This text is intended to fill the gap between texts on classical algebraic geometry and the full-blown accounts of the theory of schemes. The text focuses on interesting examples, with a minimum of machinery, to show what is happening in the field. Included is a large number of exercises, spread throughout the text. The prerequisites for reading this book are modest: a little commutative algebra and an acquaintance with algebraic varieties.
The theory of schemes is the foundation for algebraic geometry proposed
and elaborated by Alexander Grothendieck and his coworkers. It has allowed
major progress in classical areas of algebraic geometry such as invariant
theory and the moduli of curves. It integrates algebraic number theory
with algebraic geometry, fulfilling the dreams of earlier generations of
number theorists. This integration has led to proofs of some of the major
conjectures in number theory (Deligne's proof of the Weil Conjectures,
Faltings proof of the Mordell Conjecture).This book is intended to bridge
the chasm between a first course in classical algebraic geometry and a
technical treatise on schemes. It focuses on examples, and strives to show
"what is going on" behind the definitions. There are many exercises to
test and extend the reader's understanding. The prerequisites are modest:
a little commutative algebra and an acquaintance with algebraic varieties,
roughly at the level of a one-semester course. The book aims to show schemes
in relation to other geometric ideas, such as the theory of manifolds.
Some familiarity with these ideas is helpful, though not required.
