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Stable rationality and degenerations of conic bundles
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Bigazzi, Alessandro (2020) Stable rationality and degenerations of conic bundles. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3492143~S15
Abstract
The main task of the thesis is to illustrate a new techniue for establishing stable irrationality of certain conic bundles by degenerating to characteristic 2 and studying the unramified Brauer group of the degenerated conic bundle in terms of the geometry of its discriminant locus. The original material has been published in the collaborative research papers [ABBvB-19, ABBvB-18].
After introducing Voisin’s degeneration method and its various refinements, we explain the envisaged application strategy, which relies on the interaction between Chow-theoretic properties (like existence of a decomposition of the diagonal and universal triviality of zero-cycles) and other invariants; particular regard is given to the Brauer group, whose prime-to-p torsion part is universally trivial for a smooth universally CH0-trivial variety defined over fields of characteristic p. In Chapter 4 we improve this result, proving that this holds for the p-primary part as well (Theorem 4.1.1), thus extending the applicability of the Brauer group to degenerations where nontrivial classe have torsion order not coprime with the characteristic of the ground field..
We focus on studying applications of these techniques to conic bundles over fields of characteristic 6= 2 in Chapter 3: in particular, after recalling the construction of residue maps and unramified Brauer groups for low degrees, we give a geometric interpretation of these maps in terms of the discriminant locus of a conic bundle and finally prove a formula for the unramified Brauer group of a conic bundle (Theorem 3.4.15), attributed to Colliot-Thélène but not explicitly present in the literature. Furthermore, in Section 3.5 we perform a direct computation on a general conic bundle with quintic discriminant, showing that its unramified Brauer group is even trivial. This shows formally that one cannot prove stable irrationality of cubic threefold hypersurfaces with such strategy.
Finally, we extend these techniques to the case of conic bundles defined over fields of characteristic 2. After explaining how residue maps need to be re-defined in this case, we mirror the work done in Chapter 3, providing a geometric interpretation of residue maps in terms of the geometry of the discriminant locus and then establishing a formula for the unramified Brauer group in this case (Theorem 5.4.1). As an application, we run our full strategy on a particular conic bundle threefold, showing that it is not stably rational (Theorem 5.5.1) in the last Section.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Geometry, Algebraic, Brauer groups | ||||
Official Date: | March 2020 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Böhning, Christian | ||||
Format of File: | |||||
Extent: | xiv, 115 leaves : illustrations | ||||
Language: | eng |
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