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Weak Fano bundles of rank $2$ over hyperquadrics $Q^n$ of dimension $n \ge 5$

Yuta Takahashi

TL;DR

The paper addresses the classification of rank $2$ weak Fano bundles on smooth quadric hypersurfaces $Q^n$ with $n\ge 5$. It adapts the APW94 framework by proving that a normalized $\mathcal{E}$ satisfies $\mathcal{E}(n-1)$ being globally generated and by applying splitting criteria to constrain possible Chern classes. The main result states that such a bundle is either a direct sum of line bundles or, in the case $n=5$, the Cayley bundle on $Q^5$ up to twist; for $n\ge 6$, only split cases persist. This advances the understanding of rank $2$ weak Fano bundles on quadrics and highlights the Cayley bundle's special role in dimension five, with implications for broader weak Fano classifications.

Abstract

A vector bundle whose projectivization becomes a weak Fano variety is called a weak Fano bundle. We present classification results for rank 2 weak Fano bundles on higher-dimensional quadrics $Q^n$ of dimension $\ge 5$.

Weak Fano bundles of rank $2$ over hyperquadrics $Q^n$ of dimension $n \ge 5$

TL;DR

The paper addresses the classification of rank weak Fano bundles on smooth quadric hypersurfaces with . It adapts the APW94 framework by proving that a normalized satisfies being globally generated and by applying splitting criteria to constrain possible Chern classes. The main result states that such a bundle is either a direct sum of line bundles or, in the case , the Cayley bundle on up to twist; for , only split cases persist. This advances the understanding of rank weak Fano bundles on quadrics and highlights the Cayley bundle's special role in dimension five, with implications for broader weak Fano classifications.

Abstract

A vector bundle whose projectivization becomes a weak Fano variety is called a weak Fano bundle. We present classification results for rank 2 weak Fano bundles on higher-dimensional quadrics of dimension .
Paper Structure (9 sections, 16 theorems, 35 equations)

This paper contains 9 sections, 16 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chow ring of quadric hypersurfaces $Q^n$
  4. Some properties of vector bundles
  5. Splitting criterions and numerical properties
  6. Globally generated property
  7. Weak Fano bundle of rank $2$ over $Q^n$ of dimension $n\ge 5$
  8. The case $c_1({\mathcal{E}})^2\ge 4c_2({\mathcal{E}})$
  9. The case $c_1({\mathcal{E}})^2< 4c_2({\mathcal{E}})$

Key Result

Theorem 1.1

Let ${\mathcal{E}}$ be a weak Fano bundle of rank $2$ over $Q^n$ where $n\ge 5$. Then ${\mathcal{E}}$ is a direct sum of line bundles or the Cayley bundle on $Q^5$ up to twist with a line bundle (see Definition def:cayley).

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 2.1: F83
  • Proposition 2.2
  • proof
  • Proposition 2.3: MP97
  • Definition 2.4
  • Theorem 2.5: L041
  • Theorem 2.6: V82
  • Theorem 2.7: L75
  • Proposition 2.8: APW94
  • ...and 21 more