Rationality Problems in Algebraic Geometry
Rationality Problems in Algebraic Geometry

Author: Arnaud Beauville

Publisher: Springer

Published: 2016-12-06

Total Pages: 176

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.

Rational Points on Varieties
Rational Points on Varieties

Author: Bjorn Poonen

Publisher: American Mathematical Soc.

Published: 2017-12-13

Total Pages: 358

ISBN-13: 1470437732

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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.

Birational Geometry, Rational Curves, and Arithmetic
Birational Geometry, Rational Curves, and Arithmetic

Author: Fedor Bogomolov

Publisher: Springer Science & Business Media

Published: 2013-05-17

Total Pages: 324

ISBN-13: 146146482X

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​​​​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.

Algebraic Geometry for Scientists and Engineers
Algebraic Geometry for Scientists and Engineers

Author: Shreeram Shankar Abhyankar

Publisher: American Mathematical Soc.

Published: 1990

Total Pages: 311

ISBN-13: 0821815350

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Based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, this book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities.

Algebraic Geometry between Tradition and Future
Algebraic Geometry between Tradition and Future

Author: Gilberto Bini

Publisher: Springer Nature

Published: 2023-05-05

Total Pages: 371

ISBN-13: 9811982813

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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.

Surveys on Recent Developments in Algebraic Geometry
Surveys on Recent Developments in Algebraic Geometry

Author: Izzet Coskun

Publisher: American Mathematical Soc.

Published: 2017-07-12

Total Pages: 386

ISBN-13: 1470435578

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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.

Facets of Algebraic Geometry
Facets of Algebraic Geometry

Author: Paolo Aluffi

Publisher: Cambridge University Press

Published: 2022-04-07

Total Pages: 417

ISBN-13: 1108792502

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Written to honor the enduring influence of William Fulton, these articles present substantial contributions to algebraic geometry.

Algebraic Geometry
Algebraic Geometry

Author: Dr. B. Phalaksha Murthy

Publisher: RK Publication

Published: 2024-09-20

Total Pages: 326

ISBN-13: 9348020145

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Algebraic Geometry is a profound exploration of the intersection between algebra and geometry, delving into the study of geometric structures defined by polynomial equations. This book covers foundational topics such as varieties, schemes, and morphisms, bridging abstract algebraic theories with tangible geometric interpretations. Through rigorous proofs and illustrative examples, it guides readers from basic concepts to advanced topics, including cohomology, intersection theory, and moduli spaces. Ideal for mathematicians and students, Algebraic Geometry serves both as a comprehensive introduction and as a reference for deeper mathematical inquiries in geometry.

Rational and Nearly Rational Varieties
Rational and Nearly Rational Varieties

Author: János Kollár

Publisher: Cambridge University Press

Published: 2004-04-22

Total Pages: 246

ISBN-13: 9780521832076

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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.

Rationality of Varieties
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.