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Algebraic Geometry: Further study of schemes

Author: 健爾·上野

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 222

ISBN-13: 9780821813584

This is the third part of the textbook on algebraic geometry by Kenji Ueno (the first two parts were published by the AMS as Volumes 185 and 197 of this series). Here the author presents the theory of schemes and sheaves beyond introductory notions, with the goal of studying properties of schemes and coherent sheaves necessary for full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion and Zariski's main theorem. The author also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. The book contains numerous exercises and problems with solutions, which makes it (together with two previous parts) appropriate for a graduate course on algebraic geometry or for self-study.

Algebraic Geometry 1

Author: 健爾·上野

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 178

ISBN-13: 0821808621

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By studying algebraic varieties over a field, this book demonstrates how the notion of schemes is necessary in algebraic geometry. It gives a definition of schemes and describes some of their elementary properties.

The Geometry of Schemes

Author: David Eisenbud

Publisher: Springer Science & Business Media

Published: 2006-04-06

Total Pages: 265

ISBN-13: 0387226397

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Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Algebraic Geometry 3

Author: Kenji Ueno

Publisher:

Published: 1999

Total Pages: 222

ISBN-13: 9780821808627

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Algebraic Geometry I: Schemes

Author: Ulrich Görtz

Publisher: Springer Nature

Published: 2020-07-27

Total Pages: 626

ISBN-13: 3658307331

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This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.

Algebraic Geometry

Author: Ulrich Görtz

Publisher: Springer Science & Business Media

Published: 2010-08-09

Total Pages: 615

ISBN-13: 3834897221

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Basic Algebraic Geometry 2

Author: Igor Rostislavovich Shafarevich

Publisher: Springer Science & Business Media

Published: 1994

Total Pages: 292

ISBN-13: 9783540575542

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The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with Volume 1 the author has revised the text and added new material, e.g. a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as in theoretical physics.

Algebraic Geometry

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 511

ISBN-13: 1475738498

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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.

The Red Book of Varieties and Schemes

Author: David Mumford

Publisher: Springer

Published: 2004-02-24

Total Pages: 314

ISBN-13: 3540460217

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Mumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry.

A Royal Road to Algebraic Geometry

Author: Audun Holme

Publisher: Springer Science & Business Media

Published: 2011-10-06

Total Pages: 365

ISBN-13: 3642192254

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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!