Algebraic Geometry I

This is an introduction to the theory of schemes and cohomology. We plan to cover part of Chapter 2 and Chapter 3 of the textbook.

Some course materials are available during the semester at PKU disk. The password to retrieve these will be distributed in class.

You are more than welcome to send suggestions/comments about this course to the instructor.

Instructor: Zhiyu TIAN

TA: Lai SHANG

Course Announcement

• Final exam (Click to see the instruction and problems): Dec. 15 noon 12:00 to Dec. 27 noon 12:00. Read the instruction first.

• If you want to have the final grade as Pass-Fail, please send an email to me and TA before Friday Dec. 2.

• Quiz2, from Nov. 24 to Dec. 1. Use LaTex to edit your answers. Send pdf files to TA as attachment. Email subject: name student id AG1 Quiz 2.

• We will use Tencent meeting for online lectures. Tuesday lecture link Thursday lecture Link. Password for both meetings: 2022.

• 2nd Quiz scheduled on Nov. 24th-Dec. 1st.

• Quiz 1: Oct. 25- Nov. 3 10am. Due in class on Nov. 3rd. Make sure you read the instruction first.

Links for course videos

• Link for the video of the lecture 11/17 Password bDM7

• Link for the video of the lecture 11/22, Password: DR8S

• Unfortunately, there is no video for the class on 11/24. The lecture notes are available at PKU disk.

• Link for the video of the lecture 12.1, Password: Z5qD

• Link for the video of the lecture 12.6, Password: yY5G

• Link for the video of the lecture 12.8, Password: eemw

• Link for the video of the lecture 12.15, Password: rZnv

Office Hours

By appointment via email. In your email, specify a few times slots convenient for you.

Office: BICMR 77103

Check course announcements for extra office hours before mid-terms/final exams.

Prerequisite

The prerequisite list for studying algebraic geometry could go really long. But you do NOT need to have everything ready before you start this beautiful journey, as long as you are willing to learn (or accept in faith for the moment) related knowledge in the process.

But I would expect that you have prepared yourself with the following things, which means that related concepts and theorems will be used freely without further explanation in this course.

A good understanding of abstract algebra, including groups, (commutative) rings, modules, fields (e.g. the first 8 chapters book by Atiyah-Mcdonald) , and homological algebra (including basic abelian categories), especially derived functors (Hartshorne has a brief introduction in Chapter 3).

Some basic idea of varieties and such (e.g. Hilbert Nullstellensatz, Noether normalization). A brief reading of Chapter 1 Section 1-4 in Hartshorne suffices. We will not use them much. But you should know.

Some acquaintance with complex manifolds might be helpful.

Textbook and reference

We use Hartshorne's classical textbook Algebraic geometry.

You can also take a look at Mumford's red book, and Harris-Eisenbud Geometry of schemes.

Students at PKU campus can download and read these books using the above links to SpringerLink.

EGA Synopsis

A comprehensive and searchable reference is the stack project.

Homework

The exercises in Hartshorne form an essential part of the book. They provide important examples and useful results. In my opinion, anyone who has not seriously thought about these exercises should not claim that he or she has read Hartshorne.

Therefore, the homework assignment consists of the following:

• Read and think about all the exercises. Make sure you understand the conclusions and the examples.

• Solve as many problems as you can.

• In total, you should write down solutions to 60 problems this semester. The TA will check them without grading them. You should finish at least 30 problems by the first mid-term, 45 problems by the second mid-term.

On the submitted homework, make it clear which problem in Hartshorne you are solving.

You should number your solutions.

Homework assignments are to be submitted with quiz directly to TA.

Final grade

Final grades are based on homework (30%), 2 mid-terms (30%), and a take-home final exam (40%).