Algebraic Geometry and Arithmetic Curves

TRAMA

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. <BR>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<BR>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. <BR>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<BR>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. <BR>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.

SOMMARIO

1 - Some topics in commutative algebra2 - General Properties of schemes3 - Morphisms and base change4 - Some local properties5 - Coherent sheaves and Cech cohmology6 - Sheaves of differentials7 - Divisors and applications to curves8 - Birational geometry of surfaces9 - Regular surfaces10 - Reduction of algebraic curves

ALTRE INFORMAZIONI

• Condizione: Nuovo
• ISBN: 9780199202492
• Collana: Oxford Graduate Texts in Mathematics
• Dimensioni: 232 x 32.6 x 156 mm Ø 869 gr
• Formato: Brossura
• Pagine Arabe: 600
• Traduttore: Erne, Reinie