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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.

We discuss, how via limit linear series and standard facts about divisors on moduli spaces of pointed curves, one can establish a non-existence Brill–Noether results for linear series with prescribed ramification at unassigned points.

We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g8. The Kodaira dimension of S_8 is non-negative., 9 pages. Improved presentation, to appear in Advances in Mathematics