On the values of logarithmic residues along curves

Pol, Delphine

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.

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

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