Jacobian curve of singular foliations
[Courbe jacobienne de feuilletages singuliers]
Corral, Nuria1
1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros s/n, 39005 – Santander (Spain)
Annales de l'Institut Fourier, Online first, 61 p.

Nous décrivons des propriétés topologiques de la courbe jacobienne 𝒥 ℱ,𝒢 de deux feuilletages ℱ et 𝒢 en termes des invariants associés aux feuilletages. Le resultat principal donne une décomposition de la courbe jacobienne 𝒥 ℱ,𝒢 qui dépend de la similitude des feuilletages ℱ et 𝒢. Cette similitude entre les feuilletages est codifiée en termes des indices de Camacho–Sad des feuilletages avec la notion de point ou diviseur colinéaire. Notre approche permet de récupérer les résultats concernant la factorisation de la courbe jacobienne de deux courbes planes et de la courbe polaire d’une courbe ou d’un feuilletage.

Topological properties of the jacobian curve 𝒥 ℱ,𝒢 of two foliations ℱ and 𝒢 are described in terms of invariants associated to the foliations. The main result gives a decomposition of the jacobian curve 𝒥 ℱ,𝒢 which depends on how similar are the foliations ℱ and 𝒢. The similarity between foliations is codified in terms of the Camacho–Sad indices of the foliations with the notion of collinear point or divisor. Our approach allows to recover the results concerning the factorization of the jacobian curve of two plane curves and of the polar curve of a curve or a foliation.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3665
Classification : 32S65, 32S50, 14H20
Keywords: Jacobian curve, singular foliation, polar curve, Camacho–Sad index, equisingularity data.
Mot clés : Courbe jacobienne, feuilletage singulier, courbe polaire, indice de Camacho–Sad, type d’équisingularité.

Corral, Nuria 1

1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros s/n, 39005 – Santander (Spain) 
@unpublished{AIF_0__0_0_A102_0,
     author = {Corral, Nuria},
     title = {Jacobian curve of singular foliations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3665},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Corral, Nuria
TI  - Jacobian curve of singular foliations
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3665
LA  - en
ID  - AIF_0__0_0_A102_0
ER  - 
%0 Unpublished Work
%A Corral, Nuria
%T Jacobian curve of singular foliations
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3665
%G en
%F AIF_0__0_0_A102_0
Corral, Nuria. Jacobian curve of singular foliations. Annales de l'Institut Fourier, Online first, 61 p.

[1] Abhyankar, Shreeram S.; Moh, Tzuong Tsieng Newton–Puiseux expansion and generalized Tschirnhausen transformation. I, II, J. Reine Angew. Math., Volume 260 (1973), pp. 47-83 ibid. 261 (1973), p. 29–54 | DOI | MR | Zbl

[2] Alberich-Carramiñana, Maria; González-Alonso, Víctor Determining plane curve singularities from its polars, Adv. Math., Volume 287 (2016), pp. 788-822 | DOI | MR | Zbl

[3] Camacho, César; Lins Neto, Alcides; Sad, Paulo Topological invariants and equidesingularization for holomorphic vector fields, J. Differ. Geom., Volume 20 (1984) no. 1, pp. 143-174 | DOI | MR | Zbl

[4] Camacho, César; Sad, Paulo Invariant varieties through singularities of holomorphic vector fields, Ann. Math., Volume 115 (1982) no. 3, pp. 579-595 | DOI | MR | Zbl

[5] Cano, Felipe; Cerveau, Dominique; Déserti, Julie Théorie élémentaire des feuilletages holomorphes singuliers, Collection Échelles, Belin, 2013

[6] Cano, Felipe; Corral, Nuria; Mol, Rogério Local polar invariants for plane singular foliations, Expo. Math., Volume 37 (2019) no. 2, pp. 145-164 | DOI | MR | Zbl

[7] Casas-Alvero, Eduardo Singularities of plane curves, London Mathematical Society Lecture Note Series, 276, Cambridge University Press, 2000, xvi+345 pages | DOI | MR | Zbl

[8] Casas-Alvero, Eduardo Local geometry of planar analytic morphisms, Asian J. Math., Volume 11 (2007) no. 3, pp. 373-426 | DOI | MR | Zbl

[9] Corral, Nuria Sur la topologie des courbes polaires de certains feuilletages singuliers, Ann. Inst. Fourier, Volume 53 (2003) no. 3, pp. 787-814 | DOI | Numdam | MR | Zbl

[10] Corral, Nuria Infinitesimal adjunction and polar curves, Bull. Braz. Math. Soc. (N.S.), Volume 40 (2009) no. 2, pp. 181-224 | DOI | MR | Zbl

[11] Corral, Nuria Infinitesimal initial part of a singular foliation, An. Acad. Brasil. Ciênc., Volume 81 (2009) no. 4, pp. 633-640 | DOI | MR | Zbl

