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The Complete Intersection Discrepancy of a Curve II: Families of Curves

Andrei Benguş-Lasnier, Terence Gaffney, Antoni Rangachev

Abstract

We study equisingularity of families of reduced curves over smooth parameter spaces of arbitrary positive dimension, using the difference between two analytic invariants of a curve singularity: the multiplicity of its Jacobian ideal and its complete intersection discrepancy. This difference provides a fiberwise multiplicity criterion for Whitney equisingularity. We prove that Whitney equisingularity (equivalently, strong simultaneous resolution) is characterized by equidimensionality of the fibers of the exceptional locus of either the relative conormal space or the relative Nash blowup. We further show that this condition is equivalent to the emptiness of the relative polar variety of smallest dimension. In addition, we establish that the Milnor number of a reduced curve is Zariski upper semicontinuous. As an application, we show that the constancy of the Milnor number in a family of reduced curves is equivalent to its topological equisingularity.

The Complete Intersection Discrepancy of a Curve II: Families of Curves

Abstract

We study equisingularity of families of reduced curves over smooth parameter spaces of arbitrary positive dimension, using the difference between two analytic invariants of a curve singularity: the multiplicity of its Jacobian ideal and its complete intersection discrepancy. This difference provides a fiberwise multiplicity criterion for Whitney equisingularity. We prove that Whitney equisingularity (equivalently, strong simultaneous resolution) is characterized by equidimensionality of the fibers of the exceptional locus of either the relative conormal space or the relative Nash blowup. We further show that this condition is equivalent to the emptiness of the relative polar variety of smallest dimension. In addition, we establish that the Milnor number of a reduced curve is Zariski upper semicontinuous. As an application, we show that the constancy of the Milnor number in a family of reduced curves is equivalent to its topological equisingularity.
Paper Structure (16 sections, 28 theorems, 99 equations, 2 figures)

This paper contains 16 sections, 28 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conormal spaces, polar varieties, and multiplicities
  3. Conormal spaces
  4. Polar varieties and the Milnor number
  5. Integral closure and multiplicities
  6. A flatness result
  7. Proof of Theorem \ref{['main mpt']}. The relative Nash blowup
  8. Proof of Theorem \ref{['main mpt']}
  9. The relative Nash blowup
  10. A Riemann--Hurwitz formula for singular curves
  11. Zariski upper semicontinuity
  12. Equisingularity of curves
  13. Strong simultaneous resolution
  14. Topological equisingularity
  15. Whitney equisingularity
  16. ...and 1 more sections

Key Result

Theorem 1.2

We have for $y \neq 0$ in a Zariski open subset of $Y$.

Figures (2)

  • Figure 1:
  • Figure 2: Here $X_y$ has two irreducible components meeting at point. The boundary components of $(X_{y}^c)_1$ and $(X_{y}^c)_2$ are colored with red.

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 43 more