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

TL;DR

This paper investigates algebraic sets in projective space satisfying property N_{3,p}, establishing bounds on their degree and linking syzygy properties to geometric features, extending previous results on quadratic schemes.

Contribution

It provides a sharp upper bound on the length of zero-dimensional linear sections and characterizes when the degree reaches this bound for schemes with property N_{3,p}.

Findings

Degree of X is at most inom{e+2}{2} when p=e.

Equality in degree bound occurs iff X is arithmetically Cohen-Macaulay with 3-linear resolution.

Generalizes previous results from property N_{2,p} to property N_{3,p}.

Abstract

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 $i$-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 $\textbf{N}_{3,p}$. 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 $\textbf{N}_{3,p}$. In particular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is less than or equal to $\binom{e+2}{2}$, 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 $\textbf{N}_{3,p}$, $(p\leq e)$.