| |
Algebraic Geometry
| Credits |
8 credit points |
| Instructors |
Taelman, L. (Universiteit Leiden), Jong, R. de (Universiteit Leiden) |
| E-mail |
Lenny@math.leidenuniv.nl, rdejong@math.leidenuniv.nl |
| Aim |
To become acquainted with basic techniques of algebraic geometry, through the study of the proof of the Riemann Hypothesis for curves over finite fields. |
| Description |
Riemann's zeta function has a natural generalisation to zeta functions associated to finitely generated (commutative) rings, and more generally, to schemes of finite type. For nonsingular projective curves over finite fields the Riemann hypothesis has been proved by Hasse (elliptic curves) and Weil (arbitrary genus, 1940's). The case of higher dimensional varieties over finite fields (see http://en.wikipedia.org/wiki/Weil_conjectures) was proved by Deligne (1974), building on the work of Grothendieck.
In this course we will treat the case of curves over finite fields, using intersection theory on surfaces. The course will start with some explicit examples of zeta functions, including Riemann's and those of curves over finite fields. Then slowly we will develop those techniques necessary to treat Weil's proof, from Hartshorne's book `Algebraic Geometry' together with a syllabus based on a previous version of this course. Finally, we will present Weil's proof.
Our goal is to provide a good overview of Weil's proof. Obviously, it is not desirable nor possible to treat all of Hartshorne's book. |
| Organization |
Two 45 minute lectures and one 45 minute problem session, weekly. |
| Examination |
Each week, students hand in solutions to exercises that are given on the website of this course. Late homework is not accepted. At the end of the course, each student is required to take an oral exam. The deadline for the oral exams is June 30th. On the exam, students will be questioned about the homework exercises. In order to be admitted to the exam, a student has to pass at least 7 of the homework sets. Questions may be asked about all the homework exercises, not just those for which the student has received a pass. |
| Literature |
R. Hartshorne. Algebraic Geometry. Springer GTM 52. (recommended)
Bas Edixhoven and Lenny Taelman. Syllabus "Mastermath Algebraic Geometry", to be found at http://www.math.leidenuniv.nl/~lenny/AG-mastermath/. |
| Prerequisites |
The standard undergraduate algebra courses on groups, rings and fields (for more details see the three algebra syllabi (in Dutch) available at http://websites.math.leidenuniv.nl/algebra/), and some basic topology. No prior knowledge of algebraic geometry is necessary. |
|
