On Delta for parameterized Curve Singularities

TL;DR

This paper proves that the delta invariant of parameterized reduced curve singularities is semicontinuous over Noetherian base schemes, providing bounds for determinacy and addressing classification issues in positive characteristic.

Contribution

It establishes the semicontinuity of the delta invariant for families of parameterized curve singularities over arbitrary Noetherian schemes, extending previous results to a more general setting.

Findings

Delta invariant is semicontinuous in families of parameterized singularities.

Provides bounds for right-left determinacy based on delta.

Addresses classification problems in positive characteristic.

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.