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On the quasi-ordinary cuspidal foliations in (C^3,0)

Percy Fernández Sánchez, Jorge Mozo Fernández

Abstract

We study a class of holomorphic foliations in (C^3,0) that can be desingularized following the same desingularization chain that a certain quasi-ordinary surface. This intends to be a generalization to the dimension three of the cuspodal foliations in dimension two studied by different authors.

On the quasi-ordinary cuspidal foliations in (C^3,0)

Abstract

We study a class of holomorphic foliations in (C^3,0) that can be desingularized following the same desingularization chain that a certain quasi-ordinary surface. This intends to be a generalization to the dimension three of the cuspodal foliations in dimension two studied by different authors.

Paper Structure

This paper contains 5 sections, 3 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction and motivation
  2. Simple singularities of foliations and analytic classification
  3. Reduction of singularities and topology of the divisor
  4. Reduction of the separatrix to a canonical form
  5. Classification of the singularities

Key Result

Proposition 1

The foliation $\mathcal{F}_{\Omega}$ is analytically equivalent to a foliation defined by the one-form where $r=d$ if $2k\geq d$ and $r=2k$ if $2k< d$. In particular, the separatrix of the foliation $\mathcal{F}_{\Omega}$ is analytically equivalent to $S:\:z^2+(x^{p'}y^{q'})^r=0$.

Figures (1)

  • Figure 2: Coordinates $(x',s',t')$ the singular point $D_{\frac{p-1}{2}}\cap D_{\frac{p+q}{2}-2}\cap D_{\frac{p+q}{2}-1 }$

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 1