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.
Kohsuke Shibata (2026) studied this question.