  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.
