An effective bound for the gonality conjecture

TL;DR

This paper extends previous work to show that the gonality of an algebraic curve can be determined from syzygies of embeddings by any line bundle of degree at least 4g-3, broadening the applicability of the gonality conjecture.

Contribution

It provides a new bound (4g-3) for the degree of line bundles needed to detect gonality via syzygies, improving upon prior results.

Findings

Gonality can be read from syzygies with line bundle degree ≥ 4g-3

Extended the approach of Ein and Lazarsfeld to a broader class of line bundles

Established a concrete degree bound for gonality detection

Abstract

Ein and Lazarsfeld have shown that one can read off the gonality of an algebraic curve from its syzygies in the embedding defined by any one line bundle of sufficiently large degree. This note extends their approach and shows that the gonality can be detected from the syzygies of an embedding by any line bundle of degree at least 4g-3.