A lower bound for the gonality conjecture

TL;DR

This paper constructs specific k-gonal curves with particular divisors to demonstrate limitations of the vanishing statement in the Green-Lazarsfeld gonality conjecture, providing a lower bound and counterexamples.

Contribution

It presents explicit counterexamples for the gonality conjecture, establishing a lower bound where the conjecture's vanishing statement fails.

Findings

Counterexamples for the gonality conjecture's vanishing statement.

Construction of k-gonal curves with specific divisors.

Demonstration of the conjecture's limitations.

Abstract

For every integer $k \geq 3$ we construct a $k$-gonal curve $C$ along with a very ample divisor of degree $2g + k - 1$ (where $g$ is the genus of $C$) to which the vanishing statement from the Green-Lazarsfeld gonality conjecture does not apply.