On syzygies, degree, and geometric properties of projective schemes with property [formula omitted

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 Nd,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 Nd,p, d≥3. Assuming property Nd,e, we give a sharp upper bound deg⁡(X)≤(e+d−1d−1). It is natural to ask whether deg⁡(X)=(e+d−1d−1) implies that 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+22) if and only if X is ACM with a 3-linear 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.