Course Details (see here for a more detailed course description)
Meeting time: Tuesday, Thursday 10:30-11:45 (D+ Block)
Meeting location: Tuesdays in JCC 510 (Fish Bowl), Thursdays in JCC 574 (Math Library)
Text: Vakil, The rising sea: foundations of algebraic geometry, version of July 2023. See here for the archive of evolving versions of the text and the link to the blog.
Prerequisites: abstract algebra (at the level of Math 146, plus basic notions of commutative algebra), point-set topology (at the level of Math 171). Additional background in various other topics (manifolds, complex algebraic varieties, ...) would be helpful for motivation, but is not strictly required.
Office hours: by appointment; please take a look here and write by e-mail
All are welcome. However, as there are no lectures and meetings are reserved for discussion and questions on the text and exercises, it is likely that attendance will only be useful to those who are actively participating.
Exercises
Active participants will be expected to attempt, and eventually (after discussion with each other and with me) solve all of the exercises in the text. This is necessary to understand the subject fully, and also logically necessarily to follow the exposition.
Participants taking the reading course for credit will additionally be required to submit solutions to a subset of the exercises every Friday, beginning January 26, by 3:00 PM Monday by 4:00 PM. Active participants not taking the reading course for credit are also encouraged to submit solutions. Solutions should be submitted on paper (typed or hand-written) in my mailbox. Which and how many exercises to write up is up to you. We will attempt to follow the basic principle: write up whatever subset of exercises, with whatever level of detail, is most helpful to you. As needed, a more precise policy may be developed on an individual or group level as the semester progresses. I will return your solutions with feedback the following week.
Additional references
You can certainly get by completely ignoring the list below. This strategy was productive for me when I learned the subject, but for others it may be helpful to consult other sources. More may be added if you have suggestions.
Other books on schemes:
Hartshorne, Algebraic geometry. Still the "standard" reference; comprehensive, but written in far fewer words than Vakil.
Eisenbud, Harris, The geometry of schemes. A more leisurely introduction than Hartshorne.
Mumford, The Red Book of varieties and schemes. Ditto above, but with some discussion of more advanced topics too.
Algebraic geometry in the classical language of varieties (logically not necessary for us, but helpful for motivation):
Harris, Algebraic geometry: a first course. Dizzying in its beauty and volume of examples. Gives a strong sense both of the kinds of concrete geometric problems with which algebraic geometry is concerned, and of why the more flexible language of schemes is eventually needed.
Reid, Undergraduate algebraic geometry. As the name suggests, a much gentler introduction to varieties.
Griffiths-Harris, Principles of algebraic geometry. Written from a much more complex-analytic viewpoint.
Background on commutative algebra (contains much more than what we will need, but may be helpful to see the ideas isolated here):
Atiyah-MacDonald: Introduction to commutative algebra. Comprehensive but terse.
Eisenbud, Commutative algebra with a view toward Algebraic Geometry. Much less terse, more focus on examples and motivation.
Encyclopedic references:
de Jong et al., The Stacks Project. The modern universal reference on anything foundational in algebraic geometry. Individual definitions and statements can be clicked through in a modular way. Many things are written in a level of generality not needed at a first pass, so some care may be needed to extract the most relevant ideas.
Grothendieck, Éléments de Géométrie Algébrique. The original treatise on the modern theory. These days, there is not much of a reason to consult this when first learning the subject, but it seems worth mention.
Reading Schedule
These are the sections of the text to be read before our in-person meetings. The understanding is that "read" means "read and attempt all of the exercises" not starred or marked unimportant. Reading assignments will be agreed upon on a week-by-week basis at each meeting.
Thursday 1/18: 1.1, 1.2 (skip 1.2.21), 1.3 (up to exercise 1.3.M)
Tuesday 1/23: rest of 1.3, 1.4
Thursday 1/25: cont.
Tuesday 1/30: 1.5
Thursday 2/1: cont. (focus on exercises 1.4.C, 1.4.E, 1.5.C-E)
Tuesday 2/6: 1.6.1-1.6.6 (just up to the definition of exactness) and 1.6.9 (esp. exercises 1.6.G and H). The rest of the material in this section on (co-)homology will likely be important sometime in your life, but we will not need it this semester.
Thursday 2/8: 2.1-2.2 up to exercise 2.2.D
Tuesday 2/13: rest of 2.2 [meeting by zoom]
Thursday 2/15: 2.3
Tuesday 2/20: 2.4
Tuesday 2/27: 2.4 (cont.), 2.5
Thursday 2/29: cont.
Tuesday 3/5: 2.6. We will skip section 2.7, but you will need it before you read chapter 7 on morphisms of schemes.
Thursday 3/7: 3.1, 3.2 (up to 3.2.5)
Tuesday 3/12: rest of 3.2, 3.3, 3.4 up to Definition 3.4.2
Thursday 3/14: rest of 3.4, 3.5
Tuesday 3/26: rest of chapter 3; catch up as needed
Thursday 3/28: 4.1, 4.2
Tuesday 4/2: 4.3
Thursday 4/4: 4.4
Tuesday 4/9: 4.4, cont.; if you're happy with the examples, especially projective space, then move onto 4.5
Thursday 4/11: 4.5
Tuesday 4/16: finish chapter 4, start chapter 5
Thursday 4/18: 5.1, 5.2
Tuesday 4/23: 5.3
Thursday 4/25: 6.1 (up to exercise 6.1.A), skim sections 6.2-6.4.
Tuedsay 4/30 (final meeting): chapter 7 up to exercise 7.3.G. You will need to black box/skip some discussion of the inverse image sheaf, e.g. exercise 7.2.D (or read section 2.7). Look also at section 7.3.7.