Jumping numbers and ordered tree structures on the dual graph

Abstract

Let R be a two-dimensional regular local ring having an algebraically closed residue field and let $${\mathfrak{a}}$$ be a complete ideal of finite colength in R. In this article we investigate the jumping numbers of $${\mathfrak{a}}$$ by means of the dual graph of the minimal log resolution of the pair $${(X,\mathfrak{a})}$$ . Our main result is a combinatorial criterium for a positive rational number ξ to be a jumping number. In particular, we associate to each jumping number certain ordered tree structures on the dual graph.