The notions of F-regularity and F-purity for rings of characteristic p> 0, which are introduced by Hochster and Huneke [HH1] and Hochster and Roberts [HR], respectively, are now studied not only ring-theoretically but also via singularity theory. In [MS], Mehta and Srinivas studied twodimensional normal F-pure singularities and proved that such a singularity is either (a) a simple elliptic singularity with an ordinary elliptic exceptional curve,(b) a cusp singularity, or (c) a rational singularity. Moreover, they showed that a singularity of type (a) or (b) is F-pure (see also [W1] for the Gorenstein case). But being a rational singularity is not sufficient to be F-pure.

The aim of this paper is to study two-dimensional F-regular and rational F-pure singularities and to complete the classification of these singularities in terms of the dual graph of the minimal resolution and characteristic p. To do this we heavily use the result of Kei-ichi Watanabe [W3], which tells us that F-regular (resp. F-pure) normal surface singularities are log terminal (resp. log canonical). Then we can apply the classification of the dual graphs of two-dimensional log terminal and log canonical singularities [Wk], and a refinement of the method in [MS] enables us to reduce our problem to the case of graded rings [W2]. Let (A, m) be a two-dimensional Noetherian normal local ring containing an algebraically closed field k of characteristic p> 0 such that AĀm= k. Let?: X ÄY= Spec (A) be the minimal resolution of the singularity.(Note that a surface singularity has a resolution even in positive characteristic [Li2].) We call the dual graph of the exceptional divisor E=? &1 (m) simply the graph of the singularity of Y. If the graph is star-shaped with r branches, then we can associate to a branch of length la natural number d such that (&1) l d is the determinant of the intersection matrix of the