Math 631: Algebraic Geometry

Professor: David E Speyer

Fall 2014

Course meets: Monday, Wednesday, Friday 11-12; 4096 East Hall

Office Hours: 2844 East Hall, by drop in or appointment. If there is demand for it, I will schedule a regular time. The populace has demanded! I will be in either my office or the tea room from 4-6 on Tuesday afternoons. Most weeks, I can also meet earlier than that; let me know if you'd like me to.


Textbooks: Algebraic Geometry I, by Igor Shafarevich, Introduction to Commutative Algebra by Michael Atiyah and Ian MacDonald

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. Notes from algebraic geometry classes at similar levels to this one by by J. S. Milne, Karen Smith and Igor Dolgachev. The algebraic-geometry tag at mathoverflow and math.stackexchange (but see the homework policy below). At a higher level than this class: Mark Haiman's synopses of EGA, Ravi's notes and blog, the stacks project, the algebraic geometry tag at nLab.

Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should either already know or be concurrently taking commutative algebra (Math 614). Students should also know the basic definitions of topology — we won't be using any deep theorems, but we will use topological language all the time. Basic familiarity with smooth manifolds will be very helpful, as much as what we do is the hard version of things that are done more easily in a first course on manifolds. Undergraduate students intending to take this course should speak to me about your background during the first week of classes.

Anticipated topics I hope to cover the following subjects. This is simultaneously a frighteningly long list, and what many people would consider the bare minimum!

Hilbert basis theorem, Nullstellansatz, the Zariski topology, decomposition into irreducible components. Open subvarieties, localization.
Projective space, projective varieties, Grassmannians and flag varieties, the Segre and Veronese constructions.
Finite maps, Noether normalization, the many principles of conservation of number.
Dimension theory: Krull dimension, transendence degree, Hilbert polynomials.
Tangent and cotangent vector spaces, smoothness, derivations, differential forms, the algebraic Sard theorem.
Local theory of curves: Normalization and dvrs.
Global theory of curves: Divisors, global sections, Riemann-Roch and many classical applications thereof.

Student work expected: Algebraic geometry is a field which has reinvented itself multiple times, and which also stands as a model and setting for much of the rest of mathematics. It is hard to understand how anyone learns enough to work in the field in a year — but hundreds of graduate students do and you will! In order to try to get you there, this course will involve a lot of work, and a lot of kinds of work.

I will assign weekly problem sets, due on Wednesdays. See below for the homework policy.

I will also write optional problem sets on commutative algebra. These problem sets will guide you through the proofs of key commutative algebra lemmas I won't have time to prove. These are not mandatory, but I will be glad to read your solutions if you like.

I will assign readings from Shavararevich's text and other sources. I hope to work through all of Chapter 1. I hope that, by assigning readings, I can cover more material than can be covered purely in class, and can help tie together what may sometimes seem like a whirlwind of topics.

Like Karen Smith's fantastic 631 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.

Finally, I will require you all to write an expository 8-15 page paper on some subject in algebraic geometry which interests you, due Monday, December 8. Here are some ideas for paper topics, and I'd also love to hear from you about what you'd like to write about. We'll discuss planning this paper more as the term goes on.

Problem Sets

Problem Set 1 (TeX) due Wednesday September 10.

Problem Set 2 (TeX) due Wednesday September 17.

Problem Set 3 (TeX) due Wednesday September 24.

Problem Set 4 (TeX) due Wednesday October 1.

Problem Set 5 (TeX) due Wednesday October 8 Friday October 10.

Problem Set 6 (TeX) due Wednesday October 22.

Problem Set 7 (TeX) due Wednesday October 29.

Problem Set 8 (TeX) due Wednesday November 5.

Problem Set 9 (TeX) due Wednesday November 12 Friday, November 14.

Problem Set 10 (TeX) due Wednesday November 19.

Problem Set 11 (TeX) due Wednesday November 26. This is the final problem set!

• Optional problem set on noetherian rings and the Hilbert basis theorem. Problem 5 is really fun; you might enjoy doing this even if you feel that you are already comfortable with noetherianness.

• Optional problem set on Nakayama's lemma.

• Optional problem set on integral dependence: In Atiyah-MacDonald, cover up the proofs of Proposition 5.1, Corollaries 5.2, 5.3, 5.4, 5.5 and Proposition 5.6 and recreate them. Then do Exercise 8.

• Optional problem on transcendence degree: If you don't remember what the words "algebraically independent" or "transendence basis" mean, or how to prove that any two transendence bases have the same size, see Problem 5 from a Galois theory course I taught.

Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You MAY NOT post homework problems to internet fora seeking solutions. Although I participate in some such fora, I feel that they have a major tendency to be too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

Reading schedule

This will be updated as the term progresses.

