ABSTRACT
I. Algebraic Geometry 165A. Introduction 165B. Undergraduate texts 166C. Graduate texts 167D. Special topics 169II. Differential Geometry 177A. Introduction 177B. Undergraduate texts 178C. Graduate texts 179D. Special topics 182III. Conclusion 190
It is not easy to get started in algebraic geometry, as its power and strength stems overwhelmingly from its beautiful mixture of so many different tools and areas of mathematics. This can be seen in how little overlap there is between Hartshorne’s Algebraic Geometry and Griffiths and Harris’ Principles of Algebraic Geometry, even though both are excellent beginning graduate texts. The goal of algebraic geometry is to understand the geometry 165
of the zero locus of a collection of polynomials (zero loci which we will call varieties). For example, the study of x2 + y2 — 1 = 0 as a circle is an elementary example in algebraic geometry. Since x2 + y2 — 1 is a polynomial, algebra (in particular commutative algebra) is important. Since frequently the studied polynomials are allowed to have complex solutions and complex coefficients, the theory of several complex variables becomes important. Further, the study of our zero loci as topological spaces is also natural. In other words, area after area of mathematics touches, influences, and is influenced by algebraic geometry, making it difficult but worthwhile to learn.If you were starting from scratch and had a year to learn, a good starting point would be M Reid’s Undergraduate Algebraic Geometry or K Smith, L Kahanpaa, P Kekaelaeinen, and WN Traves’ An Invitation to Algebraic Geometry. The next place to go for inspiration would be to read the first few sections of D Mumford’s Curves and Their Jacobians, which is now contained as part of his The Red Book of Varieties and Schemes. Here Mumford highlights what he calls the AMAZING SYNTHESIS, which is that there are three totally distinct ways to think about complex curves. Then you should look at D Mumford’s Algebraic Geometry I: Complex Projective Varieties. You are now ready to tackle Hartshorne’s Algebraic Geometry and Griffiths and Harris’s Principles of Algbraic Geometry. To test your understanding, you should then go through the many examples in J Harris’s Algebraic Geometry: A First Course. You are now easily ready to start real work. B. Undergraduate Texts
R Bix. Conics and Cubics: A Concrete Introduction to Algebraic Geometry. New York: Springer, 1998.This book concentrates on the zero loci of second degree (conics) and third degree (cubics) two-variable polynomials. This is a true undergraduate text. Bix shows the classical fact that smooth conics (i.e., ellipses, hyperbolas, and parabolas) are all equivalent under a projective change of coordinates. He then turns to cubics, which are much more difficult and still the object of research for some of the best mathematicians living. He shows in particular how the points on a cubic form an abelian group. M Reid. Undergraduate Algebraic Geometry. London Mathematical Society Student Text, Vol. 12. New York: Cambridge University Press, 1988. This is another good text, though the “undergraduate” in the title refers to British undergraduates, who start to concentrate in mathematics at an earlier age than their U.S. counterparts. Reid starts with plane curves, shows why the natural ambient space for these curves is projective space,
and then develops some of the basic tools needed for higher dimensional varieties. His brief history of algebraic geometry is also fun to read. D Cox, J Little, and D O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. 2nd ed. New York: Springer, 1996.This is almost universally admired. This book is excellent at explaining Groebner bases, which, as the main tool for producing algorithms in algebraic geometry, have been a major theme in recent research. It might not be the best place for the rank beginner, who might wonder why these algorithms are needed and interesting. K Smith, L Kahanpaa, P Kekaelaeinen, and WN Traves. An Invitation to Algebraic Geometry. New York: Springer, 2000.This is a wonderfully intuitive book, stressing the general ideas. It would be a good place to start for any student with a firm first course in algebra that included ring theory. CG Gibson. Elementary Geometry of Algebraic Curves: An Undergraduate Introduction. New York: Cambridge University Press, 1998.This is also a good place to begin. C. Graduate Texts
There are a number of graduate texts, though the first two on the list have dominated the market for the last 25 years. R Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, Vol. 52. New York: Springer, 1977.Hartshorne’s book relies on a heavy amount of commutative algebra. Its first chapter is an overview of algebraic geometry, while chapters four and five deal with curves and surfaces, respectively. It is in chapters two and three that the heavy abstract machinery that makes much of algebraic geometry so intimidating is presented. These chapters are not easy going but vital to get a handle on the Grothendieck revolution in mathematics. While this is not the first place to learn algebraic geometry, it should be the second or third source. Certainly young budding algebraic geometers should spend time doing all of the homework exercises in Hartshorne; this is the profession’s version of “paying your dues.” P Griffiths and J Harris. Principles of Algebraic Geometry. New York: John Wiley, 1978.Griffiths and Harris take a quite different tack from Hartshorne. They concentrate on the several complex variables approach. Chapter zero in fact is
an excellent overview of the basic theory of several complex variables. In this book analytic tools are freely used, but still, there is throughout an impressive amount of geometric insight presented. One word of warning: there are a fair number of minor errors. Basically the proofs and the statements of the theorems are “morally” correct, but the details could well be wrong. It is, in spite of this, a wonderful place to learn about algebraic geometry. I Shafarevich. Basic Algebraic Geometry. 2nd ed. New York: Springer, 1994.This is another standard, long-time favorite, now split into two volumes. The first volume concentrates on the relatively concrete case of subvarieties in complex projective space (which is the natural ambient space for much of algebraic geometry). Volume II turns to schemes, the key idea introduced by Grothendieck that helped change the very language of algebraic geometry. D Mumford. Algebraic Geometry I: Complex Projective Varieties. New York: Springer, 1995.This is a good place for a graduate student to get started. One of the strengths of this book is how Mumford will give a number of definitions, one right after another, of the same object, forcing the reader to see the different reasonable ways the same thing can be viewed. D Mumford. The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. 2nd ed. Lecture Notes in Mathematics, Vol. 1358. New York: Springer, 1999.Though now a yellow book, this text was for many years only available in mimeograph form from the Harvard Mathematics department, bound in red and hence its retained title “ The Red Book.” It was prized for its clear explanation of schemes. It is an ideal second or third place to learn about schemes. (It does take that many exposures.) In this new edition there is added Mumford’s delightful book Curves and Their Jacobians, which is a wonderful place for inspiration. W Fulton. Algebraic Curves. Redwood City, CA: Addison-Wesley, 1989.This is a good brief introduction. When it was written in the late 1960s, it was the only reasonable introduction to modern algebraic geometry. It can be viewed as being written with sheaf theory in mind without ever using the word “sheaf.” R Miranda. Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, Vol. 5. Providence, RI: American Mathematical Society, 1995.This has become a popular book in recent years, emphasizing the analytic side of algebraic geometry.
