Free curves on varieties
In this thesis we study various ways in which every two general points on a variety can be connected by curves of a fixed genus, thus mimicking the notion of a rationally connected variety but for arbitrary genus. We assume the existence of a covering family of curves which dominates the product of...
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| Muut tekijät: | |
| Aineistotyyppi: | Opinnäyte |
| Kieli: | English |
| Julkaistu: |
2012
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| Aiheet: |
| _version_ | 1826316354336063488 |
|---|---|
| author | Gounelas, F |
| author2 | Flynn, EV |
| author_facet | Flynn, EV Gounelas, F |
| author_sort | Gounelas, F |
| collection | OXFORD |
| description | In this thesis we study various ways in which every two general points on a variety can be connected by curves of a fixed genus, thus mimicking the notion of a rationally connected variety but for arbitrary genus. We assume the existence of a covering family of curves which dominates the product of a variety with itself either by allowing the curves in the family to vary in moduli, or by assuming the family is trivial for some fixed curve of genus g. A suitably free curve will be one with a large unobstructed deformation space, the images of whose deformations can join any number of points on a variety. We prove that, at least in characteristic zero, the existence of such a free curve of higher genus is equivalent to the variety being rationally connected. If one restricts to the case of genus one, similar results can be obtained even allowing the curves in the family to vary in moduli. In later chapters we study algebraic properties of such varieties and discuss attempts to prove the same rational connectedness result in positive characteristic. |
| first_indexed | 2024-03-07T07:43:37Z |
| format | Thesis |
| id | oxford-uuid:3a7f6dba-fad2-4517-994e-0b51ea311df8 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:43:40Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:3a7f6dba-fad2-4517-994e-0b51ea311df82024-12-07T14:52:58ZFree curves on varietiesThesishttp://purl.org/coar/resource_type/c_db06uuid:3a7f6dba-fad2-4517-994e-0b51ea311df8Number theoryAlgebraic geometryEnglishOxford University Research Archive - Valet2012Gounelas, FFlynn, EVIn this thesis we study various ways in which every two general points on a variety can be connected by curves of a fixed genus, thus mimicking the notion of a rationally connected variety but for arbitrary genus. We assume the existence of a covering family of curves which dominates the product of a variety with itself either by allowing the curves in the family to vary in moduli, or by assuming the family is trivial for some fixed curve of genus g. A suitably free curve will be one with a large unobstructed deformation space, the images of whose deformations can join any number of points on a variety. We prove that, at least in characteristic zero, the existence of such a free curve of higher genus is equivalent to the variety being rationally connected. If one restricts to the case of genus one, similar results can be obtained even allowing the curves in the family to vary in moduli. In later chapters we study algebraic properties of such varieties and discuss attempts to prove the same rational connectedness result in positive characteristic. |
| spellingShingle | Number theory Algebraic geometry Gounelas, F Free curves on varieties |
| title | Free curves on varieties |
| title_full | Free curves on varieties |
| title_fullStr | Free curves on varieties |
| title_full_unstemmed | Free curves on varieties |
| title_short | Free curves on varieties |
| title_sort | free curves on varieties |
| topic | Number theory Algebraic geometry |
| work_keys_str_mv | AT gounelasf freecurvesonvarieties |