  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.
