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Green's Conjecture predicts than one can read off special linear series on analgebraic curve, by looking at the syzygies of its canonical embedding. Weextend Voisin's results on syzygies of K3 sections, to the case of K3 surfaceswith arbitrary Picard lattice. This, coupled with results of Voisin andHirschowitz-Ramanan, provides a complete solution to Green's Conjecture forsmooth curves on arbitrary K3 surfaces.

Green's Conjecture for curves on arbitrary K3 surfaces