This new-in-paperback edition provides a general introduction to
algebraic and arithmetic geometry, starting with the theory of schemes,
followed by applications to arithmetic surfaces and to the theory of
reduction of algebraic curves. The first part introduces basic objects
such as schemes, morphisms, base change, local properties (normality,
regularity, Zariski's Main Theorem) This is followed by the more global
aspect: coherent sheaves and a finiteness theorem for their cohomology
groups. Then follows a chapter on sheaves of differentials, dualizing
sheaves, and Grothendieck's duality theory. The first part ends with the
theorem of Riemann-Roch and its application to the study of smooth
projective curves over a field. Singular curves are treated through a
detailed study of the Picard group. The second part starts with
blowing-ups and desingularization (embedded or not) of fibered surfaces
over a Dedekind ring that leads on to intersection theory on arithmetic
surfaces. Castelnuovo's criterion is proved and also the existence of
the minimal regular model. This leads to the study of reduction of
algebraic curves. The case of elliptic curves is studied in detail. The
book concludes with the fundamental theorem of stable reduction of
Deligne-Mumford. This book is essentially self-contained, including the
necessary material on commutative algebra. The prerequisites are few,
and including many examples and approximately 600 exercises, the book is
ideal for graduate students.Download
