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This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra.

The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory.

The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.

Klaus Hulek: Universität Hannover, Hannover, Germany

Cover 1
Title 4
Copyright 5
Contents 6
Preface 8
Chapter 0. Introduction 10
Exercises 24
Chapter 1. Affine Varieties 26
§1.1. Hilbert's Nullstellensatz 26
§1.2. Polynomial Functions and Maps 46
§1.3. Rational Functions and Maps 58
Exercises 67
Chapter 2. Projective Varieties 72
§2.1. Projective Space 72
§2.2. Projective Varieties 75
§2.3. Rational Functions and Morphisms 84
Exercises 103
Chapter 3. Smooth Points and Dimension 106
§3.1. Smooth and Singular Points 106
§3.2. Algebraic Characterization of the Dimension of a Variety 112
Exercises 121
Chapter 4. Plane Cubic Curves 124
§4.1. Plane Curves 124
§4.2. Intersection Multiplicity 127
§4.3. Classification of Smooth Cubics 135
§4.4. The Group Structure of an Elliptic Curve 145
Exercises 148
Chapter 5. Cubic Surfaces 152
§5.1. The Existence of Lines on a Cubic 153
§5.2. The Configuration of the 27 Lines 160
§5.3. Rationality of Cubics 170
Exercises 172
Chapter 6. Introduction to the Theory of Curves 176
§6.1. Divisors on Curves 176
§6.2. The Degree of a Principal Divisor 181
§6.3. Bezout's Theorem 191
§6.4. Linear Systems on Curves 193
§6.5. Projective Embeddings of Curves 202
Exercises 213
Bibliography 216
§A. Books on Commutative Algebra 216
§B. Books on Algebraic Geometry 216
§C. Other Literature 217
Index 220
A 220
B 220
C 220
D 220
E 220
F 220
G 221
H 221
I 221
J 221
K 221
L 221
M 221
N 221
O 221
P 221
Q 221
R 221
S 222
T 222
U 222
V 222
W 222
Z 222
Back Cover 225
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