Title :

Ultrametric Spaces of Branches on Arborescent Singularities

Author(s) :

García Barroso, Evelia [Auteur]
Universidad de La Laguna [Tenerife - SP] [ULL]
González Pérez, Pedro [Auteur]
Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] [UCM]
Popescu-Pampu, Patrick [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Book title :

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Festschrift for Antonio Campillo on the Occasion of his 65th Birthday

Publisher :

Springer International Publishing

Publication place :

Cham

Publication date :

2018-09-19

ISBN :

978-3-319-96827-8

English keyword(s) :

Additive distance
Branch
Hierarchy
Mumford intersection number
Nef cone
Normal surface singularity
Resolution
Rooted tree
Ultrametric
Valuation

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A • B their intersection ...
Show more >Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A • B their intersection number in the sense of Mumford. If L is a fixed branch, we define U_L (A, B) = (L • A)(L • B)/(A • B) when A is different from 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.Show less >

Language :

Anglais

Audience :

Internationale

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL