Mukai bundles on Fano threefolds

TL;DR

This paper provides the first complete proof of Mukai's theorem on the existence of exceptional vector bundles on prime Fano threefolds, which is crucial for their classification and derived category decompositions.

Contribution

It offers a complete proof of Mukai's theorem using Lazarsfeld's construction, Mukai's stable bundle theory, and Brill--Noether properties, filling a gap in the literature.

Findings

Proof of Mukai's theorem established

Supports classification of prime Fano threefolds

Enables semiorthogonal decompositions in derived categories

Abstract

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our approach is based on Lazarsfeld's construction that produces vector bundles on a variety from globally generated line bundles on a divisor, on Mukai's theory of stable vector bundles on K3 surfaces, and on Brill--Noether properties of curves and (in the sense of Mukai) of K3 surfaces.