Über eine Vermutung von Azevedo
Diplomarbeit, Universität des Saarlandes, 1994
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Abstract

The thesis 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 the algebroid curves in an equisingularity class, the canonical branch always is the curve with maximum torsion of the differential module.

The thesis describes in detail a method to compute the length of torsion of the curve's differential module in terms of a parametrization for the curve. This method had been sketched in Azevedo's PhD thesis. Finally, a counterexample to Azevedo's conjecture is given.