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Constructive exceptional bundles on $\mathbb{P}^3$

Benjamin Gould

Abstract

We give a complete classification of the Chern characters of constructive exceptional vector bundles on $\mathbb{P}^3$ analogous to the work of Drézet and Le Potier on $\mathbb{P}^2$, and using this classification prove that a constructive exceptional bundle $E$ on $\mathbb{P}^3$ with $μ(E) \geq 0$ is globally generated.

Constructive exceptional bundles on $\mathbb{P}^3$

Abstract

We give a complete classification of the Chern characters of constructive exceptional vector bundles on analogous to the work of Drézet and Le Potier on , and using this classification prove that a constructive exceptional bundle on with is globally generated.
Paper Structure (16 sections, 12 theorems, 72 equations, 10 figures)

This paper contains 16 sections, 12 theorems, 72 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Exceptional bundles on $\mathbb{P}^3$
  3. Acknowledgements
  4. Preliminaries
  5. Exceptional bundles and helix theory on $\mathbb{P}^n$
  6. Exceptional bundles on $\mathbb{P}^2$
  7. Exceptional bundles and stability on $\mathbb{P}^2$
  8. Exceptional bundles on $\mathbb{P}^3$
  9. The helix condition on $\mathbb{P}^3$
  10. Exceptional bundles and stability on $\mathbb{P}^3$
  11. Graphs associated to helices on $\mathbb{P}^3$
  12. Admissible replacements
  13. The graph $\Gamma_{\mathbb{P}^3}'$
  14. Classifying constructive exceptional bundles
  15. Explicit computations
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $E$ be a constructive exceptional bundle on $\mathbb{P}^3$. There is a well-defined way to choose a distinguished constructive helix $\sigma = \sigma(E)$ containing $E$ and a full exceptional collection $(E_1, E, E_2, E_3)$ of sheaves in $\sigma$ containing $E$. For such a collection, we write $ defined inductively as and for $q \geq 0$,

Figures (10)

  • Figure 1: Mutations for Example \ref{['adm-ex']}.
  • Figure 2: Mutations for Lemma \ref{['admissible-lemma']}.
  • Figure 3: The commuting base case for Proposition \ref{['adm-replacement']}.
  • Figure 4: Extension of Figure \ref{['fig:B']} for Proposition \ref{['adm-replacement']}.
  • Figure 5: The non-commuting base case for Proposition \ref{['adm-replacement']}.
  • ...and 5 more figures

Theorems & Definitions (43)

  • Theorem 1.1: = Theorem \ref{['theorem']}
  • Theorem 1.2: = Theorem \ref{['gg']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • ...and 33 more