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The space of foliations on projective spaces in positive characteristic

Wodson Mendson, Jorge Vitório Pereira

TL;DR

This work initiates the study of foliations on projective spaces over algebraically closed fields of positive characteristic, with an emphasis on codimension one and degrees 0–2. It establishes that degree-zero foliations are largely described by linear projections, except in characteristic 2 with q=1, and degree-one foliations split into linear pull-backs and logarithmic components, with the number and nature of components depending on the characteristic; a uniform description emerges for p ≥ 5. By extending Medeiros's framework to arbitrary characteristic, the paper analyzes the $p$-curvature, introduces linear and logarithmic component families, and proves a Calvo-Andrade–type stability for generic logarithmic 1-forms in this arithmetic setting. Degree-two results are obtained via specialization from characteristic zero and a general criterion for Log components, supplemented by a Lie-algebraic construction yielding exceptional components in positive characteristic. The authors then synthesize the irreducible components of Fol^1_2 across characteristics, highlighting new phenomena in small primes and outlining open questions for finer classifications in positive characteristic.

Abstract

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces varies with the characteristic of the base field in very low degrees and establishes an arbitrary characteristic version of Calvo-Andrade's stability of generic logarithmic $1$-forms under deformation.

The space of foliations on projective spaces in positive characteristic

TL;DR

This work initiates the study of foliations on projective spaces over algebraically closed fields of positive characteristic, with an emphasis on codimension one and degrees 0–2. It establishes that degree-zero foliations are largely described by linear projections, except in characteristic 2 with q=1, and degree-one foliations split into linear pull-backs and logarithmic components, with the number and nature of components depending on the characteristic; a uniform description emerges for p ≥ 5. By extending Medeiros's framework to arbitrary characteristic, the paper analyzes the -curvature, introduces linear and logarithmic component families, and proves a Calvo-Andrade–type stability for generic logarithmic 1-forms in this arithmetic setting. Degree-two results are obtained via specialization from characteristic zero and a general criterion for Log components, supplemented by a Lie-algebraic construction yielding exceptional components in positive characteristic. The authors then synthesize the irreducible components of Fol^1_2 across characteristics, highlighting new phenomena in small primes and outlining open questions for finer classifications in positive characteristic.

Abstract

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces varies with the characteristic of the base field in very low degrees and establishes an arbitrary characteristic version of Calvo-Andrade's stability of generic logarithmic -forms under deformation.

Paper Structure

This paper contains 30 sections, 22 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Purpose and discussion of the main results
  4. Acknowledgments
  5. Basic definitions
  6. Foliations and distributions
  7. Differential forms
  8. Plücker's equations
  9. The schemes $\textsf{Dist}^{q}_{d}(\mathbb P_{{\mathsf k}}^{n})$ and $\textsf{Fol}^{q}_{d}(\mathbb P_{{\mathsf k}}^{n})$
  10. Foliations of degree zero
  11. Euler's relation
  12. Foliations defined by closed projective polynomial 1-forms
  13. Locally decomposable differential forms with linear coefficients
  14. Foliations of degree zero
  15. Foliations of degree one
  16. ...and 15 more sections

Key Result

Theorem 1

Let $r\ge 2$ and $1 \le d_1 \le \ldots \le d_r$ be positive integers. Let ${\mathsf k}$ be an algebraically closed field of characteristic $p>0$. Assume that one of the following conditions holds: Then the set $\textsf{Log}_{(d_1, \ldots, d_r)}(\mathbb P_{{\mathsf k}}^{n})$ is an irreducible component of $\textsf{Fol}^{1}_{d}(\mathbb P_{{\mathsf k}}^{n})$ (with its reduced scheme structure) for $

Theorems & Definitions (45)

  • Theorem 1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['T:Medeiros']}
  • ...and 35 more