Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of pullbacks of foliations on weighted projective spaces

Javier Gargiulo Acea, Ariel Molinuevo, Federico Quallbrunn, Sebastián Lucas Velazquez

TL;DR

This work establishes a stability-type theorem for foliations on projective spaces that are pullbacks of foliations with split tangent sheaf on weighted projective spaces, under precise non-positivity and codimension conditions. By showing that every first-order deformation of such a pullback is the pullback of a base deformation (up to a controlled map deformation), the authors construct numerous irreducible components of the moduli spaces of codimension-1 foliations on P^n and prove generic reducedness along these components. The results unify and extend prior stability phenomena, including foliations induced by Lie group actions and logarithmic foliations, and encompass pullbacks from toric varieties as a broader source of stable families. Overall, the paper provides a versatile toolkit for generating and understanding stable components in spaces of foliations ^1(^n, d) via weighted-projective pullbacks.

Abstract

We show a stability-type theorem for foliations on projective spaces which arise as pullbacks of foliations with a split tangent sheaf on weighted projective spaces. As a consequence, we will be able to construct many irreducible components of the corresponding spaces of foliations, most of them being previously unknown. This result also provides an alternative and unified proof for the stability of other families of foliations.

Stability of pullbacks of foliations on weighted projective spaces

TL;DR

This work establishes a stability-type theorem for foliations on projective spaces that are pullbacks of foliations with split tangent sheaf on weighted projective spaces, under precise non-positivity and codimension conditions. By showing that every first-order deformation of such a pullback is the pullback of a base deformation (up to a controlled map deformation), the authors construct numerous irreducible components of the moduli spaces of codimension-1 foliations on P^n and prove generic reducedness along these components. The results unify and extend prior stability phenomena, including foliations induced by Lie group actions and logarithmic foliations, and encompass pullbacks from toric varieties as a broader source of stable families. Overall, the paper provides a versatile toolkit for generating and understanding stable components in spaces of foliations ^1(^n, d) via weighted-projective pullbacks.

Abstract

We show a stability-type theorem for foliations on projective spaces which arise as pullbacks of foliations with a split tangent sheaf on weighted projective spaces. As a consequence, we will be able to construct many irreducible components of the corresponding spaces of foliations, most of them being previously unknown. This result also provides an alternative and unified proof for the stability of other families of foliations.
Paper Structure (9 sections, 19 theorems, 76 equations, 1 table)

This paper contains 9 sections, 19 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Preliminaries
  4. Idea of the proof
  5. Proof of main Theorem.
  6. Applications
  7. Pullbacks of foliations on $\mathbb{P}^2(\overline{e})$
  8. Pullbacks of foliations induced by Lie group actions
  9. A word on pullbacks from toric varieties

Key Result

Theorem 1

Let $X=\mathbb{P}^{m}(\overline{e})$ be a weighted projective space, $\delta\in \mathrm{Cl}(X)$ and $\alpha \in \mathbb{P} H^0(X,\widehat{\Omega}^1_X(\delta))$ defining a foliation with split tangent sheaf with non-positive splitting and such that $d\alpha$ vanishes in codimension greater than $2$. as sections of $\Omega^1_{\mathbb{P}^n\times \Sigma|\Sigma}(k\delta)$, where $\Sigma=\mathrm{Spec}(

Theorems & Definitions (45)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1: Main Theorem
  • Corollary 2
  • Corollary 3
  • Definition 2.1
  • ...and 35 more