# Algebraic Geometry Reading Seminar

This is the homepage for the Algebraic Geometry Reading Seminar focused on Robin Hartshorne's textbook (here is PDF copy).

This seminar is co-organized by Alex Wang (awaw@uw.edu) and myself (atawfeek@uw.edu) at the University of Washington.

This seminar ran for the 2022 - 2023 academic year.

## Structure:

### Meetings:

Our two meetings each week take the following forms:

Discussion day:

The first half of the meeting will be dedicated to a volunteer summarizing the contents of a section.

The remainder of the meeting will be dedicated to discussing the section together -- collectively working through the details of proofs, statements, and examples in the reading.

Problem session day:

We will have selected some exercises from the end of the section in Hartshorne.

These will be posted here on the website under the corresponding problem solving day.

You are welcome to look at them and try them before Wednesday, but that will not be expected of you.

We will meet all together at the start of the session in the graduate lounge, then split off into groups of 3 - 4 to work on the problems with eachother. If you arrive late, feel free to join any of the on-going groups!

Prior to discussion meetings:

the non-speakers should try to read (or at very least skim through) the section before the meeting, and

the speaker should try their to give the section a thorough read-through (perhaps noting difficulties encountered to discuss them as a group).

After discussions, people are welcome (and encouraged!) to then begin TeXing up solutions to problems in the shared Overleaf file. Details on this below!

If you have thoughts on any of these, please let me (atawfeek@uw.edu) or Alex Wang (awaw@uw.edu) know! Again, although Alex and I are happy to take charge of the logistics behind the seminar -- we believe that the actual reading seminar is run by all of us collectively. So if you have comments on the structure, please let us know!

## Class Summaries:

During/after our Monday class meetings (discussion days), we will have someone summarize (the scribe) what we talked about that day.

The AG Summaries document may be accessed by emailing one of the above organizers.

## TeXing Solutions to Exercises:

After our group problem solving meetings on Wednesday, we plan to TeX up our solutions to problems on a shared Overleaf project.

Please email me (atawfeek@uw.edu) or Alex Wang (awaw@uw.edu) to receive the invite link to view and edit the document.

## Autumn 2022 Schedule

Below are very brief summaries of days and what to expect during future meetings (prefaced by "Plan").

Please see the AG Summaries document (above) for a more thorough summary.

### Week 2 (of autumn quarter):

Monday, October 3rd

First day of seminar. Structure and expectations discussed. Alex Wang lectured on start of Section 1.1 (Affine Varieties).

Between Monday and Wednesday, everyone was encouraged to take a look/skim through rest of Section 1.1.

Wednesday, October 5th

Discuss the remainder of Section 1.1.

Work in groups through selected exercises in Section 1.1.

Exercises: 1.3 - 1.8 and 1.12

### Week 3:

Monday, October 10th

Cameron Wright led the discussion on Section 1.2 (Projective Varieties).

Wednesday, October 12th

Work in small (3-4) person groups on selected exercises in Section 1.2.

Exercises: 2.2 - 2.7 and 2.14

### Week 4:

Monday, October 17th

Nelson Niu led the discussion on Section 1.3 (Morphisms).

Wednesday, October 19th

Worked in small (3-4) person groups on selected exercises in Section 1.3.

Exercises: 3.2, 3.3, 3.10, 3.11, 3.12, 3.15, 3.17, 3.20

### Week 5:

Monday, October 24th

Andrew Tawfeek led the discussion on Section 1.4 (Rational Functions).

Wednesday, October 26th

Work in small (3-4) person groups on selected exercises in Section 1.4.

Exercises: 4.1 - 4.6

### Week 6:

Monday, October 31st

Andrew Tawfeek continued leading the discussion on Section 1.4 (Blow Ups).

Jackson Morris led the discussion on Section 1.5 (Nonsingular Varieties).

Wednesday, November 2nd

Work in small (3-4) person groups on selected exercises in Section 1.4 and 1.5.

