Torsion of Differential Modules of Quasihomogeneous Algebroid Curves
Ann. Univ. Sarav. Vol 9 No.4, 1999

Abstract

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

R. Berger conjectured (in his paper "On the Torsion of the Differential Module of a Curve Singularity", Arch. Math Vol. 50, 1988) that the length of the torsion submodule of such a differential module does not increase if the curve is replaced by a larger ring still contained in the integral closure of the curve.

The paper at hand develops an algorithm to compute the length of torsion for such differential modules in terms of the value semigroup of the curve. This algorithm is used to compute a counterexample to Berger's conjecture.

Since the larger ring constructed in the counterexample is strictly contained in the first quadratic transform of the curve, the following variation of the conjecture might still hold:

Conjecture: The length of the torsion submodule of the differential module decreases if the curve is replaced by its first quadratic transformation.

As a second application of the algorithm we examine this conjecture for curves of small multiplicity: We show that it is decidable by a computer whether the conjecture holds for all curves of a given, fixed multiplicity. For curves of multiplicity n=2,3 the corresponding computation can be done by hand. For n=4 it has been done with the aid of a computer program.
The conjecture remains open for n >= 5 .