Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rank two weak Fano bundles on del Pezzo threefolds of degree five

Takeru Fukuoka, Wahei Hara, Daizo Ishikawa

TL;DR

This work completes the classification of rank two weak Fano bundles on del Pezzo threefolds of Picard rank one by treating the degree five case with an explicit, resolution-based description. It uses the full strong exceptional collection on X, together with mutations, to derive concrete resolutions for all indecomposable rank two weak Fano bundles, organized into eight types with detailed Chern class data. The paper then analyzes moduli spaces of these bundles, showing irreducibility and smoothness in key cases, and reveals Brauer obstructions that prevent fineness in the c2=4 case, while establishing fineness only for c2=3. It also proves that for deg ≤ 2, all rank two weak Fano bundles split, completing the classification on del Pezzo 3-folds of Picard rank one. The results highlight deep connections between derived category techniques, instanton bundles, and quiver representations (5-Kronecker) in understanding moduli and positivity properties of vector bundles on Fano threefolds.

Abstract

This paper classifies rank two vector bundles on a del Pezzo threefold $X$ of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on $X$, and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result give a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one.

Rank two weak Fano bundles on del Pezzo threefolds of degree five

TL;DR

This work completes the classification of rank two weak Fano bundles on del Pezzo threefolds of Picard rank one by treating the degree five case with an explicit, resolution-based description. It uses the full strong exceptional collection on X, together with mutations, to derive concrete resolutions for all indecomposable rank two weak Fano bundles, organized into eight types with detailed Chern class data. The paper then analyzes moduli spaces of these bundles, showing irreducibility and smoothness in key cases, and reveals Brauer obstructions that prevent fineness in the c2=4 case, while establishing fineness only for c2=3. It also proves that for deg ≤ 2, all rank two weak Fano bundles split, completing the classification on del Pezzo 3-folds of Picard rank one. The results highlight deep connections between derived category techniques, instanton bundles, and quiver representations (5-Kronecker) in understanding moduli and positivity properties of vector bundles on Fano threefolds.

Abstract

This paper classifies rank two vector bundles on a del Pezzo threefold of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on , and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result give a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one.

Paper Structure

This paper contains 28 sections, 24 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Known Results
  4. Main Results
  5. Classification on a del Pezzo threefold of degree $5$.
  6. Moduli spaces
  7. Classifications on del Pezzo $3$-folds of degree $\leq 2$
  8. Strategy for Proofs
  9. Strategy for Proof of Theorem \ref{['mainthm-resol']}
  10. Plan of Proof of Theorem \ref{['mainthm-moduli']}
  11. Organization of this article
  12. Preliminaries
  13. Weak Fano bundles on del Pezzo threefolds of Picard rank one
  14. Derived categories
  15. Rank $2$ weak Fano bundles with odd first Chern classes
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be a del Pezzo $3$-fold of degree $5$. Then, every weak Fano bundle $\mathcal{E}$ is isomorphic to one of the following up to twisting with a line bundle. In the above results, we regard the numerical class groups of $\mathbb{Z}$-coefficients $N^{1}(X)_{\mathbb{Z}}$ and $N^{2}(X)_{\mathbb{Z}}$ as $\mathbb{Z}$ by taking the effective classes generating these class groups. Furthermore, on a

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 50 more