The Delta Invariant and Simultaneous Normalization for Families of Isolated Non-Normal Singularities

TL;DR

This paper introduces a modified delta invariant for isolated non-normal singularities, demonstrating its upper semicontinuity in flat families and linking its constancy to fiberwise normalization, generalizing prior results.

Contribution

It defines a new delta invariant for non-normal singularities of any dimension, extending existing theories and improving algorithms for computing curve genus.

Findings

Delta invariant is upper semicontinuous in flat families.

Fiberwise normalization occurs iff the delta invariant is locally constant.

Results generalize Teissier and Chiang-Hsieh--Lipman's work for reduced curve singularities.

Abstract

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points and prove that it behaves upper semicontinuous in flat families parametrized by an arbitrary principal ideal domain. Moreover, if the fibers contain no isolated points, then the familly admits a fiberwise normalization iff the delta invariant is locally constant. The results generalize results by Teissier and Chiang-Hsieh--Lipman for families of reduced curve singularities and provide possible improvements for algorithms to compute the genus of a curve.