Effective gonality theorem on weight-one syzygies of algebraic curves

TL;DR

This paper proves the gonality conjecture for algebraic curves by establishing an effective vanishing theorem for weight-one syzygies, providing the best possible degree bounds and resolving a long-standing open problem.

Contribution

The paper introduces an effective vanishing theorem that confirms the gonality conjecture for specific degree bounds, completing the conjecture's proof.

Findings

Gonality conjecture holds if deg L ≥ 2g + gon(C)

The conjecture also holds if deg L = 2g + gon(C) - 1 and C is not a plane curve

The results are optimal, fully resolving the conjecture

Abstract

In 1986, Green-Lazarsfeld raised the gonality conjecture asserting that the gonality $\operatorname{gon}(C)$ of a smooth projective curve $C$ of genus $g\geq 2$ can be read off from weight-one syzygies of a sufficiently positive line bundle $L$ on $C$, and also proposed possible least degree of such a line bundle. In 2015, Ein-Lazarsfeld proved the conjecture when $\operatorname{deg} L$ is sufficiently large, but the effective part of the conjecture remained widely open and was reformulated explicitly by Farkas-Kemeny. In this paper, we establish an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if $\operatorname{deg} L \geq 2g+\operatorname{gon}(C)$ or $\operatorname{deg} L = 2g+\operatorname{gon}(C)-1$ and $C$ is not a plane curve. As Castryck observed that the gonality conjecture may not hold for a plane curve when $\operatorname{deg} L = 2g+\operatorname{gon}(C)-1$, our theorem is the best possible and thus gives a complete answer to the gonality conjecture.