Ultrametric spaces of branches on arborescent singularities

Abstract

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any resolution of it is a tree. Whenever A,B are distinct branches on S, we denote by A⋅B their intersection number in the sense of Mumford. If L is a fixed branch, we define U_L(A,B)=(L⋅A)(L⋅B)/(A⋅B) when A≠B and U_L(A,A)=0 otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then UL is an ultrametric on the set of branches of S different from L. We compute the maximum of U_L, which gives an analog of a theorem of Teissier. We show that U_L encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.