings with a rational singularity

TL;DR

This paper explores conditions under which two-dimensional graded rings with rational singularities are also F-rational, focusing on the role of multiplicity and identifying cases with small multiplicity.

Contribution

It provides criteria linking F-rationality to multiplicity in two-dimensional graded rings with rational singularities, including specific conditions for small multiplicity cases.

Findings

F-rationality depends on multiplicity in these rings

Characterization of F-rationality for small multiplicity cases

Clarifies the relationship between rational singularities and F-rationality

Abstract

It is known that a two-dimensional $F$-rational ring has a rational singularity. However a two-dimensional ring with a rational singularity is not $F$-rational in general. In this paper, we investigate $F$-rationality of a two-dimensional graded ring with a rational singularity in terms of the multiplicity. Moreover, we determine when a two-dimensional graded ring with a rational singularity and a small multiplicity is $F$-rational.