Higher-Dimensional Algebraic Geometry

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Author: Olivier Debarre

Publisher: Springer Science & Business Media

ISBN: 147575406X

Category: Mathematics

Page: 234

View: 5442

The classification theory of algebraic varieties is the focus of this book. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The authors goal is to provide an easily accessible introduction to the subject. The book starts with preparatory and standard definitions and results, then moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Moris minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well-organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction.

Complex Algebraic Surfaces

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Author: Arnaud Beauville

Publisher: Cambridge University Press

ISBN: 9780521498425

Category: Mathematics

Page: 132

View: 9188

The classification of algebraic surfaces is an intricate and fascinating branch of mathematics, developed over more than a century and still an active area of research today. In this book, Professor Beauville gives a lucid and concise account of the subject, expressed simply in the language of modern topology and sheaf theory, and accessible to any budding geometer. A chapter on preliminary material ensures that this volume is self-contained while the exercises succeed both in giving the flavor of the classical subject, and in equipping the reader with the techniques needed for research. The book is aimed at graduate students in geometry and topology.

Singularities of the Minimal Model Program

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Author: János Kollár,Sándor Kovács

Publisher: Cambridge University Press

ISBN: 1107035341

Category: Mathematics

Page: 370

View: 4581

An authoritative reference and the first comprehensive treatment of the singularities of the minimal model program.

Rational Curves on Algebraic Varieties

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Author: Janos Kollar

Publisher: Springer Science & Business Media

ISBN: 3662032767

Category: Mathematics

Page: 321

View: 801

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.

Algebraic Geometry and Commutative Algebra

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Author: Siegfried Bosch

Publisher: Springer Science & Business Media

ISBN: 1447148290

Category: Mathematics

Page: 504

View: 8890

Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck’s schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly seemed to be far away from geometry, like algebraic number theory. The new techniques paved the way to spectacular progress such as the proof of Fermat’s Last Theorem by Wiles and Taylor. The scheme-theoretic approach to algebraic geometry is explained for non-experts. More advanced readers can use the book to broaden their view on the subject. A separate part deals with the necessary prerequisites from commutative algebra. On a whole, the book provides a very accessible and self-contained introduction to algebraic geometry, up to a quite advanced level. Every chapter of the book is preceded by a motivating introduction with an informal discussion of the contents. Typical examples and an abundance of exercises illustrate each section. This way the book is an excellent solution for learning by yourself or for complementing knowledge that is already present. It can equally be used as a convenient source for courses and seminars or as supplemental literature.

Algebraic Geometry over the Complex Numbers

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Author: Donu Arapura

Publisher: Springer Science & Business Media

ISBN: 1461418097

Category: Mathematics

Page: 329

View: 8136

This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry. The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.

Introduction to the Mori Program

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Author: Kenji Matsuki

Publisher: Springer Science & Business Media

ISBN: 147575602X

Category: Mathematics

Page: 478

View: 3363

Mori's Program is a fusion of the so-called Minimal Model Program and the IItaka Program toward the biregular and/or birational classification of higher dimensional algebraic varieties. The author presents this theory in an easy and understandable way with lots of background motivation. Prerequisites are those covered in Hartshorne's book "Algebraic Geometry." This is the first book in this extremely important and active field of research and will become a key resource for graduate students wanting to get into the area.

Geometric Invariant Theory

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Author: David Mumford,John Fogarty,Frances Kirwan

Publisher: Springer Science & Business Media

ISBN: 9783540569633

Category: Mathematics

Page: 292

View: 6089

"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Algebraic Geometry

An Introduction to Birational Geometry of Algebraic Varieties

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Author: S. Iitaka

Publisher: Springer

ISBN: 9781461381211

Category: Mathematics

Page: 357

View: 3809

The aim of this book is to introduce the reader to the geometric theory of algebraic varieties, in particular to the birational geometry of algebraic varieties. This volume grew out of the author's book in Japanese published in 3 volumes by Iwanami, Tokyo, in 1977. While writing this English version, the author has tried to rearrange and rewrite the original material so that even beginners can read it easily without referring to other books, such as textbooks on commutative algebra. The reader is only expected to know the definition of Noetherin rings and the statement of the Hilbert basis theorem. The new chapters 1, 2, and 10 have been expanded. In particular, the exposition of D-dimension theory, although shorter, is more complete than in the old version. However, to keep the book of manageable size, the latter parts of Chapters 6, 9, and 11 have been removed. I thank Mr. A. Sevenster for encouraging me to write this new version, and Professors K. K. Kubota in Kentucky and P. M. H. Wilson in Cam bridge for their careful and critical reading of the English manuscripts and typescripts. I held seminars based on the material in this book at The University of Tokyo, where a large number of valuable comments and suggestions were given by students Iwamiya, Kawamata, Norimatsu, Tobita, Tsushima, Maeda, Sakamoto, Tsunoda, Chou, Fujiwara, Suzuki, and Matsuda.