Exercises: 5.1 - 5.3, 5.6 - 5.10

### Week 7:

Monday, November 7th

Jackson Morris continued leading the discussion on Section 1.5 (Competitions).

Natasha Crepeau led the discussion introducing valuation rings.

Wednesday, November 9th

Work in small (3-4) person groups on previously selected exercises in Section 1.5.

### Week 8:

Monday, November 14th

Cameron Wright led the discussion on Section 1.6 (Nonsingular Curves).

Wednesday, November 16th

Work in small (3-4) person groups on selected exercises in Section 1.6.

Exercises: 6.1, 6.2, 6.4, 6.6

### Week 9:

Monday, November 21st

Lorenzo Bottiglione led the discussion on Section 1.7 (Intersections in Projective Space).

Wednesday, November 23rd

Work in small (3-4) person groups on selected exercises in Section 1.7.

Exercises: 7.1 - 7.4

### Week 10:

Monday, November 28th

Work in small (3-4) person groups on exercises from Chapter 1.

Wednesday, November 30th

Work in small (3-4) person groups on exercises from Chapter 1.

### Week 11:

Monday, December 5th

Work in small (3-4) person groups on exercises from Chapter 1.

Wednesday, December 7th

Work in small (3-4) person groups on exercises from Chapter 1.

## Winter 2023 Schedule:

Below is a very brief summary of the day and what to expect during future meetings.

Please see the AG Summaries document (above) for a more thorough summary.

### Week 2 (of winter quarter):

Monday, January 9th

First day of seminar. Nelson Niu lectured on start of Section 2.1 (Sheaves).

Wednesday, January 11th

Work in small (3-4) person groups on selected exercises in Section II.1.

Exercises: 1.1 - 1.9, 1.18, 1.22

### Week 3:

Monday, January 16th No meeting for holiday (Martin Luther King Jr. Day).

Wednesday, January 18th

As we had not met this week for a discussion, choose whichever of the two options below be suit your interests.

Building intuition on sheaves:

Problems involving concrete sheaves to work with:

Exercise 1.17 (skyscraper sheaves)

Exercise 1.21(a-d) (examples of sheaves on varieties, connects to Chapter 1 content)

Problems required for future exercises this quarter:

Exercise 1.15 (sheaf hom: used for Ext groups of sheaves)

Exercise 1.16 (flasque sheaves: used for computing cohomology)

Exercise 1.19 (extending a sheaf by zero: a helpful short exact sequence of sheaves)

Exercise 1.20 (subsheaf with supports: helpful for computing "the measure of the obstruction of extending a function from an open set to an entire space")

Moving into studying schemes:

We recommending skimming the section up until the definition of schemes (page 74) and try following exercises:

Exercise 2.1 (scheme associated to localizing a ring)

Exercise 2.2 (an open subset of a scheme is a scheme)

Exercise 2.3 (reduced schemes)

### Week 4:

Monday, January 23rd

Andrew Tawfeek will lead discussion on Section II.2 (Schemes).

Wednesday, January 25th

Work in small (3-4) person groups on selected exercises in Section II.2.

Exercises:

Exercises 2.1, 2.2, 2.4 (basic properties of schemes)

Exercises 2.5, 2.6, 2.10, and 2.11 (examples of schemes)

Exercise 2.7 (residue field at a point)

Exercise 2.8 (Zariski tangent spaces)

Exercise 2.9 (generic points)

Exercise 2.3 (reduced schemes)

Exercise 2.13 (quasi-compact schemes)

Exercise 2.12 (gluing schemes)

Exercise 2.15 (relation between classical varieties and schemes)

### Week 5:

Monday, January 30th

Alex Wang will lead discussion on Section II.3 (First Properties of Schemes).

Wednesday, February 1st

Work in small (3-4) person groups on selected exercises in Section II.2.

### Week 6:

Monday, February 6th

Work in small (3-4) person groups on selected exercises in Section II.2 and II.3.

Wednesday, February 8th

Work in small (3-4) person groups on selected exercises in Section II.2 and II.3.

