Ebook: Algebraic Curves: An Introduction to Algebraic Geometry

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 [email protected].

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