New evidence for Green's conjecture on syzygies of canonical curves

Abstract

What we call the generic Green's conjecture predicts what are the numbers of syzygies of the generic canonical curve of genus g. Green and Lazarsfeld have observed that curves with nonmaximal Clifford index have extra syzygies and we call specific Green's conjecture the stronger prediction that the curves which have the numbers of syzygies expected for generic curves are precisely those with maximal Clifford index. In this note, we prove that, as stated above, the generic and specific Green's conjectures for canonical curves are equivalent at least when g is odd.