2019/2020
Algebraic Geometry: Language of Schemes
Category 'Best Course for Career Development'
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
Where:
Faculty of Mathematics
When:
1, 2 module
Instructors:
Vadim Vologodsky
Language:
English
ECTS credits:
6
Contact hours:
60
Course Syllabus
Abstract
The course will cover most of "Algebraic Geometry" by Harshorne. Additional topics may include: general Riemann-Roch theorem, The Hilbert scheme and its application to the existence theorems, a proof of the Weil conjectures for curves over finite fields, rational curves on Fano varietis ("bend and break trick")
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