Algebraic Varieties: Minimal Models and Finite Generation
Algebraic Varieties: Minimal Models and Finite Generation

Author: Yujiro Kawamata

Publisher: Cambridge University Press

Published: 2024-06-30

Total Pages: 263

ISBN-13: 1009344676

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The finite generation theorem is a major achievement of modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic zero is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar-Cascini-Hacon-McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend and break method, vanishing theorems, positivity theorems and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

Classification of Higher Dimensional Algebraic Varieties
Classification of Higher Dimensional Algebraic Varieties

Author: Christopher D. Hacon

Publisher: Springer Science & Business Media

Published: 2011-02-02

Total Pages: 206

ISBN-13: 3034602901

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Higher Dimensional Algebraic Geometry presents recent advances in the classification of complex projective varieties. Recent results in the minimal model program are discussed, and an introduction to the theory of moduli spaces is presented.

Classification of Higher Dimensional Algebraic Varieties: Compact moduli spaces of canonically polarized varieties
Classification of Higher Dimensional Algebraic Varieties: Compact moduli spaces of canonically polarized varieties

Author: Christopher D. Hacon

Publisher:

Published: 2010

Total Pages: 208

ISBN-13: 9781280391460

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This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof of the theorems on the existence of flips, on the existence of minimal models for varieties of log general type and of the finite generation of the canonical ring. The second part is an introduction to the theory of moduli spaces. It includes topics such as representing and moduli functors, Hilbert schemes, the boundedness, local closedness and separatedness of moduli spaces and the boundedness for varieties of general type. The book is aimed at advanced graduate students and researchers in algebraic geometry.

Classification of Algebraic Varieties
Classification of Algebraic Varieties

Author: Carel Faber

Publisher: European Mathematical Society

Published: 2011

Total Pages: 356

ISBN-13: 9783037190074

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Fascinating and surprising developments are taking place in the classification of algebraic varieties. The work of Hacon and McKernan and many others is causing a wave of breakthroughs in the minimal model program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the field. Inspired by this exciting progress, the editors organized a meeting at Schiermonnikoog and invited leading experts to write papers about the recent developments. The result is the present volume, a lively testimony to the sudden advances that originate from these new ideas. This volume will be of interest to a wide range of pure mathematicians, but will appeal especially to algebraic and analytic geometers.

Polynomial Methods and Incidence Theory
Polynomial Methods and Incidence Theory

Author: Adam Sheffer

Publisher: Cambridge University Press

Published: 2022-03-24

Total Pages: 263

ISBN-13: 1108832490

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A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.

Complex Algebraic Surfaces
Complex Algebraic Surfaces

Author: Arnaud Beauville

Publisher: Cambridge University Press

Published: 1996-06-28

Total Pages: 148

ISBN-13: 9780521498425

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Developed over more than a century, and still an active area of research today, the classification of algebraic surfaces is an intricate and fascinating branch of mathematics. In this book Professor BeauviIle gives a lucid and concise account of the subject, following the strategy of F. Enriques, but expressed simply in the language of modern topology and sheaf theory, so as to be accessible to any budding geometer. This volume is self contained and the exercises succeed both in giving the flavour of the extraordinary wealth of examples in the classical subject, and in equipping the reader with most of the techniques needed for research.

Algebraic Varieties
Algebraic Varieties

Author: G. Kempf

Publisher: Cambridge University Press

Published: 1993-09-09

Total Pages: 180

ISBN-13: 9780521426138

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An introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint.

Snowbird Lectures in Algebraic Geometry
Snowbird Lectures in Algebraic Geometry

Author: Ravi Vakil

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 202

ISBN-13: 0821837192

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A significant part of the 2004 Summer Research Conference on Algebraic Geometry (Snowbird, UT) was devoted to lectures introducing the participants, in particular, graduate students and recent Ph.D.'s, to a wide swathe of algebraic geometry and giving them a working familiarity with exciting, rapidly developing parts of the field. One of the main goals of the organizers was to allow the participants to broaden their horizons beyond the narrow area in which they are working. A fine selection of topics and a noteworthy list of contributors made the resulting collection of articles a useful resource for everyone interested in getting acquainted with the modern topic of algebraic geometry. The book consists of ten articles covering, among others, the following topics: the minimal model program, derived categories of sheaves on algebraic varieties, Kobayashi hyperbolicity, groupoids and quotients in algebraic geometry, rigid analytic varieties, and equivariant cohomology. Suitable for independent study, this unique volume is intended for graduate students and researchers interested in algebraic geometry.

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.