Green's Conjecture for curves on arbitrary K3 surfaces

TL;DR

This paper proves Green's Conjecture for smooth curves on any K3 surface by extending previous results and analyzing syzygies related to canonical embeddings.

Contribution

It generalizes Voisin's results to K3 surfaces with arbitrary Picard lattices, providing a complete proof of Green's Conjecture in this setting.

Findings

Complete proof of Green's Conjecture for curves on arbitrary K3 surfaces.

Extension of syzygy analysis to K3 surfaces with arbitrary Picard lattices.

Integration of Voisin and Hirschowitz-Ramanan results to solve the conjecture.

Abstract

Green's Conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin's results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz-Ramanan, provides a complete solution to Green's Conjecture for smooth curves on arbitrary K3 surfaces.