Linear syzygies of curves with prescribed gonality

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 + 1 2 satisfies Schreyer's Conjecture, that is, b g − k , 1 ( C , ω C ) = 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.