Koszul property for points in projective spaces

Abstract

A graded $K$-algebra $R$ is said to be Koszul if the minimal $R$-free graded resolution of $K$ is linear. In this paper we study the Koszul property of the homogeneous coordinate ring $R$ of a set of $s$ points in the complex projective space $\boldsymbol P^n$. Kempf proved that $R$ is Koszul if $s\leq 2n$ and the points are in general linear position. If the coordinates of the points are algebraically independent over $\boldsymbol Q$, then we prove that $R$ is Koszul if and only if $s\le 1 +n + n^2/4$. If $s\le 2n$ and the points are in linear general position, then we show that there exists a system of coordinates $x_0,\dots,x_n$ of $\boldsymbol P^n$ such that all the ideals $(x_0,x_1,\dots,x_i)$ with $0\le i \le n$ have a linear $R$-free resolution.