Families of ICIS with constant total Milnor number

TL;DR

This paper proves that families of ICIS with constant total Milnor number do not experience singularity coalescence, extending known results for hypersurfaces and applying monodromy and Lefschetz number techniques.

Contribution

It extends the classical coalescence result to ICIS and introduces new applications for functions on ICIS using monodromy and Lefschetz number analysis.

Findings

No coalescence of singularities in ICIS with constant Milnor number

Lefschetz number of monodromy is zero for singular ICIS

Applications to families of functions on ICIS

Abstract

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well known result of Gabrielov, Lazzeri and L\^e for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.