Geometric Koszul complexes, syzygies of K3 surfaces and the Tango bundle

TL;DR

This paper introduces new geometric constructions related to Koszul complexes and syzygies of K3 surfaces, providing a shorter proof of Voisin's Green's conjecture for even genus K3s.

Contribution

It constructs representations of Koszul complexes on Grassmannians and offers a more concise proof of Voisin's result for certain K3 surfaces.

Findings

Representation of Koszul complexes on Grassmann varieties

Shorter proof of Green's conjecture for even genus K3 surfaces

New geometric tools for studying syzygies of K3 surfaces

Abstract

A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul cohomology on the Hilbert scheme of points on the surface. In this note we construct representations of the Koszul complex on Grassmann varieties; Voisin's setup arises as the inverse image of one of the maps. Using a different map, we give a substantially shorter proof of Voisin's result for K3 surfaces of even sectional genus.