2019/2020
Введение в римановы поверхности
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Львовский Сергей Михайлович
Язык:
английский
Кредиты:
6
Контактные часы:
72
Course Syllabus
Abstract
The aim of the course is to demonstrate how some of the key ideas of algebraic geometry work, using the approach that does not require a hard technical introduction. With this aim in mind, the course is mainly concentrated on compact Riemann surfaces. Assuming Riemann's existing theorem without proof (in the form that any compact Riemann surfacecan be represented as a ramified covering of the extended complex plane corrseponding to an algebraic equation), we prove Riemann——Roch theorem using adèles (after André Weil) and give some basic examples and results from the theory of algebraic curves.
Expected Learning Outcomes
- Having basic definitions introduced
- The student will learn to work, using Riemann-Roch theorem, with divisors and linear systems on Riemann surfaces
- The student will be able to compute genera of compact Riemann surfaces defined as ramified coverings of the projective line or, alternatively, as plane curves, maybe with simple singularities.
- The student will learn to find (in the simplest situations) points of finite order on Jacobians.
Course Contents
- Definitions. Compact Riemann surfaces associated to algebraic equations.
- Differentials, residues, divisors. Genus of a compact Riemann surface.
- Compact Riemann surfaces associated to smooth and nodal plane curves. Poincaré residue.
- Riemann’s existence theorem (without proof). Riemann – Roch theorem.
- Linear systems and line bundles
- Theta divisor. Torelli theorem.
- Jacobian variety. Abel and Jacobi theorems.
- Canonical curves. Castelnuovo–De Franchis theorem.
Bibliography
Recommended Core Bibliography
- Gunning, R. C. (1972). Lectures on Riemann Surfaces : Jacobi Varieties. Princeton, New Jersey: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=948671
Recommended Additional Bibliography
- Harder, G. (2011). Lectures on Algebraic Geometry I : Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces. [Place of publication not identified]: Springer Spektrum. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1245198
- Martin Schlichenmaier. (1998). Sugawara construction for HIGHER GENUS RIEMANN SURFACES. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.A3D332B0
- Schlichenmaier, M. (1998). Sugawara Construction for Higher Genus Riemann Surfaces. https://doi.org/10.1016/S0034-4877(99)80041-X
- Verjovsky, A., Gomez-Mont, X., Cornalba, M., & International Centre for Theoretical Physics. (1989). Lectures On Riemann Surfaces - Proceedings Of The College On Riemann Surfaces. Singapore: World Scientific. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=689728