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 $A\cdot B$ their intersection number in the sense of Mumford. If $L$ is a fixed branch, we define $U_L(A,B)=(L\cdot A)(L\cdot B)(A\cdot B{)}^{-1}$ when $A\ne B$ and $U_L(A,A)=0$ otherwise. We generalize a theorem of Płoski concerning smooth germs of surfaces, by proving that whenever $S$ is arborescent, then $U_L$ 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.

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 corrected

References in corpus (2)

• Variations on inversion theorems for Newton-Puiseux series
• Ultrametric spaces of branches on arborescent singularities

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