**Algebraic Curves: An Introduction to Algebraic Geometry**

**ISBN/ASIN**: *0805330828,0201510103,9780805330823,9780201510102* | *2008* | *English* | *pdf* | *132/132* pages | *0.81* Mb**Author**: *William Fulton* | **Edition**: *3*

Preface

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 now to 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 at wfulton@umich.edu.

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 them gain

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.

PREFACE

First Preface, 1969

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.

We have assumed that the reader is familiar with some basic properties of rings,

ideals, and polynomials, such as is often covered in a one-semester course in modern

algebra; additional commutative algebra is developed in later sections. Chapter

1 begins with a summary of the facts we need from algebra. The rest of the chapter

is concerned with basic properties of affine algebraic sets; we have given Zariski’s

proof of the important Nullstellensatz.

The coordinate ring, function field, and local rings of an affine variety are studied

in Chapter 2. As in any modern treatment of algebraic geometry, they play a fundamental

role in our preparation. The general study of affine and projective varieties

is continued in Chapters 4 and 6, but only as far as necessary for our study of curves.

Chapter 3 considers affine plane curves. The classical definition of the multiplicity

of a point on a curve is shown to depend only on the local ring of the curve at the

point. The intersection number of two plane curves at a point is characterized by its

properties, and a definition in terms of a certain residue class ring of a local ring is

shown to have these properties. Bézout’s Theorem and Max Noether’s Fundamental

Theorem are the subject of Chapter 5. (Anyone familiar with the cohomology of

projective varieties will recognize that this cohomology is implicit in our proofs.)

In Chapter 7 the nonsingular model of a curve is constructed by means of blowing

up points, and the correspondence between algebraic function fields on one

variable and nonsingular projective curves is established. In the concluding chapter

the algebraic approach of Chevalley is combined with the geometric reasoning of

Brill and Noether to prove the Riemann-Roch Theorem.

These notes are from a course taught to Juniors at Brandeis University in 1967–

68. The course was repeated (assuming all the algebra) to a group of graduate students

during the intensive week at the end of the Spring semester. We have retained

an essential feature of these courses by including several hundred problems. The results

of the starred problems are used freely in the text, while the others range from

exercises to applications and extensions of the theory.

From Chapter 3 on, k denotes a fixed algebraically closed field. Whenever convenient

(including without comment many of the problems) we have assumed k to

be of characteristic zero. The minor adjustments necessary to extend the theory to

arbitrary characteristic are discussed in an appendix.

Thanks are due to Richard Weiss, a student in the course, for sharing the task

of writing the notes. He corrected many errors and improved the clarity of the text.

Professor PaulMonsky provided several helpful suggestions as I taught the course.

“Je n’ai jamais été assez loin pour bien sentir l’application de l’algèbre à la géométrie.

Je n’ai mois point cette manière d’opérer sans voir ce qu’on fait, et il me sembloit que

résoudre un probleme de géométrie par les équations, c’étoit jouer un air en tournant

une manivelle. La premiere fois que je trouvai par le calcul que le carré d’un

binôme étoit composé du carré de chacune de ses parties, et du double produit de

l’une par l’autre, malgré la justesse de ma multiplication, je n’en voulus rien croire

jusqu’à ce que j’eusse fai la figure. Ce n’étoit pas que je n’eusse un grand goût pour

l’algèbre en n’y considérant que la quantité abstraite; mais appliquée a l’étendue, je

voulois voir l’opération sur les lignes; autrement je n’y comprenois plus rien.”

Les Confessions de J.-J. Rousseau