Title :

The Combinatorics of Plane Curve Singularities
How Newton Polygons Blossom into Lotuses

Author(s) :

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

Book title :

Handbook of Geometry and Topology of Singularities I

Publisher :

Springer International Publishing

Publication place :

Cham

Publication date :

2020-10-25

English keyword(s) :

Blow ups
Branch
Characteristic exponents
Dual graph
Eggers-Wall tree
Embedded resolution
Enriques diagram
Intersection numbers
Newton polygon
Newton-Puiseux series
Plane curve singularity
Proximity relation
Resolution of singularities
Toric geometry
Toroidal embedding
Tropicalization
Valuation
Valuative tree

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs (C, o) of complex analytic curves contained in a smooth complex analytic ...
Show more >This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs (C, o) of complex analytic curves contained in a smooth complex analytic surface S. The embedded topological type of such a pair (S, C) is usually defined to be that of the oriented link obtained by intersecting C with a sufficiently small oriented Euclidean sphere centered at the point o, defined once a system of local coordinates (x, y) was chosen on the germ (S, o). If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of (S, C). One may define it by looking either at the Newton-Puiseux series associated to C relative to a generic local coordinate system (x, y), or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ (C, o) or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of (C, o) by successive toric modifications.Show less >

Language :

Anglais

Audience :

Internationale

Popular science :

Non

ANR Project :

Géométrie Lipschitz des singularités

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL