This book is based on
one-semester courses
given at Harvard in 1984，
at Brown in 1985， and at
Harvard in 1988. It is
intended to be， as the
title suggests， a first
introduction to the subject.
Even so， a few words are
in order about the
purposes of the book.
Algebraic geometry has
developed tremendously
over the last century.
During the 19th century，
the subject was practiced
on a relatively concrete，
down-to-earth level; the
main objects of study were
projective varieties， and
the techniques for the
most part were grounded
in geometric constructions.
This approach flourished
during the middle of the
century and reached its
culmination in the work of
the Italian school around
the end of the 19th and
the beginning of the 20th
centuries.Ultimately， the
subject was pushed
beyond the limits of its
foundations： by the end
of its period the Italian
school had progressed to
the point where the
language and techniques
of the subject could no
longer serve to express or
carry out the ideas of its
best practitioners.
本书为英文版。
目录
Preface
Acknowledgments
Using This Book
PART Ⅰ：EXAMPLES OF VARIETIES AND MAPS
LECTURE 1 Affine and Projective Varieties
A Note About Our Field
Affine Space andAffineVarieties
Projective Space and Projective Varieties
Linear Spaces
Finite Sets
Hypersurfaces
Analytic Subvarieties and Submanifolds
The Twisted Cubic
Rational Normal Curves
Determinantai Representation of the Rational Normal Curve
Another Parametrization of the Rational Normal Curve
The Family of Plane Conics
A Synthetic Construction of the Rational Normal Curve
0ther Rational Curves
Varieties Defined over Subfields of K
A Note on Dimension，Smoothness，and Degree
LECTURE 2 Regular Functions and Maps
The Zariski Topology
Regular Functions on an Affine Variety
Projective Varieties
Regular Maps
The Veronese Map
Determinantal ReDresentatiOn of Veronese Varieties
Subvarieties of Veronese Varieties
The Segre Maps
Subvarieties of Segre Varieties
Products of Varieties
Graphs
Fiber Products
Combinations of Veronese and Segre Maps
LECTURE 3 Cones，Projections，and More About Products
Cones
Quadrics
Projections
M0re Cones
More Projections
Constructible Sets
LECTURE 4 Families and Parameter Spaces
Families of Varieties
The Universal Hyperplane
The Universal Hyperplane Section
Parameter Spaces of Hypersurfaces
Universal Families of Hypersurfaces
A Family of Lines
LECTURE 5 Ideals of Varieties，Irreducible Decomposition，and the Nullstellensatz
