green's canonical syzygy conjecture for generic curves of odd genus

Abstract

we prove in this paper the green conjecture for generic curves of odd genus. that is, we prove the vanishing $k_{k,1}(x,k_x)=0$ for x a generic curve of genus $2k+1$. this completes our previous work, where the green conjecture for generic curves of genus g with fixed gonality d was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus. the case of generic curves of odd genus was considered as especially important, since hirschowitz and ramanan proved that if the conjecture is true for the generic curve of odd genus, then the locus of jumping syzygies is exactly the locus of exceptional gonality, as predicted by green's conjecture. thus, our result combined with the hirschowitz–ramanan result is a strong confirmation of green's conjecture.