## Math 416/816: Algebraic Curves and Riemann Surfaces

Lectures:   Are in Slot 14 (Tuesday 11:30-12:20, Wednesday 1:30-2:20, and Friday 12:30-1:20), in Room 110.

Office Hours:   The hour following class (Tuesday 12:30--1:20, Wednesday 2:30-3:20, and Friday 1:30-2:20) or by appointment.

Riemann Surfaces were introduced in the 19th century to help resolve strange formulae and explain mysterious behaviour arising from problems in function theory. Riemann's initial discription was confusing to many people and the attempt to understand these objects and his wonderful new ideas led to the growth or creation of many fields of mathematics: topology, the theory of manifolds, potential theory, and algebra to name a few.

The idea of a Riemann surface is now a central one in mathematics, and appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory.

Because of the many points of view on Riemann surfaces, and the large number of other parts of mathematics that they interact with, there are a lot of different approaches, and topics we could cover.

My goal is to touch on several of these areas and show how they relate. In particular, I would like to concentrate on the interaction between the complex analytic, classical geometric, and commutative algebraic points of view.

The organization of the course will largely follow the natural flow of ideas between these subjects.

We will first try to understand Riemann surfaces as complex analytic manifolds. We want to understand the basic topological properties of these surfaces, and of the maps between them.

We will then look at curves as submanifolds of complex manifolds, in particular as submanifolds of the projective plane. This will lead to the beautiful classical theorems in the plane: Bezout's theorem, and the adjunction formula.

The description as subvarieties of the projective plane will allow us to treat the curves purely algebraically. We'll study how to describe the curves using only algebraic objects like rings and ideals, and how to express maps between curves in the language of rings.

Finally, we'll largely return to the manifold point of view and understand the final fundamental theorem about algebraic curves: the Riemann-Roch theorem, which governs the geometry of curves in subtle ways. We will also look at applications of this theorem to geometric questions.

Anything we have to learn, we learn by the actual doing of it.

-- Aristotle, Nicomachean Ethics, Book II.

As with many things in mathematics, most of the ideas involved in studying Riemann surfaces are not that complicated, and our difficulties with them are mostly psychological: we we have to go through an awkward period of confusion before realizing how simple and straightforward the ideas were ''all along''.

The usual cure for the confusion brought on by abstract ideas and new definitions is to do a few concrete computations. In order to encourage everyone to do computations, I'm putting most of the grading weight of the course on the homework. The grading scheme is

 Homework 75% Final 25%

The homework questions will be designed to give you a chance to play with some of the concepts we've been talking about in class. For those who want more practice problems, verifying uncompleted or omitted calculations from the lectures is a good place to start.

### Homework Assignments

Assignment 1: Practice with coordinate charts:   .ps   .pdf  .dvi
Assignment 2: Branched covers and the Riemann-Hurwitz formula:   .ps   .pdf  .dvi
Assignment 3: Homogeneous and nonhomogeneous polynomials:   .ps   .pdf  .dvi
Assignment 4: Verifying details of the proof of Bezout's theorem:   .ps   .pdf  .dvi

Final Exam: The exam is due Wednesday, December 17, 2003 .ps

### Lecture Notes

I was going to attempt to put sketchy TeX versions of the lecture notes here, but I now realize that I will never manage to get this done. Instead I'm just going to use it to outline the lecture topics.

The plan for future lectures is flexible -- I'll be moving it around a bit depending on how things are going in class.

• Lecture 1 (Tuesday, September 9): A brief historical introduction to the problems motivating the introduction of the Riemann surface, a (large) list of other parts of mathematics where a Riemann surface is considered a basic object, and an outline of the course.
• Lecture 2 (Wednesday, September 10): Some basic facts from topology (open and closed sets, compact sets). Underlying possible shapes of a Riemann surface.
• Lecture 3 (Friday, September 12): The definition of a two-dimensional real manifold, with example. Functions on a real manifold.
• Lecture 4 (Tuesday, September 16): The definition of a one dimensional complex manifold, with example.
• Lecture 5 (Wednesday, September 17): Functions on manifolds. Complex manifolds by glueing.
• Lecture 6 (Friday, September 19): More about manifolds by glueing. Elliptic Curves.
• Lecture 7 (Tuesday, September 23): Definition of a map between complex manifolds and examples.
• Lecture 8 (Wednesday, September 24): Topology of maps between Riemann Surfaces I.
• Lecture 9 (Friday, September 26): Topology of maps between Riemann Surfaces II.
• Lecture 10 (Tuesday, September 30): Topology of maps between Riemann Surfaces III.
• Lecture 11 (Wednesday, October 1): The Riemann-Hurwitz formula.
• Lecture 12 (Friday, October 3): Introduction to Projective Space.
• Lecture 13 (Tuesday, October 7): Catch-up lecture: Simple consequences of the Riemann-Hurwitz formula, and more about projective space.
• Lecture 14 (Wednesday, October 8): Submanifolds of a complex manifold: Curves in projective space.
• Lecture 15 (Friday, October 10): Submanifolds of projective space part II.
• Lecture 16 (Tuesday, October 14): The genus of degree d plane curves Part I.
• Lecture 17 (Wednesday, October 15): The genus of degree d plane curves Part II.
• Lecture 18 (Friday, October 17): The genus of a degree d plane curve Part III.
• Lecture 19 (Tuesday, October 21): Some discussion of intersection multiplicity. The group law of an Elliptic curve part I.
• Lecture 20 (Wednesday, October 22): The group law of an Elliptic curve part II.
• Lecture 21 (Friday, October 24): Flex points and bitangent lines.
• Lecture 22 (Tuesday, October 28): End of Flex point computation.
• Lecture 23 (Wednesday, October 29): Resultants.
• Lecture 24 (Friday, October 31): Bezout's Theorem.
• Lecture 25 (Tuesday, November 4): Riemann's description of a Riemann Surface.
• Lecture 26 (Wednesday, November 5): Introduction to algebraic ideas of Riemann Surfaces.
• Lecture 27 (Friday, November 7): Algebraic Curves and coordinate patches.
• Lecture 28 (Tuesday, November 11): Homomorphisms between rings of algebraic functions.
• Lecture 29 (Wednesday, November 12): Maps between affine curves and algebraic consequences.
• Lecture 30 (Friday, November 14): Maximal ideals, DVR's, and ramification.
• Lecture 31 (Tuesday, November 18): Vector bundles on manifolds, examples.
• Lecture 32 (Wednesday, November 19): --- Class Cancelled ---
• Lecture 33 (Friday, November 21): Vector bundles via transition functions.
• Lecture 34 (Tuesday, November 25): Line bundles, global sections, and examples.
• Lecture 35 (Wednesday, November 26): The Riemann-Roch theorem.
• Lecture 36 (Friday, November 29): Some consequences of the Riemann-Roch theorem.