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

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 $N_{d,p}(d\ge 2,p\ge 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\mathrm{ND}<!--FUNCTION\; APPLICATION-->(\ell )$ 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 $\mathrm{ND}<!--FUNCTION\; APPLICATION-->(\ell )$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first 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 $N_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithmetically Cohen–Macaulay as having property $N_{d,e}$ and condition $\mathrm{ND}<!--FUNCTION\; APPLICATION-->(d-1)$ 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.