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Dicritical foliations and semiroots of plane branches

Nuria Corral, Marcelo E. Hernandes, Maria Elenice R. Hernandes

Abstract

In this work we describe dicritical foliations in $(\mathbb{C}^2,0)$ at a triple point of the resolution dual graph of an analytic plane branch $\mathcal{C}$ using its semiroots. In particular, we obtain a constructive method to present a one-parameter family $\mathcal{C}_{u}$ of separatrices for such foliations. As a by-product we relate the contact order between a special member of $\mathcal{C}_{u}$ and $\mathcal{C}$ with analytic discrete invariants of plane branches.

Dicritical foliations and semiroots of plane branches

Abstract

In this work we describe dicritical foliations in at a triple point of the resolution dual graph of an analytic plane branch using its semiroots. In particular, we obtain a constructive method to present a one-parameter family of separatrices for such foliations. As a by-product we relate the contact order between a special member of and with analytic discrete invariants of plane branches.
Paper Structure (6 sections, 15 theorems, 111 equations)

This paper contains 6 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Semiroots and dicritical foliations
  4. Analytical invariants of $\mathcal{C}_F$ and dicritical foliations
  5. The Zariski invariant of $\mathcal{C}_F$
  6. Technical Lemmas

Key Result

Lemma 3.1

Given any $n,m\in\mathbb{Z}_{>0}$ and $\omega\in\Omega^1$, there exist $H_1,H_2\in\mathbb{C}\{x,y\}$ such that

Theorems & Definitions (25)

  • Lemma 3.1: Azevedo
  • Example 3.2
  • Proposition 3.3: Zariski, Zariski
  • Proposition 3.4: see Abh or Popescu
  • Remark 3.5
  • Example 3.6
  • Theorem 3.7
  • Corollary 3.8
  • Remark 3.9
  • Example 3.10
  • ...and 15 more