The valuative tree
| dc.creator | Favre, Charles | |
| dc.creator | Jonsson, Mattias | |
| dc.date | 2002-10-17 | |
| dc.date | 2003-11-26 | |
| dc.date.accessioned | 2026-07-07T04:52:04Z | |
| dc.date.available | 2026-07-07T04:52:04Z | |
| dc.description | 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. | |
| dc.description | 206 pages, 21 figures | |
| dc.identifier | https://arxiv.org/abs/math/0210265 | |
| dc.identifier | http://arxiv.org/abs/math/0210265 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65330 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14H20; 13A18; 54F50 | |
| dc.title | The valuative tree | |
| dc.type | text |