Ultrametric properties for valuation spaces of normal surface singularities

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Date(s) - 14/03/2019
14 h 00 min - 15 h 00 min

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Let (X,x_0) be a normal surface singularity, and denote by B_X the set of irreducible curves (branches) at (X,x_0).
Consider the functional u_L(A,B)=(L · A) (L · B) / (A · B), where L,A,B are branches.
In a joint work with E. García Barroso, P. González Pérez and P. Popescu Pampu, we show that u_L defines an (extended) ultrametric distance on B_X for a (any) branch L if and only if (X,x_0) is arborescent: the dual graph of any good resolution is a tree.
The proof relies on intersection properties of exceptional divisors, obtained in collaboration with W. Gignac.
I will present this result, and an analogous statement on the space V_X of (rank-1 normalized semi-)valuations at (X,x_0).
If time allows, I will also present a topological condition on dual graphs (resp., valuation spaces) to ensure that u_L is an ultrametric on a given subset of B_X (resp., V_X).


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