Remarks on the multiplicity of normal surface star-shaped singularities

When we consider a star-shaped surface singularity (W, w) as filtered ring with the associated graded ring G = R(E,D), we study the problem what conditions on R = R(E,D) needs to satisfy in order for the equality e(m, OW,w) = e(R+, R) holds. Here, we will give two types of judgment methods. The first is Theorem 2, where the statement is given by the conditions of maximal ideal cycle M about minimal antinef property and about freeness of linear system by M. In the second method, we use author’s previous multiplicity theory studied as a filtered ring to a star-shaped singularity. The condition in Theorem 3 is given as the comparison between deg x1 deg x2 deg D and e(R+, R).