Johnstone, E
(2017)
On the birational geometry of singular Fano varieties.
PhD thesis, University of Liverpool.
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Abstract
This thesis investigates the birational geometry of a class of higher dimensional Fano varieties of index 1 with quadratic hypersurface singularities. The main investigating question is, what structures of a rationally connected fibre space can these varieties have? Two cases are considered: double covers over a hypersurface of degree two, known as double quadrics and double covers over a hypersurface of degree three, known as double cubics. This thesis extends the study of double quadrics and cubics, first studied in the non-singular case by Iskovskikh and Pukhlikov, by showing that these varieties have the property of birational superrigidity, under certain conditions on the singularities of the branch divisor. This implies, amongst other things, that these varieties admit no non-trivial structures of a rationally connected fibre space and are thus non-rational. Additionally, the group of birational automorphisms coincides with the group of regular automorphisms. This is shown using the ``Method of maximal singularities" of Iskovskikh and Manin, expanded upon by Pukhlikov and others, in conjunction with the connectedness principal of Shokurov and Kollar. These results are then used to give a lower bound on the codimension of the set of all double quadrics (and double cubics) which are either not factorial or not birationally superrigid, in the style of the joint work of Pukhlikov and Eckl on Fano hypersurfaces. Such a result has applications to the study of varieties which admit a fibration into double quadrics or cubics.
| Item Type: | Thesis (PhD) |
|---|---|
| Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 23 Aug 2017 10:22 |
| Last Modified: | 19 Jan 2023 07:02 |
| DOI: | 10.17638/03008126 |
| Supervisors: |
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| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3008126 |
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