Geometry of Schemes
Resources
We intend to cover the first half of “The Geometry of Schemes” by Eisenbud and Harris, linked below. I’ve slowed down the pace in order to make room for Vakil’s online AG course, which I think would be nice to do concurrently. If you’d like to be kept updated, please join the mailing list which I think you may be able to do at the bottom of this page.
Schedule
| Date | Leader | Reading | Problems |
|---|---|---|---|
| Week 1 (May 29, Jun 2) | John Cobb | I.1 Affine Schemes | 5, 7, 9, 10, 20, 21 |
| Week 2 (Jun 5, Jun 9) | Tejasi Bhatnagar | I.2 Schemes in General | 24, 25, 30, 31, 34, 32, 37, 44, 45 |
| Week 3 (Jun 12, Jun 16) | John Cobb | I.3 Relative Schemes | 46, 47, 48, 50, 51 |
| Week 4 (Jun 19, Jun 23) | Daniel Levitin | II.1-II.2 Reduced Schemes | 2, 3, 6, 7, (maybe 8) |
| Week 5 (Jun 26, Jun 30) | Ivan Aidun | II.3.1-II.3.4 Nonreduced Schemes | 11, 15, 14, 22 |
| Week 6 (Jul 3, Jul 7) | John Cobb | II.3.4-II.3.5 Nonreduced Schemes | 25, 31, 33, 35 |
| Week 7 (Jul 10) | Daniel Erman Q&A | Break | Come up with questions! |
| Week 8 (Jul 14, Jul 17) | Ivan Aidun | II.4 Arithmetic Schemes | 36, 38, 40, 44 |
| Week 9 (Jul 21, Jul 24, Jul 28) | Tejasi Bhatnagar | III.1-III.2.3 Attributes of Morphisms | 1, 2, 3, 4, 6, 12, 15 |
From here, the group has decided that it will be more worthwhile at this point to pick up with another book after taking our qualifying exams, likely Vakil’s notes.
Pushed off schedule: III.2.4-III.2.8 Proj of a Graded Ring, III.3.1-III.3.4 Invariants of Projective Schemes, III.3.5-III.3.6 Invariants of Projective Schemes & IV.1 Flexes of Plane Curves
If we have the time, covering V.1-V-2 Images and Resultants or VI.1 The Functor of Points would be good.
General Meeting Structure
We will meet Tuesdays at 1:00pm and Fridays at 1:00pm (CST) via Zoom for about an hour, with the following goals in mind:
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Fridays: We discuss the reading of the week. A different person will be the “leader” each week, who directs the meeting that day. The focus would be to quickly highlight (around 20 minutes max) the theorems and definitions that seem the most relevant, preferably alongside prepared examples or counterexamples. Then, we collectively discuss questions we thought of while reading. Note that being in charge doesn’t mean you’re expected to know everything – You can have lots of questions too. The last part of the meeting would be choosing exercises related to the reading. Ideally everyone has glanced at the exercises and has suggestions on what we might want to do. Then the leader divides up the exercises and assigns them to the group members.
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Tuesdays: We discuss the exercises done over the weekend. Each person will have been assigned exercises from the week before. Everyone should still look at as many exercises as they can, but prioritize the ones assigned to them. That way between the group, we’ll have done the whole set. In this meeting, people can ask questions about exercises they thought were difficult, get feedback on solutions, and present solutions they thought were especially interesting. Whoever was in charge on the previous Friday will run this meeting as well.
Obviously two meetings every week can be a lot, so no one should feel like they have to attend every single meeting. However, I think it will work best if everyone comes as much as possible.
Google Forum
While going through the settings of our new google group, I found a way to embed a whole forum system. I’ve included it below in case it turns out to be helpful.