2021/2022
Научно-исследовательский семинар "Алгебраическая геометрия 1"
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Финкельберг Михаил Владленович
Язык:
английский
Кредиты:
3
Контактные часы:
30
Course Syllabus
Abstract
DESCRIPTION:During the 50 years after Grothendieck, the algebraic geometry became the common language of Physics and Mathematics, just like the functional analysis used to be during the previous 50 years. The course will cover most of «Algebraic Geometry» by Hartshorne. Additional topics may include: the Hilbert scheme and its application to the existence theorems, a proof of the Weil conjectures for curves over finitefields, Intersection theory.PREREQUISITES:1. Our basic courses of the 1st and 2nd years. Thus, this course is for the bachelor studentsat least in their 3rd year. However, I know some 2nd year students and even prospective 1st year studentsthat can take this course. 2. Introduction to the category theory and homological algebra by A. Gorodentsev3. Some commutative algebra will be very helpful (e.g. the book by Atiyah – Macdonald is more than enough).However, we will recall some commutative algebra along the way, and if you are willing to believe some results and catch up, there is no need to take a Commutative algebra course beforehand
Expected Learning Outcomes
- Get prepared for research in Algebraic Geometry and related areas such as Geometric Representation Theory and Number Theory
Course Contents
- Review of commutative algebra
- Schemes, fiber products
- Proper morphisms, valuation criteria
- Coherent sheaves
- Divisors, the Picard group
- The case of curves
- Differentials, smooth morphisms
- Cohomology of coherent sheaves
- Serre duality
- Riemann - Roch theorem
- Applications to counting points over finite field
- Introduction to the deformation theory with applications to rational curves on Fano varieties
Bibliography
Recommended Core Bibliography
- Hartshorne, R., & American Mathematical Society. (1975). Algebraic Geometry, Arcata 1974 : [proceedings]. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=772699
Recommended Additional Bibliography
- Dolgachev, I. (2012). Classical Algebraic Geometry : A Modern View. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=473170