RELATIONS BETWEEN MULTIPLICITY AND DIVISOR CLASS GROUP FOR RATIONAL SURFACE SINGULARITIES

Abstract

In this paper we prove that a rational surface singularity with divisor class group ℤ/(2) is a rational double point. This generalizes a result by Brieskorn: if the divisor class group of a rational singularity is trivial then it is the E8 singularity [3]. We also prove several inequalities involving the integers e, δ, mi, [Formula: see text], where [Formula: see text] is the fundamental cycle. The proof of this result uses ideas from Minkowski's theory of reduction of positive-definite quadratic forms. We also give some interesting counterexamples to some of the related questions in this context.