Jinhyung Park, 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 can be read off from weight-one syzygies of a sufficiently positive line bundle L, and also proposed possible least degree of L, that is 2g+gon(C)-1. 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 a few years ago. We show an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if deg(L) is at least 2g+gon(C) or equal to 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, this result is the best possible and thus gives a complete answer to the gonality conjecture. This is joint work with Wenbo Niu.