MATH 510: Riemann Surfaces and Algebraic Curves (Spring 2016)

Instructor: James Pascaleff

Email: jpascale@illinois.edu
Office: 341B Illini Hall
Office hours: Thursday 2:00–4:00 pm

Lectures

MWF Noon–12:50 pm in 345 Altgeld Hall.

Prerequisites

MATH 500 (Abstract Algebra I) and MATH 542 (Complex Variables I)

Text

Rick Miranda, Algebraic Curves and Riemann Surfaces. Volume 5 of Graduate Studies in Mathematics, published by the American Mathematical Society.

Grading

UPDATED 2016-02-05 Fri: Grades will be based on homework, which will be collected every two weeks, one midterm exam, and a final exam.

Exams

Midterm exam: Take-home, assigned March 7, due March 14.
Final exam: 1:30-4:30 pm, Thursday, May 12.

This take-home midterm is due Monday, March 14, in class: Midterm.

The final exam will be held 1:30-4:30 pm, Thursday, May 12 in the normal classroom. You will be permitted to use your textbook and notes. Final Exam.

Date	Lecture notes
W Jan 20	1. Introduction; complex charts
F Jan 22	2. Atlases; real and complex manifolds
M Jan 25	3. Topological surfaces; examples
W Jan 27	4. Affine plane curves
F Jan 29	5. Projective curves
M Feb 1	6. Holomorphic functions
W Feb 3	7. Examples of meromorphic functions
F Feb 5	8. Meromorphic functions on smooth projective curves
M Feb 8	9. Holomorphic maps
W Feb 10	10. Multiplicies and degrees
F Feb 12	11. Euler characteristic and Riemann-Hurwitz formula
M Feb 15	12. Lines, conics, hyperelliptic curves
W Feb 17	13. Hyperelliptic curves (notes above)
F Feb 19	14. Maps between complex tori
M Feb 22	15. Plugging holes, resolving nodes
W Feb 24	16. Nodes (notes above)
F Feb 26	17. Monomial singularities
M Feb 29	18. Quotients
W Mar 2	19. Quotients (notes above)
F Mar 4	20. Covering spaces and monodromy
M Mar 7	21. Monodromy II
W Mar 9	22. Monodromy III (notes above)
F Mar 11	23. Monodromy IV, differential forms (notes above and below)
M Mar 14	24. Differential forms
W Mar 16	25. Differential forms II
F Mar 18	26. Differential forms III (notes above)
M Mar 21	Spring vacation
W Mar 23	Spring vacation
F Mar 25	Spring vacation
M Mar 28	27. Differential forms IV (notes above)
W Mar 30	28. Integration
F Apr 1	29. Integration II
M Apr 4	30. Integration III
W Apr 6	31. Divisors
F Apr 8	No class
M Apr 11	32. Linear equivalence
W Apr 13	33. Spaces of meromorphic functions and forms associated to a divisor
F Apr 15	34. Divisors and maps to projective space
M Apr 18	35. Divisors and maps II
W Apr 20	36. Divisors and maps III
F Apr 22	37. Divisors and maps IV
M Apr 25	38. Riemann-Roch
W Apr 27	39. The canonical map
F Apr 29	40. Finding equations for projective curves
M May 2	41. Jacobians and Abel's Theorem
W May 4	42. Abel-Jacobi maps

• Homework 1, Due Wednesday, February 3.
• Homework 2, Due Wednesday, February 17.
• Homework 3, Due Wednesday, March 2.
• Homework 4, Due Wednesday, April 13.
• Homework 5, Due Wednesday, April 27.
• Homework 6, Due Wednesday, May 4.

Riemann surfaces are obtained by gluing together patches of the complex plane by holomorphic maps, whereas algebraic curves are one-dimensional shapes defined by polynomial equations, such as conic sections , the cusp , or say the Klein quartic . The theories of (compact) Riemann surfaces and (complex smooth projective) algebraic curves are equivalent in a precise sense. This means we can study the same objects using both complex analysis and abstract algebra.

After defining these objects carefully and showing how to construct examples, some of the questions we will discuss are: When are two Riemann surfaces isomorphic? Can they be classified? In what ways can an algebraic curve be embedded in a larger ambient space, such as the plane? The two key tools are Abel's theory of contour integration on Riemann surfaces, and the Riemann-Roch theorem, which helps one to find meromorphic functions with certain properties.

This is one of the most beautiful areas of mathematics (at least in the instructor's opinion), and it serves as an entry point to both algebraic geometry and complex analytic geometry.