A Riemann surface is a complex manifold of dimension 1; as such, it is defined without reference to an ambient space. However, if it also compact, such a surface can be obtained as the set of zeros of a set of homogeneous polynomials, and becomes an algebraic curve described by explicit equations in a projective space.

This course aims at being an introduction to some aspects of algebraic and complex differential geometry, concentrating on the dimension 1 case, where many examples, computations and constructions can be carried out in a very concrete way.

**Pre-requisits**: One variable complex analysis (the needed background from geometry, algebra and topology will be developed during the course).

**Program:**

• 1. *Topology of algebraic curves*

• 2. *Sheaf Cohomology*

• 3. *Compact Riemann surfaces*

• 4. *Moduli spaces*

**Bibliography**: (The main reference will be [M]):

[M] R. Miranda, *Algebraic curves and Riemann surfaces*, Amer. Math. Soc., 1994

[K] F. Kirwan, *Complex algebraic curves*, London Math. Soc., ST 23, 1992

[F] O. Forster, *Lectures on Riemann surfaces*, Springer Verlag, 1981

[FK] H. Farkas & I. Kra, *Riemann surfaces*, Springer Verlag, 1980

**Lecturer**: Carlos Florentino