On syzygies, degree, and geometric properties of projective schemes with property N-3,N-p

Abstract

Let X be a reduced, but not necessarily irreducible closed subscheme of codimension e in a projective space. One says that X satisfies property N-d,N-p (d >= 2) if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree < d i for 0 <= i <= p (see [10] for details). Much attention has been paid to linear syzygies of quadratic schemes (d = 2) and their geometric interpretations (cf. [1,9, 15-17]). However, not very much is actually known about algebraic sets satisfying property N-d,N-p, d >= 3. Assuming property N-d,N-e, we give a sharp upper bound deg (X) <= ((e+d-1)(d-1)). It is natural to ask whether deg(X) = ((e+d-1)(d-1)) implies that e) X is arithmetically Cohen-Macaulay (ACM) with a d-linear resolution. In case of d = 3, by using the elimination mapping cone sequence and the generic initial ideal theory, we show that deg(X) = ((e+2)(2)) if and only if X is ACM with a 3-linear 2 resolution. This is a generalization of the results of Eisenbud et al. (d = 2) [9,10]. We also give more general inequality concerning the length of the finite intersection of X with a linear space of not necessary complementary dimension in terms of graded Betti numbers. Concrete examples are given to explain our results.