“Weakly” Elliptic Gorenstein Singularities of Surfaces

Let p be a singularity of a normal two–dimensional analytic surface X. In general π : M → X will denote a resolution of (X, p), and A = π(p). It is well–known that the dual resolution graph of the germ (X, p) determines the topology of the germ (X, p). Indeed, the (real) link of (X, p) can be reconstructed from the resolution graph as a plumbing. Conversely, by a result of W. Neumann [15], the topology of (X, p) determines the (minimal) resolution graph. Therefore, if an invariant of (X, p) depends only on the dual resolution graph, we say that it is a “topological invariant”. For example, the self– intersection Z num of Artin’s (fundamental) cycle Znum, or the Euler–characteristic χ(D) of any cycle D supported by A, can be determined from the graph. Now, it is fascinating to investigate if an invariant, a priori defined from the analytic structure of (X, p), is topological or not. In this article, we ask this question for the geometric genus pg = h (M,OM ) and the Hilbert–Samuel function of (X, p), in particular for the multiplicity mult(X, p) and the embedding dimension emb dim(X, p) = dimmp/m 2 p. In general, these invariants are not topological, but if we restrict our study to some special classes, then they can be determined from the graph. The first result of this type was obtained by M. Artin [1, 2] for rational singularities. He proved that they can be characterized topologically: