Students taking this course should be at a level to benefit from all
of these seminars. You will probably not understand every detail of
them, but you should have enough background to understand the point of
what is going on and to note down concepts to look up or ask people
(including me!) about afterwards.
Office hours:
Tuesdays, 4:00 PM-5:30 PM and Fridays 1:00-2:30 PM in East Hall 2844. I'll probably have to miss the Friday date a few times.
Intended Level: This is not a first course in algebraic
geometry. Many of you took Math 631 with Aaron Pixton last term, which covered chapoters 1-5 and 7-13 of Vakil's book. I am certainly glad to review individual concepts from last term — mathematical study involves frequent circling back! — but you should be largely familiar with this material.
Anticipated topics I am committed to covering chapters 6, 14, 15, 16 and 18 well; this gives you enough to understand sheaf cohomology and its basic uses. I would very much like to also cover chapter 19, on algebraic curves, and I think that will be doable. Perhaps naively, I think that there will be enough time to do more than that. Ideally, I'd like to get to Serre duality, which would mean pursuing Chapters 21, 23, 24 and 29 but, if I see that this is undoable, I may switch goals, and talk a bit about algebraic de Rham cohomology, or do more on the theory of curves, instead.
Problem Sets: I will assign weekly problem sets, due
on Thursday nights on Gradescope. The first assignment will be due on Thursday the 25th. See below for the homework policy. Most of my problems will be taken from Vakil's book.
Reading: I will assign weekly reading from Vakil's book, and will send out a quick poll, due on Friday nights to gather questions about the reading each week. The first poll will be due on Friday the 19th. I will check the polls each weekend and use them to guide my teaching in the week that follows.
Blog: I will maintain a blog about the readings in Vakil's book, hosted here. I will be glad to engage with comments on the blog.
I hope that, just as Vakil's blog has done, this can be a valuable place to discuss the readings and algebraic geometry more generally.
Tentative schedule and reading assignments
A nontrivial line bundle over the circle
Reading assignments are marked with •, future lecture topics are marked with ○, past lecture topics are marked with ✓.
I definitely do not plan to cover every topic from the reading in class. I'll try to focus on topics where I think I can give extra perspective, or where people seem particularly confused. I will check the poll results first thing every weekend and modify my lecture accordingly.
✓ January 11: Review of classical motivation and of the definition of a scheme
• January 16-19: please read Chapters 6.1-6.4 in FOAG. Complete the poll by January 19.
✓ January 16: Quasicoherent sheaves.
✓ January 18: Coherent sheaves.
• January 22-26: please read Chapter 14.1-1.4.3 in FOAG. Complete the poll by January 26.
✓ January 23: Vector bundles.
✓ January 25: Vector bundles are locally free sheaves.
• January 29-February 2: This week, we finish off Chapter 14 and start Chapter 15. Please read 14.4 (Nakayama's lemma, fiber and rank) and 14.7 (the quasi-coherent sheaf corresponding to a graded module) and Chapter 15.1-2. Complete the poll by February 2.
✓ January 30: Nakayama's lemma; from graded modules to quasi-coherent sheaves.
✓ February 1: Using line bundles to get maps to projective space.
• February 5-9: Please go back read/reread Chapter 13.5; it will be used in essential ways in Chapters 15.4-15.5. Then read 15.3-15.6. I plan to spend signficant time on 15.4-15.5 and a fair bit of time on 15.6; they are difficult and important. I view 15.3 as more of a technical point that can wait until we need it; if it doesn't get requested in the poll, it may not show up in class for a while.
✓ February 6: Cartier divisors, Weil divisors and line bundles on a normal affine scheme
✓ February 8: Dedekind domains, regular rings and dvrs; the algebraic Hartog's property
• February 12-16: Please read sections 16.1 and 16.2. I do plan to cover the starred section 16.2.5, I thought it was very clarifying.
✓ February 13: The Weil and Cartier divisor class groups. Brief mention of the ring of sections and the module of sections (Section 15.6).
