Bounds on the Torsion Subgroups of Néron–Severi Groups
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 subscrip...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2022
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| Online Access: | https://hdl.handle.net/1721.1/139463 |
| _version_ | 1826199526941130752 |
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| author | Kweon, Hyuk Jun |
| author2 | Poonen, Bjorn |
| author_facet | Poonen, Bjorn Kweon, Hyuk Jun |
| author_sort | Kweon, Hyuk Jun |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:21:22Z |
| format | Thesis |
| id | mit-1721.1/139463 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:21:22Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1394632022-01-15T03:42:08Z Bounds on the Torsion Subgroups of Néron–Severi Groups Kweon, Hyuk Jun Poonen, Bjorn Massachusetts Institute of Technology. Department of Mathematics 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. 2022-01-14T15:12:54Z 2022-01-14T15:12:54Z 2021-06 2021-05-25T12:47:03.558Z Thesis https://hdl.handle.net/1721.1/139463 0000-0002-3056-1306 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Kweon, Hyuk Jun Bounds on the Torsion Subgroups of Néron–Severi Groups |
| title | Bounds on the Torsion Subgroups of Néron–Severi
Groups |
| title_full | Bounds on the Torsion Subgroups of Néron–Severi
Groups |
| title_fullStr | Bounds on the Torsion Subgroups of Néron–Severi
Groups |
| title_full_unstemmed | Bounds on the Torsion Subgroups of Néron–Severi
Groups |
| title_short | Bounds on the Torsion Subgroups of Néron–Severi
Groups |
| title_sort | bounds on the torsion subgroups of neron severi groups |
| url | https://hdl.handle.net/1721.1/139463 |
| work_keys_str_mv | AT kweonhyukjun boundsonthetorsionsubgroupsofneronseverigroups |