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For an algebraic set $X$ (union of varieties) embedded in projective space,we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-thsyzygies of the homogeneous coordinate ring are generated by elements of degree$< d+i$ for $0\le i\le p$ (see \cite{EGHP2} for details). Much attention hasbeen paid to linear syzygies of quadratic schemes $(d=2)$ and their geometricinterpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}).However, not very much is actually known about the case satisfying property$\textbf{N}_{3,p}$. In this paper, we give a sharp upper bound on the maximallength of a zero-dimensional linear section of $X$ in terms of graded Bettinumbers (Theorem 1.2 (a)) when $X$ satisfies property $\textbf{N}_{3,p}$. Inparticular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is lessthan or equal to $\binom{e+2}{2}$, and equality holds if and only if $X$ isarithmetically Cohen-Maucalay with $3$-linear resolution (Theorem 1.2 (b)).This is a generalization of the results of Eisenbud et al. (\cite{EGHP1,EGHP2})to the case of $\textbf{N}_{3,p}$, $(p\leq e)$.

On Syzygies, degree, and geometric properties of projective schemes with property $\textbf{N}_{3,p}$