2024/12/07 by Anand Deopurkar, Deopurkar, Anand, Jayan Mukherjee +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2412.05500
We study the syzygies of the canonical embedding of a ribbon \widetildeC on a curve C of genus g ≥ 1. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus pa on a general curve of genus g with pa ≥ max\3g+7, 6g-4\. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for C and it implies Green's conjecture for all double covers C' of C with g(C') ≥ \textrmmax\3g+2, 6g-4\. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to C, which may be of independent interest.