On the values of logarithmic residues along curves

TL;DR

This paper extends the symmetry property of semigroups to fractional ideals of Gorenstein curves, analyzing the values of logarithmic residues and their relation to Jacobian ideals and Kähler differentials, with applications to curve classification.

Contribution

It generalizes the symmetry property to fractional ideals and explores the behavior of logarithmic residues in deformations of plane curves.

Findings

Symmetry of value sets for fractional ideals in Gorenstein curves.

Relation between logarithmic residues and Jacobian ideals.

Behavior of residues under equisingular deformations.

Abstract

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 Kahler 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.