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).

• 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

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