On a conjecture of Azevedo
Archiv der Mathematik, Vol 64, p. 188-198, 1995

Abstract

The paper is concerned with plane irreducible algebroid curves over an algebraically closed field k of characteristic 0.

In his paper "Characterization of Plane Algebroid Curves whose Module of Differentials has Maximum Torsion" (Proc. Nat. Acad. Sci. Vol. 56, 1966), Zariski proved that the differential module of such a curve has "maximum torsion" if and only if it is a curve, which can be parametrized by

X=t ⁿ , Y= t ^m for relatively prime natural numbers n,m .

As an algebroid curve of this particular type is the canonical branch of its equisingularity class; Alberto de Azevedo conjectured (in his PhD thesis "The Jacobian Ideal of a Plane Algebroid Curve", Purdue 1967) that of all algebroid curves forming an equisingularity class, the canonical branch always is the curve with maximum torsion of the differential module.

The paper at hand gives a counterexample to Azevedo's conjecture. The paper is based on results from my diploma thesis; however the computation of the length of torsion of the differential module from a curve's parametrization is done more elegantly. Instead of the method used in Azevedo's PhD thesis, the elegant method described by Zariski in "Le problème des modules pur les branches planes" (Nouvelle édition, Paris, 1986) is used.