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 improve the degree bound to 4g-4 with two exceptional cases.