[12] Corral, Nuria Polar pencil of curves and foliations, Équations différentielles et singularités. En l’honneur de J. M. Aroca (Cano, F.; Loray, F.; Moralez-Ruiz, J. J.; Sad, P.; Spivakovsky, M., eds.) (Astérisque), Société Mathématique de France, 2009 no. 323 | Numdam | Zbl

[13] García Barroso, Evelia R. Sur les courbes polaires d’une courbe plane réduite, Proc. Lond. Math. Soc., Volume 81 (2000) no. 1, pp. 1-28 | DOI | MR | Zbl

[14] García Barroso, Evelia R.; Gwoździewicz, Janusz On the approximate Jacobian Newton diagrams of an irreducible plane curve, J. Math. Soc. Japan, Volume 65 (2013) no. 1, pp. 169-182 | DOI | MR | Zbl

[15] Genzmer, Yohann; Mol, Rogério Local polar invariants and the Poincaré problem in the dicritical case, J. Math. Soc. Japan, Volume 70 (2018) no. 4, pp. 1419-1451 | DOI | MR | Zbl

[16] Gómez-Martínez, Oziel Foliaciones dicríticas en la realización de invariantes analíticos de curvas singulares, Ph. D. Thesis, Universidad Nacional Autónoma de México (2021)

[17] Gwoździewicz, Janusz; Ploski, Arkadiusz On the approximate roots of polynomials, Ann. Pol. Math., Volume 60 (1995) no. 3, pp. 199-210 | DOI | MR | Zbl

[18] Hefez, Abramo; Hernandes, Marcelo E.; Iglesias, Mauro F. H. On the factorization of the polar of a plane branch, Singularities and foliations. geometry, topology and applications (Springer Proceedings in Mathematics & Statistics), Volume 222, Springer, 2018, pp. 347-362 | DOI | MR | Zbl

[19] Kuo, Tzee-Char; Parusiński, Adam On Puiseux roots of Jacobians, Proc. Japan Acad., Ser. A, Volume 78 (2002) no. 5, pp. 55-59 | MR | Zbl

[20] Kuo, Tzee-Char; Parusiński, Adam Newton–Puiseux roots of Jacobian determinants, J. Algebr. Geom., Volume 13 (2004) no. 3, pp. 579-601 | DOI | MR | Zbl

[21] Lê, Dung Trang; Michel, Françoise; Weber, Claude Sur le comportement des polaires associées aux germes de courbes planes, Compos. Math., Volume 72 (1989) no. 1, pp. 87-113 | Numdam | MR | Zbl

[22] Mattei, Jean-Francois; Salem, Eliane Modules formels locaux de feuilletages holomorphes (2004) (https://arxiv.org/abs/math/0402256)

[23] Maugendre, Hélène Discriminant d’un germe (g,f):(C 2 ,0)→(C 2 ,0) et quotients de contact dans la résolution de f·g, Ann. Fac. Sci. Toulouse, Math., Volume 7 (1998) no. 3, pp. 497-525 | DOI | MR | Zbl

[24] Maugendre, Hélène Discriminant of a germ Φ:(C 2 ,0)→(C 2 ,0) and Seifert fibred manifolds, J. Lond. Math. Soc., Volume 59 (1999) no. 1, pp. 207-226 | DOI | MR | Zbl

[25] Merle, Michel Invariants polaires des courbes planes, Invent. Math., Volume 41 (1977) no. 2, pp. 103-111 | DOI | MR | Zbl

[26] Ortiz-Bobadilla, Laura; Rosales-González, Ernesto; Voronin, Sergei M. Rigidity theorems for generic holomorphic germs of dicritic foliations and vector fields in (ℂ 2 ,0), Mosc. Math. J., Volume 5 (2005) no. 1, pp. 171-206 | DOI | MR | Zbl

[27] Paul, Emmanuel Classification topologique des germes de formes logarithmiques génériques, Ann. Inst. Fourier, Volume 39 (1989) no. 4, pp. 909-927 | DOI | Numdam | MR | Zbl

[28] Paul, Emmanuel Cycles évanescents d’une fonction de Liouville de type f 1 λ 1 ⋯f p λ p , Ann. Inst. Fourier, Volume 45 (1995) no. 1, pp. 31-63 | DOI | MR | Zbl

[29] Popescu-Pampu, Patrick Approximate roots, Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999) (Fields Institute Communications), Volume 33, American Mathematical Society, 2003, pp. 285-321 | MR | Zbl

[30] Rouillé, Patrick Théorème de Merle: cas des 1-formes de type courbes généralisées, Bol. Soc. Bras. Mat., Nova Sér., Volume 30 (1999) no. 3, pp. 293-314 | DOI | MR | Zbl

[31] Saravia, Nancy E. Curva polar de una foliación asociada a sus raíces aproximadas, Ph. D. Thesis, Pontificia Universidad Católica del Perú (2018)

[32] Seidenberg, Abraham Reduction of singularities of the differential equation Ady=Bdx, Am. J. Math., Volume 90 (1968), pp. 248-269 | DOI | MR | Zbl

Cité par Sources :