# Riemann Surfaces and Algebraic Curves (Math 510, Fall 2019)

## Basic information

• Lectures: TR 11:00am–12:20pm in 447 Altgeld Hall.
• Course website: http://jpascale.pages.math.illinois.edu/510fa19
• Instructor: James Pascaleff
• Email: jpascale@illinois.edu; Office: 341B Illini Hall; Phone: (217) 244-7277.
• Office hours: W 1:00pm–3:00pm.
• Prerequisites: Math 542, Complex Variables I.
• Textbook: Rick Miranda, Algebraic Curves and Riemann Surfaces. Volume 5 of Graduate Studies in Mathematics, published by the American Mathematical Society.

## Content

This course will develop the theory of Riemann surfaces from complex-analytic foundations. All of the basic notions of complex analysis on Riemann surfaces (holomorphic and meromorphic functions, contour integrals) will be introduced, as well as various standard classes of examples.

The latter part of the course will specialize to compact Riemann surfaces. This special class is essentially the same as the class of projective complex algebraic curves in algebraic geometry. We will develop the theory of divisors and the Riemann-Roch theorem. These give a great deal of information about spaces of meromorphic functions and holomorphic maps on Riemann surfaces.

## Policies

• Assessment: Grades will be based on regular homework assignments (50%), one midterm exam (22%), and the final exam (28%).
• Homework: Collaboration on homework assignments is encouraged, but it is very important that you write up your solutions yourself, and that you understand completely everything that you write. Late homework: I understand that graduate students have many demands on their time. If you are unable to turn in your homework on the due date, you should email me (JP) before the due date to ask for an extension. I anticipate granting all reasonable requests.
• Exams: The midterm exam will be held during the regular class period on Tuesday, October 15. It will cover the material up through lecture 12 and homework 5. The final exam date will be held on Friday, December 13, 8:00am–11:00am, in 447 Altgeld Hall. You will be allowed to use the textbook and your notes for the exams.

## Homework assignments

• Homework 1 Due Tuesday, Sept. 10: Miranda, Problems I.1.A, I.1.G, I.1.H, I.1.I, I.2.C, I.2.D, I.2.E, I.2.G, I.2.H.
• Homework 2 Due Tuesday, Sept. 17: Miranda, Problems I.3.C, I.3.D, I.3.E, II.1.A, II.1.C, II.1.F, II.1.I.
• Homework 3 Due Tuesday, Sept. 24: Miranda, Problems II.2.{B,C,D,E,F}, II.3.{F,G,H,K}. Hint for II.2.F: To integrate $$\theta'/\theta$$ around a fundamental parallelogram, use the translation properties of $$\theta$$ to get some cancellation.
• Homework 4 Due Tuesday, Oct. 1: Miranda, Problems II.4.{C,G,I,J}, IV.1.{A,B,C,D}. Note: You may take for granted the fact that an irreducible affine or projective plane curve is connected.
• Homework 5 Due Tuesday, Oct. 8: Miranda, Problems IV.2.{B,C,F,H,I}, IV.3.{C,E,F,G}.
• Homework 6 Due Tuesday, Oct. 22: Miranda, Problems V.1.{A,E,G,H,J}, V.2.{C,D,F}.
• Homework 7 Due Tuesday, Oct. 29: Miranda, Problems V.3.{C,D,G,H,I}, V.4.{A,B,H}.
• Homework 8 Due Tuesday, Nov. 5: Miranda, Problems V.4.{C,E,J,K}, VI.1.{A,C,F,J}.
• Homework 9 Due Tuesday, Nov. 12: Miranda, Problems VI.2.{E,F,G,J}, VI.3.{B,C,D,G}. Recall that what Miranda denotes $$T[D](X)$$ is isomorphic to the space $$R/R(D)$$ from the lecture notes.
• Homework 10 Due Tuesday, Nov. 19: Miranda, Problems VI.3.{H,J}, VII.1.{B,C,G,H,I}.
• Homework 11 Due Tuesday, Dec. 3: Miranda, Problems VIII.1.A, VIII.1.C (give a plausible argument that any loop on $$X$$ is homologous to a combination of the stated classes), VIII.2.{A,C}.
• Homework 12 Due Tuesday, Dec. 10: Miranda, Problems VIII.4.{A,B,C,D,E}.

## Detailed schedule

Topics for future lectures may change as the course progresses.

 Date Lecture Remarks [2019-08-27 Tue] No class. [2019-08-29 Thu] 1. Background: topological spaces and holomorphic functions. Lecture by Prof. Dodd [2019-09-03 Tue] 2. Atlases and the definition of a Riemann surface. Prof. Pascaleff [2019-09-05 Thu] 3. Examples of Riemann surfaces. [2019-09-10 Tue] 4. Complex tori, projective varieties. Homework 1 due. [2019-09-12 Thu] 5. Holomorphic and meromorphic functions on Riemann surfaces. [2019-09-17 Tue] 6. Examples of meromorphic functions. Homework 2 due. [2019-09-19 Thu] 7. Holomorphic maps between Riemann surfaces. [2019-09-24 Tue] 8. Riemann-Hurwitz formula. Homework 3 due. [2019-09-26 Thu] 9. Hyperelliptic curves; differential forms. [2019-10-01 Tue] 10. Operations on differential forms. Homework 4 due. [2019-10-03 Thu] 11. Integration on Riemann surfaces. [2019-10-08 Tue] 12. Stokes’ theorem and the residue theorem. Homework 5 due. [2019-10-10 Thu] 13. Divisors. [2019-10-15 Tue] Midterm Exam. Solutions. [2019-10-17 Thu] 14. Linear equivalence of divisors. [2019-10-22 Tue] 15. Spaces of functions and forms associated to divisors. Homework 6 due. [2019-10-24 Thu] 16. Divisors and maps to projective spaces I. [2019-10-29 Tue] 17. Divisors and maps to projective spaces II. Homework 7 due. [2019-10-31 Thu] 18. Algebraic curves and Mittag-Leffler problems. [2019-11-05 Tue] 19. Weak Riemann-Roch, finiteness of the function field. Homework 8 due. [2019-11-07 Thu] 20. Finite dimensionality of $$H^1(D)$$. Textbook supplement. [2019-11-12 Tue] 21. Serre duality and full Riemann-Roch. Homework 9 due. [2019-11-14 Thu] 22. Applications of Riemann-Roch. [2019-11-19 Tue] 23. The canonical map. Homework 10 due. [2019-11-21 Thu] 24. Homology, periods, the Jacobian, and the Abel-Jacobi map. [2019-11-26 Tue] Fall Break. [2019-11-28 Thu] Fall Break. [2019-12-03 Tue] 25. Easy direction of Abel’s theorem, properties of the periods. Homework 11 due. [2019-12-05 Thu] 26. Hard direction of Abel’s theorem, more properties of periods. [2019-12-10 Tue] 27. Abel’s theorem in genus one, $$\mathcal{M}_g$$ and $$\mathcal{A}_g$$. Homework 12 due. [2019-12-13 Fri] Final exam to be posted Friday, December 13 at 8:00am Due by 5pm on Monday, December 16.