(Landelijk Mastercollege Riemannoppervlakken)

This course gives an introduction to the theory of Riemann surfaces and algebraic curves. After treating sheaves and their cohomology, differential forms and residues we intend to prove the theorem of Riemann-Roch and Serre-duality. After that we discuss coverings of Riemann surfaces, the Hurwitz-Zeuthen formula and hyperelliptic Riemann surfaces. Finally, we shall treat Jacobian varieties the Abel-Jacobi map and we end with algebraic curves.

Besides attending the lectures students are supposed to work on exercises. Examination is either oral or written. Also a project is possible.

Prerequisites: a healthy knowledge of elementary complex function theory and topology.

Literature:

O. Forster: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York-Berlin,

Further literature will be given during the course.

Time and place: Wednesday, 14:00-16:45, at the VU (see Mastermath.nl for precise locations). Starts Wednesday, September 11, 2013.

See also the website mastermath .

Exercises for week 38: Ch. 1: 1.2, 1.3, 1.4, 1.5, 2.3, 2.4, 2.5. Treated in week 38: Riemann surface, holomorphic map, cateory, functor, sheaf of holomorphic functions, singularities, meromorphic functions, identity thm, maximum principle.

Topics for weeks 39: complex tori, Weierstrass P-function, field of elliptic functions, algebraicity of complex tori of dim 1, fundamental group, ramification, liftings of maps. Exercises: 3.1-3.3, 4.1-4.5

Treated in week 40: degree of a proper map, universal covering space, Galois cover, Exercises: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7.

Treated in week 41: Sheaves, construction of Riemann surfaces associated to function field extensions. Exercises: 6.1, 6.2, 8.1., 8.2.

Treated in week 42: Differential forms on Riemann surfaces. Exercises: 9.1, 9.2, 9.3, 10.2,

Treated in week 43: cohomology Exercices:

Treated in week 45: Dolbeault lemma Exercises:

Treated in week 46: Cohomological calculations Exercises: 12.1, 12.2, 12.3, 13.1, 13.2,

Treated in week 47: Riemann-Roch Exercises: 16.1, 16.2, 16.3, 16.4, 16.5,

Treated in week 48: Serre duality Exercises: 16.1, 16.2, 16.3, 16.4, 16.5,

Treated in week 49: Applications of Riemann Roch; Hurwitz-Zeuthen; the Jacobian of a Riemann surface Exercises: 17.1, 17.4, 17.5, 17.7,

Treated in week 50: Linear systems, the Theta divisor. The canonical map. Low genera. Lecture notes for the last lectures: notes

Topics for a project: project list

Here is an example of an exam: Test Exam

with here a solution solution .

Here is the Exam of January 8, 2014 and a solution .