# 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. The final exam date is yet to be determined by the registrar.

## 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.

## Detailed schedule

Topics for future lectures may change as the course progresses.

Date | Lecture | Remarks |

No class. | ||

1. Background: topological spaces and holomorphic functions. | Lecture by Prof. Dodd | |

2. Atlases and the definition of a Riemann surface. | Prof. Pascaleff | |

3. Examples of Riemann surfaces. | ||

4. Complex tori, projective varieties. | Homework 1 due. | |

5. Holomorphic and meromorphic functions on Riemann surfaces. | ||

6. Examples of meromorphic functions. | Homework 2 due. | |

7. Holomorphic maps between Riemann surfaces. | ||

Riemann-Hurwitz formula. | Homework 3 due. | |

Differential forms. | ||

Integration on Riemann sufaces. | ||

Divisors and linear equivalence. | ||

Spaces of functions and forms associated to divisors. | ||

Divisors and maps to projective spaces. | ||

Midterm Exam. |
||

Algebraic curves. | ||

Riemann-Roch theorem and Serre duality I. | ||

Riemann-Roch theorem and Serre duality II. | ||

Applications of Riemann-Roch I. | ||

Applications of Riemann-Roch II. | ||

Applications of Riemann-Roch III. | ||

Homology, periods, the Jacobian of a compact Riemann surface. | ||

Abel-Jacobi map. | ||

Trace operations. | ||

Abel’s theorem. | ||

Abel’s theorem in genus one, elliptic curves. | ||

Fall Break. |
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Fall Break. |
||

Towards the moduli space of Riemann surfaces. | ||