Abstract
We prove two statements concerning the linear strand of the minimal free resolution of a k-gonal curve C of genus g. Firstly, we show that a general curve C of genus g of non-maximal gonality k≤g+12 satisfies Schreyer's Conjecture, that is, bg−k,1(C,ωC)=g−k. This statement goes beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the (k−1)-dimensional scroll containing C. Secondly, we prove an optimal effective version of the Gonality Conjecture for general k-gonal curves, which makes more precise the (asymptotic) Gonality Conjecture proved by Ein–Lazarsfeld and improves results of Rathmann.
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