{"id":{"repo_id":"iastate","oai_identifier":"oai:dr.lib.iastate.edu:20.500.12876/EzR2odYz"},"canonical_url":"https://search.dev.ndltd.org/etd/iastate/oai:dr.lib.iastate.edu:20.500.12876/EzR2odYz","repository":{"repo_id":"iastate","name":"Iowa State University","base_url":"https://dr.lib.iastate.edu/server/oai/request"},"display":{"title":"Generic lines in projective space and the Koszul property","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.","abstract_html":"In this thesis, we study the Koszul property of the homogeneous coordinate ring of a generic collection of $m$ lines in <span class=\"etd-inline-math\">\\mathbb{P}<sup>n</sup></span> and the homogeneous coordinate ring of a collection of lines in general linear position in <span class=\"etd-inline-math\">\\mathbb{P}<sup>n</sup>.</span> In particular, we show that if $\\mathcal{M}$ is a collection of $m$ lines in general linear position in <span class=\"etd-inline-math\">\\mathbb{P}<sup>n</sup></span> 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 <span class=\"etd-inline-math\">\\mathbb{P}<sup>n</sup></span> 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 &gt; \\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.","abstract_has_math":true,"creators":["Rice, Joshua Andrew"],"institution":"Iowa State University","degree_name":"Doctor of Philosophy","degree_level":"dissertation","degree_discipline":"Mathematics","degree_department":"Department of Mathematics","school":null,"contributors":[],"advisors":["McCullough, Jason","Song, Sung-Yell","Basak, Tathagata","Hartwig, Jonas","Smith, Jonathan"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-24T02:38:57Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dr.lib.iastate.edu/handle/20.500.12876/EzR2odYz","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["McCullough, Jason","Song, Sung-Yell","Basak, Tathagata","Hartwig, Jonas","Smith, Jonathan"]},{"key":"dc:contributor.department","label":"Department","values":["Department of Mathematics"]},{"key":"dc:creator","label":"Author","values":["Rice, Joshua Andrew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-06-20T22:18:49Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2023-06-20T22:18:49Z"]},{"key":"dc:date.issued","label":"Date","values":["2023-05"]},{"key":"dc:type","label":"Dc Type","values":["dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Iowa State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://dr.lib.iastate.edu/handle/20.500.12876/EzR2odYz"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["PDF"]},{"key":"dc:title","label":"Title","values":["Generic lines in projective space and the Koszul property"]}]}],"canonical_facts":{"dc:contributor.advisor":["McCullough, Jason","Song, Sung-Yell","Basak, Tathagata","Hartwig, Jonas","Smith, Jonathan"],"dc:contributor.department":["Department of Mathematics"],"dc:creator":["Rice, Joshua Andrew"],"dc:date.accessioned":["2023-06-20T22:18:49Z"],"dc:date.available":["2023-06-20T22:18:49Z"],"dc:date.issued":["2023-05"],"dc:description.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."],"dc:format.mimetype":["PDF"],"dc:identifier.uri":["https://dr.lib.iastate.edu/handle/20.500.12876/EzR2odYz"],"dc:language.iso":["en"],"dc:title":["Generic lines in projective space and the Koszul property"],"dc:type":["dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["dissertation"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Iowa State University"]},"updated_at":"2026-07-24T02:38:57Z"}