On Delta for parameterized Curve Singularities

DOI10.1016/J.JALGEBRA.2022.10.021zbMATH Open1511.13011arXiv2101.01784MaRDI QIDQ6357657

Publication date: 5 January 2021

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.

Mathematics Subject Classification ID

Integral closure of commutative rings and ideals (13B22) Étale and flat extensions; Henselization; Artin approximation (13B40) Galois theory and commutative ring extensions (13B05) Secant varieties, tensor rank, varieties of sums of powers (14N07)

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