ON KOSZUL RINGS, SYZYGIES, AND ABELIAN VARIETIES

Abstract

In this paper we prove the following results: let X be a complex torus and M a line bundle on X; if M satisfies Property N ₁ of Green, then, for every q ≥ 1, the line bundle M^q satisfies Property N_q (precisely we prove a more general result concerning not necessarily linear syzygies); this is a particular case of a conjecture formulated in [1]; finally we prove that, if M satisfies Property N ₀ and s ≥ 2, then the ring ⊕ _n H ⁰(Y, M^sn ) is Koszul.

ACKNOWLEDGMENTS

I warmly thank G. Pareschi for an useful conversation and in particular for explaining me how Lazarsfeld simplified the proof of the main result of Kempf' work [9].