The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture

TL;DR

This paper proves the Green-Lazarsfeld secant conjecture for a broad class of algebraic curves, linking syzygies to secant varieties using classical Brill-Noether theory and vanishing criteria.

Contribution

It extends the Green-Lazarsfeld secant conjecture to curves with Clifford index at least two, not bielliptic, and for specific line bundles, utilizing Martens-Mumford results and a new vanishing approach.

Findings

Proved the secant conjecture for curves with Clifford index ≥ 2

Connected syzygies of curves to their secant varieties

Introduced a vanishing criterion based on symmetric products

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.