Riemann Surfaces (SdR), basic definitions: complex maps and complex structures. First examples. Recalls on the classification of compact surfaces. Genus of a compact SdR.
Complex projective spaces.
Affine and projective algebraic curves. Associated SDR.
Holomorphic and meromorphic functions on a SdR.
Holomorphic maps between SdR. Multiplicity of a point.
The sum of the orders of a meromorphic function is zero.
Maps between compact SDRs. Degree.
Euler characteristic and triangulations. Hurwitz theorem (topological proof).
Automorphisms and group actions on a SdR. Monodromy.
Hyperelliptic surfaces.
Integration on SdR: Differential forms. Operations on forms. Lemmas of Poincare and Dolbeaux. Stokes' theorem. Residue theorem.
Divisors and meromorphic functions.
Complex vector bundles on a SdR and to correspondence between divisors and vector bundles of rank 1.
Linear equivalence of divisors, spaces of forms and functions associated with a divisor. divisors and maps in the projective space. Very ample divisors. Algebraic curves.
Riemann Roch theorem, proof.
Applications of Riemann Roch: a SdR is an algebraic curve. Each algebraic curve is projective. Clifford's theorem. Canonical map, hyperelliptic curves. Geometric form of Riemann-Roch. Calculation of the Riemann parameters.
Degree of projective curves. Monodromy of the hyperplane divisors. Lemma of uniform position, and Lemma of general position. Bound of Castelnuovo. Maximal genus curves. Inflection points and Weierstrass points.
Homology, periods and Jacobians.
The Abel-Jacobi map. Proof of Abel's Theorem.

Bundles on curves. Introduction to Cech cohomology. Picard group of invertible sheaves and correspondence between divisors and invertible sheaves.