On the number of generators of the torsion module of differentials

Abstract:

In this paper we study the (minimum) global number of generators of the torsion module of differentials of affine hypersurfaces with only isolated singularities. We show that for reduced plane curves the torsion module of differentials can be generated by at most two elements, whereas for higher codimensions there is no universal upper bound. We then proceed to give explicit examples. In particular (when $N \geq 5$) , we give examples of a reduced hypersurface with a single isolated singularity at the origin in $\mathbf {A}^{N}_{K}$ that require \[ \frac {N!}{2} + N(N-1)/2\] generators for the torsion module, $\operatorname {Torsion}(\Omega _{A/K}^{N-1})$.