Elementary Algebraic Geometry

by Hulek, Klaus; Verrill, Helena

• ISBN13: 9780821829523
• ISBN10: 0821829521
• Format: Paperback
• Copyright: 2003-02-01
• Publisher: Amer Mathematical Society

Summary

This is a true 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.

Table of Contents

Preface	vii
Introduction	1	(16)
Exercise	15	(2)
Affine Varieties	17	(46)
Hilbert's Nullstellensatz	17	(20)
Polynomial Functions and Maps	37	(12)
Rational Functions and Maps	49	(14)
Exercises	58	(5)
Projective Varieties	63	(34)
Projective Space	63	(3)
Projective Varieties	66	(9)
Rational Functions and Morphisms	75	(22)
Exercises	94	(3)
Smooth Points and Dimension	97	(18)
Smooth and Singular Points	97	(6)
Algebraic Characterization of the Dimension of a Variety	103	(12)
Exercises	112	(3)
Plane Cubic Curves	115	(28)
Plane Curves	115	(3)
Intersection Multiplicity	118	(8)
Classification of Smooth Cubics	126	(10)
The Group Structure of an Elliptic Curve	136	(7)
Exercises	139	(4)
Cubic Surfaces	143	(24)
The Existence of Lines on a Cubic	144	(7)
The Configuration of the 27 Lines	151	(10)
Rationality of Cubics	161	(6)
Exercises	163	(4)
Introduction to the Theory of Curves	167	(40)
Divisors on Curves	167	(5)
The Degree of a Principal Divisor	172	(10)
Bezout's Theorem	182	(2)
Linear Systems on Curves	184	(9)
Projective Embeddings of Curves	193	(14)
Exercises	204	(3)
Bibliography	207	(4)
Books on Commutative Algebra	207	(1)
Books on Algebraic Geometry	207	(1)
Other Literature	208	(3)
Index	211