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

Adam Parusiński, Laurenţiu Păunescu

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.

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

Abstract

We associate to every analytic surface singularity in , not necessarily isolated, an invariant and show that an analytic family of such singularities , , is generically Zariski equisingular if and only if is constant. The invariant, that we call the multiplicity sequence of , takes into account the multiplicities of the successive discriminants of by generic corank one projections.
Paper Structure (6 sections, 14 theorems, 15 equations)

This paper contains 6 sections, 14 theorems, 15 equations.

Table of Contents

  1. Zariski equisingularity of surface singularities in $\mathbb{C}^3$
  2. Nested uniformly transverse system of coordinates
  3. $\nu$-transverse Zariski equisingular families
  4. The multiplicity sequence
  5. multiplicity sequence for isolated singularities
  6. Proof of Main Theorem

Key Result

Theorem 2

Consider the germ of an analytic family of surface singularities in $\mathbb{C}^3$ where $t\in (\mathbb{C}^l,0)$ is considered as a parameter. Then the following conditions are equivalent:

Theorems & Definitions (28)

  • Remark 1
  • Theorem 2
  • Corollary 3
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Lemma 1.4: PP24
  • Theorem 1.5: PP24
  • Theorem 1.6: PP24
  • Definition 1.7
  • ...and 18 more