Algebraic Curves WS 2022/23

Algebraic Curves - WS 2022/23

Lectures: Tuesday, 12:30-14. Raum S09 (C605).
Thursday, 12:30-14. Raum S09 (C605).

Exercise sessions: Thursday, 10-12. Raum S07.

Online meeting: Sometimes the lectures and the exercise sessions will be held on Zoom here.

Exam: Für die Zulassung der Prüfung sind 60% von den gesamten Punkten den Aufgaben erforderlich.
To access the final exam one needs to obtain 60% of the points in the exercise sheets.

Aktuelles - Announcements

The time and place for the exercise sessions has changed in 2023, look up the the new information above.

Wer an den Lehrveranstaltung teilnehmen möchte, sollte seine Daten bereits im Anmeldesystem URM eintragen.
Der Kurs ist auf Englisch abgehalten.
Die Übungen beginnen in der zweiten Vorlesungswoche.
Prof. Schaetzle haltet in diesem Semester ein Kurs über analytische aspekte von riemannschen Flächen.

Interested participants should register at URM.
The course will be held in English.
The exercises sessions will start from the second week of lectures.
Prof. Schaetzle is offering in this semester a course over the analytical aspects of Riemann surfaces.

Inhalt - Content

Der Kurs ist eine Einführung in die Theorie der algebraischen Kurven und riemannschen Flächen. Diese sind wunderbare Objekte, die an der Kreuzung von Algebra, Geometrie und Analysis liegen. Auf der einen Seite sind sie komplexe Mannifältigkeiten von Dimension eins, auf der anderen Seite sind sie algebraische Varietäten, die durch Polynomgleichungen beschrieben werden können. Außerdem sind sie in der Mathematik allgegenwärtig, von diophantischen Gleichungen in der Zahlentheorie, bis Wasserwellen in mathematischen Physik und Translationsflächen in dynamischen Systemen.

Detallierter Inhalt des Kurses: Sätze von Riemann-Hurwitz und Riemann-Roch, meromorphe Funktionen und deren Nullstellen und Pole, Divisoren und Geradenbündeln, ebene Kurven, elliptische Kurven, abelsche Integrale und der Satz von Abel-Jacobi. Wenn die Zeit reicht, können wir weitere Themen betrachten, z.B. Jacobi-Varietäten, kanonische Kurven, Modulrämen, das Schottky-Problem und tropische Kurven.

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.

References : Notes from the lectures will be uploaded below. We will not follow exactly any reference, but we will get inspiration from:

D. Agostini und R. Sinn, Notes for a similar course in Leipzig.
R. Cavalieri und E. Miles, Riemann Surfaces and Algebraic Curves. Cambridge University Press.
G. Fischer, Ebene Algebraische Kurven. Springer.
W. Fulton, Algebraic Curves. Available online.
A. Gathmann, Plane algebraic curves. Available online.
R. Miranda, Algebraic Curves and Riemann Surfaces . American Mathematical Society.

Other beautiful references on this topic 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.
F. Kirwan, Complex Algebraic Curves. Cambridge University Press.
H. Markwig, Notes from a similar course in Tübingen.

Kursprotokoll - Course log

A short description of each lecture and the relative notes are available here.

18 October 2022: Introduction, abelian integrals. Notes.
20 October 2022: Affine algebraic sets, Zariski topology, Nullstellensatz, irreducible algebraic sets. Notes.
25 October 2022: Plane affine curves, projective algebraic sets, projective plane curves. Notes.
27 October 2022: Intersection multiplicity. Notes.
3 November 2022: Algorithm to compute the intersection multiplicity, Bezout's Theorem. Notes (slightly different to what I have done in the lecture).
8 November 2022: Tangent spaces, smooth and singular curves, transversality. Notes.
10 November 2022: Multiplicity of a singularity, tangent cone, relation with intersection multiplicity. Notes.
15 November 2022: Complex manifolds, Riemann surfaces and examples, hyperelliptic curves. Notes.
17 November 2022: Holomorphic maps of Riemann surfaces and their properties. Notes.
22 November 2022: Maps of compact Riemann surfaces, degree, linear projection of a plane curve Notes.
24 November 2022: Topology of Riemann surfaces, genus, Riemann-Hurwitz, genus of plane curves. Notes.
29 November 2022: Meromorphic functions, meromorphic functions and maps to the projective line. Notes.
1 December 2022: Order of a meromorphic function, meromorphic functions on the projective line and plane curves. Notes.
6 December 2022: Divisors, degree, linear equivalence, Picard group. Case of projective line and plane curves. Notes.
8 December 2022: Complex tori, maps between them, upper half-space, theta function. Notes.
13 December 2022: Zeroes of theta function, divisors on complex tori, Abel's theorem. Notes.
15 December 2022: Linear systems of divisors - I. Notes.
20 December 2022: Linear systems of divisors - II. Notes.
22 December 2022: Linear systems and maps to projective spaces - I. Notes.
10 January 2023: Linear systems and maps to projective spaces - II. Notes.
12 January 2023: Holomorphic and meromorphic differential forms. Notes.
17 January 2023: Holomorphic forms on the projective line and complex tori, canonical divisors. Notes.
19 January 2023: Riemann-Roch and applications. Degree of the canonical divisor, equality of topological genus with geometric genus. Notes.

Aufgaben - Exercises

The exercises can be worked together in group of at most three people.

Sheet 0: this will be discussed in class on October 26.
Sheet 1: please upload your solutions to URM or send them by email by November 2. Solutions.
Sheet 2: please upload your solutions to URM or send them by email by November 10.
Sheet 3: please upload your solutions to URM or send them by email by November 16.
Sheet 4: please upload your solutions to URM or send them by email by November 23.
Sheet 5: please upload your solutions to URM or send them by email by November 30.
Sheet 6: please upload your solutions to URM or send them by email by December 7.
Sheet 7: please upload your solutions to URM or send them by email by December 14.
Sheet 8: please upload your solutions to URM or send them by email by December 21.
Sheet 9: please upload your solutions to URM or send them by email by January 19.
Sheet 10: please upload your solutions to URM or send them by email by January 26.