Algebraic Geometry

Fall Semester 2025


Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] wustl.edu
Office Hours: Monday 3-4, Thursday 2-3
Course Outline:

In this graduate-level course, we will study the solution sets of polynomial equations, mainly over the complex numbers, and primarily concentrating on algebraic curves and surfaces in projective space. In addition to the obvious algebraic perspective (polynomial rings, Zariski topology), the topological and complex analytic points of view (e.g. 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, the Riemann-Roch formula and Abel's theorem, birational maps and invertible sheaves, and cubic, abelian, and K3 surfaces. Individualized projects will cover additional material of interest to the students.

Prerequisites: Math 5031-5032 or permission of the instructor.
Class and Exam Schedule:

Lectures are on Monday, Wednesday, and Friday, from 2-2:50 PM, in Crow 205. The first class is on Monday Aug. 25 and the last class is on Friday Dec. 5; we have off Sep. 1 (Labor day), Oct. 6 (Fall break), and Nov. 26 and 28 (Thanksgiving).

The midterm exam will be replaced by 2-3 in-class quizzes. The final exam will be replaced by individualized final projects, with individual student presentations to be done during reading and finals weeks.
Assignments:

Each chapter of the text has several exercises, probably too many overall. So here I've listed what are probably the most important ones to look at. Problems in bold are to write up and hand in. They serve mainly to check your understanding, and you are encouraged to attempt other problems in the text. There is no separate grader for this course.

Note: something like 3.5 means Chapter 3 Exercise 5.

Week 1: 1.1, 1.5, 2.1, 2.3, 3.1, 3.2, 3.5, 3.6, 4.1, 4.4
Week 2: 5.2, 5.6, 5.7, 6.2, 6.4, 6.5, 6.7, 7.1, 7.2, 7.3, 7.4, 7.5(b)
Week 3: 8.2, 8.3, 9.1, 9.3, 10.3, 10.6, 11.1, 11.2
Week 4: 12.1, 12.3, 12.4, 12.5, 12.6, 13.1, 13.2, 13.3, 13.4, 13.6, 14.1, 14.4, 14.6, 14.7
Week 5: 15.1, 15.3, 15.5, 16.3, 16.6, 17.3, 17.4
Week 6: 18.1, 18.4, 19.2, 19.4
Week 7: 20.4, 20.6, 20.7, 21.5
Week 8: 1.2, 22.1, 23.6, 23.4, 25.1, 25.5
Week 9: 24.2, 26.2, 26.3, 26.4, 26.5, 27.3
Week 10: 28.2, 28.4, 28.5, 28.6, 29.5
Week 11: 30.1, 30.3, 31.1, 31.2, 31.3
Week 12: 32.1, 32.2, 32.3, 33.1
Week 13: 34.4, 34.5, 34.6, 35.1, 35.4
Bold problems for Weeks 11-13 are due Friday Dec. 5. This is the last HW. You may hand them in to me directly or e-mail them to me, whichever is more convenient.
Lecture Notes:

On canvas I have posted an official syllabus and the course text. The latter will be regularly updated as I write and edit. The lectures will not cover this material linearly. For convenient access I have posted old versions of some chapters below, as well as titles of planned sections. These will not be updated.

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. Quadric and cubic surfaces
33. The 27 lines on a cubic surface
34. Abelian varieties and complex tori
35. Theta functions and period maps
36. Polarized tori as projective varieties
37. K3 surfaces and Calabi-Yau varieties
38. K3 surfaces and the irrationality of ζ(3)
39. Invitation to Hodge theory
Grading Policy:

Grades will be regularly updated on Canvas, where you will also find a link to the official course syllabus. Your final grade for the semester is determined as follows: attendance 15%, HW 20%, quizzes 15%, final project 50%. Attendance means that you usually come to class (as opposed to usually not coming to class) and get in touch with me when you plan to be absent for relevant course material.

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 (for undergrads only) is that you must get at least a C- to earn a "Pass".

These links take 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, including GenAI. (I cannot strictly prohibit the use of the latter, but I'd strongly discourage it.)
The picture in the background represents (the real points of) a Clebsch cubic surface, with its 27 lines displayed.