Brauer Groups And Obstruction Problems
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Brauer Groups And Obstruction Problems
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Author : Asher Auel
language : en
Publisher: Birkhäuser
Release Date : 2017-03-02
Brauer Groups And Obstruction Problems written by Asher Auel and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-02 with Mathematics categories.
The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory. Contributors: · Nicolas Addington · Benjamin Antieau · Kenneth Ascher · Asher Auel · Fedor Bogomolov · Jean-Louis Colliot-Thélène · Krishna Dasaratha · Brendan Hassett · Colin Ingalls · Martí Lahoz · Emanuele Macrì · Kelly McKinnie · Andrew Obus · Ekin Ozman · Raman Parimala · Alexander Perry · Alena Pirutka · Justin Sawon · Alexei N. Skorobogatov · Paolo Stellari · Sho Tanimoto · Hugh Thomas · Yuri Tschinkel · Anthony Várilly-Alvarado · Bianca Viray · Rong Zhou
The Brauer Grothendieck Group
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Author : Jean-Louis Colliot-Thélène
language : en
Publisher: Springer Nature
Release Date : 2021-07-30
The Brauer Grothendieck Group written by Jean-Louis Colliot-Thélène and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-30 with Mathematics categories.
This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer–Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong’s proof of Gabber’s theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer–Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer–Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.
Brauer Type Embedding Problems
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Author : Arne Ledet
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
Brauer Type Embedding Problems written by Arne Ledet and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
This monograph is concerned with Galois theoretical embedding problems of so-called Brauer type with a focus on 2-groups and on finding explicit criteria for solvability and explicit constructions of the solutions. The advantage of considering Brauer type embedding problems is their comparatively simple condition for solvability in the form of an obstruction in the Brauer group of the ground field. The book presupposes knowledge of classical Galois theory and the attendant algebra. Before considering questions of reducing the embedding problems and reformulating the solvability criteria, the author provides the necessary theory of Brauer groups, group cohomology and quadratic forms. The book will be suitable for students seeking an introduction to embedding problems and inverse Galois theory. It will also be a useful reference for researchers in the field.
Unramified Brauer Group And Its Applications
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Author : Sergey Gorchinskiy
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-09-10
Unramified Brauer Group And Its Applications written by Sergey Gorchinskiy and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-10 with Mathematics categories.
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
Rationality Problems In Algebraic Geometry
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Author : Arnaud Beauville
language : en
Publisher: Springer
Release Date : 2016-12-06
Rationality Problems In Algebraic Geometry written by Arnaud Beauville and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-06 with Mathematics categories.
Providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. It offers a comprehensive introduction to this fascinating topic, and will certainly become an essential reference for anybody working in the field. Rationality problems are of fundamental importance both in algebra and algebraic geometry. Historically, rationality problems motivated significant developments in the theory of abelian integrals, Riemann surfaces and the Abel–Jacobi map, among other areas, and they have strong links with modern notions such as moduli spaces, Hodge theory, algebraic cycles and derived categories. This text is aimed at researchers and graduate students in algebraic geometry.
Algebraic Geometry Salt Lake City 2015
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Author : Richard Thomas
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-06-01
Algebraic Geometry Salt Lake City 2015 written by Richard Thomas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-01 with Mathematics categories.
This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.
Geometry Over Nonclosed Fields
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Author : Fedor Bogomolov
language : en
Publisher: Springer
Release Date : 2017-02-09
Geometry Over Nonclosed Fields written by Fedor Bogomolov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-09 with Mathematics categories.
Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.
Birational Geometry Of Hypersurfaces
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Author : Andreas Hochenegger
language : en
Publisher: Springer Nature
Release Date : 2019-10-08
Birational Geometry Of Hypersurfaces written by Andreas Hochenegger and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-08 with Mathematics categories.
Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results. The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side. Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.
Rational Points And Arithmetic Of Fundamental Groups
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Author : Jakob Stix
language : en
Publisher: Springer
Release Date : 2012-10-19
Rational Points And Arithmetic Of Fundamental Groups written by Jakob Stix and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-10-19 with Mathematics categories.
The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.
Inverse Galois Theory
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Author : Gunter Malle
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Inverse Galois Theory written by Gunter Malle and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Inverse Galois Theory is concerned with the question of which finite groups occur as Galois Groups over a given field. In particular, this includes the question of the structure and the representations of the absolute Galois group of K and also the question about its finite epimorphic images, the so-called inverse problem of Galois theory. In all these areas important progress was made in the last few years. The aim of the book is to give a consistent and reasonably complete survey of these results, with the main emphasis on the rigidity method and its applications. Among others the monograph presents the most successful known existence theorems and construction methods for Galois extensions and solutions of embedding problems combined with a collection of the existing Galois realizations.