# Combinatorics of rational singularities

### Lê Dũng Tráng

Université de Provence, Marseille, France### Meral Tosun

U.N.A.M., Cuernavaca, Mexico

## Abstract

A normal surface singularity is rational if and only if the dual intersection graph of a desingularization satisfies some combinatorial properties. In fact, the graphs defined in this way are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency greater than 3 in the dual intersection tree of the minimal desingularization of a rational singularity of multiplicity m greater than 3 is at most m - 2.

## Cite this article

Lê Dũng Tráng, Meral Tosun, Combinatorics of rational singularities. Comment. Math. Helv. 79 (2004), no. 3, pp. 582–604

DOI 10.1007/S00014-004-0808-Y