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Rational and Nearly Rational Varieties

Author: János Kollár

Publisher: Cambridge University Press

Published: 2004-04-22

Total Pages: 246

ISBN-13: 9780521832076

The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. In many applications where algebraic varieties appear in mathematics and the sciences, we see rational ones emerging as the most interesting examples. The authors have given an elementary treatment of rationality questions using a mix of classical and modern methods. Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems.

Rational Curves on Algebraic Varieties

Author: Janos Kollar

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 330

ISBN-13: 3662032767

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The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

Rational Points on Varieties

Author: Bjorn Poonen

Publisher: American Mathematical Society

Published: 2023-08-10

Total Pages: 357

ISBN-13: 1470474581

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This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University

Rational and Nearly Rational Varieties ICM Edition

Author: Kollar

Publisher:

Published: 2010-07-23

Total Pages:

ISBN-13: 9780521168878

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Rationality of Varieties

Author: Gavril Farkas

Publisher: Springer Nature

Published: 2021-10-19

Total Pages: 433

ISBN-13: 3030754219

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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.

Rational Points on Algebraic Varieties

Author: Emmanuel Peyre

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 455

ISBN-13: 3034883684

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This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.

Rationality Problems in Algebraic Geometry

Author: Arnaud Beauville

Publisher: Springer

Published: 2016-12-06

Total Pages: 170

ISBN-13: 3319462091

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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.

Torsors and Rational Points

Author: Alexei Skorobogatov

Publisher: Cambridge University Press

Published: 2001-07-05

Total Pages: 197

ISBN-13: 0521802377

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This book, first published in 2001, is a complete and coherent exposition of the theory and applications of torsors to rational points.

Predictably Rational?

Author: Richard B. McKenzie

Publisher: Springer Science & Business Media

Published: 2009-10-21

Total Pages: 319

ISBN-13: 3642015867

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Mainstream economists everywhere exhibit an "irrational passion for dispassionate rationality." Behavioral economists, and long-time critic of mainstream economics suggests that people in mainstrean economic models "can think like Albert Einstein, store as much memory as IBM’s Big Blue, and exercise the will power of Mahatma Gandhi," suggesting that such a view of real world modern homo sapiens is simply wrongheaded. Indeed, Thaler and other behavioral economists and psychology have documented a variety of ways in which real-world people fall far short of mainstream economists' idealized economic actor, perfectly rational homo economicus. Behavioral economist Daniel Ariely has concluded that real-world people not only exhibit an array of decision-making frailties and biases, they are "predictably irrational," a position now shared by so many behavioral economists, psychologists, sociologists, and evolutionary biologists that a defense of the core rationality premise of modedrn economics is demanded.

Rational Ritual

Author: Michael Suk-Young Chwe

Publisher: Princeton University Press

Published: 2013-04-28

Total Pages: 152

ISBN-13: 0691158282

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"Why do beer commercials dominate Super Bowl advertising? How do political ceremonies establish authority? Why were circular forms favored for public festivals during the French Revolution? This book answers these questions using a single concept: common knowledge. Game theory shows that in order to coordinate its actions, a group of people must form "common knowledge." Each person wants to participate only if others also participate. Members must have knowledge of each other, knowledge of that knowledge, and so on. Michael Chwe applies this insight, with striking erudition, to analyze a range of rituals across history and cultures. He shows that public ceremonies are powerful not simply because they transmit meaning from a central source to each audience member but because they let audience members know what other members know. In a new afterword, Chwe delves into new applications of common knowledge, both in the real world and in experiments, and considers how generating common knowledge has become easier in the digital age." -- From the jacket.