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The gonality conjecture, proved by Ein--Lazarsfeld, asserts that the gonalityof a nonsingular projective curve of genus $g$ can be detected from itssyzygies 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 atleast 4g-3 would work in the gonality theorem. In this note, we improve thedegree bound to 4g-4 with two exceptional cases.

A note on an effective bound for the gonality conjecture