An upper bound of geometric genus of normal two-dimensional star-shaped singularities

For normal two-dimensional star-shaped singularity, we show an upper bound of the geometric genus. The bound is given by pa(Z0) times some numerical bound defined related to a-invariant of the normalized of associated graded ring of the centeral filtration of star-shaped singularity (Theorem 1). Further we give a sufficient condition for a numerical Gorenstein star-shaped singularity to be Gorenstein by attaining sufficiently large value with respect to our bound (Theorem 3). We also study a criterion for a normal graded ring R = R(E,D) to be numerical Q-Gorenstein in terms of Pinkham-Demazure’s data E,D (Theorem 2).