Algebraic Surfaces in Positive Characteristics

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&#x2013;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&#x2013;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.<b>Contents:</b> <ul><li><b><i>Forms of the Affine Line:</i></b><ul><li>Picard Scheme and Jacobian Variety</li><li>Forms of the Affine Line</li><li>Groups of Russell Type</li><li>Hyperelliptic Forms of the Affine Line</li><li>Automorphisms</li><li>Divisor Class Groups</li></ul></li><li><b><i>Purely Inseparable and Artin&#x2013;Schreier Coverings:</i></b><ul><li>Vector Fields and Infinitesimal Group Schemes</li><li>Zariski Surfaces</li><li>Quasi-Elliptic or Quasi-Hyperelliptic Fibrations</li><li>Mordell-Weil Groupps of Quasi-Elliptic or Quasi-Hyperelliptic Surfaces</li><li>Artin-Schreier Coverings</li><li>Higher Derivations</li><li>Unified <i>p</i>-Group Scheme</li></ul></li><li><b><i>Rational Double Points:</i></b><ul><li>Basics on Rational Double Points</li><li>Deformation of Rational Double Points</li><li>Open Problems on Rational Double Points in Positive Characteristics</li></ul></li></ul><br><b>Readership:</b> Graduate students and researchers in the fields of Algebraic Geometry, Fields and Rings, and Commutative Algebra.Form of the Affine Line;Zariski Surface;Quasi-Elliptic Fibration;Mordell–Weil Group;Artin–Schreier Covering;Higher Derivation;Unified p-group Scheme;Rational Double Point;Versal Deformation;Equisingular Locus0<b>Key Features:</b><ul><li>The mainstreams of arguments are explained, followed by computations</li><li>Several concrete examples are given to elucidate the stated results</li><li>All in all, the present book is more for practice than learning a general theory</li></ul>