RELATIONS BETWEEN MULTIPLICITY AND DIVISOR CLASS GROUP FOR RATIONAL SURFACE SINGULARITIES

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 E(8) singularity [3]. We also prove several inequalities involving the integers e, delta, m(i), C(i)(2), where Z = Sigma(i)m(i)C(i) 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.