Generic Lines in Projective Space and the Koszul Property

Abstract

In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in ${\mathbb{P}}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in ${\mathbb{P}}^n.$ We show that if $\mathcal{M}$ is a collection of $m$ lines in general linear position in ${\mathbb{P}}^n$ with $2m\le n+1$ and $R$ is the coordinate ring of $\mathcal{M},$ then $R$ is Koszul. Further, if $\mathcal{M}$ is a generic collection of $m$ lines in ${\mathbb{P}}^n$ and $R$ is the coordinate ring of $\mathcal{M}$ with $m$ even and $m+1\le n$ or $m$ is odd and $m+2\le n,$ then $R$ is Koszul. Lastly, we show if $\mathcal{M}$ is a generic collection of $m$ lines such that \[ m > \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\] then $R$ is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n\le 6$ or $m\le 6$. We also determine the Castelnuovo-Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.