Difference varieties and the Green-Lazarsfeld Secant Conjecture

TL;DR

This paper proves the Green-Lazarsfeld Secant Conjecture for certain algebraic curves, extending Green's Conjecture on syzygies to broader line bundle cases, specifically when the non-(p+1)-very ample line bundles form a divisor.

Contribution

It establishes the Green-Lazarsfeld Secant Conjecture for genus g curves in the divisorial case, advancing understanding of syzygies and line bundle properties.

Findings

Proves the conjecture for curves where non-(p+1)-very ample line bundles form a divisor.

Extends Green's Conjecture to the more general Green-Lazarsfeld Secant Conjecture.

Provides new insights into the geometry of line bundles on algebraic curves.

Abstract

The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. We establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial case, that is, when the line bundles that fail to be (p+1)-very ample form a divisor in the Jacobian of the curve.