no. 2
On the values of logarithmic residues along curves
[Sur les multi-valuations des résidus logarithmiques le long des courbes]
Pol, Delphine1
1 LAREMA, UMR CNRS 6093, Université d’Angers, Département de Mathématiques, 2 Boulevard Lavoisier 49045 Angers Cedex 01 (France)
Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 725-766.

On considère un germe de courbe réduit, éventuellement réductible. F. Delgado de la Mata a montré qu’une telle courbe est Gorenstein si et seulement si son semigroupe des multi-valuations est symétrique. Nous étendons ici cette propriété de symétrie à tout idéal fractionnaire d’une courbe Gorenstein. Nous nous intéressons ensuite à l’ensemble des multi-valuations du module des résidus logarithmiques d’une courbe plane ou intersection complète, qui détermine et est déterminé par les multi-valuations de l’idéal jacobien grâce à notre théorème de symétrie. De plus, nous donnons la relation avec les différentielles de Kähler, qui sont utilisées dans la classification analytique des branches planes. Nous étudions aussi le comportement des résidus logarithmiques dans une déformation équisingulière de courbe plane.

We consider the germ of a reduced curve, possibly reducible. F. Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of a Gorenstein curve. We then focus on the set of values of the module of logarithmic residues along plane curves or complete intersection curves, which determines and is determined by the values of the Jacobian ideal thanks to our symmetry theorem. Moreover, we give the relation with Kähler differentials, which are used in the analytic classification of plane branches. We also study the behaviour of logarithmic residues in an equisingular deformation of a plane curve.

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DOI : 10.5802/aif.3176
Classification : 14H20, 14B07, 32A27
Keywords: logarithmic residues, duality, Gorenstein curves, values, equisingular deformations
Mot clés : résidus logarithmiques, dualité, courbes Gorenstein, multi-valuations, déformations équisingulière

Pol, Delphine 1

1 LAREMA, UMR CNRS 6093, Université d’Angers, Département de Mathématiques, 2 Boulevard Lavoisier 49045 Angers Cedex 01 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Pol, Delphine. On the values of logarithmic residues along curves. Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 725-766. doi : 10.5802/aif.3176. https://aif.centre-mersenne.org/articles/10.5802/aif.3176/

[1] Aleksandrov, Aleksandr G.; Tsikh, Avgust K. Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière, C. R. Math. Acad. Sci. Paris, Volume 333 (2001) no. 11, pp. 973-978 | DOI | Zbl

[2] Aleksandrov, Alexandr. G. Nonisolated Saito singularities, Mat. Sb., N. Ser., Volume 137(179) (1988) no. 4(12), pp. 554-567 | Zbl

[3] Aleksandrov, Alexandr G. Logarithmic differential forms, torsion differentials and residue, Complex Var. Theory Appl., Volume 50 (2005) no. 7-11, pp. 777-802 | Zbl

[4] Aleksandrov, Alexandr. G. Multidimensional residue theory and the logarithmic de Rham complex, J. Singul., Volume 5 (2012), pp. 1-18 | Zbl

[5] Barlet, Daniel Le faisceau ω X · sur un espace analytique X de dimension pure, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) (Lecture Notes in Math.), Volume 670, Springer, 1978, pp. 187-204 | Zbl

[6] Briançon, Joël; Geandier, Françoise; Maisonobe, Philippe Déformation d’une singularité isolée d’hypersurface et polynômes de Bernstein, Bull. Soc. Math. Fr., Volume 120 (1992) no. 1, pp. 15-49 | DOI | Zbl

[7] 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 | Zbl

[8] Campillo, Antonio; Delgado, Félix; Gusein-Zade, Sabir M. The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math. J., Volume 117 (2003) no. 1, pp. 125-156 | DOI | Zbl

[9] Cassou-Noguès, Pierrette; Płoski, Arkadiusz Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math. (2011) no. 49, pp. 9-34 | Zbl

[10] Eisenbud, David Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995, xvi+785 pages | Zbl

[11] Faber, Eleonore Towards transversality of singular varieties: splayed divisors, Publ. Res. Inst. Math. Sci., Volume 49 (2013) no. 3, pp. 393-412 | DOI | Zbl

[12] Granger, Michel; Schulze, Mathias Normal crossing properties of complex hypersurfaces via logarithmic residues, Compos. Math., Volume 150 (2014) no. 9, pp. 1607-1622 | DOI | Zbl

[13] Greuel, Gert-Martin; Lossen, Cristoph; Shustin, Eugenii Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, 2007, xii+471 pages | Zbl

[14] Hefez, Abramo; Hernandes, Marcelo E. Standard bases for local rings of branches and their modules of differentials, J. Symb. Comput., Volume 42 (2007) no. 1-2, pp. 178-191 | DOI | 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 | Zbl

[16] Hefez, Abramo; Hernandes, Marcelo E.; Hernandes, Maria E. Rodrigues The analytic classification of plane curves with two branches, Math. Z., Volume 279 (2015) no. 1-2, pp. 509-520 | DOI | Zbl

[17] de Jong, Theo; Pfister, Gerhard Local analytic geometry, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2000, xii+382 pages (Basic theory and applications) | Zbl

[18] Korell, Philipp; Schulze, Mathias; Tozzo, Laura Duality on value semigroups (2017) (https://arxiv.org/abs/1510.04072, to appear in J. Commut. Algebra)

[19] Kunz, Ernst The value-semigroup of a one-dimensional Gorenstein ring, Proc. Am. Math. Soc., Volume 25 (1970), pp. 748-751 | DOI | 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 | Zbl

[21] Delgado de la Mata, Félix Gorenstein curves and symmetry of the semigroup of values, Manuscr. Math., Volume 61 (1988) no. 3, pp. 285-296 | DOI | Zbl

[22] Michler, Ruth Torsion of differentials of hypersurfaces with isolated singularities, J. Pure Appl. Algebra, Volume 104 (1995) no. 1, pp. 81-88 | DOI | Zbl

[23] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press; University of Tokyo Press, 1968, iii+122 pages | Zbl

[24] Piene, Ragni Ideals associated to a desingularization, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) (Lecture Notes in Math.), Volume 732, Springer, 1979, pp. 503-517 | Zbl

[25] Pol, Delphine Logarithmic residues along plane curves, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 4, pp. 345-349 | DOI | Zbl

[26] Pol, Delphine Characterizations of freeness for Cohen-Macaulay spaces (2016) (https://arxiv.org/abs/1512.06778v2)

[27] Pol, Delphine Singularités libres, formes et résidus logarithmiques, Université d’Angers (France) (2016) (Ph. D. Thesis)

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

[29] Schulze, Mathias On Saito’s normal crossing condition, J. Singul., Volume 14 (2016), pp. 124-147 | Zbl

[30] Teissier, Bernard The hunting of invariants in the geometry of discriminants, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 565-678 | Zbl

[31] Torielli, Michele Deformations of free and linear free divisors, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2097-2136 | DOI | Zbl

[32] Zariski, Oscar Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Nat. Acad. Sci. U.S.A., Volume 56 (1966), pp. 781-786 | DOI | Zbl

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

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