Masayoshi Miyanishi - Böcker

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"".

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.

This volume is the proceedings of the workshop on 'Algebraic Varieties and Automorphism Groups' which was held at RIMS, Kyoto University during July 7-11, 2014.The volume consists of fifteen original research articles by specialists in fast developing areas of algebraic geometry including various aspects of algebraic varieties (singularities, toric surfaces, weak positivity of log canonical pairs, Fano threefolds with cylinders, etc.), algebraic group actions on algebraic varieties (actions of additive group, unipotent group, semi-abelian variety, etc.), automorphism groups of affine varieties and Cremona groups.Highly recommended to researchers and graduate students interested in research trends in affine and projective algebraic geometry and algebraic group actions.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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.