**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.

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

**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 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

• 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

• Problem Set 10 (TeX) due Wednesday November 19.

• Problem Set 11 (TeX) due Wednesday November 26.

• 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.

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

Optionally, skim

Read

Read Enrique Arrondo's Another Elementary Proof of the Nullstellensatz by

Read

Read

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

Read

WARNING: In some places, Shavarevich seems to take the statement "

Shavarevich's discussion of finiteness is also missing a subtle point. Let

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:

LetReadAbe an integrally closed domain,BanAalgebra which is finitely generated and torsion-free as anA-module and θ an element ofB. MapA[t] → Bby sendingttoθ. The kernel of this map is a principal ideal generated by a monic polynomialg(t).

Read

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

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

On Friday we talk about Grassmannians! Reread

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

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

Read

Read

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

Optional reading: I'll be using the following lemma in class on Monday:

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})

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

I intend to do a shamefully brief job talking about normalization. To make up for this, please skim

I also plan to be very brief about divisors in dimensions > 1;. Please skim

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

Optional other sources for Riemann-Roch: I'll pass out copies of the relevant pages of 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.

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