Lazarsfeld-Mukai bundles and applications

TL;DR

This paper surveys Lazarsfeld-Mukai bundles, highlighting their applications in algebraic geometry, including classification of Mukai manifolds, Brill-Noether theory, and syzygies of K3 sections, with a demonstration of techniques through a proof of Reid's result.

Contribution

It provides a comprehensive overview of Lazarsfeld-Mukai bundles and demonstrates their utility in various geometric problems, including a new proof of Reid's elliptic pencils result.

Findings

Lazarsfeld-Mukai bundles are versatile tools in algebraic geometry.

Applications include classification of Mukai manifolds and Brill-Noether theory.

A short proof of Reid's elliptic pencils existence is presented.

Abstract

We survey the development of the notion of Lazarsfeld-Mukai bundles together with various applications, from the classification of Mukai manifolds to Brill-Noether theory and syzygies of $K3$ sections. To see these techniques at work, we present a short proof of a result of M. Reid on the existence of elliptic pencils.