Generic lines in projective space and the Koszul property
Date
2023-05
Authors
Rice, Joshua Andrew
Major Professor
Advisor
McCullough, Jason
Song, Sung-Yell
Basak, Tathagata
Hartwig, Jonas
Smith, Jonathan
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Research Projects
Journal Issue
Series
Department
Mathematics
Abstract
In this thesis, we study the Koszul property of the homogeneous coordinate ring of a generic collection of $m$ lines in $\mathbb{P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb{P}^n.$ In particular, 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 collections of lines in general linear position.