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Planar curve singularities relative to a smooth boundary

Nobuyoshi Takahashi

TL;DR

The paper develops a logarithmic deformation framework for planar curve singularities relative to a smooth boundary, introducing the log equianalytic and log equisingular ideals I^{ea}_{log D}(C) and I^{es}_{log D}(C) and proving a chain of inclusions among tangent-space ideals. It establishes smooth, formally semiuniversal log deformation spaces and explicit log-semiuniversal families, linking log deformations to classical ones via divisors on the boundary. A key result is the invariant τ^{es}_{log D}(C), related to τ^{es}(C ∪ D) by τ^{es}_{log D}(C) = τ^{es}(C ∪ D) − (2w−1), enabling a codimension-based classification. The paper culminates with a detailed catalog of low-codimension singularities (τ^{es}_{log D}(C) ≤ 3) for w ≥ 8 when δ(C) ≤ 3, providing explicit normal forms and associated ideals, illustrating the structured interaction between curve singularities and boundary data in log geometry.

Abstract

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove results on the inclusion relations between ideals describing different deformations. A classification is given of singularities expected to appear in codimension $\leq 3$ in a general family with $w\geq 8$.

Planar curve singularities relative to a smooth boundary

TL;DR

The paper develops a logarithmic deformation framework for planar curve singularities relative to a smooth boundary, introducing the log equianalytic and log equisingular ideals I^{ea}_{log D}(C) and I^{es}_{log D}(C) and proving a chain of inclusions among tangent-space ideals. It establishes smooth, formally semiuniversal log deformation spaces and explicit log-semiuniversal families, linking log deformations to classical ones via divisors on the boundary. A key result is the invariant τ^{es}_{log D}(C), related to τ^{es}(C ∪ D) by τ^{es}_{log D}(C) = τ^{es}(C ∪ D) − (2w−1), enabling a codimension-based classification. The paper culminates with a detailed catalog of low-codimension singularities (τ^{es}_{log D}(C) ≤ 3) for w ≥ 8 when δ(C) ≤ 3, providing explicit normal forms and associated ideals, illustrating the structured interaction between curve singularities and boundary data in log geometry.

Abstract

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number with the boundary, and prove results on the inclusion relations between ideals describing different deformations. A classification is given of singularities expected to appear in codimension in a general family with .

Paper Structure

This paper contains 9 sections, 32 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Deformations of planar curves
  3. Equigeneric, equiclassical and equisingular deformations
  4. Deformations of planar curves in log surfaces
  5. Equisingular deformations in the log setting
  6. Inclusion relations between ideals
  7. Singularities with ${\tau^{\mathrm{es}}_{\log{D}}}(C)\leq 3$
  8. Case $\delta(C)=2$
  9. Case $\delta(C)=3$

Key Result

Theorem 1.1

Let $(S, D)$ be a formal smooth surface pair and $C\subset S$ a reduced algebroid curve not containing $D$, and let $w=C.D$. Then the functor ${\underline{\mathcal{D}ef}}_{C\to ({{S}\supset{D}\supset{P}})}^w$ of deformations with the constant intersection multiplicity $w$ has a smooth formal semiuni

Theorems & Definitions (76)

  • Theorem 1.1: Corollary \ref{['cor_versal_log']}
  • Theorem 1.2: Theorem \ref{['thm_inclusion']}
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Schlessinger1968
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 66 more