
Math 2050: Algebraic geometry, Fall 2019
MWF 22:50 Kassar House 105
Instructor: Melody Chan
office: Kassar House 311
course email: melody_chan@brown.edu
course website: http://www.math.brown.edu/~mtchan/2019Fall_2050.html
office hours, Mondays 1:102, Fridays 4:155, and by appointment
Course description and goals
This is the first semester of a yearlong graduate course in algebraic geometry. We will cover the foundations of varieties and schemes. Please read Section 0.1 What is algebraic geometry? of Gathmann's notes for a preview of what we will study, and why.
The recommended prerequisites are commutative algebra at the level of Math 25102520, including familiarity with rings and modules, tensor product and localization, various finiteness conditions, flatness... Some knowledge of general topology is also necessary, and a basic familiarity with manifolds will also be very helpful for understanding what is going on. If you have any questions about prerequisites, please let me know.
Algebraic geometry is a rigorous, beautiful subject. The only way to learn it is to spend lots of time engaging with the material. I will expect lots of work on the problem sets, and a level of rigor at least at the level of Math 2520. But I will try to make sure that the work you put in will be well worth it.
Textbooks
Andreas Gathmann, Algebraic geometry, course notes linked here.
Qing Liu, Algebraic geometry and arithmetic curves, 2006 paperback edition (available to read online.)
Individual chapters of the previous 2002 edition may be downloaded in PDF.
The author maintains a list of errata here.
This book is also available at the bookstore for $85 new, $63.75 used.
The last time I taught this course I taught from Liu as the main textbook. This time, I may try to shift the focus of the course largely towards what is covered in Gathmann's notes. The exact balance is yet to be determined.
Other useful references
David Eisenbud and Joe Harris, Geometry of schemes (available online). This is a great book for some supplementary examples, exercises, and intuition.
Ravi Vakil, The rising sea: Foundations of algebraic geoemtry (available online). This is a great learnityourself pathway into the subject, full of exercises to work out.
Joe Harris, Algebraic geometry: a first course (available online). Full of great examples. Classical perspective, no schemes.
Grading
Weekly problem sets posted here, typically due once a week on Fridays, at the beginning of class in hard copy (LaTeX strongly preferred) and stapled. No late problem sets will be accepted. The problem sets are the most important component of the course.
Optional short inclass presentation and writeup, in the second half of the course. This is optional but highly recommended. Your presentation grade replaces 1.5 lowest problem set grades.
No final exam.
Problem sets
All problem sets in one PDF
Schedule
We meet during reading week; the last day of class is Wednesday December 11.
I am out of town Sept 913. Class is cancelled on September 9 only. On September 11 and 13 there will be guest lectures by Joe Silverman and Jonathan Wise.
Collaboration
You are encouraged to collaborate with other students in the class on your homework, although I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)
How much time will this class take?
Roughly speaking, you should expect to spend twelve hours every week outside of class, including attending office hours, reviewing class material and doing problem sets. In addition to three hours of class every week, I estimate a total of 15*13 = 195 hours of time spent on this class.
Accommodations for students with disabilities
Please contact me as early in the semester as possible so that we may arrange reasonable accommodations for a disability. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 4018639588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html
