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Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. Weare 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 upto $p$-th step in the minimal free resolution and the other is a new notion$\mathrm{ND}(\ell)$ which generalizes the classical "being nondegenerate" tothe condition that requires a general finite linear section not to be containedin any hypersurface of degree $\ell$.  First, we introduce condition $\mathrm{ND}(\ell)$ and consider examples andbasic properties deduced from the notion. Next we prove sharp upper bounds onthe graded Betti numbers of the first non-trivial strand of syzygies, whichgeneralize results in the quadratic case to higher degree case, and providecharacterizations for the extremal cases. Further, after regarding someconsequences 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 thisresult, we obtain a syzygetic rigidity theorem which suggests a naturalgeneralization of syzygetic rigidity on $2$-regularity due toEisenbud-Green-Hulek-Popescu to a general $d$-regularity.

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