Rationality Problems In Algebraic Geometry

Author : Arnaud Beauville,Brendan Hassett,Alexander Kuznetsov,Alessandro Verra
Publisher : Springer
Page : 170 pages
File Size : 50,5 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.

Author : Fedor Bogomolov,Yuri Tschinkel
Publisher : Springer Science & Business Media
Page : 314 pages
File Size : 46,9 Mb
Release : 2009-11-03
Category : Mathematics
ISBN : 9780817649340

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Cohomological and Geometric Approaches to Rationality Problems by Fedor Bogomolov,Yuri Tschinkel Pdf

Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. Such advances has led to many interdisciplinary applications to algebraic geometry. This comprehensive book consists of surveys of research papers by leading specialists in the field and gives indications for future research in rationality problems. Topics discussed include the rationality of quotient spaces, cohomological invariants of quasi-simple Lie type groups, rationality of the moduli space of curves, and rational points on algebraic varieties. This volume is intended for researchers, mathematicians, and graduate students interested in algebraic geometry, and specifically in rationality problems. Contributors: F. Bogomolov; T. Petrov; Y. Tschinkel; Ch. Böhning; G. Catanese; I. Cheltsov; J. Park; N. Hoffmann; S. J. Hu; M. C. Kang; L. Katzarkov; Y. Prokhorov; A. Pukhlikov

Author : Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman
Publisher : Springer Nature
Page : 433 pages
File Size : 43,8 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.

Author : Leonard Roth
Publisher : Springer Science & Business Media
Page : 150 pages
File Size : 44,5 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783642855313

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Algebraic Threefolds by Leonard Roth Pdf

Author : Károly Jr. Böröczky,János Kollár,Szamuely Tamas
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 54,7 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9783662051238

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Higher Dimensional Varieties and Rational Points by Károly Jr. Böröczky,János Kollár,Szamuely Tamas Pdf

Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "Higher Dimensional Varieties and Rational Points" held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.

Author : Bjorn Poonen
Publisher : American Mathematical Society
Page : 357 pages
File Size : 52,6 Mb
Release : 2023-08-10
Category : Mathematics
ISBN : 9781470474584

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Rational Points on Varieties by Bjorn Poonen Pdf

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

Author : Thomas A. Garrity
Publisher : American Mathematical Soc.
Page : 363 pages
File Size : 47,8 Mb
Release : 2013-02-01
Category : Mathematics
ISBN : 9780821893968

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Algebraic Geometry by Thomas A. Garrity Pdf

Algebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of ex

Author : Emmanuel Peyre,Yuri Tschinkel
Publisher : Birkhäuser
Page : 455 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883689

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Rational Points on Algebraic Varieties by Emmanuel Peyre,Yuri Tschinkel Pdf

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.

Author : National Research Council (U.S.). Committee on Rational Transformations,Virgil Snyder
Publisher : American Mathematical Soc.
Page : 518 pages
File Size : 46,7 Mb
Release : 1970
Category : Mathematics
ISBN : 0828401896

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Selected Topics in Algebraic Geometry by National Research Council (U.S.). Committee on Rational Transformations,Virgil Snyder Pdf

This book resulted from two reports (published in 1928 and 1932) of the Committee on Rational Transformations, established by the National Research Council. The purpose of the reports was to give a comprehensive survey of the literature on the subject. Each chapter is regarded as a separate unit that can be read independently.

Author : Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman
Publisher : Unknown
Page : 0 pages
File Size : 54,6 Mb
Release : 2021
Category : Electronic
ISBN : 3030754227

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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.

Author : Izzet Coskun,Tommaso de Fernex,Angela Gibney
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 52,5 Mb
Release : 2017-07-12
Category : $K$-theory -- Higher algebraic $K$-theory -- $Q$- and plus-constructions
ISBN : 9781470435578

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Surveys on Recent Developments in Algebraic Geometry by Izzet Coskun,Tommaso de Fernex,Angela Gibney Pdf

The algebraic geometry community has a tradition of running a summer research institute every ten years. During these influential meetings a large number of mathematicians from around the world convene to overview the developments of the past decade and to outline the most fundamental and far-reaching problems for the next. The meeting is preceded by a Bootcamp aimed at graduate students and young researchers. This volume collects ten surveys that grew out of the Bootcamp, held July 6–10, 2015, at University of Utah, Salt Lake City, Utah. These papers give succinct and thorough introductions to some of the most important and exciting developments in algebraic geometry in the last decade. Included are descriptions of the striking advances in the Minimal Model Program, moduli spaces, derived categories, Bridgeland stability, motivic homotopy theory, methods in characteristic and Hodge theory. Surveys contain many examples, exercises and open problems, which will make this volume an invaluable and enduring resource for researchers looking for new directions.

Author : Anonim
Publisher : Unknown
Page : 484 pages
File Size : 42,6 Mb
Release : 1970
Category : Electronic
ISBN : OCLC:488447224

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Selected Topics in Algebraic Geometry by Anonim Pdf

Author : Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel
Publisher : Springer Science & Business Media
Page : 324 pages
File Size : 55,8 Mb
Release : 2013-05-17
Category : Mathematics
ISBN : 9781461464822

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Birational Geometry, Rational Curves, and Arithmetic by Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel Pdf

​​​​This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.

Author : Gilberto Bini
Publisher : Springer Nature
Page : 371 pages
File Size : 50,8 Mb
Release : 2023-05-05
Category : Mathematics
ISBN : 9789811982811

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Algebraic Geometry between Tradition and Future by Gilberto Bini Pdf

An incredible season for algebraic geometry flourished in Italy between 1860, when Luigi Cremona was assigned the chair of Geometria Superiore in Bologna, and 1959, when Francesco Severi published the last volume of the treatise on algebraic systems over a surface and an algebraic variety. This century-long season has had a prominent influence on the evolution of complex algebraic geometry - both at the national and international levels - and still inspires modern research in the area. "Algebraic geometry in Italy between tradition and future" is a collection of contributions aiming at presenting some of these powerful ideas and their connection to contemporary and, if possible, future developments, such as Cremonian transformations, birational classification of high-dimensional varieties starting from Gino Fano, the life and works of Guido Castelnuovo, Francesco Severi's mathematical library, etc. The presentation is enriched by the viewpoint of various researchers of the history of mathematics, who describe the cultural milieu and tell about the bios of some of the most famous mathematicians of those times.

Author : N. E. Hurt
Publisher : Unknown
Page : 372 pages
File Size : 42,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 9401702527

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Many Rational Points by N. E. Hurt Pdf