  We prove two statements concerning the linear strand of the minimal free
resolution of a curve of fixed gonality. Firstly, we show that a general curve
C of genus g of non-maximal gonality k\leq (g+1)/2 satisfies Schreyer's
Conjecture, that is, b_{g-k,1}(C,K_C)=g-k, and all its highest order linear
syzygies are of Eagon-Northcott type. This is a statement going 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 formulate an optimal effective version of the
Gonality Conjecture and prove it for general k-gonal curves. This generalizes
the asymptotic Gonality Conjecture proved by Ein-Lazarsfeld and improves
results of Rathmann in the case where C is a general curve of fixed gonality.
