Effective gonality theorem on weight-one syzygies of algebraic curves

In 1986, Green-Lazarsfeld raised the gonality conjecture asserting that the gonality gon (C) of a smooth projective curve C of genus g 2 can be read off from weight-one syzygies of a sufficiently positive line bundle L on C, and also proposed possible least degree of such a line bundle. In 2015, Ein-Lazarsfeld proved the conjecture when deg L is sufficiently large, but the effective part of the conjecture remained widely open and was reformulated explicitly by Farkas-Kemeny. In this paper, we establish an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if deg L 2g+gon (C) or deg L = 2g+gon (C) -1 and C is not a plane curve. As Castryck observed that the gonality conjecture may not hold for a plane curve when deg L = 2g+gon (C) -1, our theorem is the best possible and thus gives a complete answer to the gonality conjecture.