MATH 767
Complex Algebraic Curves — an Introduction to Algebraic Geometry
(Fall 2003)

Instructor: Michael Stillman

Meeting Time & Room   

The theory of complex algebraic curves and Riemann surfaces is a beautiful theory at the crossroads of several fields surrounding algebraic geometry: complex analysis, commutative algebra, geometry, and topology. In this course, we will cover roughly the material in chapters 1-8 of Miranda's book. We will get to the point as quickly as possible where we can state and prove the Riemann-Roch theorem, and then we will spend the rest of the semester applying the Riemann-Roch theorem to obtain deep results in the algebraic geometry and function theory of curves. Riemann-Roch is a great theorem, it has many applications, and also allows one to obtain function theory results and algebraic results using geometry (and vice versa).

Along the way, we will see many examples of curves and Riemann surfaces. This course will attempt to be reasonably self contained: we will state carefully results that we will use without proof. Miranda's book also has a good selection of exercises.

The main objects of study are plane algebraic curves (i.e. the solution set of f(x,y) = 0), and Riemann surfaces, as well as holomorphic and meromorphic functions on Riemann surfaces and maps between them. There are several approaches one may take to develop the theory: one can do it completely algebraically, or one may use complex analysis. We will use complex analysis, as it allows one to get deeper into the field with less background.

Tentative list of topics:

• Affine and projective plane algebraic curves
• Riemann surfaces
• Holomorphic and meromorphic functions
• Holomorphic maps between Riemann surfaces
  Local description of maps, degree of a map, Riemann-Hurwitz theorem
• Differentials and integration
  Residue theorem
• Divisors, maps to projective space
• Riemann-Roch theorem, and its proof
• Applications
  • to curves of low genus
  • to curves of high degree
  • equivalence of the three definitions of genus
  • canonical curve
  • Clifford's theorem, hyperelliptic curves
  • Geometric form of Riemann-Roch
  • Weierstrass points, flexes, bitangents, etc.
• Monodromy and uniform position
  • connectedness of algebraic curves
  • normalization of algebraic curves
  • uniform position lemma for a generic hyperplane section
• Abel's theorem and the Jacobian of a curve

Textbook: R. Miranda: Algebraic curves and Riemann surfaces

Prerequisites: A course in complex analysis, a small amount of algebraic topology is useful, but not essential, and a small amount of commutative algebra is useful as well.

Last modified: August 25, 2003