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Rank two Weak Fano bundles on Fano threefolds of Picard rank one

Takeru Fukuoka, Wahei Hara, Daizo Ishikawa

TL;DR

The paper advances the classification of rank two weak Fano bundles on Fano threefolds with Picard rank one, focusing on the odd index cases where X is either the quadric threefold Q^3 or index one. It delivers a complete numerical and geometric classification on Q^3, including explicit resolutions for each candidate, and analyzes the moduli space of the (-1,4) case via quiver representations, showing it is irreducible, smooth, and fine of dimension 18. For index-one Fano threefolds, it separates the even and odd c1 cases: when c1 is even, all weak Fano rank-two bundles are split; when c1 is odd, the classification includes split, globally generated bundles with controlled c2, and indecomposable ones whose existence is established using elliptic normal curves and Hartshorne–Serre constructions. An embedding theorem links weak Fano bundles with c1(F)=c1(X) to Grassmannian embeddings, except in a few geometric exceptions, connecting the classification to linear sections of homogeneous varieties and Gushel–Mukai structures. The work combines Hilbert schemes of conics, Noether–Lefschetz theory, Brill–Noether theory on K3s, Serre constructions, and derived-category methods to produce a cohesive, geometry-grounded catalog of rank two weak Fano bundles on these Fano threefolds.

Abstract

We classify rank two vector bundles on a Fano threefold of Picard rank one whose projectivizations are weak Fano. We also prove the existence of examples for each case of the classification result. Our classification includes detailed resolutions of them on the quadric threefold.

Rank two Weak Fano bundles on Fano threefolds of Picard rank one

TL;DR

The paper advances the classification of rank two weak Fano bundles on Fano threefolds with Picard rank one, focusing on the odd index cases where X is either the quadric threefold Q^3 or index one. It delivers a complete numerical and geometric classification on Q^3, including explicit resolutions for each candidate, and analyzes the moduli space of the (-1,4) case via quiver representations, showing it is irreducible, smooth, and fine of dimension 18. For index-one Fano threefolds, it separates the even and odd c1 cases: when c1 is even, all weak Fano rank-two bundles are split; when c1 is odd, the classification includes split, globally generated bundles with controlled c2, and indecomposable ones whose existence is established using elliptic normal curves and Hartshorne–Serre constructions. An embedding theorem links weak Fano bundles with c1(F)=c1(X) to Grassmannian embeddings, except in a few geometric exceptions, connecting the classification to linear sections of homogeneous varieties and Gushel–Mukai structures. The work combines Hilbert schemes of conics, Noether–Lefschetz theory, Brill–Noether theory on K3s, Serre constructions, and derived-category methods to produce a cohesive, geometry-grounded catalog of rank two weak Fano bundles on these Fano threefolds.

Abstract

We classify rank two vector bundles on a Fano threefold of Picard rank one whose projectivizations are weak Fano. We also prove the existence of examples for each case of the classification result. Our classification includes detailed resolutions of them on the quadric threefold.
Paper Structure (32 sections, 36 theorems, 69 equations)

This paper contains 32 sections, 36 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Classification on $\mathbb{Q}^{3}$
  3. Application to the moduli space
  4. Classification on a Fano threefold whose Fano index is one
  5. Geometric feature of rank two weak Fano bundles
  6. Outline of our classification
  7. Outline of our proof of Theorem \ref{['main-emb']}
  8. Organization of this paper
  9. Weak Fano bundles and Hilbert scheme of conics
  10. An observation for $(2,1)$-type crepant contractions from almost Fano fourfolds
  11. Positivity of weak Fano bundles along conics
  12. Noether--Lefschetz theorems for weak Fano varieties
  13. Preliminaries for K3 surfaces
  14. Ravindra--Srinivas--Grothendieck--Noether--Lefschetz theorem
  15. Numerical Classification on $\mathbb{Q}^{3}$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a normalized rank two vector bundle on $\mathbb{Q}^{3}$. Then $\mathcal{E}$ is weak Fano if and only if $\mathcal{E}$ is isomorphic to one of the following. Furthermore, there exist examples for each case (i) -- (vi). Among the above results, only $\mathcal{O}_{\mathbb{Q}^{3}}(-1) \oplus \mathcal{O}_{\mathbb{Q}^{3}}(1)$, $\mathcal{O}_{\mathbb{Q}^{3}}(-1) \oplus \mathcal{O}_{\

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2: discriminant divisors
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • ...and 80 more