{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139463"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139463","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Bounds on the Torsion Subgroups of Néron–Severi Groups","abstract":"Let 𝑋 ⤷ Pʳ be a smooth projective variety defined by homogeneous polynomials of degree ≤ 𝑑 over an algebraically closed field 𝑘. Let Pic 𝑋 be the Picard scheme of 𝑋, and let Pic⁰ 𝑋 be the identity component of Pic 𝑋. The Néron–Severi group scheme of 𝑋 is defined by NS 𝑋 = (Pic 𝑋)/(Pic⁰ 𝑋)ᵣₑ subscript d, and the Néron–Severi group of 𝑋 is defined by NS 𝑋 = (NS 𝑋)(𝑘). We give an explicit upper bound on the order of the finite group (NS 𝑋)ₜₒᵣ and the finite group scheme (NS 𝑋)ₜₒᵣ in terms of 𝑑 and 𝑟. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of 𝑋 and the finite group [mathematical equation]. We also show that (NS 𝑋)ₜₒᵣ is generated by (deg 𝑋 − 1)(deg 𝑋 − 2) elements in various situations.","abstract_html":"Let 𝑋 ⤷ Pʳ be a smooth projective variety defined by homogeneous polynomials of degree ≤ 𝑑 over an algebraically closed field 𝑘. Let Pic 𝑋 be the Picard scheme of 𝑋, and let Pic⁰ 𝑋 be the identity component of Pic 𝑋. The Néron–Severi group scheme of 𝑋 is defined by NS 𝑋 = (Pic 𝑋)/(Pic⁰ 𝑋)ᵣₑ subscript d, and the Néron–Severi group of 𝑋 is defined by NS 𝑋 = (NS 𝑋)(𝑘). We give an explicit upper bound on the order of the finite group (NS 𝑋)ₜₒᵣ and the finite group scheme (NS 𝑋)ₜₒᵣ in terms of 𝑑 and 𝑟. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of 𝑋 and the finite group [mathematical equation]. We also show that (NS 𝑋)ₜₒᵣ is generated by (deg 𝑋 − 1)(deg 𝑋 − 2) elements in various situations.","abstract_has_math":false,"creators":["Kweon, Hyuk Jun"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Poonen, Bjorn"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:21:24Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/139463","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Poonen, Bjorn"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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Let Pic 𝑋 be the Picard scheme of 𝑋, and let Pic⁰ 𝑋 be the identity component of Pic 𝑋. The Néron–Severi group scheme of 𝑋 is defined by NS 𝑋 = (Pic 𝑋)/(Pic⁰ 𝑋)ᵣₑ subscript d, and the Néron–Severi group of 𝑋 is defined by NS 𝑋 = (NS 𝑋)(𝑘). We give an explicit upper bound on the order of the finite group (NS 𝑋)ₜₒᵣ and the finite group scheme (NS 𝑋)ₜₒᵣ in terms of 𝑑 and 𝑟. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of 𝑋 and the finite group [mathematical equation]. We also show that (NS 𝑋)ₜₒᵣ is generated by (deg 𝑋 − 1)(deg 𝑋 − 2) elements in various situations."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Bounds on the Torsion Subgroups of Néron–Severi Groups"]}]}],"canonical_facts":{"dc:contributor.advisor":["Poonen, Bjorn"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Kweon, Hyuk Jun"],"dc:date.accessioned":["2022-01-14T15:12:54Z"],"dc:date.available":["2022-01-14T15:12:54Z"],"dc:date.issued":["2021-06"],"dc:description.abstract":["Let 𝑋 ⤷ Pʳ be a smooth projective variety defined by homogeneous polynomials of degree ≤ 𝑑 over an algebraically closed field 𝑘. Let Pic 𝑋 be the Picard scheme of 𝑋, and let Pic⁰ 𝑋 be the identity component of Pic 𝑋. The Néron–Severi group scheme of 𝑋 is defined by NS 𝑋 = (Pic 𝑋)/(Pic⁰ 𝑋)ᵣₑ subscript d, and the Néron–Severi group of 𝑋 is defined by NS 𝑋 = (NS 𝑋)(𝑘). We give an explicit upper bound on the order of the finite group (NS 𝑋)ₜₒᵣ and the finite group scheme (NS 𝑋)ₜₒᵣ in terms of 𝑑 and 𝑟. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of 𝑋 and the finite group [mathematical equation]. We also show that (NS 𝑋)ₜₒᵣ is generated by (deg 𝑋 − 1)(deg 𝑋 − 2) elements in various situations."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139463"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Bounds on the Torsion Subgroups of Néron–Severi Groups"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:21:24Z"}