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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.
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.
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:
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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