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Green’s conjecture for curves on arbitrary K3 surfaces

Published online by Cambridge University Press:  15 February 2011

Marian Aprodu
Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, RO-014700 Bucharest, Romania (email: Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, Sector 1, RO-010702 Bucharest, Romania
Gavril Farkas
Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin, Germany (email:
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Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

Research Article
Copyright © Foundation Compositio Mathematica 2011


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