The Saito module and the moduli of a germ of curve in ( 2 ,0).
Annales de l'Institut Fourier, Online first, 55 p.

This article proposes to study the moduli space of a germ of curve S in the complex plane, that is to say the equisingularity class of S up to analytical equivalence relation. The first part is devoted to proving that this last quotient can be endowed with a reasonable, yet not canonical, complex structure. The second part deals with the computation of its generic dimension in terms of topological invariants of S. It can be obtained from the study of the valuations of the Saito module of S, Der(logS), i.e. the module of vector fields tangent to S.

Cet article est une étude des espaces de module d’une courbe S dans le plan complexe, c’est-à-dire, de la classe d’équisingularité de S modulo la relation d’équivalence analytique. La première partie établit l’existence d’une structure non canonique de variété complexe sur ce quotient. La seconde partie se consacre au calcul de sa dimension générique à partir de la donnée d’invariants primitifs topologiques de S. Ce calcul est le fruit de l’étude des valuations des champs de vecteurs tangents à S.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3655
Classification: 32S65, 14H20, 14Q05
Keywords: Complex curves, Foliations, Vector fields, Zariski problem.
Mot clés : Courbes complexes, feuilletages, champs de vecteurs, problem de Zariski.

Genzmer, Yohann 1

1 Institut de Mathématiques de Toulouse, 118, route de Narbonne, 31062 Toulouse Cedex 09 (France)
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Genzmer, Yohann. The Saito module and the moduli of a germ of curve in $(\mathbb{C}^{2},0)$.. Annales de l'Institut Fourier, Online first, 55 p.

[1] Briançon, Joël; Granger, Michel; Maisonobe, Philippe Le nombre de modules du germe de courbe plane x a +y b =0, Math. Ann., Volume 279 (1988) no. 3, pp. 535-551 | DOI | MR | Zbl

[2] Carvalho, E.; Hernandes, Marcelo E. Standard bases for fractional ideals of the local ring of an algebroid curve, J. Algebra, Volume 551 (2020), pp. 342-361 | DOI | MR | Zbl

[3] Cerveau, Domonique; Mattei, Jean-François Formes intégrables holomorphes singulières, Astérisque, 97, Société Mathématique de France, 1982, 193 pages | Numdam | MR | Zbl

[4] Delorme, Charles Sur les modules des singularités des courbes planes, Bull. Soc. Math. Fr., Volume 106 (1978) no. 4, pp. 417-446 | DOI | Numdam | MR | Zbl

[5] Dulac, Henri Recherches sur les points singuliers des équations différentielles, J. Éc. Polytech., Math., Volume 9 (1904) no. 2, pp. 1-125 | Zbl

[6] Ebey, Sherwood The classification of singular points of algebraic curves, Trans. Am. Math. Soc., Volume 118 (1965), pp. 454-471 | DOI | MR | Zbl

[7] Genzmer, Yohann Dimension of the Moduli Space of a Germ of Curve in 2, Int. Math. Res. Not. (2022) no. 5, pp. 3805-3859 | DOI | MR | Zbl

[8] Genzmer, Yohann Number of moduli for a union of smooth curves in ( 2 ,0), J. Symb. Comput., Volume 113 (2022), pp. 148-170 | DOI | MR | Zbl

[9] Genzmer, Yohann; Hernandes, Marcelo E. On the Saito basis and the Tjurina number for plane branches, Trans. Am. Math. Soc., Volume 373 (2020) no. 5, pp. 3693-3707 | DOI | MR | Zbl

[10] Genzmer, Yohann; Paul, Emmanuel Normal forms of foliations and curves defined by a function with a generic tangent cone, Mosc. Math. J., Volume 11 (2011) no. 1, p. 41-72, 181 | DOI | MR | Zbl

[11] Genzmer, Yohann; Paul, Emmanuel Moduli spaces for topologically quasi-homogeneous functions, J. Singul., Volume 14 (2016), pp. 3-33 | DOI | MR | Zbl

[12] Gómez-Mont, Xavier The transverse dynamics of a holomorphic flow, Ann. Math., Volume 127 (1988) no. 1, pp. 49-92 | DOI | MR | Zbl

[13] Granger, Jean-Michel Sur un espace de modules de germe de courbe plane, Bull. Sci. Math., Volume 103 (1979) no. 1, pp. 3-16 | MR | Zbl

[14] Hefez, Abramo; Hernandes, Marcelo E. Analytic classification of plane branches up to multiplicity 4, J. Symb. Comput., Volume 44 (2009) no. 6, pp. 626-634 | DOI | MR | Zbl

[15] Hefez, Abramo; Hernandes, Marcelo E. The analytic classification of plane branches, Bull. Lond. Math. Soc., Volume 43 (2011) no. 2, pp. 289-298 | DOI | MR | Zbl

[16] Hefez, Abramo; Hernandes, Marcelo E. Algorithms for the implementation of the analytic classification of plane branches, J. Symb. Comput., Volume 50 (2013), pp. 308-313 | DOI | MR | Zbl

[17] Hernandes, Marcelo E.; de Carvalho, E. The value semiring of an algebroid curve, Commun. Algebra, Volume 48 (2020) no. 8, pp. 3275-3284 | DOI | Zbl

[18] Hertling, Claus Formules pour la multiplicité et le nombre de Milnor d’un feuilletage sur ( 2 ,0), Ann. Fac. Sci. Toulouse, Math., Volume 9 (2000) no. 4, pp. 655-670 | DOI | MR | Zbl

[19] Laudal, Olav A.; Martin, Bernd; Pfister, Gerhard Moduli of plane curve singularities with C * -action, Singularities (Warsaw, 1985) (Banach Center Publications), Volume 20, PWN, Warsaw, 1988, pp. 255-278 | MR | Zbl

[20] Delgado de la Mata, Félix The semigroup of values of a curve singularity with several branches, Manuscr. Math., Volume 59 (1987) no. 3, pp. 347-374 | DOI | MR | Zbl

[21] Mattei, Jean-François Quasi-homogénéité et équiréductibilité de feuilletages holomorphes en dimension deux, Géométrie complexe et systèmes dynamiques (Astérisque), Société Mathématique de France, 2000 no. 261, pp. xix, 253-276 | Numdam | MR | Zbl

[22] Mattei, Jean-François; Moussu, Robert Holonomie et intégrales premières, Ann. Sci. Éc. Norm. Supér., Volume 13 (1980) no. 4, pp. 469-523 | DOI | Numdam | MR | Zbl

[23] Saito, Kyoji Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 2, pp. 265-291 | MR | Zbl

[24] Siu, Yum-Tong Every Stein subvariety admits a Stein neighborhood, Invent. Math., Volume 38 (1976) no. 1, pp. 89-100 | MR | Zbl

[25] Waldi, Rolf Wertehalbgruppe und Singularität einer ebenen algebraischen Kurve, Dissertation, Regensburg (1972)

[26] Zariski, Oscar Le problème des modules pour les branches planes. Course given at the Centre de Mathématiques de l’École Polytechnique, Paris, October–November 1973, With an appendix by Bernard Teissier, Hermann, 1986, x+212 pages | MR | Zbl

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