A note on an effective bound for the gonality conjecture

The gonality conjecture, proved by Ein–Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus g can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least $4g-3$ would work in the gonality theorem. In this note, we develop a new method to improve the degree bound to $4g-4$ with two exceptional cases.