2023/2024
Toric Varieties
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Karine Kuyumzhiyan
Language:
English
ECTS credits:
6
Contact hours:
72
Course Syllabus
Abstract
This is an introduction to the theory of toric varieties, which are algebraic manifolds obtained from convex polytopes by means of some wonderful explicit geometric construction. For example, the standard n-simplex gives in this way the projective space of dimension n. The advantage of the pass from polytopes to toric varieties is that the crucial combinatorial and geometric properties of polytopes predetermine the key properties of the corresponding varieties and vice versa. Almost all essential algebraic, geometric, and topological characteristics of a toric variety are explicitly computable in terms of its polytope. This makes the toric varieties very suitable for testing algebro-geometric and topological conjectures, seeking examples and counterexamples, etc.
Learning Objectives
- The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk.
Expected Learning Outcomes
- Successful participants imporve their presentation skills and prepare for participation in research projects in the subject area.
Course Contents
- Affine and projective toric varieties, orbit - cone correspondence, automorphisms of affine toric varieties and locally nilpotent derivations, resolution of singularities in dimension 2 and continued fractions.
- Divisors on toric varieties.
- Cohomology groups of nonsingular toric varieties
Assessment Elements
- листкиДля сдачи листка необходимо сдать достаточное число задач из листка (6 из первого, 8 из второго, 6 из третьего, 6 из четвёртого)
- Письменный экзамен по курсу
Interim Assessment
- 2023/2024 4th module0.4 * Письменный экзамен по курсу + 0.3 * листки + 0.3 * листки
Bibliography
Recommended Core Bibliography
- Cox, D. A., Little, J. B., & Schenck, H. K. (2011). Toric Varieties. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971087
Recommended Additional Bibliography
- Fulton, W. (1993). Introduction to Toric Varieties. (AM-131), Volume 131. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1432979