Generic lines in projective space and the Koszul property

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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

dissertation

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Department of Mathematics

Grantor

Iowa State University

Year dc:date.issued

2023

Author dc:creator

• Rice, Joshua Andrew

Advisors dc:contributor.advisor

• McCullough, Jason
• Song, Sung-Yell
• Basak, Tathagata
• Hartwig, Jonas
• Smith, Jonathan

Language dc:language.iso

en

Repository record dc:identifier.uri

https://dr.lib.iastate.edu/handle/20.500.12876/EzR2odYz

OAI identifier oai:identifier

oai:dr.lib.iastate.edu:20.500.12876/EzR2odYz