The Delta Invariant and Fiberwise Normalization for Families of isolated Non-Normal Singularities

Gert-Martin Greuel and Gerhard Pfister

Journal of Singularities
volume 25 (2022), 173-196

Received: 12 May 2021. Received in revised form: 26 February 2022.

DOI: 10.5427/jsing.2022.25j

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Abstract:

We prove the semicontinuity of the delta invariant in a family of schemes or analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular for families of generically reduced curves. We define and use a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points. Our results generalize results by Teissier and Chiang-Hsieh--Lipman for families of reduced curve singularities. The base ring for our families can be an arbitrary PID such that our semicontinuity result provides possible improvements for algorithms to compute the genus of a curve.

2020 Mathematical Subject Classification:

13B22, 13B40, 14B05, 14B07

Key words and phrases:

Isolated non-normal singularity, delta invariant, semicontinuity, generically reduced curves, simultaneous normalization

Author(s) information:

Gert-Martin Greuel
University of Kaiserslautern
Department of Mathematics
Erwin-Schroedinger Str.
67663 Kaiserslautern, Germany
greuel@mathematik.uni-kl.de

Gerhard Pfister
University of Kaiserslautern
Department of Mathematics
Erwin-Schroedinger Str.
67663 Kaiserslautern, Germany
pfister@mathematik.uni-kl.de