Multi-Harnack smoothings of real plane branches

[Lissages multi-Harnack de branches planes réelles]

González Pérez, Pedro Daniel ; Risler, Jean-Jacques

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 143-184 / Harvested from Numdam

Résumé

Soit $\Delta \subset \mathbf {R}^2$ 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.

Abstract

Let $\Delta \subset \mathbf {R}^2$ 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.

Détail

Publié le : 2010-01-01

MR 2583267 | Zbl 1194.14042

DOI : https://doi.org/10.24033/asens.2118
Classification:  14P25,  14H20,  14M25
Mots clés: lissification d'une singularité, courbe algébrique réelle, courbes de harnack

Bibliographie

[1] N. A'Campo & M. Oka, Geometry of plane curves via Tschirnhausen resolution tower, Osaka J. Math. 33 (1996), 1003-1033. | Zbl 0904.14014

[2] V. I. ArnolʼD, Some open problems in the theory of singularities, in Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., 1983, 57-69. | Zbl 0519.58019

[3] F. Bihan, Viro method for the construction of real complete intersections, Adv. Math. 169 (2002), 177-186. | Zbl 1048.14035

[4] L. Brusotti, Curve generatrici e curve aggregate nella costruzione di curve piane d'ordine assegnato dotate del massimo numero di circuiti, Rend. Circ. Mat. Palermo 42 (1917), 138-144. | JFM 46.0950.02

[5] M. Forsberg, M. Passare & A. Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. Math. 151 (2000), 45-70. | Zbl 1002.32018

[6] W. Fulton, Introduction to toric varieties, Annals of Math. Studies 131, Princeton Univ. Press, 1993. | Zbl 0813.14039

[7] E. García Barroso & B. Teissier, Concentration multi-échelles de courbure dans des fibres de Milnor, Comment. Math. Helv. 74 (1999), 398-418. | MR 1710694 | Zbl 0956.32028

[8] I. M. Gel'Fand, M. M. Kapranov & A. V. Zelevinsky, Discriminants, resultants and multi-dimensional determinants, Birkhäuser, 1994. | MR 1264417 | Zbl 0827.14036

[9] R. Goldin & B. Teissier, Resolving singularities of plane analytic branches with one toric morphism, in Resolution of singularities (Obergurgl, 1997), Progr. Math. 181, Birkhäuser, 2000, 315-340. | MR 1748626 | Zbl 0995.14002

[10] P. D. González Pérez, Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canad. J. Math. 52 (2000), 348-368. | Zbl 0970.14027

[11] P. D. González Pérez, Approximate roots, toric resolutions and deformations of a plane branch, to appear in J. of the Math. Soc. of Japan. | MR 2648070 | Zbl 1258.14039

[12] A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), 51-61, English transl. Functional Anal. Appl. 12 (1978), 38-46. | MR 487230 | Zbl 0406.14035

[13] I. Itenberg, Viro’s method and $T$-curves, in Algorithms in algebraic geometry and applications (Santander, 1994), Progr. Math. 143, Birkhäuser, 1996, 177-192. | MR 1414451 | Zbl 0879.14029

[14] I. Itenberg, Amibes de variétés algébriques et dénombrement de courbes (d'après G. Mikhalkin), Séminaire Bourbaki 2002/03, exposé no 921, Astérisque 294 (2004), 335-361. | Numdam | MR 2111649 | Zbl 1059.14067

[15] I. Itenberg, G. Mikhalkin & E. Shustin, Tropical algebraic geometry, Oberwolfach Seminars 35, Birkhäuser, 2007. | MR 2292729 | Zbl 1162.14300

[16] I. Itenberg & O. Y. Viro, Patchworking algebraic curves disproves the Ragsdale conjecture, Math. Intelligencer 18 (1996), 19-28. | MR 1413249 | Zbl 0876.14017

[17] V. M. Kharlamov, S. Y. Orevkov & E. Shustin, Singularity which has no $M$-smoothing, in The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun. 24, Amer. Math. Soc., 1999, 273-309. | MR 1733581 | Zbl 0978.14048

