**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.

- 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

Skim

Read

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

Please read

Read

Read

Read

Read

Read

Read

At this point, I disagree with Shavarevich's ordering. Ship ahead 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:

LetIf 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 characteristicXbe an irreducibled-dimensional subvariety ink. We can choose a Noether normalization^{n}X → kso that the field extension^{d}Frac(X)/k(xis separable._{1}, …, x_{d})

Read

Read

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.