Geometric Syzygies of Canonical Curves of even Genus lying on a K3-Surface

TL;DR

This paper describes the last nonzero syzygy space of even genus canonical curves on K3 surfaces, showing geometric syzygies span all syzygies, thus generalizing Green's conjecture.

Contribution

It provides a geometric description of syzygies for canonical curves on K3 surfaces and proves a natural generalization of Green's conjecture.

Findings

Last nonzero syzygy space identified as a k-2-uple embedded projective space

Geometric syzygies form a non-degenerate configuration of rational normal curves

Green's conjecture is generalized to this setting

Abstract

Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^1_{k+1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+1}. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in this case.