What is this course about?
As is reflected by its title, the central notion of this course is that of a moduli space of curves. It has variants in several contexts: algebro-geometric, complex-analytic, differential-geometric and combinatorial. We will touch on all of them.
Literature
The first few lectures will be based on my brief survey paper A mini course on moduli spaces of curves. Recommended books are Harris-Morrison: Moduli of Curves and Arbarello-Cornalba-Griffiths: Geometry of Algebraic Curves, vol. I and (especially) vol. II.
Course notes
I will produce lecture notes that keep up with the course (at least, I try). These notes can be downloaded here (last update Jan. 16 2012). Please inform me of any inaccuracies that you are aware of.
Place and time
On Mondays 15:20-16:55 and Thursdays 9:50-11:25, starting Sept. 15, Lecture Building 4, Room 4406.
Office hour
On Thursdays after class (at 11:25) in class room. You can also see me by appointment in my office, which is room 251 in the Mathematical Sciences Center (Jin Chun Yuan West Building).
Home work and grading
Each week I give a few exercises. These must be handed in in English to my teaching assistent Dali Shen (which you may do in paper or in the form of a PDF file sent to D.Shen@uu.nl), by Monday afternoon the following week, just before the course begins. He will grade them. Your final grade will be determined by these homework grades.
Course log and home work
Sept. 15: Mapping class groups, conformal structures.
Sept. 19: (Homework: Exerc. 1) Rough classification of Riemann surfaces. Geometry of upper half plane.
Sept. 22: Teichmüller space and mapping class group for (g,n)=(0,3) and (1,1). Characterization of right-angled hexagons.
Sept. 26: (Homework: Exerc. 2,3) Geodesic shearing, geodesic representation of isotopy classes, pairs of pants.
Sept. 29: Graph attached to a pair of pants, hyperbolic pair of pants, Fenchel-Nielsen parametrization.
Oct. 10: (Homework: Exerc. 4,5) Fenchel-Nielsen in the presence of cusps.
Oct. 13: Quadratic differentials: local normal form, associated metric and foliations.
Oct. 17: Jenkins-Strebel differentials, ribbon graphs.
Oct. 20: Metrized ribbon graphs.
Oct. 24: (Homework: Exerc. 8,11,12) Ribbon graph category and associated thickened Teichmüller space.
Oct. 31: Ideal triangulation of Teichmüller space and homotopical implications, Dolbeault cohomology.
Nov. 3: (Homework: Exerc. 14,15) Review Riemann-Roch and Serre duality. Notion of orbifold.
Nov. 7: Heuristic determination of tangent space of Teichmüller space. Family of P-pointed Riemann surfaces, (universal) deformation of a P-pointed Riemann surface.
Nov. 10: (Homework: Exerc. 16,17,18) Kodaira-Spencer map, basic properties of a universal deformation, construction of universal family over Teichmüller space in the complex-analytic setting.
Nov. 14: Moduli space via Hilbert schemes, Weil-Petersson metric.
Nov. 21: Teichmüller space as a symplectic manifold. Curve complex, permissible metrics.
Nov. 24: Harvey bordification. Example of (g,n)=(1,1).
Nov. 28: (Homework: Exerc. 21,22) (1,1)-Example cont’d, stable pointed curve, deformation of a node.
Dec. 1: Local deformation theory of a stable pointed curve.
Dec. 5: (Homework: Exerc. 23,24) Moduli space of stable pointed curves, its relation to Harvey’s bordification. Basics of GIT and the projective structure on the Deligne-Mumford compactification.
Dec. 8: The functor from the category of stable weighted graphs to Deligne-Mumford orbifolds, brief review of group cohomology.
Dec. 12: (Homework: Exerc. 25,26) Moduli spaces of curves as virtual classifying spaces, review of Gysin maps, tautological algebra, psi classes.
Dec. 15: Kappa classes, restriction properties of the psi classes and the kappa classes. Harer’s stability.
Dec. 19: Exercise session run by Dali Shen.
Dec. 22: No class.
Dec. 26: Lambda classes, Faber conjectures, Teichmüller space associated to a central extension of a mapping class group.
Dec. 29: Harer’s stability continued, Hopf structure on stable cohomology, Mumford’s conjecture, group theoretical characterization of mapping class groups.