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 UL(A,B)=(L⋅A)(L⋅B)(A⋅B)−1 when A=B and UL(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 UL, which gives an analog of a theorem of Teissier. We show that
UL 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.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has
a new section 4.3, accompanied by 2 new figures. Several passages were
clarified and the typos discovered in the meantime were correcte