✓ February 15: Global generation, and the "variant of the qcqs lemma" (Section 15.4.17 in the February 17 version)
• February 19-23: I'm going to skip past Cech cohomology and try to do some concrete work with curves. Please read Chapter 16.3 and 19.1.
✓ February 20: Definition of an ample line bundle (using starred section 16.2.5); ample line bundles and global geneartion.
✓ February 22: Ample line bundles and projective embeddings. No reading over spring break! But I'll start into Chapter 16.3 when I get back.
• March 4-8: Please read Chapter 16.3. In addition, this would be a good time to reread section 10.3, on finite morphisms.
✓ March 5: We'll discuss projective embeddings of curves. This makes the category of curves very concrete.
✓ March 7: We'll talk about finite maps, especially between curves. Degree of a map, in general, is in Section 10.3.
• March 11-15: Please read section 18.1, which discusses the properties which we want sheaf cohomology to have.
✓ March 12: I proved that, if X → Y is finite, and Y is projective, then X is projective. This completes the proof of Theorem 16.3.3, identifying the category of proper curves with the category of projective curves.
✓ March 14: I discussed what I sometimes call the "weak Riemann-Roch theorem", which is as close as we can get to Riemann-Roch without mentioning cohomology. This isn't in the book, so I wrote a blog post.
• March 18-22: Please read section 18.2, which defines Cech cohomology.
✓ March 19: I will preview sheaf cohomology and discuss its properties.
✓ March 21: I will define Cech cohomology and give examples.
• March 25-29: No new reading, but I'll be using a lot more homological algebra than I have before. If you are rusty on the Snake Lemma (Example 1.7.5) or the long exact sequence coming from a short exact sequence of complexes (Theorem 1.6.8), this is a good time to review them.
✓ March 26: I will prove key properties of Cech cohomology.
✓ March 28: I will compute the cohomology groups of projective space (Theorem 18.1.2). Ideally, I will also prove Grothendieck's coherence theorem (Theorem 18.1.3.(i)) and Serre vanishing (Theorem 18.1.3.(ii)).
• April 1-5: Please read sections 18.4-18.6. (You don't need to read the starred parts of 18.5.)
✓ April 2: I will prove the homological Riemann-Roch theorem (Theorem 18.4.B), define Hilbert polynomials (Section 18.6) and compute some examples.
✓ April 4: I will preview Serre duality, and some of its simple consequences (Section 18.5). Then I'll start talking about curves (Chapter 19).
• April 8-12 Please read Chapter 19.1-19.7 and Chapter 21.1-21.4. I realize that's a lot, but I'm going to try to talk about a lot of it.
✓ April 9: I'll work through as many of the fun computations and examples from Chapter 19 as I can. Unfortunately, I only have time for one day here.
✓ April 11: I'll try to give a rapid overview of differentials, and get to the Riemann-Hurwitz formula, which is a crucial result for understanding curves.
• April 15-18: No new reading assigned. However, I wrote some lecture notes on Riemann-Roch and Serre Duality for curves a number of years ago you might find them useful.
○ April 16: I'll talk about residues of differential forms, and Serre duality, on curves.
○ April 18 (last day): I think there is an appetite to hear about algebraic de Rham cohomology, and it's one of my favorite topics, so I'll try to say something about it.
Problem Sets
Homework will be due most Thursdays, through Gradescope. If you have not yet recieved a Gradescope invitation, please ask me.
I always post homework a minimum of 7 days before it is due, and I aim to post it 14 days ahead. During that 7-14 day interval, I may edit problems based on class. After that point, I will only edit problems to fix errors or to remove problems. (Fair warning, though, my students usually do find a fair number of errors! I appreciate your careful reading!)
Homework Policy:
You are welcome to work together with your classmates 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 do not intend for you to need to consult other sources, printed or online. If you do consult such, you should be looking for better/other expositions of the material, not solutions to specific problems. Math problems are often called "exercises"; note that you cannot get stronger by watching someone else exercise!
You MAY NOT post homework problems to internet fora seeking solutions. Although I know of cases where such fora are valuable, and I participate in some, 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.
No reading over spring break! But I'll start into Chapter 16.3 when I get back.