2002/10/17 by Charles Favre, Mattias Jonsson, Favre, Charles +1 · 3 citations
Mathematics · #13A18 #14H20 #54F50 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13A18 #msc:14H20 #msc:54F50
paper · pdf · doi:10.48550/arxiv.math/0210265
206 pages, 21 figures
arxiv created 2003/11/26 · arxiv updated 2009/11/30
We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by minv(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.