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The Saito vector field of a germ of complex plane curve

Yohann Genzmer

Abstract

In this article, we prove that an algorithm introduced by the author in a previous work and giving the generic dimension of the moduli space of a germ of curve in the complex plane that is the union of smooth curves, can be used identically to find this dimension for any kind of germ of plane curve.

The Saito vector field of a germ of complex plane curve

Abstract

In this article, we prove that an algorithm introduced by the author in a previous work and giving the generic dimension of the moduli space of a germ of curve in the complex plane that is the union of smooth curves, can be used identically to find this dimension for any kind of germ of plane curve.
Paper Structure (10 sections, 4 theorems, 158 equations, 7 figures, 1 table)

This paper contains 10 sections, 4 theorems, 158 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Saito dicriticity of an ordered numbered tree.
  3. Saito foliations of a germ of curve and its deformation.
  4. Semi-local models for Saito foliations.
  5. The dicritical model.
  6. The non dicritical model.
  7. Gluing local models.
  8. Deformation of $\mathcal{F}$.
  9. Generic dimension of the moduli space $\textup{Mod}\left(C\right)$
  10. Examples

Key Result

Theorem 1

Let $C$ be a germ of curve in $\left(\mathbb{C}^{2},0\right)$ generic in its moduli space. Let $X$ be Saito for $C$. Let $\mathbb{A}$ be the dual tree of the desingularization process $E$ of $C.$ We number a vertex $s$ of $\mathbb{A}$ by the number of singular points of the strict transform $X^{E}$

Figures (7)

  • Figure 2.1: Inductive construction of the partial order $\leq$ on $\mathbb{A}$.
  • Figure 2.3: An ordered tree, its numbering and its dicriticity.
  • Figure 2.4: The cusp tree and its Saito dicriticity
  • Figure 2.5: Unique admissible choice of $\Delta=\left(\Delta_{1},\Delta_{2}\right).$
  • Figure 2.6: Mixed branch stopping from being mixed at the $N^{th}$ vertex.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem
  • Definition 1
  • Example 2
  • Theorem 3
  • Example 4
  • Example 5
  • proof : Proof of Theorem \ref{['Theoreme.Un']}.
  • Definition 6
  • Proposition 7
  • proof
  • ...and 5 more