Math 631: Algebraic Geometry
Professor: David E Speyer
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.
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,
Smith and Igor Dolgachev. The algebraic-geometry tag at mathoverflow
(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
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
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
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
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.
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
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 (email@example.com) 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.
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
solutions, and must disclose this help. I am, of course, glad to
- Problem Set 1 (TeX), due Friday September 14
- Problem Set 2 (TeX), due Friday September 21
- Problem Set 3 (TeX), due Wednesday September 26
- Problem Set 4 (TeX), due Wednesday October 3
- Problem Set 5 (TeX), due Wednesday October 10
- Problem Set 6 (TeX), due Wednesday
October 24 (extra week due
to Fall Break) due Wednesday October 31, because dimension
theory is hard
- Problem Set 7 (TeX), due Wednesday November 7
- Problem Set 8 (TeX), due Wednesday November 14
- Problem Set 9 (TeX), due Wednesday November 21
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
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
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
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
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
Read Chapter II.3 by Monday, November 19.
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.