Multi-Harnack smoothings of real plane branches

González Pérez, Pedro Daniel; Risler, Jean-Jacques

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 1, pp. 143-184.

Soit $\Delta \subset {𝐑}^2$ un polygone convexe à sommets entiers ; G. Mikhalkin a défini les « courbes de Harnack » (définies par un polynôme de support contenu dans $\Delta$ et plongées dans la surface torique correspondante) et montré leur existence (via la « méthode du patchwork de Viro ») ainsi que l’unicité de leur type topologique plongé (qui est determiné par $\Delta$). Le but de cet article est de montrer un résultat analogue pour la lissification (smoothing) d’un germe de branche réelle plane $(C,O)$ analytique réelle. On définit pour cela une classe de smoothings dite « Multi-Harnack » à l’aide de la résolution des singularités constituée d’une suite de $g$ éclatements toriques, si $g$ est le nombre de paires de Puiseux de la branche $(C,O)$. Un smoothing multi-Harnack est réalisé de la manière suivante : à chaque étape de la résolution (en commençant par la dernière) et de manière successive, un smoothing « De Harnack » (au sens de Mikhalkin) intermédiaire est obtenu par la méthode de Viro. On montre alors l'unicité du type topologique de tels smoothings. De plus, on peut supposer ces smoothings « multi-semi-quasi homogènes »  ; on montre alors que des propriétés métriques (« multi-taille » des ovales) de tels smoothings sont caractérisées en fonction de la classe d’équisingularité de $(C,O)$ et que réciproquement ces tailles caractérisent la classe d’équisingularité de la branche.

Let $\Delta \subset {𝐑}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion of Harnack curves, a class of real algebraic curves, defined by polynomials supported on $\Delta$ and contained in the corresponding toric surface. He proved their existence, via Viro's patchworking method, 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 of multi-Harnack smoothings 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 suitable Harnack 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.

Classification : 14P25, 14H20, 14M25
Keywords: smoothings of singularities, real algebraic curves, harnack curves
Mot clés : lissification d'une singularité, courbe algébrique réelle, courbes de harnack

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