The Martens–Mumford theorem and the Green–Lazarsfeld secant conjecture

Abstract

The Green–Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill–Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.