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The valuative tree | Charles Favre
; Mattias Jonsson
; | Date: |
17 Oct 2002 | Subject: | Commutative Algebra; Algebraic Geometry MSC-class: 14H20; 13A18; 54F50 | math.AC math.AG | Abstract: | We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by min{v(x),v(y)}=1. It has a natural structure of an R-tree, induced by the order relation v is less than v’ iff v(f) is less than v’(f) for all f. It can also be metrized, endowing it with a metric tree structure. From the algebraic point of view, these structures are obtained by taking a suitable quotient of the Riemann-Zariski variety of C[[x,y]], in order to force it to be a Hausdorff topological space. The tree structure on V also provides an identification of valuations with balls of irreducible curves in a natural ultrametric. We show that the dual graphs of all sequences of blow-ups patch together, yielding an R-tree naturally isomorphic to V. Altogether, this gives many different approaches to the valuative tree V. We then describe a natural Laplace operator on V. It associates to (special) functions of V a complex Borel measure. Using this operator, we show how measures on the valuative tree can be used to encode naturally both integrally closed ideals in R and cohomology classes of the local analog of voute etoilee over the complex plane. | Source: | arXiv, math.AC/0210265 | Services: | Forum | Review | PDF | Favorites |
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