### Week 7:

Monday, February 13th

Haocheng Cai will lead discussion on Section II.4 (Proper and Separated Morphisms).

Wednesday, February 15th

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

### Week 8:

Monday, February 20th No meeting for holiday (Presidents' Day).

Wednesday, February 22nd

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

### Week 9:

Monday, February 27th

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

Exercises: (thanks to Haoming Ning)

Section II.2:

Exercises 2.1 - 2.13, 2.15 (these are the previously mentioned ones above)

Exercises 2.16 - 2.18 (introduce important techniques needed in reading/working through sections II.3 and II.4)

Exercises 2.19 (helpful for understanding disconnected spaces)

Section II.3:

Exercises 3.1 - 3.4 (helpful for gaining intuition, though not necessary to do all)

Exercises 3.5 - 3.14, 3.20 - 3.22 (relevant for future content, can postpone 3.7, 3.8, 3.12, 3.22)

Exercises 3.16 - 3.19 (builds towards Chevalley's Theorem -- though not immediately useful)

Section II.4:

Exercises 4.1 - 4.6, 4.8 - 4.9 (building general intuition/will be used later, can postpone 4.5 and 4.9 if necessary)

Exercise 4.7 (important and interesting example for those interested in field extensions and descent (number theorist-minded people!))

Exercise 4.12 (good example to work through)

Exercises 4.10 - 4.11 (important statements to know for future content -- though not necessary to prove now)

Wednesday, March 1st

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

### Week 10:

Monday, March 6th

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

Wednesday, March 8th

Work in small (3-4) person groups on selected exercises in Section II.2 - II.4.

## Spring 2023 Schedule:

Below is a very brief summary of the day and what to expect during future meetings.

Please see the AG Summaries document (above) for a more thorough summary.

### Week 1:

Wednesday, March 29th

Alex Wang will lecture on start of Section II.5 (Sheaves of Modules).

Friday, March 31st

Work in small (3-4) person groups on selected exercises in Section II.5.

### Week 2:

Wednesday, April 5th

Ting Gong will continue lecture on Section II.5 (Sheaves of Modules).

Friday, April 7th

Work in small (3-4) person groups on selected exercises in Section II.5.

### Week 3:

Wednesday, April 12th

Andrew Tawfeek will lecture on remainder of Section II.5 (Sheaves of Modules).

Friday, April 14th

Work in small (3-4) person groups on selected exercises in Section II.5.

### Week 4:

Wednesday, April 19th

Haoming Ning will lecture on Section II.6 (Divisors).

Friday, April 21st

Work in small (3-4) person groups on selected exercises in Section II.6.

### Week 5:

Wednesday, April 26th

Haoming Ning will continue lecture on Section II.6 (Divisors).

Friday, April 28th

Work in small (3-4) person groups on selected exercises in Section II.6.

### Week 6:

Wednesday, May 3rd

Haoming Ning will continue lecture on Section II.6 (Divisors).

Friday, May 5th

Work in small (3-4) person groups on selected exercises in Section II.6.

### Week 7:

Wednesday, May 10th

Haoming Ning will lecture on Section II.7 (Projective Morphisms).

Friday, May 12th

Work in small (3-4) person groups on selected exercises in Section II.7.

### Week 8:

Wednesday, May 17th

Haoming Ning will finish lecture on Section II.7 (Projective Morphisms).

Friday, May 19th

Work in small (3-4) person groups on selected exercises in Section II.7.

### Week 9:

Wednesday, May 24th

Lorenzo Bottiglione will lecture on Section II.8 (Differentials).

Friday, May 26th

Work in small (3-4) person groups on selected exercises in Section II.8.

### Week 10:

Wednesday, May 31st

Ting Gong will finish lecture on Section II.8 (Differentials).

Friday, June 2nd

Work in small (3-4) person groups on selected exercises in Section II.8.

## Communication:

We run the hartshorne@uw.edu mailing list and updates are sent over this.

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