Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

TL;DR

This paper introduces a new invariant called the multiplicity sequence for analytic surface singularities in ^3, demonstrating that constancy of this invariant characterizes Zariski equisingularity in families.

Contribution

The paper defines the multiplicity sequence invariant for surface singularities and proves its constancy characterizes Zariski equisingularity in analytic families.

Findings

The invariant mult^* (V) is well-defined for all surface singularities in ^3.

Constancy of mult^* (V_t) in a family implies Zariski equisingularity.

Mult^* (V) captures multiplicities of successive discriminants via generic projections.

Abstract

We associate to every analytic surface singularity $(V,0)$ in $(\mathbb C^3,0)$, not necessarily isolated, an invariant $mult^* (V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in (\mathbb C^l,0)$, is generically Zariski equisingular if and only if $mult^* (V_t)$ is constant. The invariant, that we call the multiplicity sequence of $V$, takes into account the multiplicities of the successive discriminants of $V$ by generic corank one projections.