Algebraic Geometry and Arithmetic Curves

Algebraic Geometry and Arithmetic Curves
Author: Qing Liu
Publisher: Oxford University Press
Total Pages: 593
Release: 2006-06-29
Genre: Mathematics
ISBN: 0191547808

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 desingularisation (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 funadmental 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.

Basic Mathematics

Basic Mathematics
Author: Serge Lang
Publisher:
Total Pages: 475
Release: 1988-01
Genre: Mathematics
ISBN: 9783540967873

A Mathematician's Survival Guide

A Mathematician's Survival Guide
Author: Steven George Krantz
Publisher: American Mathematical Soc.
Total Pages: 242
Release: 2003
Genre: Education
ISBN: 082183455X

"When you are a young mathematician, graduate school marks the first step toward a career in mathematics. During this period, you will make important decisions which will affect the rest of your career. This book is a detailed guide to help you navigate graduate school and the years that follow. -- Publisher description.