About the course
Curves - Geometry and moduli: The course aims to provide
an introduction to moduli spaces of curves, deformation theory and
Geometric Invariant Theory. Possible/tentative topics are the
following:
- Curves: Riemann-Roch, Serre duality, Riemann-Hurwitz formula (crash
course)
- Hyperelliptic curves, the canonical embedding, Clifford's theorem
- Curves of low genus. Rudiments of Brill-Noether theory.
- Parameter spaces, moduli spaces, Hilbert scheme
- Tangent space, dimension of the Hilbert scheme
- Deformation theory, Kodaira-Spencer maps
- Constructions of the moduli space of curves
- The ring of tautological classes
- Background on GIT
- The Deligne-Mumford compactification of the moduli space
- Moduli spaces of stable maps, quantum cohomology, Gromov-Witten
invariants
References
Basic references for the course are
-
Moduli of Curves, J. Harris and I. Morrison, GTM 187, Springer
- Geometric invariant theory, D. Mumford, J. Fogarty, and F. Kirwan, 3rd ed.Springer
- Geometry of algebraic curves,
Geometry of Algebraic Curves, E. Arbarello, M. Cornalba, Ph. Griffiths, J. Harris.
Among other useful references
- Lectures on invariant theory, I. Dolgachev, Cambridge 2003. Download PDF version.
- An Introduction to Invariants and Moduli, Sh. Mukai, Cambridge 2003.
- A conjectural description of the tautological ring of the moduli space of curves, C. Faber, math.AG/9711218
- The geometry of moduli spaces of shaves, D. Huybrechts, M. Lehn, Aspects of Math. 31, Vieweg