Algebraic Geometry II

This course is an introduction to the theory of schemes and cohomology and applications. We plan to cover Serre duality, cohomology and base change, theorem on formal functions, basics of algebraic curves and surfaces. Depending on the remaining time, we will also cover other selected topics, such as Hilbert schemes and moduli spaces, positivity, Grothendieck-Riemann-Roch, etc..

Some course materials are available 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: Ziqi Guo

Covid-19 information

• Be responsible. If you have Covid-19 symptoms, or you are in close contact with someone infected, do not come to the classroom, stay at home and watch the video instead.

• Wearing a mask is recommended.

Course Announcement

• Final exam: from 8:00 Monday June 12th to 12:30 Wednesday June 21. Click here for the problems. Submit to your solutions to TA in person at his office (Building 19, Room 107) between 12:30-13:00 on June 21st. You may use the textbook (Hartshorne), your notes, and lecture notes, and watch the recorded lecture videos. Any other form of assistance during the exam is strictly prohibited.

• Send a proof that you have finished your homework assignment to TA by email no later than June 16th.

• Reading group on resolution of singularities: Thursday 10:10-12:00 BICMR 77201, starting from May 11th. To attend, please register via this link. If you are interested in giving a talk, contact me.

• Second Quiz: May 8th 17:00 -May 15 15:00. Hand in your solutions in class in person, or ask a classmate to do it for you, or make an arrangement with TA if you cannot come to the class.

• The second quiz is to be finished in consecutive 24 hours, at any time during the exam period (May 8 17:00 -May 15 15:00), at any place of your choice. Click here to read the problems.

• First Quiz: Apr. 3 17:00 -Apr. 10 15:00. Hand in your solutions in class in person, or ask a classmate to do it for you, or make an arrangement with TA if you cannot come to the class.

• The quiz is to be finished in consecutive 3 hours, at any time during the exam period (Apr. 3 17:00 -Apr. 10 15:00), at any place of your choice. Click here to read the problems.

• Email TA a proof of the homework problems you have solved. Email title: student ID name AG2 homework. You should have finished at least 25 problems.

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

A good understanding of Chapter 2 Sections 1-6, Chapter 3, Sections 1-5, 8 of Hartshorne's textbook.

Textbook and reference

For the first part, we use Hartshorne's classical textbook Algebraic geometry. I will also provide some notes for things not covered in Hartshorne.

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 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.

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.

Take-home Quiz

We will have 2 take-home quizzes during the semester, to help the student to check how well they understand the course material. These quizzes will be graded. More specific guidelines will be discussed in Class.

Final grade

Final grades are based on homework (30%), quizzes (30%) and a take-home exam (40%).