Jet scheme and simple isolated hypersurface singularities

Abstract

Let m >= 2 be a positive integer. Let (H(f), o) be the analytic germ of hypersurface attached to a weighted-homogeneous polynomial f is an element of C[x(1), . . . , x(m)]. In this article, we mainly characterize the simpleness of any isolated analytic quasi-homogeneous complex hypersurface singularity (H, o) in C-m in terms of the associated jet schemes. We also compute the dimension of the jet scheme L-n(H)(o), formed by the n-jets centered at o, for every positive integer n >> 0, which is the crucial technical part of our simpleness criterion.