Syllabus

Course Description

This course introduces techniques for studying intersection theory on moduli spaces such as homogeneous varieties, the Deligne-Mumford moduli space of stable curves and the Kontsevich moduli spaces of stable maps. The course emphasizes how one can deduce global geometric properties of moduli spaces and the objects they parameterize using intersection theory.

The topics include:

  1. Littlewood-Richardson rules for Grassmannians,
  2. Basic results about the divisor theory and cohomology of Mg due to Harer, Zagier, Arbarello and Cornalba,
  3. A study of Brill-Noether theory and the theory of limit linear series in order to prove that Mg is of general type if g is greater than or equal to 24,
  4. Recent developments due to Farkas, Gibney, Keel, Khosla and Morrison about the ample and effective cones of Mg,
  5. The Gromov-Witten invariants of simple homogeneous varieties, and
  6. The ample and effective cones of Kontsevich moduli spaces.

Prerequisites

Algebraic Geometry (18.725). This is a first year graduate class in algebraic geometry at the level of the second and third chapters of R. Hartshorne (Algebraic geometry. New York, NY: Springer-Verlag, 1977). Familiarity with algebraic topology helpful.

Required Text

There is no required text for this course. However, there are many recommended readings.

Grading

Since there are no problem sets or exams for this course, the grade is based primarily on participation in the class.