[18] V. M. Kharlamov & J.-J. Risler, Blowing-up construction of maximal smoothings of real plane curve singularities, in Real analytic and algebraic geometry (Trento, 1992), de Gruyter, 1995, 169-188. | MR 1320318 | Zbl 0861.14022

[19] V. M. Kharlamov, J.-J. Risler & E. Shustin, Maximal smoothings of real plane curve singular points, in Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc., 2001, 167-195. | MR 1819188 | Zbl 0988.14025

[20] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31. | MR 419433 | Zbl 0328.32007

[21] D.-T. Lê & M. Oka, On resolution complexity of plane curves, Kodai Math. J. 18 (1995), 1-36. | MR 1317003 | Zbl 0844.14010

[22] G. Mikhalkin, Real algebraic curves, the moment map and amoebas, Ann. of Math. 151 (2000), 309-326. | MR 1745011 | Zbl 1073.14555

[23] G. Mikhalkin & H. Rullgård, Amoebas of maximal area, Int. Math. Res. Not. 2001 (2001), 441-451. | MR 1829380 | Zbl 0994.14032

[24] J. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton Univ. Press, 1968. | MR 239612 | Zbl 0184.48405

[25] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse Math. Grenzg. 15, Springer, 1988. | MR 922894 | Zbl 0628.52002

[26] M. Oka, Geometry of plane curves via toroidal resolution, in Algebraic geometry and singularities (La Rábida, 1991), Progr. Math. 134, Birkhäuser, 1996, 95-121. | MR 1395177 | Zbl 0857.14014

[27] M. Oka, Non-degenerate complete intersection singularity, Actualités Mathématiques., Hermann, 1997. | MR 1483897 | Zbl 0930.14034

[28] M. Passare & H. Rullgård, Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope, Duke Math. J. 121 (2004), 481-507. | MR 2040284 | Zbl 1043.32001

[29] P. Popescu-Pampu, Approximate roots, in Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999), Fields Inst. Commun. 33, Amer. Math. Soc., 2003, 285-321. | MR 2018562 | Zbl 1036.13017

[30] J.-J. Risler, Un analogue local du théorème de Harnack, Invent. Math. 89 (1987), 119-137. | MR 892188 | Zbl 0672.14020

[31] J.-J. Risler, Construction d'hypersurfaces réelles (d'après Viro), Séminaire Bourbaki 1992/93, exposé no 763, Astérisque 216 (1993), 69-86. | Numdam | MR 1246393 | Zbl 0824.14045

[32] E. Shustin & I. Tyomkin, Patchworking singular algebraic curves. I, Israel J. Math. 151 (2006), 125-144. | MR 2214120 | Zbl 1128.14019

[33] E. Shustin & I. Tyomkin, Patchworking singular algebraic curves. II, Israel J. Math. 151 (2006), 145-166. | MR 2214121 | Zbl 1128.14020

[34] B. Sturmfels, Viro's theorem for complete intersections, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21 (1994), 377-386. | Numdam | MR 1310632 | Zbl 0826.14032

[35] O. Y. Viro, Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, in Proceedings of the Leningrad International Topological Conference (Leningrad 1982), Nauka, 1983, 149-197. | Zbl 0605.14021

[36] O. Y. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, in Topology (Leningrad, 1982), Lecture Notes in Math. 1060, Springer, 1984, 187-200. | MR 770238 | Zbl 0576.14031

[37] O. Y. Viro, Real plane algebraic curves: constructions with controlled topology, Algebra i Analiz 1 (1989), 1-73, English translation: Leningrad Math. J., 1 (1990), 1059-1134. | MR 1036837 | Zbl 0732.14026

[38] O. Y. Viro, Patchworking real algebraic varieties, U.U.D.M. Report 42 (Uppsala University, 1994).

[39] O. Zariski, Le problème des modules pour les branches planes, Hermann, 1986. | MR 861277 | Zbl 0592.14010