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
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
Eisenbud: "The geometry of syzygies"
Matsumura: "Commutative Ring Theory"
Vasconcelos: "Computational methods in Commutative
Weibel: "Homological Algebra"
Hartshorne: "Algebraic Geometry" (algebraic)
Griffiths/Harris: "Principles of Algebraic Geometry"
Okonek, Schneider, Spindler : "Vector Bundles on Complex
Harris : A second course in Algebraic Geometry
Beauville: "Complex Algebraic Surfaces"
Friedman: "Algebraic Surfaces and Vector
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:
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
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.
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!