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

Abstract

For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property ${\text{N}}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies 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 has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}). However, not very much is actually known about the case satisfying property ${\text{N}}_{3,p}$. In this paper, we give a sharp upper bound on the maximal length of a zero-dimensional linear section of $X$ in terms of graded Betti numbers (Theorem 1.2 (a)) when $X$ satisfies property ${\text{N}}_{3,p}$. In particular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is less than or equal to $\left(\genfrac{}{}{0ex}{}{e+2}{2}\right)$, and equality holds if and only if $X$ is arithmetically 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 ${\text{N}}_{3,p}$, $(p\le e)$.

14 pages

References in corpus (1)

• Koszul homology and syzygies of Veronese subalgebras

Cited by in corpus (1)

• Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes