We propose a new approach to study Koszul cohomology of general g d’s on general curves via degenerations and deformations. As an application of this new method, we prove projective normality for general line bundles L on general curves with h(L) ≤ 1 and degree greater than roughly 5 4 g. This improves the bound in a classical theorem of Green and Lazarsfeld. Also we give some partial results on vanishing of higher syzygies for line bundles with h(L) = 1.

On the Generic vanishing of certain Koszul cohomology groups

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On the Generic vanishing of certain Koszul cohomology groups