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A Frobenius characterization of rational singularity in 2-dimensional graded rings

A ring R is said to be F-rational if, for every prime P in R, the local ring R P has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if R is a 2-dimensional graded ring with an isolated singularity at the irrelevant maximal ideal m, then the following are equivalent: (1) R has a rational singularity at m. (2) R is F-rational. (3) a(R) < 0. Here a(R) (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module H m (R). The proof of this result relies heavily on the properties of derivations of R, and suggests further questions in that direction; paradigmatically, if one knows that D(a) satisfies a certain property for every derivation D, what can one conclude about the original ring element a?