Algebraic Geometry and Arithmetic Curves
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...
Search in google:
This book is a general introduction to 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. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises. Booknews A textbook describing the foundation of the geometry of arithmetic surfaces and the theory of stable reduction for undergraduate or graduate students and mathematicians who are not specialists in the field but have a typical undergraduate background in mathematics. Annotation c. Book News, Inc., Portland, OR
1Some topics in commutative algebra11.1Tensor products11.2Flatness61.3Formal completion152General properties of schemes262.1Spectrum of a ring262.2Ringed topological spaces332.3Schemes412.4Reduced schemes and integral schemes592.5Dimension673Morphisms and base change783.1The technique of base change783.2Applications to algebraic varieties873.3Some global properties of morphisms994Some local properties1154.1Normal schemes1154.2Regular schemes1264.3Flat morphisms and smooth morphisms1354.4Zariski's 'Main Theorem' and applications1495Coherent sheaves and Cech cohomology1575.1Coherent sheaves on a scheme1575.2Cech cohomology1785.3Cohomology of projective schemes1956Sheaves of differentials2106.1Kahler differentials2106.2Differential study of smooth morphisms2206.3Local complete intersection2276.4Duality theory2367Divisors and applications to curves2527.1Cartier divisors2527.2Weil divisors2677.3Riemann-Roch theorem2757.4Algebraic curves2847.5Singular curves, structure of Pic[superscript [degree]](X)3038Birational geometry of surfaces3178.1Blowing-ups3178.2Excellent schemes3328.3Fibered surfaces3479Regular surfaces3759.1Intersection theory on a regular surface3769.2Intersection and morphisms3949.3Minimal surfaces4119.4Applications to contraction; canonical model42910Reduction of algebraic curves45410.1Models and reductions45410.2Reduction of elliptic curves48310.3Stable reduction of algebraic curves50510.4Deligne-Mumford theorem532Bibliography557Index562
\ From The CriticsA textbook describing the foundation of the geometry of arithmetic surfaces and the theory of stable reduction for undergraduate or graduate students and mathematicians who are not specialists in the field but have a typical undergraduate background in mathematics. Annotation c. Book News, Inc., Portland, OR\ \
