Brauer Groups And Obstruction Problems

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Brauer Groups and Obstruction Problems

Author : Asher Auel,Brendan Hassett,Anthony Várilly-Alvarado,Bianca Viray
Publisher : Birkhäuser
Page : 247 pages
File Size : 49,6 Mb
Release : 2017-03-02
Category : Mathematics
ISBN : 9783319468525

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Brauer Groups and Obstruction Problems by Asher Auel,Brendan Hassett,Anthony Várilly-Alvarado,Bianca Viray Pdf

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

Brauer Type Embedding Problems

Author : Arne Ledet
Publisher : American Mathematical Soc.
Page : 183 pages
File Size : 41,5 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821837269

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Brauer Type Embedding Problems by Arne Ledet Pdf

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. 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.

The Brauer–Grothendieck Group

Author : Jean-Louis Colliot-Thélène,Alexei N. Skorobogatov
Publisher : Springer Nature
Page : 450 pages
File Size : 55,8 Mb
Release : 2021-07-30
Category : Mathematics
ISBN : 9783030742485

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The Brauer–Grothendieck Group by Jean-Louis Colliot-Thélène,Alexei N. Skorobogatov Pdf

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.

Algebraic Geometry

Author : Richard Thomas
Publisher : American Mathematical Soc.
Page : 635 pages
File Size : 49,6 Mb
Release : 2018-06-01
Category : Geometry, Algebraic
ISBN : 9781470435783

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Algebraic Geometry by Richard Thomas Pdf

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.

Brauer Type Embedding Problems

Author : Arne Ledet
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 41,5 Mb
Release : 2024-06-30
Category : Mathematics
ISBN : 0821871803

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Brauer Type Embedding Problems by Arne Ledet Pdf

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. 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.

Geometry Over Nonclosed Fields

Author : Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel
Publisher : Springer
Page : 261 pages
File Size : 45,8 Mb
Release : 2017-02-09
Category : Mathematics
ISBN : 9783319497631

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Geometry Over Nonclosed Fields by Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel Pdf

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.

Recent Developments in Algebraic Geometry

Author : Hamid Abban,Gavin Brown,Alexander Kasprzyk,Shigefumi Mori
Publisher : Cambridge University Press
Page : 368 pages
File Size : 48,8 Mb
Release : 2022-09-30
Category : Mathematics
ISBN : 9781009190824

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Recent Developments in Algebraic Geometry by Hamid Abban,Gavin Brown,Alexander Kasprzyk,Shigefumi Mori Pdf

Written in celebration of Miles Reid's 70th birthday, this illuminating volume contains 11 papers by leading mathematicians in and around algebraic geometry, broadly related to the themes and interests of Reid's varied career. Just as in Reid's own scientific output, some of the papers give comprehensive accounts of the state of the art of foundational matters, while others give expositions of subject areas or techniques in concrete terms. Reid has been one of the major expositors of algebraic geometry and a great influence on many in this field – this book hopes to inspire a new generation of graduate students and researchers in his tradition.

Birational Geometry of Hypersurfaces

Author : Andreas Hochenegger,Manfred Lehn,Paolo Stellari
Publisher : Springer Nature
Page : 297 pages
File Size : 51,7 Mb
Release : 2019-10-08
Category : Mathematics
ISBN : 9783030186388

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Birational Geometry of Hypersurfaces by Andreas Hochenegger,Manfred Lehn,Paolo Stellari Pdf

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.

Unramified Brauer Group and Its Applications

Author : Sergey Gorchinskiy,Constantin Shramov
Publisher : American Mathematical Soc.
Page : 200 pages
File Size : 45,8 Mb
Release : 2018-09-10
Category : Associative algebras
ISBN : 9781470440725

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Unramified Brauer Group and Its Applications by Sergey Gorchinskiy,Constantin Shramov Pdf

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

Author : Arnaud Beauville,Brendan Hassett,Alexander Kuznetsov,Alessandro Verra
Publisher : Springer
Page : 170 pages
File Size : 46,7 Mb
Release : 2016-12-06
Category : Mathematics
ISBN : 9783319462097

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Rationality Problems in Algebraic Geometry by Arnaud Beauville,Brendan Hassett,Alexander Kuznetsov,Alessandro Verra Pdf

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.

Rationality of Varieties

Author : Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman
Publisher : Springer Nature
Page : 433 pages
File Size : 45,6 Mb
Release : 2021-10-19
Category : Mathematics
ISBN : 9783030754211

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Rationality of Varieties by Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman Pdf

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields. The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.

Generic Polynomials

Author : Christian U. Jensen,Arne Ledet,Noriko Yui
Publisher : Cambridge University Press
Page : 272 pages
File Size : 43,9 Mb
Release : 2002-12-09
Category : Mathematics
ISBN : 0521819989

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Generic Polynomials by Christian U. Jensen,Arne Ledet,Noriko Yui Pdf

Table of contents

Rational Points and Arithmetic of Fundamental Groups

Author : Jakob Stix
Publisher : Springer
Page : 257 pages
File Size : 53,8 Mb
Release : 2012-10-19
Category : Mathematics
ISBN : 9783642306747

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Rational Points and Arithmetic of Fundamental Groups by Jakob Stix Pdf

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.

Number Theory III

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 41,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642582271

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Number Theory III by Serge Lang Pdf

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in sights. Fermat's last theorem occupies an intermediate position. Al though it is not proved, it is not an isolated problem any more.