Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves - WS 2020/21

Instructors : Daniele Agostini and Rainer Sinn.

Office hours: By appointment.

Lectures: Wednesday 15:15 - 16:45.

Exercise Sessions: Wednesday, 11:00 - 12:30.

Virtual meeting: The course will take place on Zoom. Please send an email to Daniele Agostini to get the videobroadcast link.

Important announcements

From November 4 the course will take place on Zoom.
The course at the Universität Leipzig consists only of the lectures on Wednesday 15:15-16:45. However everybody is strongly encouraged to attend also the exercise sessions on Wednesday 11:00-12:30.

Course description

The course will be a first introduction to Riemann surfaces and algebraic curves. These are beautiful objects which sit at the intersection of algebra, geometry and analysis. Indeed, on one side these are complex manifolds of dimension one, and on the other they are varieties defined as a zero locus of polynomial equations. Furthermore, they are ubiquitous throughout mathematics, from diophantine equations in number theory to water waves in mathematical physics and Teichmüller theory in dynamical systems.

We will aim to cover the theorems of Riemann-Hurwitz and Riemann-Roch, meromorphic functions and their zeroes and poles, plane curves and elliptic curves, abelian integrals, the theorem of Abel-Jacobi and the construction of Jacobian varieties. Time permitting, we might touch upon further topics such as canonical curves, moduli spaces, the Schottky problem and tropical curves.

Prerequisites: abstract algebra and familiarity with differential or algebraic geometry.

References : Notes for some of the lectures will appear below. We will not follow exactly any particular book, but the main inspirations for the course will be

R. Cavalieri and E. Miles, Riemann Surfaces and Algebraic Curves. Cambridge University Press.
W. Fulton, Algebraic Curves. Available online.
F. Kirwan, Complex Algebraic Curves. Cambridge University Press.
R. Miranda, Algebraic Curves and Riemann Surfaces . American Mathematical Society.

There are many other beautiful references for this topic. Some of them are:

E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves I. Springer.
O. Forster, Riemannsche Flächen. Springer.
P. Griffiths and J. Harris, Principles of algebraic geometry. Wiley.
J. Jost, Compact Riemann Surfaces. Springer.

Course log

A brief description of each lecture's content, together with some notes, will appear here.

28 October 2020, 15-17. Presentation of the course. Abelian integrals. Manifolds. Holomorphic functions. Notes.
4 November 2020, 15-17. Riemann surfaces. Examples: the projective line, affine and projective plane curves. Holomorphic maps between Riemann surfaces. The degree of a map. Notes.
11 November 2020, 15-17. The topology of a Riemann surface, the topological genus, Riemann-Hurwitz formula. Genus of a smooth plane curve. Meromorphic functions, zeroes, poles. Notes.
25 November 2020, 15-17. Meromorphic functions as maps to the projective line. Divisors, linear equivalence, ramifification divisors, branch divisors, intersection divisors. Notes.
2 December 2020, 15-17. Jacobi theta functions, quasiperiodicity, zeroes. Meromorphic functions on complex tori. Notes.
9 December 2020, 15-17. Linear systems: global sections associated to a divisor, examples in genus zero, divisors of negative degree, degree zero and degree one. Notes.
13 January 2021, 15-17. Linear systems and maps to projective space. Base-point-free linear systems. Notes.
20 January 2021, 15-17. Differentials, canonical divisors, pullbacks and Riemann-Hurwitz. Riemann-Roch and some consequences. Notes.
27 January 2021, 15-17. Projective geometry of algebraic curves, degree. Classification of curves of small genera. Geometric Riemann-Roch. Cayley-Bacharach theorem. Notes.
03 February 2021, 15-17.Jacobians, Abel-Jacobi map and Abel's theorem. Notes.

Exercise sheets

The exercises will appear below and will be discussed together in the session of Wednesday morning at the MPI-MiS.

Sheet 1: to be discussed on Wednesday 4 November.
Sheet 2: to be discussed on Wednesday 11 November.
Sheet 3: to be discussed on Wednesday 25 November.
Bonus Sheet 1: to be discussed (maybe) on Wednesday 25 November.
Sheet 4: to be discussed on Wednesday 2 December.
Sheet 5: to be discussed on Wednesday 9 December.
Sheet 6: to be discussed on Wednesday 13 January. Solutions.
Sheet 7: to be discussed on Wednesday 20 January. Solutions.
Sheet 8: to be discussed on Wednesday 27 December. Solutions.
Sheet 9: to be discussed on Wednesday 3 February.

Last modified: Mon Apr 12 21:14:14 CEST 2021