The Rising Sea: Vakil Summer Reading Group (Summer 2021)

RRS

Resources

We intend to cover parts I-IV of Vakil’s notes. This will leave us in a good place to continue on during the semester to Part V on Quasicoherent Sheaves and some of the extra chapters at the end (27 lines), which will give a different perspective on the material taught in the second semester of algebraic geometry here at UW Madison.

Also, we are choosing to focus on Problems and are using the helpful write-up’s of Ravi Vakil to guide our attention. Here is a link.

Schedule

Date Leader Reading Problems
Tuesday (05/18) Ivan 1.1-1.3 Categories and Functors N/A
Thursday (05/20) Ivan 1.4-1.6 (Co)Limits, Adjoints, Ab Categories N/A
Tuesday (05/25) Ivan 2.1-2.4 Intro to Sheaves 1.1-1.3
Thursday (05/27) Ivan 2.5-2.7 Sheaf constructions 1.4-1.6
Tuesday (06/01) Ivan 3.1-3.3 Intro to Schemes 2.1-2.4
Thursday (06/03) Ivan 3.4-3.7 Scheme Constructions 2.5-2.7
BREAK      
Thursday (06/10) John 4.1-4.3 Structure Sheaves and General Schemes  
Tuesday (06/15) John 4.4-4.5 + catch up Projective Schemes and Proj  
Thursday (06/17) Ivan 5.1-5.4 Properties of Schemes 4.1-4.3 (4.1.A, 4.3.A, 4.3.F, 4.3.G)
Tuesday (06/22)   5.5 + catch up Associated Points and Primes 4.5 + catch up (4.4.D, 4.5.E)
Thursday (06/24)   6.1-6.4 Morphisms of Schemes 5.1-5.4 (5.1.E, 5.2.F, 5.3.C, 5.3.H, 5.4.H, 5.4.J if you are Ivan)
Tuesday (06/29)   6.5 + catch up Rational Maps from Reduced Schemes 5.5 + catch up
Thursday (07/01)   7.1-7.3 Useful classes of morphisms of schemes 6.1-6.4
Tuesday (07/06)   7.4 + catch up Images of Morphisms: Chevalley’s Theorem and Elimination Theory 6.5 + catch up
Thursday (07/08)   8.1-8.3 Closed Embeddings and Related Notions 7.1-7.3
Tuesday (07/13)   8.4 + catch up Effective Cartier Divisors, Regular Sequences and Regular Embeddings 7.4 + catch up
BREAK      
Tuesday (08/04)   9.1-9.3 Intro to Fibered Products 8.1-8.3
Thursday (08/06)   9.4, 9.6, 9.7 Base Changes, Products, Normalization 8.4 + catch up
Tuesday (08/11)   10.1-10.2 Separated Morphisms 9.1-9.3
Thursday (08/13)   10.3 + catch up Proper Morphisms 9.4, 9.6, 9.7
Tuesday (08/18)   11.1-11.2 Dimension and Noether Normalization 10.1-10.2
Thursday (08/20)   11.3-11.4 Codimension 1 and Dimension of Fibers 10.3 + catch up
Tuesday (08/25)   12.1-12.3 Zariski Tangent Spaces, Regularity and Smoothness 11.1-11.2
Thursday (08/27)   12.4-12.6 Bertini’s Theorem, DVRs, Smooth and Étale Morphisms 11.3-11.4
Tuesday (08/31)   N/A 12.1-12.3
Thursday (09/02)   N/A 12.4-12.6

General Meeting Structure

We’ll meet twice a week on Tuesday and Thursday via zoom. Each meeting we’ll read about half of a chapter from The Rising Sea. We’ll have a volunteer each time to be the discussion leader.  The role of the discussion leader will be to summarize the main ideas from the section, and can include any thoughts they had about the section as sort of discussion fodder.  Importantly, the discussion leader should not lecture.  The point of this time is to work out difficulties we had with the reading together, not give an in-depth recap.  I’ll be the discussion leader for the first meeting.

We’ll need people to volunteer to lead at least two discussions.  You can see roughly what topics we will cover when and sign up to be a discussion leader here.  As you will see, we intend to skip starred and double starred sections of the text.  If you’re interested in reading those sections you can do that on your own and tell us about it when we discuss…

The majority of the time each meeting will be spent discussing problems, since a lot of understanding the content of this book lies in doing the problems.  John and I have found that we really want a long time to meditate on problems in Vakil in order to get the most out of them.  So, we have chosen to stagger the problems from the readings by a week: e.g. we’ll talk about problems from section 1.1-1.3 a week after we discuss those sections.  Because of this, the first week’s meetings are likely to be quite short.

We will not assign particular exercises, so everyone should feel free to do whatever piques their interest.  We have enough people that we’re probabilistically guaranteed good coverage, so the exercises you don’t get to someone else probably did.

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.