Optionally, skim Chapter I.1. The point here is not to follow all the details: Imagine the author as an exuberant, perhaps slightly drunken mathematician, who can't wait to burble out all of his favorite examples to you, and you'll have the right attitude about this chapter.

Read Chapter I.2 by Monday, September 8.

Read Enrique Arrondo's Another Elementary Proof of the Nullstellensatz by Friday, September 12; this is the proof I plan to do in class. Optionally, also read the introduction and first section of Zariski's A new proof of Hilbert's Nullstellensatz. These are the two briefest proofs I know. Neither of them is truly constructive; to make this theorem computationally effective (to my knowledge) requires developing significant portions of the theory of Gröbner bases.

Read Chapter 1.3.1 and I.3.2 by Friday, September 19. I don't think I'll talk about rational maps much; they are certainly an important notion but they are not quite as important as the other material, and I think they are hard to do right. If you are rusty on your commutative algebra, you might want to also look through Chapter 3 of Atiyah and Macdonald, on rings of fractions.

Read Chapter I.4 by Friday, September 26. Again, I'm not such a fan of Shavarevich's approach to rational functions. You want to focus on the definition of projective space, the relationship between projective space and homogenous ideals, and how to build regular functions on projective space as ratios of homogenous polynomials. There are roughly three ways to do any of the basic things you want to do in projective space: Work with graded rings, work with invariant functions on affine space, or work with affine covers. Shavarevich generally uses the first, I'll try to present all three. Once you get good at thinking about projective space, you will move between them without thinking.

Optional reading suggestion: If you'd like to see ringed spaces and sheafs of functions defined carefully in general, and a careful explanation of why affine and projective varieties are examples of ringed spaces, you might like to look at Chapters 3a-g and 6a-g in Milne's Alegbraic Geometry notes.

Read Chapter I.5.1 and I.5.2 by Monday, September 29. Karen tells me that people find products confusing, so let me try to help. The points of X × Y are the product of the points of X and the points of Y, just as you'd hope. Intuitively, the ring of regular functions on X × Y is the ring generated by the regular functions in the x coordinates and the regular functions in the y coordinates. That's fine for a definition of global regular functions, but awkward for local ones, because open sets of the form U × V aren't a basis of the topology on X × Y. We can solve this problem in three ways (like most foundational problems in projective varieties). Shavarevch uses the Segre embedding, which I personally find the least intuitive of the three, and this might be the cause of the trouble. The theorem that projective maps have closed image are is one of the most beautiful and surprising results in an early course on algebraic geometry; I'm looking forward to helping you appreciate it.

Read Chapter I.5.3 by Monday, October 6. This short chapter, in some sense, answers the question of why enumerative problems have well defined answers in algebraic geometry. I'll try to expand on this at length. You'll want to be absolutely solid on integral ring extensions, so look at the First Section of Chapter 5 in Atiyah-Macdonald if you have any doubts. We'll also be using Nakayama's lemma, so you might want to read pages 21 and 22. At least glance at the optional problem set on Nakayama's lemma.

WARNING: In some places, Shavarevich seems to take the statement "f:Y → X is finite" to imply that f has dense image (and is thus surjective, since we will show that, if f is finite, then f(Y) is closed.) For example, see his statements of Noether normalization (Theorems 9 and 10, Chapter I.5.4). This is very non-standard. I checked EGA, Hartshorne, Milne, Vakil and the stacks project, and none of them do this. For all those authors, and for us, a map MaxSpec B → MaxSpec A of affine varieties is finite if B is a finitely generated A-module; we do not require that A injects into B.

Shavarevich's discussion of finiteness is also missing a subtle point. Let f: Y → X be a regular map of quasi-projective varieties. Shavarevich (Definition I.5.3.2) defines f to be finite if there is a cover of X by open affines U for which f(-1)(U) is affine and f(-1)(U)→U is finite. But conditions that are checked locally should still remain true if we refine the open cover, and that isn't clear in this case! For a weak, but good enough, statement along these lines, see Problem 1, Problem Set 6. In fact, much more is true: If V is any open affine in X, then f(-1)(V) will be affine and f(-1)(V)→V will be finite. If you'd like to read a proof, see Proposition 8.21 in Milne. (This is completely optional.)

Completely optional: I have realized that I might want the following lemma on Friday, October 10, and will definitely want it on Friday, October 17. You can either read the proof (a few paragraphs) or just believe it:
Let A be an integrally closed domain, B an A algebra which is finitely generated and torsion-free as an A-module and θ an element of B. Map A[t] → B by sending t to θ. The kernel of this map is a principal ideal generated by a monic polynomial g(t).
Read Chapter I.5.4 by Wednesday, October 15. Noether normalization will be a key technical tool in many of our proofs, and has a strong geometric intuition behind it. I'll want a stronger version than Shavarevich proves, so please also read Mel Hochster's note. I learned to call Shavarevich's version "Noether normalization" and Hochster's version "relative Noether normalization".

