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The Gauss map of a projective foliation

Claudia R. Alcántara, Dominique Cerveau, Alcides Lins Neto

TL;DR

<p>The paper studies the Gauss map of codimension-one holomorphic foliations on complex projective space, with a focus on the threefold case $\mathbb{P}^3$ and the birationality of the Gauss map. It establishes that the maximal topological degree of the Gauss map for degree-$d$ foliations on $\mathbb{P}^3$ is $d^2$, and when achieved, the foliation admits a rational first integral of the form $F/L^{d+1}$ with $\deg F=d+1$ and $\deg L=1$. It provides birationality criteria in terms of singular data via the quantity $\sum_V \mu(\mathcal F,V)\deg V$ and plane-restriction arguments, and applies these to logarithmic and monomial foliations, including a detailed study of exceptional components such as $\mathcal E(2,3;3)$ and $\mathcal E(3,4;4)$. The work also describes methods to construct and invert Gauss maps and presents several explicit examples illustrating when birationality holds or fails, highlighting the rich structure of the moduli spaces of foliations.>

Abstract

In this paper, we study the Gauss map of a holomorphic codimension one foliation on the projective space $\mathbb{P}^n$, $n\ge 2$, mainly the case $n=3$. Among other things, we will investigate the case where the Gauss map is birational.

The Gauss map of a projective foliation

TL;DR

<p>The paper studies the Gauss map of codimension-one holomorphic foliations on complex projective space, with a focus on the threefold case and the birationality of the Gauss map. It establishes that the maximal topological degree of the Gauss map for degree- foliations on is , and when achieved, the foliation admits a rational first integral of the form with and . It provides birationality criteria in terms of singular data via the quantity and plane-restriction arguments, and applies these to logarithmic and monomial foliations, including a detailed study of exceptional components such as and . The work also describes methods to construct and invert Gauss maps and presents several explicit examples illustrating when birationality holds or fails, highlighting the rich structure of the moduli spaces of foliations.>

Abstract

In this paper, we study the Gauss map of a holomorphic codimension one foliation on the projective space , , mainly the case . Among other things, we will investigate the case where the Gauss map is birational.
Paper Structure (24 sections, 21 theorems, 171 equations)

This paper contains 24 sections, 21 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Basic properties, Results, and Examples
  3. Problems
  4. Proofs
  5. Proof of Theorem \ref{['pM']}
  6. Proof of Proposition \ref{['p1']}
  7. Proof of Proposition \ref{['p2']}
  8. Proof of Theorem \ref{['tnp']}
  9. Proofs of Theorem \ref{['p4']} and of Corollaries \ref{['c']}, \ref{['c12']} and \ref{['c14']}.
  10. Proof of Theorem \ref{['p4']}
  11. Proof of corollary \ref{['c']}
  12. Proof of corollary \ref{['c12']}.
  13. Proof of corollary \ref{['new']}
  14. Proof of corollary \ref{['c14']}
  15. Proof of Theorem \ref{['pmon']}
  16. ...and 9 more sections

Key Result

Proposition 1

If the Gauss map of a foliation $\mathcal{F}$ on $\mathbb{P}^3$ is not dominant then, either $\mathcal{F}$ is a linear pull-back of a foliation on $\mathbb{P}^2$, or $\mathcal{F}$ has a rational first integral $F$, which can be written in homogeneous coordinates as where $P$, $Q$ and $R$ are homogeneous polynomials such that $dg(P)+1=dg(Q)=dg(R)$.

Theorems & Definitions (53)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Proposition 1
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • Remark 6
  • ...and 43 more