Generic Lines in Projective Space and the Koszul Property

TL;DR

This paper investigates when the coordinate rings of collections of lines in projective space are Koszul, providing conditions for both generic and general linear position cases, and characterizing the property for small parameters.

Contribution

It offers new criteria for the Koszul property of coordinate rings of lines in projective space, including complete characterizations for small cases and bounds for non-Koszul instances.

Findings

If lines are in general linear position with 2m ≤ n+1, the coordinate ring is Koszul.

For generic lines with certain m and n relations, the coordinate ring is Koszul.

A threshold is established beyond which the coordinate ring is not Koszul.

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 $2m \leq n+1$ 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 $m +1\leq n$ or $m$ is odd and $m +2\leq n,$ 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 $n \leq 6$ or $m \leq 6$. 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.