Classification of Higher Dimensional Algebraic Varieties

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Author: Christopher D. Hacon,Sándor Kovács

Publisher: Springer Science & Business Media

ISBN: 3034602901

Category: Mathematics

Page: 220

View: 7339

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.

Algebraic Geometry

An Introduction

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Author: Daniel Perrin

Publisher: Springer Science & Business Media

ISBN: 9781848000568

Category: Mathematics

Page: 263

View: 8097

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.

Hodge Theory and Complex Algebraic Geometry I:

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Author: Claire Voisin

Publisher: Cambridge University Press

ISBN: 9781139437691

Category: Mathematics

Page: N.A

View: 9101

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Undergraduate Algebraic Geometry

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Author: Miles Reid

Publisher: Cambridge University Press

ISBN: 9780521356626

Category: Mathematics

Page: 129

View: 7799

This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time.

Positivity in algebraic geometry 2

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Author: R.K. Lazarsfeld

Publisher: Springer Science & Business Media

ISBN: 9783540225348

Category: Mathematics

Page: 385

View: 7315

This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".

Algebraic Geometry: From algebraic varieties to schemes

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Author: 健爾·上野

Publisher: American Mathematical Soc.

ISBN: 0821808621

Category: Mathematics

Page: 154

View: 1762

This is the first of three volumes on algebraic geometry. The second volume, Algebraic Geometry 2: Sheaves and Cohomology, is available from the AMS as Volume 197 in the Translations of Mathematical Monographs series. Early in the 20th century, algebraic geometry underwent a significant overhaul, as mathematicians, notably Zariski, introduced a much stronger emphasis on algebra and rigor into the subject. This was followed by another fundamental change in the 1960s with Grothendieck's introduction of schemes. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject. The book begins with a description of the standard theory of algebraic varieties. Then, sheaves are introduced and studied, using as few prerequisites as possible. Once sheaf theory has been well understood, the next step is to see that an affine scheme can be defined in terms of a sheaf over the prime spectrum of a ring. By studying algebraic varieties over a field, Ueno demonstrates how the notion of schemes is necessary in algebraic geometry. This first volume gives a definition of schemes and describes some of their elementary properties. It is then possible, with only a little additional work, to discover their usefulness. Further properties of schemes will be discussed in the second volume. Ueno's book is a self-contained introduction to this important circle of ideas, assuming only a knowledge of basic notions from abstract algebra (such as prime ideals). It is suitable as a text for an introductory course on algebraic geometry.

Algebraic Curves and Riemann Surfaces

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Author: Rick Miranda

Publisher: American Mathematical Soc.

ISBN: 0821802682

Category: Mathematics

Page: 390

View: 7517

The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.

Algebraic Topology

A First Course

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Author: William Fulton

Publisher: Springer Science & Business Media

ISBN: 1461241804

Category: Mathematics

Page: 430

View: 6589

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups

A Royal Road to Algebraic Geometry

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Author: Audun Holme

Publisher: Springer Science & Business Media

ISBN: 9783642192258

Category: Mathematics

Page: 366

View: 3384

This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements. Euclid is said to have answered: “There is no royal road to geometry!” The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained. The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!

Conics and Cubics

A Concrete Introduction to Algebraic Curves

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Author: Robert Bix

Publisher: Springer Science & Business Media

ISBN: 1475729758

Category: Mathematics

Page: 292

View: 6768

Algebraic curves are the graphs of polynomial equations in two vari 3 ables, such as y3 + 5xy2 = x + 2xy. By focusing on curves of degree at most 3-lines, conics, and cubics-this book aims to fill the gap between the familiar subject of analytic geometry and the general study of alge braic curves. This text is designed for a one-semester class that serves both as a a geometry course for mathematics majors in general and as a sequel to college geometry for teachers of secondary school mathe matics. The only prerequisite is first-year calculus. On the one hand, this book can serve as a text for an undergraduate geometry course for all mathematics majors. Algebraic geometry unites algebra, geometry, topology, and analysis, and it is one of the most exciting areas of modem mathematics. Unfortunately, the subject is not easily accessible, and most introductory courses require a prohibitive amount of mathematical machinery. We avoid this problem by focusing on curves of degree at most 3. This keeps the results tangible and the proofs natural. It lets us emphasize the power of two fundamental ideas, homogeneous coordinates and intersection multiplicities.