Although algebraic geometry is a highly developed and thriving field of mathematics, it is notoriously difficult for the beginner to make his way into the subject. There are several texts on an undergraduate level that give an excellent treatment of the classical theory of plane curves, but these do not prepare the student adequately for modern algebraic geometry.

On the other hand, most books with a modern approach demand considerable background in algebra and topology, often the equivalent of a year or more of graduate study. The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. (First Preface, 1969)

Third Preface, 2008
This text has been out of print for several years, with the author holding copyrights. Since I continue to hear from young algebraic geometers who used this as their first text, I am glad nowto make this edition available without charge to anyone interested. I am most grateful to Kwankyu Lee for making a careful LaTeX version, which was the basis of this edition; thanks also to Eugene Eisenstein for help with the graphics.

As in 1989, I have managed to resist making sweeping changes. I thank all who have sent corrections to earlier versions, especially Grzegorz Bobi´nski for the most recent and thorough list. It is inevitable that this conversion has introduced some new errors, and I and future readers will be grateful if you will send any errors you find to me.

Second Preface, 1989
When this book first appeared, there were few texts available to a novice in modern algebraic geometry. Since then many introductory treatises have appeared, including excellent texts by Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz, Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.

The past two decades have also seen a good deal of growth in our understanding of the topics covered in this text: linear series on curves, intersection theory, and the Riemann-Roch problem. It has been tempting to rewrite the book to reflect this progress, but it does not seem possible to do so without abandoning its elementary character and destroying its original purpose: to introduce students with a little algebra background to a few of the ideas of algebraic geometry and to help themgain some appreciation both for algebraic geometry and for origins and applications of many of the notions of commutative algebra. If working through the book and its exercises helps prepare a reader for any of the texts mentioned above, that will be an added benefit.

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Hardcover: 226 pages
Author: William Fulton
Publisher: Addison Wesley Publishing Company (March 1989)
Language: English
ISBN-10: 0201510103
ISBN-13: 978-0201510102

Contents
Preface i
1 Affine Algebraic Sets 1
1.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Affine Space and Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Ideal of a Set of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The Hilbert Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Irreducible Components of an Algebraic Set . . . . . . . . . . . . . . . . 7
1.6 Algebraic Subsets of the Plane . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Modules; Finiteness Conditions . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Integral Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.10 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Affine Varieties 17
2.1 Coordinate Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 PolynomialMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Coordinate Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Rational Functions and Local Rings . . . . . . . . . . . . . . . . . . . . . 20
2.5 Discrete Valuation Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Direct Products of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Operations with Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9 Ideals with a Finite Number of Zeros . . . . . . . . . . . . . . . . . . . . . 26
2.10 QuotientModules and Exact Sequences . . . . . . . . . . . . . . . . . . . 27
2.11 FreeModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Local Properties of Plane Curves 31
3.1 Multiple Points and Tangent Lines . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Multiplicities and Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Intersection Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Projective Varieties 43
4.1 Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Projective Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Affine and Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Multiprojective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Projective Plane Curves 53
5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Linear Systems of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Bézout’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Multiple Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Max Noether’s Fundamental Theorem . . . . . . . . . . . . . . . . . . . . 60
5.6 Applications of Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 62
6 Varieties,Morphisms, and RationalMaps 67
6.1 The Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Morphisms of Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 Products and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 Algebraic Function Fields and Dimension of Varieties . . . . . . . . . . 75
6.6 RationalMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 Resolution of Singularities 81
7.1 RationalMaps of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Blowing up a Point in A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.3 Blowing up Points in P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.4 Quadratic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5 NonsingularModels of Curves . . . . . . . . . . . . . . . . . . . . . . . . 92
8 Riemann-Roch Theorem 97
8.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 The Vector Spaces L(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Riemann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.4 Derivations and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.5 Canonical Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.6 Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A Nonzero Characteristic 113
B Suggestions for Further Reading 115
C Notation 117

About William Fulton:

William Fulton (born 1939) is an American algebraic geometer. He received his undergraduate degree from Brown University in 1961 and his doctorate from Princeton University in 1966. Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago. He is currently (2007) a professor at the University of Michigan.

Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory. In 1996 he received the Steele Prize for mathematical exposition for his text Intersection Theory.[1] Fulton is a member of the U. S. National Academy of Sciences. (wikipedia.org)