Read Chapter I.6.1 and I.6.2 by Friday, October 17. If you get to I.6.1 by Wednesday, that would be great. Theorem 5 is the technical key that makes dimension theory work; I plan to spend Friday trying to unpack and motivate the proof.

If you never learned/don't remember about transendence degree, pages 109-113 of Milne's Field Theory textbook are pretty good. See Problem 5 on this old problem set of mine for a quick version (solution here, or solve it yourself!)

Read Chapter I.6.3 by Wednesday, October 22. My current plan is not to cover I.6.4 on any specific day but mention bits of it as they come up, so consider that optional.

WARNING: The Corollary to Theorem I.6.3.7 is false. See my answer here.

On Friday we talk about Grassmannians! Reread Example 1 in Chapter I.4.1 by Friday, October 24. If you are not comfortable with exterior algebra, find a source you like and become comfortable by Friday.

On Monday and Wednesday, October 27 and 29, we'll talk about Hilbert series and prove Bezout's theorem in the plane. Look over the first section of Atiyah-Macdonald, Chapter 11 by Monday, October 27, to refresh your memory for Hilbert series. Optionally, read the rest of Chapter 11 if you want to see the last two weeks of lectures compressed into six pages!

Optional reading: this discussion on Mathoverflow talks about the subtleties of proving Bezout's theorem in higher dimensional projective spaces.

Read Chapter II.1.1 through II.1.4 by Monday, November 3. It's kind of amazing that we got this far without ever needing to mention a local ring — it just always seemed easier to localize the specific elements I cared about. But we should deal with it at some point, and Shavarevich decides this is the point. Of course, Chapter 3 of Atiyah-Macdonald can help you with algebraic background.

Read Chapter II.2.1 and II.2.2 by Friday, November 7. If you have forgotten/never learned about regular rings, you might want to turn back to Chapter 11 of Atiyah-Macdonald.

At this point, I disagree with Shavarevich's ordering. Ship ahead to Chapter III.5 and read it by Monday, November 10. Basically, II.1 and II.2 talk about tangent and cotangent spaces, while III.5 talks about tangent and cotangent bundles; I find it unnatural to separate them.

Optional reading: I'll be using the following lemma in class on Monday:
Let X be an irreducible d-dimensional subvariety in kn. We can choose a Noether normalization X → kd so that the field extension Frac(X)/k(x1, …, xd) is separable.
If you'd like to read a proof, here is a quick one. Many of you have probably gotten through mathematics thinking about inseparable field extensions as "that issue in characteristic p that I can ignore". To be honest, you can think of them that way in this course too. But, if you're starting to feel guilty about it, Keith Conrad has some good notes.

We're going to be talking about derivations and Kahler differentials, an algebraic subject which Atiyah-Macdonald omits. I'll spend some class and problem set time to try to make up for it, but I'm also trying to find you a good free reference to serve as a supplement. The best I've found so far is Chapter 14 of Pete Clark's field theory notes or Chapter 12 of the CRing Project. Clark does a great job saying exactly what I want about derivations of fields, but doesn't talk much about rings and doesn't get to differentials; the CRing Project says far more than I need. If you find a better source, please let me know!

Read Chapter II.6.1 and II.6.2 by Friday, November 14. This will bring us to the end of our discussion of differentials.

I intend to do a shamefully brief job talking about normalization. To make up for this, please skim Chapter II.5.1-3 by Monday, November 17 if possible, or soon thereafter if that is unreasonable.

I also plan to be very brief about divisors in dimensions > 1;. Please skim Chapter III.1 and read Chapter III.2 by Monday, November 24.

I have written up notes on the proof of the Riemann-Roch theorem which I plan to present. Please read the first four sections by Monday, December 1 and finish reading them by Monday, December 8.

Optional other sources for Riemann-Roch: I'll pass out copies of the relevant pages of Serre's Algebraic Groups and Class Fields. Ravi Vakil's presentation is very similar to Serre's.

I also really like the first eight lectures or so of Joe Harris's course (notes by Atanas Atanasov). In particular, the fake proof at the end of Lecture 1 is something really cool that I haven't seen anywhere else. One should note, however, that the note taker doesn't discuss the biggest gap in the proof. In Lecture 7.2, this is discussed again and the main flaw is pointed out.

The daily update

When it is your turn to write the Update, download the TeX file, uncomment your date, and edit in a paragraph long summary of what happened that day. The goal is to get the key definitions and theorems; you certainly do not need to record everything. Then e-mail it back to me. The deadline for editing the update is 24 hours after the lecture. I will, in turn, proofread and edit your entries in the next 24 hours and post them back to this webpage, so that the class always has a good record of what we have covered.

If you have forgetten when you are scheduled to write the update, you can check here.