Riemann Surfaces and Algebraic Curves

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.

Description of the first lecture's

• 28 October 2020: Presentation of the course. Abelian integrals. Manifolds. Holomorphic functions.
• 4 November 2020:Riemann surfaces. Examples: the projective line, affine and projective plane curves. Holomorphic maps between Riemann surfaces. The degree of a map
• 11 November 2020: The topology of a Riemann surface, the topological genus, Riemann-Hurwitz formula. Genus of a smooth plane curve. Meromorphic functions, zeroes, poles
• 25 November 2020: Meromorphic functions as maps to the projective line. Divisors, linear equivalence, ramifification divisors, branch divisors, intersection divisors
• 2 December 2020: Jacobi theta functions, quasiperiodicity, zeroes. Meromorphic functions on complex tori

References
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.

Date and time info
Wednesday 15:15 - 16:45 (lectures), Wednesday 11:00 - 12:30 (exercise sessions)

Keywords
Riemann surfaces, algebraic curves

Prerequisites
abstract algebra and familiarity with differential or algebraic geometry