Math 632: Algebraic Geometry

Professor: David E Speyer

Winter 2015

Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

Gathering time It is extremely difficult to learn this material alone. In order to give people a place and time to meet their fellow students working on algebraic geometry, Tuesday afternoons in the tea room. There is a schedule problem: based on class polling, most people become free after 4 PM, I will almost always need to leave by 5, and 4-5 is the colloquium. I will come to the tea room between 4 and 5 unless I am going to the colloquium, but I hope that you will also be able to help each other.

Webpage: http://www.math.lsa.umich.edu/~speyer/632

Textbooks: Algebraic Geometry, by Robin Hartshorne. I also strongly recommend Foundations of Algebraic Geometry by Ravi Vakil.

The Update: A day by day summary of the course, written by the students and edited by me.

Other valuable online sources: Mel Hochster's commutative algebra notes. The algebraic-geometry tag at mathoverflow and math.stackexchange (but see the homework policy below). Mark Haiman's synopses of EGA, Ravi's notes and blog, the stacks project, the algebraic geometry tag at nLab.

Intended Level: This is not a first course in algebraic geometry. Many of you took Math 631 with me last term, and I am glad to see you back. Whether through my course or another, you should already have some geometric intuition for varieties before you work with schemes.

The commutative algebra prerequisite will also be much more serious this term. Last term, I made an effort to avoid significant use of homological algebra, localization, completion or tensor products, and it was often possible to do so by some clever planning. That is no longer possible. You should have taken Math 614 or the equivalent before taking this class.

Anticipated topics The primary goals of this construction are schemes and sheaf cohomology — that is to say, Chapters 2 and 3 of Hartshorne. I hope to have time to say a bit about K₀-theory, which is one of my favorite subjects, and about algebraic de Rham theory. Ultimately, though, the two main technical subjects are large enough that there will not be time for another large topic.

Problem Sets: I will assign weekly problem sets, due on Wednesdays. See below for the homework policy. Unlike last term, most of my problem sets will be taken from Hartshorne's book.

Reading:I think Hartshorne's book is best used as a reference after understanding the big picture. I will once again assign reading, but you should focus on reading the statements of theorems and definitions, and only on reading those theorems that you want to read the proofs of. It is very easy to get lost in the details with Hartshorne.

The obvious assertion quizzes: The foundations of scheme theory involve an extraordinary number of plausible assertions, each of which takes only a few minutes to check. I will maintain a file of such assertions and, on each Wednesday, I will challenge you to prove your choice of one of three of them. My tentative plan is that assertions from k weeks ago will appear with probability proportional to 2^-k; I may change this formula if it produces absurd results.

Update Like Karen Smith's fantastic 632 classes, this class will have a Daily Update covering what has happened in class each day. Unlike Karen, I will insist that you write it! I will have you take turns writing a one or two paragraph summary of the key definitions and theorems of the day. (I will be glad to help you with this.) These will be concatenated into a single document, which I hope will serve as a valuable record.

Schedule and reading

Problem Sets

The daily update