On Delta for parameterized Curve Singularities

Abstract

We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes.