F-rationality of two-dimensional graded ringswith a rational singularity

It is known that a two-dimensional [Formula: see text]-rational ring has a rational singularity. However, a two-dimensional ring with a rational singularity is not [Formula: see text]-rational in general. In this paper, we investigate [Formula: see text]-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 [Formula: see text]-rational.