Masayoshi Miyanishi's Books

Open Algebraic Surfaces

By: Masayoshi Miyanishi

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Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic background in algebraic geometry. This volume is a continuation of the work presented in the author's previous publication, Algebraic Geometry, Volume 136 in the AMS series, Translations of Mathematical Monographs.

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Page Count

269

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821805045

ISBN 13

9780821805046

Algebraic Geometry

By: Masayoshi Miyanishi

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Algebraic Geometry

By: Masayoshi Miyanishi

Students often find, in setting out to study algebraic geometry, that most of the serious textbooks on the subject require knowledge of ring theory, field theory, local rings, and transcendental field extensions, and even sheaf theory. Often the expected background goes well beyond college mathematics. This book, aimed at senior undergraduates and graduate students, grew out of Miyanishi's attempt to lead students to an understanding of algebraic surfaces while presenting thenecessary background along the way. Originally published in Japanese in 1990, it presents a self-contained introduction to the fundamentals of algebraic geometry. This book begins with background on commutative algebras, sheaf theory, and related cohomology theory. The next part introduces schemes andalgebraic varieties, the basic language of algebraic geometry. The last section brings readers to a point at which they can start to learn about the classification of algebraic surfaces.

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268

Publisher

American Mathematical Soc.

ISBN 10

082188770X

ISBN 13

9780821887707

Affine Space Fibrations

By: Rajendra V. Gurjar,Kayo Masuda,Masayoshi Miyanishi

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Affine Space Fibrations

By: Rajendra V. Gurjar,Kayo Masuda,Masayoshi Miyanishi

Affine algebraic geometry has progressed remarkably in the last half a century, and its central topics are affine spaces and affine space fibrations. This authoritative book is aimed at graduate students and researchers alike, and studies the geometry and topology of morphisms of algebraic varieties whose general fibers are isomorphic to the affine space while describing structures of algebraic varieties with such affine space fibrations.

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Page Count

275

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110577429

ISBN 13

9783110577426

Algebraic Surfaces In Positive Characteristics: Purely Inseparable Phenomena In Curves And Surfaces

By: Masayoshi Miyanishi,Hiroyuki Ito

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Algebraic Surfaces In Positive Characteristics: Purely Inseparable Phenomena In Curves And Surfaces

By: Masayoshi Miyanishi,Hiroyuki Ito

Customarily, the framework of algebraic geometry has been worked over an algebraically closed field of characteristic zero, say, over the complex number field. However, over a field of positive characteristics, many unpredictable phenomena arise where analyses will lead to further developments.In the present book, we consider first the forms of the affine line or the additive group, classification of such forms and detailed analysis. The forms of the affine line considered over the function field of an algebraic curve define the algebraic surfaces with fibrations by curves with moving singularities. These fibrations are investigated via the Mordell-Weil groups, which are originally introduced for elliptic fibrations.This is the first book which explains the phenomena arising from purely inseparable coverings and Artin-Schreier coverings. In most cases, the base surfaces are rational, hence the covering surfaces are unirational. There exists a vast, unexplored world of unirational surfaces. In this book, we explain the Frobenius sandwiches as examples of unirational surfaces.Rational double points in positive characteristics are treated in detail with concrete computations. These kinds of computations are not found in current literature. Readers, by following the computations line after line, will not only understand the peculiar phenomena in positive characteristics, but also understand what are crucial in computations. This type of experience will lead the readers to find the unsolved problems by themselves.

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456

Publisher

World Scientific

Published Date

ISBN 10

9811215227

ISBN 13

9789811215223

Affine Algebraic Geometry: Geometry Of Polynomial Rings

By: Masayoshi Miyanishi

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Affine Algebraic Geometry: Geometry Of Polynomial Rings

By: Masayoshi Miyanishi

Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar-Moh-Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata.Readers will find the book covers vast basic material on an extremely rigorous level:

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Page Count

441

Publisher

World Scientific

Published Date

ISBN 10

981128010X

ISBN 13

9789811280108

Mathematics of Fractals

By: Masaya Yamaguchi,Masayoshi Hata,Jun Kigami

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Mathematics of Fractals

By: Masaya Yamaguchi,Masayoshi Hata,Jun Kigami

This book aims at providing a handy explanation of the notions behind the self-similar sets called "fractals" and "chaotic dynamical systems". The authors emphasize the beautiful relationship between fractal functions (such as Weierstrass's) and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems. These functions are shown to be in a sense unique solutions of certain boundary problems. The last chapter of the book treats harmonic functions on fractal sets.

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