## Algebraic Curves and their Moduli

### Claudio Fontanari

Historical perspectives from Apollonius to Riemann. Linear series and Brill-Noether number. Elliptic curves and moduli spaces. Deligne-Mumford compactification and inductive computation of its rational cohomology.

Divisors and line bundles on complex manifolds. Weil and Cartier divisors on schemes. Morphisms to projective spaces.

Dualizing sheaf and Serre duality. The Riemann-Roch Theorem and applications.

Hurwitz Theorem and applications (in characteristic zero). Every curve can be embedded into P3. Canonical model.

Geometric version of the Riemann-Roch Theorem. Clifford's Theorem. Canonical sheaf of projective spaces. Adjunction formula and genus of plane curves. Castelnuovo's Theorem. Blowing-up and Del Pezzo surfaces. Halphen's Theorem (a' la Gruson-Peskine).

Construction of the moduli space as a quotient of the Hilbert scheme. Geometric description of the boundary of the Deligne-Mumford compactification.

Stable cohomology of the moduli space and Mumford's conjecture (now Madsen-Weiss Theorem). Ample and effective divisors and related conjectures by Faber, Fulton, Harris and Morrison.

Theory of special linear series. The original approach by Brill and Noether and the existence theorem by Kempf and Kleiman-Laksov. The variational approach by Castelnuovo and the non-existence theorem by Griffiths-Harris.

### References

- P. Griffiths and J. Harris: Principles of Algebraic Geometry. Wiley, 1978.
- J. Harris: Curves and Their Moduli. Proceedings of Symposia in Pure Mathematics 46 (1987), 99--143.
- R. Hartshorne: Algebraic Geometry. Springer, 1978.
- E. Sernesi: A brief introduction to algebraic curves. www.uniroma3.it/users/sernesi (2008).