On Syzygies of Projective

In this article, we give a criterion for an embedding of a projective variety to be deened by quadratic equations and for it to have linear syzygies. Our criterion is intrinsic in nature and implies that embedding corresponding to a suuciently high power of any ample line bundle will have linear syzygies up to a given order. Introduction. Let X be a projective variety. The conditions that guarantee projective normality and quadratic generations of the ideal deening the embedding of X were studied classically. These results of Mumford et al. were considered by Mark Green as the rst step towards understanding the higher syzygies. We say that a line bundle L on X satisses Property N p if the ideal deening the embedding of X in the complete linear system of L is generated by quadratic equations and has linear syzygies till p th stage (see 2]). In 3], Green introduced the notion of Koszul cohomology and proved that vanishing of certain higher Koszul cohomology is equivalent to the Property N p. In 4], it was proved that the required higher Koszul cohomology groups vanish for suuciently ample line bundles ((4, Theorem 3.2]). However, the proof given in 4] makes a tacit assumption that the ideal sheaf of the diagonal embedding of X in its twofold product X 2 is locally free which is not true. If X is a smooth projective variety, using diierent methods, L. Ein and R. Lazarsfeld obtained an eeective bound on the power of L required for satisfying Property N p (see 2]). In this article, we give a criterion for a line bundle L to have Property N p in terms of vanishing of certain rst cohomology. Our method does not assume the smoothness of X. Using this criterion, we see that suuciently high power of L will have property N p for xed p. Therefore, we also see that higher Koszul cohomology of a suuciently high power of an ample line bundle vanish. This answers the question 5.13 of 3]. Here, we would like to mention that when X is a smooth variety, a vanishing theorem of M. Nori can also be used to answer this question ((8, Proposition 3.4]). Our method does not give an eeective bound on the required power of an ample line bundle to have certain linear syzygies. However, the vanishing conditions of our criterion are explicit in nature …