# 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 myself (atawfeek@uw.edu) and Alex Wang (awaw@uw.edu) at the University of Washington.

Our meeting times for Winter 2022 are as follows:

Mondays 4PM - 5PM in Communications (CMU) B-006, and

Wednesdays 3PM - 4:30PM in the mathematics graduate lounge (where we form groups).

## Structure:

### Meetings:

Monday meetings will be designated as a 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.

Wednesday meetings will be designated as a problem session day.

We will have selected some exercises from the end of the section in Hartshorne (through conversations with AG faculty).

These will be posted here on the website under the corresponding problem solving day (below in the schedule).

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

For the first half of the seminar, we'll split off into groups to collaborate with eachother on the problems.

For the later half, we'll re-assemble and discuss our solutions/approaches with one another.

Prior to Monday meetings (modulo first week):

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 Wednesdays, 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.

Here is AG Summaries document, which is synced and update every time the Overleaf file is edited.

## 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 (prefaced by "Plan").

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

### Week 7:

Monday, February 13th

Wednesday, February 15th

### Week 8:

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

Wednesday, February 22nd

### Week 9:

Monday, February 27th

Wednesday, March 1st

### Week 10:

Monday, March 6th

Wednesday, March 8th

## Communication:

### Mailing List:

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

Please click here if you would like to enter your email and subscribe to the mailing list.