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
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 fundamental theorem of stable reduction
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.
"Although other books do offer a fast passage to modern number theory, ... only Liu provides a systematic development of algebraic geometry aimed at arithmetic."--Choice
Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of differentials 7. Divisors and applications to curves 8. Birational geometry of surfaces 9. Regular surfaces 10. Reduction of algebraic curves Bibilography Index