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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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