On Delta for parameterized Curve Singularities

Gert-Martin Greuel; Gerhard Pfister

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arXiv · 2101.01784

On Delta for parameterized Curve Singularities

Gert-Martin Greuel·Gerhard Pfister

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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.

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Gert-Martin Greuel, Gerhard Pfister. 2021-01-05. On Delta for parameterized Curve Singularities. https://arxiv.org/abs/2101.01784

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