On the first nontrivial strand of syzygies of projective schemes and condition nd(ℓ)

Abstract

Let X ⊂ Pn+e be any n-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: One is the property Nd,p (d ≥ 2, p ≥ 1), which means that X is d-regular up to p-th step in the minimal free resolution and the other is a new notion ND(ℓ) which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained in any hypersurface of degree ℓ. First, we introduce condition ND(ℓ) and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first nontrivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property Nd,p, we characterize the resolution of X to be d-linear arithmetically Cohen-Macaulay as having property Nd,e and condition ND(d-1) at the same time. From this result, we obtain a syzygetic rigidity theorem which suggests a natural generalization of syzygetic rigidity on 2-regularity due to Eisenbud, Green, Hulek and Popescu to a general d-regularity. © 2023 MSP (Mathematical Sciences Publishers.)