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Download (1570Kb) A Thesis Submitted for the Degree of PhD at the University of Warwick Permanent WRAP URL: http://wrap.warwick.ac.uk/148579 Copyright and reuse: This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. For more information, please contact the WRAP Team at: [email protected] warwick.ac.uk/lib-publications Stable rationality and degenerations of conic bundles Author: Supervisor: Alessandro Bigazzi Christian Böhning Thesis submitted to the University of Warwick for the degree of Doctor of Philosophy Department of Mathematics March 2020 Contents Acknowledgments. iii Declaration. v Abstract. vii Introduction. ix Chapter 1. Nearly rational varieties and the stable Lüroth problem. 1 1.1. Rationality and unirationality. 1 1.2. The stable Lüroth problem. 3 1.3. Currently known results. 5 Chapter 2. Chow-theoretic invariants and the degeneration method. 11 2.1. Decomposition of the diagonal and zero-cycles. 11 2.2. Mild desingularisation 13 2.3. The degeneration method. 14 2.4. Further generalisation: semi-stable degeneration method. 16 Chapter 3. Brauer groups, unramified invariants and applications to conic bundles. 25 3.1. Brauer group of fields. 25 3.2. Brauer group of schemes. 33 3.3. Residue maps and unramified cohomology. 38 3.4. Conic bundles. 44 3.5. The case of cubic threefold hypersurfaces. 56 Chapter 4. Purity and universal triviality for the p-torsion of the Brauer group in characteristic p. 65 4.1. Introduction. 65 4.2. Unramified elements and purity. 66 4.3. A variant of Weil reciprocity and a pairing. 70 4.4. Proof of universal triviality of the Brauer group. 75 Chapter 5. The unramified Brauer group of conic bundle threefolds in characteristic 2. 77 5.1. Preliminaries on conic bundles in characteristic 2. 77 5.2. Residue maps in characteristic 2. 84 5.3. Residue maps and conic bundles in characteristic 2. 89 i ii CONTENTS 5.4. A formula for the unramified Brauer group of a conic bundle threefold in characteristic 2. 96 5.5. A stably irrational conic bundle threefold. 98 Bibliography 111 Acknowledgments. I would like to express my sincere gratitude to my supervisor, Dr. Chris- tian Bõhning, for the continuous support during my Ph.D. study, for his patience, motivation, and immense knowledge but especially for his ment- orship, which helped me broadening my understanding of mathematics. I could not have imagined having a better advisor and mentor for my Ph.D. I also want to thank all my family for believing in me and offering constant support: none of my achievements would have ever been realisable without their help. Finally, I am indebted with my wife Claudia for her infinite patience during these years of intense study and work. No matter how far away from earth my mind was, she has never left my side. Coventry, 21 − 03 − 2020 Addendum: I wish to extend my gratitude to my two examiners, who have carefully read my work and offered a much welcome external point of view on it. In particular, I am grateful to Professor Miles Reid for the really helpful notes on the usage of the English language in mathematics and to Professor Nicholas Shepherd-Barron for his deep, inquisitive questions that allowed me to demonstrate the cohesion between all chapters of this thesis. Coventry, 21 − 6 − 2020 iii Declaration. I hereby confirm that this thesis has been composed by myself and has not been submitted in any previous application for any degree. Each part of the thesis which is not product of my own work has been acknowledged with appropriate citations. Parts of this thesis are a result of collaborative research, whose results have been published in the papers [ABBvB-19, ABBvB-18]. v Abstract. The main task of the thesis is to illustrate a new techniue for establishing stable irrationality of certain conic bundles by degenerating to character- istic 2 and studying the unramified Brauer group of the degenerated conic bundle in terms of the geometry of its discriminant locus. The original ma- terial has been published in the collaborative research papers [ABBvB-19, ABBvB-18]. After introducing Voisin’s degeneration method and its various refine- ments, we explain the envisaged application strategy, which relies on the interaction between Chow-theoretic properties (like existence of a decom- position 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 non- trivial 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. vii Introduction. This thesis investigates stable rationality of a class of rationally connec- ted varieties that admit a structure of conic bundle through application of a degeneration-type argument introduced by Claire Voisin in 2013. In this respect, much of the effort is devoted to develop computable invariants that obstruct a Chow-theoretic property included in this degeneration method; this is achieved through a number of techniques stemming from disparate areas of algebra, arithmetic and K-theory; we also highlight our original contribution in the particular case of degenerations to conic bundles defined over fields of characteristic 2. Some of these conic bundles might be not unirational either. The rela- tionship between unirationality (and more generally rational connectedness), stable rationality and rationality has been studied since the second half of the XIX century, the first milestone of the problem dating back to the proof of the famous Lüroth’s Theorem, which in modern language establishes the equivalence of rationality and unirationality for algebraic varieties of dimen- sion 1. The efforts of the Italian school of algebraic geometry to replicate the result for varieties of dimension 2 were successful thanks to the work of Castelnuovo and Enriques but did not manage to give a satisfactory an- swer for higher dimensional varieties, whose geometry was too sophisticated for the heuristic arguments of the era: some incorrect attempts by Fano and Enriques are recorded in [Roth-55, Chapter V, Section 9]; nonetheless, Roth himself also proposed a new counter-example which was later proved to be invalid. It was only in the years 1971-72 that three indisputable classes of counter-examples finally appeared: these are described in the influential papers [AM-72, CG-72, IM-71] and were obtained with three different techniques. However, it was soon clear that each of these techniques was only effective if applied to very specific examples and either gave little in- formation in more general settings (the Artin-Mumford invariant), or it was not generalisable to higher dimensions (the intermediate Jacobian by Clem- ens and Griffiths), or it was extremely hard to work with (the birational rigidity approach by Iskovskikh and Manin). The mathematical community began therefore to examine more closely the intermediate notion of stable rationality, giving rise to the so-called stable Lüroth problem − that is to say, distinguishing unirational, stably rational varieties from those which are unirational but not stably rational. The ix x INTRODUCTION. motivation came from the promising Artin and Mumford example of unir- ational threefold which is not only irrational, but even stably irrational; the invariant used to show stable irrationality of a (smooth) variety X was the torsion subgroup in the third singular cohomology, which has to vanish for any stably rational variety. It was clear at the time that this subgroup could be identified with the Brauer group of X itself, which in turn embeds into the Brauer group of the function field k(X); this latter object elicited much interest for various reasons: first of all, it was much easier to ma- nipulate, since it could be identified with some Galois cohomology group − an abelian group. And, on a second instance, if it were possible to charac- terise the image of Br(X) inside Br k(X) in function-field theoretic terms, such characterisation would have been already invariant up to birational isomorphism.
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