On the first non-trivial strand of syzygies of projective schemes and Condition ${\mathrm ND}(l)$

TL;DR

This paper introduces a new geometric condition called ND(ℓ), explores its properties, and establishes bounds and characterizations for syzygies and regularity of projective schemes, extending classical results to higher degrees.

Contribution

It defines the ND(ℓ) condition, generalizes bounds on syzygies for higher degrees, and characterizes when a scheme's resolution is d-linear and Cohen-Macaulay.

Findings

Sharp upper bounds on graded Betti numbers for higher degree syzygies

Characterization of d-linear Cohen-Macaulay resolutions via ND(ℓ) and N_{d,p}

A generalized syzygetic rigidity theorem for d-regular schemes

Abstract

Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\mathrm{ND}(\ell)$ 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 $\ell$. First, we introduce condition $\mathrm{ND}(\ell)$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first non-trivial 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 $\mathbf{N}_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithmetically Cohen-Macaulay as having property $\mathbf{N}_{d,e}$ and condition $\mathrm{ND}(d-1)$ 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-Popescu to a general $d$-regularity.