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Poonen, Bjorn Kweon, Hyuk Jun Massachusetts Institute of Technology. Department of Mathematics 2022-01-14T15:12:54Z 2022-01-14T15:12:54Z 2021-06 2021-05-25T12:47:03.558Z https://hdl.handle.net/1721.1/139463 0000-0002-3056-1306 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. Ph.D. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ Bounds on the Torsion Subgroups of Néron–Severi Groups Thesis application/pdf Doctoral Doctor of Philosophy Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="60062b6a-4b56-4918-a4ed-cb9d29d3e9d5"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Bounds on the Torsion Subgroups of Néron–Severi Groups</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2021-06</PublicationDate> <Authors> <Author> <DisplayName>Kweon, Hyuk Jun</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://rightsstatements.org/page/InC-EDU/1.0/</License> <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> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>