Algebraic Geometry

Spring Semester 2022


Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] wustl.edu
(Zoom) Office Hours: 8-8:50pm M (Zoom), 3-3:50 W/F
Course Outline:

We will study the solution sets of polynomial equations, mainly over the complex numbers, and primarily concentrating on algebraic curves in the projective plane. In addition to the obvious algebraic perspective (polynomial rings), the topological and complex analytic points of view (by viewing algebraic curves as Riemann surfaces) will be developed throughout. Topics covered include differential forms and the genus formula, group laws on cubic curves, periods and Picard-Fuchs equations, and the Riemann-Roch formula and Abel's theorem.

Prerequisites: Math 310, 429, and 430, or permission of the instructor. (We will also make use of elementary 1-variable complex analysis, at the level of Math 416, like Rouche's theorem, Weierstrass M-test, power series, etc.)
Class and Exam Schedule:

Lectures are on Monday, Wednesday, and Friday, from 9-9:50 AM, in Weil 010. (Our first two weeks, however, are on Zoom.) The first class is on Wednesday Jan. 19 and the last class is on Friday Apr. 29; Spring Break is March 13-19.

Midterm Exam 1: take-home, due Friday March 25
Final Exam: Friday May 6, 8-10 AM, in Weil 010

The final exam info is what is posted by the registrar. I may change this to a take-home exam, final project, or combination.
Assignments:

These will be due, via PDF upload to Canvas, on Tuesdays at 5pm. Solutions will also be posted on Canvas and may include students' work. I encourage you to visit my office hours to discuss problem sets, and to form study groups to discuss the more difficult problems (though solutions must be written up independently).

HW #1 (due Tuesday Feb. 1): Ch. 1: #1,2,choose one of {3,4,5}; Ch. 2: #1,2,3,choose 4 or 5; Ch. 3: #1,2,5,choose one from {3,4,6}; Ch. 4: #1,3,4,choose 2 or 5
HW #2 (due Tuesday Feb. 8): Ch. 5: #2,4,6,7; Ch. 6: #2,4,5,7; Ch. 7: #1,2
HW #3 (due Tuesday Feb. 15): Ch. 7: #3,4,5; Ch. 8: #1,choose one more; Ch. 9: #1; Ch. 10: #1,2
HW #4 (due Tuesday Feb. 22): Ch. 11: #1,2; Ch. 12: #1-6; Ch. 13: #1,2(a)
HW #5 (due Tuesday Mar. 1): Ch. 13: #3,4; Ch. 14: #1,2,4,6,7; Ch. 15: #1,3
HW #6 (due Thursday Mar. 10): Ch. 17: #1; Ch. 18: #1,4; Ch. 19: #2,4; Ch. 20: #1,2,7
HW #7 (due Thursday Apr. 7): Ch. 23: #6; Ch. 25: #1; Ch. 26: #1,2,3,4
HW #8 (due Monday Apr. 25): Ch. 21: #5; Ch. 27: #3; Ch. 28: #2,4; Ch. 31: #1; Ch. 32: #8,9
Grader: Xiaojiang Cheng
Lecture Notes:

These will be posted below as they are written, and form the text of the course. If you would like to have an alternative point of view on the material, try these notes by Miles Reid.

I. Introduction and Motivation
1. Two theorems on conics in the plane
2. Riemann surfaces and algebraic curves
3. The normalization theorem
4. Lines, conics, and duality
II. General definitions and results
5. Complex manifolds and algebraic varieties
6. More on projective algebraic varieties
7. Smooth varieties as complex manifolds
8. The connectedness of algebraic curves
9. Hilbert's Nullstellensatz
10. Local analytic factorization of polynomials
11. Proof of normalization theorem (A)
12. Intersections of curves
13. Meromorphic 1-forms on a Riemann surface
14. The genus formula
15. Some applications of Bezout
16. Genera of singular curves
III. Cubic curves
17. The singular cubic
18. Putting a nonsingular cubic in standard form
19. Canonical normalization of the Weierstrass cubic
20. Group law on the nonsingular cubic
21. Abel's Theorem for elliptic curves
22. The Poncelet problem
23. Periods of familes of elliptic curves
24. Elliptic curves over finite fields
IV. Higher degree
25. The algebraicity of global analytic objects
26. The Riemann-Roch theorem
27. Applications of Riemann-Roch, I
28. Applications of Riemann-Roch, II
29. Proof of normalization theorem (B)
30. Abel's Theorem, I
31. Abel's Theorem, II
V. Higher dimension
32. Cubic surfaces
33. Abelian varieties
Grading Policy:

Homework and examination grades will be regularly updated on Canvas. Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework.

Grades are typically curved in a course like this but will never be less than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

The Pass/Fail policy is that you must get at least a C- to earn a "Pass".

This link takes you to the standard university policies on academic integrity. All work submitted under your name is expected to be your own; as mentioned above, please make sure to document any ideas that come from another source.
The picture in the background represents (the real points of) a Clebsch cubic surface, with its 27 lines displayed.