# Math 631: Algebraic Geometry

### Fall 2018

Course meets: Monday, Wednesday, Friday 11-12; 3866 East Hall

Office Hours: 2844 East Hall, Tuesdays and Wednesday, 2:30-4:00. I am also glad to make appointments to meet at other times.

Webpage: http://www.math.lsa.umich.edu/~speyer/631

Textbooks: Algebraic Geometry I, by Igor Shafarevich.

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 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. Back in 2014, I had my students write the update; you can see their great work here.

This year, I expect to follow a fairly similar calendar. I will require students to take terms writing an updated version of the notes. You are free to do this by editing the notes from the previous course or starting from scratch, but you should have spent the effort to go read and think through everything you put in. Most students found both writing and reading these notes to be extremely valuable.

Finally, I will require you all to either write an expository 8-15 page paper, or to prepare a 30-50 minute talk on some subject in algebraic geometry which interests you.

You may give a talk in the student algebraic geometry seminar, which meets Thursdays at 4-5 PM, to satisfy this requirement. Please confer with Devlin Mallory (malloryd@umich.edu) regarding a date and tell me when to show up so I can see your presentation. If you are unable to schedule a talk in the seminar (and you won't all fit), I will find another time you can speak.

Papers will be due December 7. I will schedule times for talks once I know how much interest there is. 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.

The talks have now been scheduled! Here is the list of speakers. Please enjoy as many as you can.

## Problem Sets

Problem Sets will generally be due on Wednesdays. However, due to the timing of the Jewish High Holidays this year, the first two problem sets will be due on Fridays.
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.

This will be updated as the term progresses.

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 Friday, September 7.

We will stay in Chapter I.2 for the week September 10-14. We'll be talking about the Nullstellansatz and what it tells us about closed sets (I.2.1) in the first part of the week and we'll hopefully get to regular functions (I.2.2) and maps (I.2.3) by Friday. If you didn't read these last week, please look at them during the week of September 10-14.

Please read Chapter I.3.1 by Wednesday September 19. I expect to either get to irreducible components on that date, or on Friday the 21st.

Read Chapter I.4 by Monday, September 24, focusing on the discussion of projective varieties (as opposed to rational functions). 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.

Read Chapter I.5.1 and I.5.2 by Friday, September 28. I hope to reach projective varieties that day, although it might have to wait for the following week. Karen Smith 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 and I.5.4 by Friday, October 12. There are some natural results about finite maps which Shafarevich unfortunately doesn't prove; I've added references to the update file in the October 8 notes. Last time I caught this, I got hung up a long time on basic properties of finite maps, so this time I am trying to only do the essentials.

Read Chapter I.6.1 and I.6.2 by Friday, October 19. Theorem 5 in Section 6.2 is the techinical key which will unlock dimension theory, and is probably one of the longer arguments to date.

Read Chapter I.6.3 by Wednesday, October 24. This concludes reading Chapter I, and the main theorems on dimension theory. I plan to stay in dimension theory a little longer to talk about Hilbert polynomials, Bezout's theorem, and Grassmannians. Warning: The Corollary to Theorem I.6.3.7 is false, although the Theorem is true; see the discussion here.

Read Chapter II.1.1 through II.1.3 by Monday, November 5. 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.

At this point, I disagree with Shavarevich's ordering. Ship ahead to Chapter III.5 and read it all four parts by Wednesday, November 7. 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.

I won't assign Chapter II.1.4 because I think I can cover it better, but it touches on some of the same material I hope to do Friday. I'm not planning to talk about the tangent cone, from Chapter II.1.5.

Optional reading: I'll be using the following lemma in class sometime around November 9:
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.

Read Chapter II.2.1 and II.2.2 by Monday, November 12.

Read Chapter II.3 by Monday, November 19.

Optional Reading Sometime around November 19, I'll want to use the fact that a regular local ring is a UFD, and I've decided it would be too big a sidetrack to prove it. If you'd like to read a proof, there is a very nice exposition in Section 13 of these notes. They also contain a proof of a result that I quoted without proof earlier: A regular domain is a UFD if and only if every codimension one prime is principal.

Optional reading I intend to be shocking brief about normal varieties. I will prove that smooth varieties are normal, but I won't explain that most of what we do with smooth varieties only needs the normal hypothesis. If you'd lke to fill this gap, look at Chapter II.5 in Shavarevich.

On Monday, November 19, we set up the vocabulary to start talking about divisors, and will stay on this topic until the middle of the following week. Please read Chapter III.1.1-III.1.5 by Wednesday, November 28.

After that, we will start talking about curves. Shavarevich's curve material is spread out through Chapter III.2, III.3, and III.6.5 and III.6.6. I'll probably hit this material in a different order, but these might be good places to look when I'm confusing.

## The update

This class has a wonderful daily notes file called The Update! It is written by the students and edited by me.

I plan for the first three classes to be hevaily interactive and will not assign scribes for them. By time those classes are done, we'll hopefully have a schedule set up for the first half of the term.

When it is your turn to write the update, download the template and write roughly 0.5 to 1.5 pages describing what was covered that day. You may use the 2014 file as a source. (If you are going to be compiling the 2014 file, you'll also want to download the various image files it includes. While I am at it, here is the source of the 2018 file and its images)

The deadline to e-mail me the notes is 24 hours after class, and I will endeavor to always incorporate your notes to the master file before the next class.

Please edit the template rather than editing the main file. I have found that doing all the incorporation into the main file myself reduces LaTeX issues.

If you've forgotten when you are scheduled to write the update, you can check the schedule.