Combinatorics of rational singularities
Let \((X,x)\) be a germ of a complex analytic normal surface and \(X'\to X\) a desingularization, with exceptional divisor whose irreducible components are curves \(E_1,\dots,E_n\). We associate to this a dual intersection graph, whose vertices are the curves \(E_i\), and distinct vertices \(E_i\) and \(E_j\) are linked by the intersection number \((E_i, E_j)\) of edges. To each vertex \(E_i\) we associate the weight \((w_i,g_i)\), \(w_i= -(E_i,E_i)\), \(g_i=\) genus of \(E_i\). The present paper is concerned with the study of weighted graphs that are associated to rational surface singularities. These are necessarily trees, with \(g_i=0\) for all \(i\). They can be defined in purely graph-theoretical terms, without mentioning rational singularities. In this article many properties of these trees are discussed, some already known, most being new. Some of the topics covered are: (a) different characterizations of rational trees, (b) a proof of a conjecture of Spivakovsky (a tree obtained from a rational one by increasing the weights \(w_i\) is still rational), (c) the notion of glueing trees at suitable vertices, and conditions that insure that the glueing of rational trees is still rational, (d) the notions of the complexity \(c\) of the multiplicity \(m\) of a rational trees (both are integers), and a theorem showing that necessarily \(c\leq m-2\), (e) the definition and properties of the so-called rational minimal, non-singular and sandwich trees. The presentation of the article is very clear and precise.
scientific article; zbMATH DE number 2113906