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Distributions with Unstable Tangent Sheaf on $\mathbb{P}^3$

Pedro Barbassa

Abstract

We study codimension one distributions on the projective three-space, focusing on cases where the tangent sheaf of the distribution is nonsplit and unstable. We relate the order of nonstability to the degree of the induced subfoliation by curves, showing that the order of nonstability is bounded. Moreover, we classify the tangent sheaf of the codimension one distributions that admit a subfoliation by curves of degree 1. In other words, assuming the sheaf is nonsplit, we classify the situations in which the tangent sheaf attains the maximal possible order of nonstability.

Distributions with Unstable Tangent Sheaf on $\mathbb{P}^3$

Abstract

We study codimension one distributions on the projective three-space, focusing on cases where the tangent sheaf of the distribution is nonsplit and unstable. We relate the order of nonstability to the degree of the induced subfoliation by curves, showing that the order of nonstability is bounded. Moreover, we classify the tangent sheaf of the codimension one distributions that admit a subfoliation by curves of degree 1. In other words, assuming the sheaf is nonsplit, we classify the situations in which the tangent sheaf attains the maximal possible order of nonstability.
Paper Structure (9 sections, 13 theorems, 84 equations)

This paper contains 9 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributions and Foliations on Smooth Varieties
  4. Stability and Nonstability of Reflexive Sheaves
  5. Distributions and Foliations on ${\mathbb{P}^{3}}$
  6. Foliation by Curves on ${\mathbb{P}^{3}}$
  7. Codimension One Distributions on ${\mathbb{P}^{3}}$
  8. Codimension One Distributions with Unstable Tangent Sheaf
  9. Final Remarks

Key Result

Lemma 1

Let $\mathcal{F}$ be a codimension one distribution of degree $d$ on ${\mathbb{P}^{3}}$ such that $T_\mathcal{F}$ is nonsplit. Then $T_\mathcal{F}$ is unstable if and only if $d \geq 3$ and where $\varepsilon \in \{0,1\}$ and $\varepsilon \equiv d (\textnormal{mod }2)$. Moreover, $T_\mathcal{F}$ will be unstable of order $\frac{d+ \varepsilon}{2} - t_{\mathcal{F}}$.

Theorems & Definitions (38)

  • Lemma : \ref{['lemmaintro']}
  • Definition 2.1
  • Remark
  • Remark
  • Definition 2.2
  • Remark
  • Definition 2.3
  • Remark
  • Lemma 2.4: RH2
  • Definition 2.5
  • ...and 28 more