Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez; Jean-Jacques Risler

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 1, page 143-184
  • ISSN: 0012-9593

Abstract

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Let Δ ⊂ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class ofmulti-Harnacksmoothings of ( C , 0 ) by passing through a resolution of singularities of ( C , 0 ) consisting of g monomial maps (where g is the number of characteristic pairs of the branch). A multi-Harnack smoothing is a g -parametrical deformation which arises as the result of a sequence, beginning at the last step of the resolution, consisting of a suitableHarnack smoothing (in terms of Mikhalkin’s definition) followed by the corresponding monomial blow down. We prove then the unicity of the topological type of a multi-Harnack smoothing. In addition, the multi-Harnack smoothings can be seen as multi-semi-quasi-homogeneous in terms of the parameters. Using this property we analyze the asymptotic multi-scales of the ovals of a multi-Harnack smoothing. We prove that these scales characterize and are characterized by the equisingularity class of the branch.

How to cite

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González Pérez, Pedro Daniel, and Risler, Jean-Jacques. "Multi-Harnack smoothings of real plane branches." Annales scientifiques de l'École Normale Supérieure 43.1 (2010): 143-184. <http://eudml.org/doc/272195>.

@article{GonzálezPérez2010,
abstract = {Let $\Delta \subset \mathbf \{R\}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on $\Delta $ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by $\Delta $). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C,0)$. We introduce the class ofmulti-Harnacksmoothings of $(C,0)$ by passing through a resolution of singularities of $(C,0)$ consisting of $g$ monomial maps (where $g$ is the number of characteristic pairs of the branch). A multi-Harnack smoothing is a $g$-parametrical deformation which arises as the result of a sequence, beginning at the last step of the resolution, consisting of a suitableHarnack smoothing (in terms of Mikhalkin’s definition) followed by the corresponding monomial blow down. We prove then the unicity of the topological type of a multi-Harnack smoothing. In addition, the multi-Harnack smoothings can be seen as multi-semi-quasi-homogeneous in terms of the parameters. Using this property we analyze the asymptotic multi-scales of the ovals of a multi-Harnack smoothing. We prove that these scales characterize and are characterized by the equisingularity class of the branch.},
author = {González Pérez, Pedro Daniel, Risler, Jean-Jacques},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {smoothings of singularities; real algebraic curves; harnack curves},
language = {eng},
number = {1},
pages = {143-184},
publisher = {Société mathématique de France},
title = {Multi-Harnack smoothings of real plane branches},
url = {http://eudml.org/doc/272195},
volume = {43},
year = {2010},
}

TY - JOUR
AU - González Pérez, Pedro Daniel
AU - Risler, Jean-Jacques
TI - Multi-Harnack smoothings of real plane branches
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 1
SP - 143
EP - 184
AB - Let $\Delta \subset \mathbf {R}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on $\Delta $ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by $\Delta $). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C,0)$. We introduce the class ofmulti-Harnacksmoothings of $(C,0)$ by passing through a resolution of singularities of $(C,0)$ consisting of $g$ monomial maps (where $g$ is the number of characteristic pairs of the branch). A multi-Harnack smoothing is a $g$-parametrical deformation which arises as the result of a sequence, beginning at the last step of the resolution, consisting of a suitableHarnack smoothing (in terms of Mikhalkin’s definition) followed by the corresponding monomial blow down. We prove then the unicity of the topological type of a multi-Harnack smoothing. In addition, the multi-Harnack smoothings can be seen as multi-semi-quasi-homogeneous in terms of the parameters. Using this property we analyze the asymptotic multi-scales of the ovals of a multi-Harnack smoothing. We prove that these scales characterize and are characterized by the equisingularity class of the branch.
LA - eng
KW - smoothings of singularities; real algebraic curves; harnack curves
UR - http://eudml.org/doc/272195
ER -

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