Graduate Studies in Commutative Algebra and Algebraic Geometry

Some time ago, a graduate student interested in commutative algebra and algebraic geometry asked me for input on what to read and how to choose a thesis advisor. For what they are worth, here are some thoughts on the subject.

I. What to study

Here are my top books for a beginning graduate student interested in commutative algebra and algebraic geometry.

Cox, Little, O'Shea: "Ideals, varieties, and algorithms".
This is a Springer UTM, and is a nice introduction to the area; I taught an undergraduate class out of it with great success. It is full of examples and an easy, fun read.

Atiyah-Macdonald: "Commutative Algebra"
A very approachable introduction to the important ideas of commutative algebra , and a good lead in to the book of Eisenbud (see below).

Miranda: "Algebraic curves and Riemann surfaces"
Probably the most comprehensive introduction to curves, it looks at the subject from all angles. Miranda notes in the introduction that the book grew out of a course he has taught 5 times; this shows in the smooth presentation.

Griffiths: "Introduction to Algebraic Curves"
A complex analytic introduction to algebraic curves, and one of my favorite elementary books. You need only a class in one complex variable (or read, say, Churchill and Brown); it is amazing how far one can go with elementary tools.

Harris: "Algebraic Geometry-A first course"
This book is a logical follow-up to Cox-Little-O'Shea, and is also very example driven. Lots of good exercises.

Schenck: "Computational algebraic geometry"
Notes from an undergraduate class I taught at Harvard; basics of commutative algebra and Grobner bases, plus a quick intro to homological algebra (Ext and Tor) and a bit of sheaf cohomology.

For the student interested in connections to discrete geometry and combinatorics, the following two texts are a good start:

Stanley: "Commutative Algebra and Combinatorics"
This book explores the beautiful connections between commutative algebra and combinatorics. Also, gives a nice introduction to homological algebra, with many concrete examples of things like free resolutions, as well as your favorite derived functors (Ext, Tor and local cohomology).

Ziegler: "Lectures on Polytopes"
This book pairs well with Stanley's book above; it is another book with lots of examples and an informal, easy style. Great book for a grad run seminar in the summer (I did this a few years back), and useful if you will be doing anything with toric varieties.

OK, so now you've covered all the above, and need some more background/reference books. Here is the advanced reading list:

Commutative and Homological Algebra

Eisenbud: "Commutative Algebra with a view toward Algebraic Geometry"
Eisenbud: "The geometry of syzygies"
Matsumura: "Commutative Ring Theory"
Vasconcelos: "Computational methods in Commutative Algebra and Algebraic Geometry"
Weibel: "Homological Algebra"

Algebraic Geometry

Hartshorne: "Algebraic Geometry" (algebraic)
Griffiths/Harris: "Principles of Algebraic Geometry" (analytic)
Okonek, Schneider, Spindler : "Vector Bundles on Complex Projective Spaces"
Harris : A second course in Algebraic Geometry


Beauville: "Complex Algebraic Surfaces"
Friedman: "Algebraic Surfaces and Vector Bundles"
Miranda: An Overview of Algebraic Surfaces, in "Algebraic Geometry", edited by Sinan Sertoz.
Reid: "Chapters on Algebraic Surfaces", in "Complex Algebraic Geometry", edited by Janos Kollar


Cox, Little, O'Schenck: Toric Varieties. This is an AMS Graduate studies in math book. The first 9 chapters of the book each begin with an introductory section reviewing the basic commutative algebra/algebraic geometry background. The second portion of the book treats advanced topics (Hirzebruch-Riemann-Roch, Equivariant Cohomology, Toric GIT, Secondary fan, Toric minimal model program, Toric resolutions).
Danilov: The geometry of toric varieties. A nice overview, and one of the watersheds in the theory (from 1978).
Ewald: "Combinatorial Convexity and Algebraic Geometry"
Fulton: "Toric Varieties"
Sturmfels: "Grobner Bases and Convex Polytopes"

II. The Thesis Advisor

And now, some remarks about one of the most important decisions of your grad career: finding a thesis advisor. I think the most important considerations are those below:

1) Compatibility.
It is really important that you and your advisor get along well; grad school is difficult enough without having a chilly (or worse, hostile) relationship with your advisor. The best time to figure this out is BEFORE you ask someone if they'll work with you. One way to determine this is by taking a course with the person, and seeing how you interact when you go to office hours to ask questions. It goes without saying that you should make sure you do a good job in the class, because if you do poorly, it is likely that the potential advisor will not be interested in taking you on as a student.

2) Track record and reputation.
It is important that your advisor be respected as a scholar. You will quickly find that one of the first questions mathematicians ask a graduate student is: "and who is your advisor?". You may also want to find out how many theses the professor has directed, and how many students have dropped out (this should be done tactfully, of course, by talking to former students or the director of graduate studies).

3) Good thesis topic.
Most professors supervising dissertations have a set of problems that they think might make an appropriate thesis; you should ask about this, 'cause getting a bad problem can cause you to waste years of your life.

III. Studying with me

In keeping with the remarks above, I ask that students take two classes with me before we discuss advisor possiblities. My students need to have a broad background, and I expect them to pass comps in the areas of algebra (500-501), complex analysis (542), algebraic topology (525-526), and differentiable manifolds (518-519). Students should also take the introductory classes in commutative and homological algebra (502, 505), and algebraic geometry (510-511). These classes are prerequisites for all reading courses.

IV. Disclaimer

The remarks above reflect my own tastes and are obviously not right for everyone. And my remarks about choosing an advisor are also what I would do in hindsight; I was fortunate enough to luck into a fantastic advisor by chance, which illustrates the aphorism that (sometimes) it is better to be lucky than to be good!

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