2020/2021

Research Seminar "Geometric Introduction to Algebraic Geometry"

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: Alexander S. Tikhomirov

Language: English

ECTS credits: 6

Contact hours: 60

Course Syllabus

Abstract

Algebraic geometry studies geometric loci looking locally as a solution set for a system of polynomial equations on an affine space. The main feature of this subject is that it provides an algebraic explanation to various geometric properties of the figures and at the same time gives geometric intuition to purely algebraic constructions. It plays an important role in many are as of mathematics and the or etical physics, and provides the most visual and elegant tools to express all aspects of the interaction between different branches of mathematical knowledge. The course gives the flavor of the subject by presenting example sand applications of the ideas of algebraic geometry, as well as a first discussion of its technical tools. Рrerequisites: first year of undergraduate study (algebra, calculus, geometry, topology).

Learning Objectives

• Students will be competent in basic constructions of algebraic geometry that will allow them to start studying more advanced courses in algebraic geometry and apply their knowledge to study the courses like algebraic groups, algebraic curves and surfaces.

Expected Learning Outcomes

• Students will learn basic constructions, theorems of algebraic geometry. Also they will gain sufficient package of examples of algebraic varieties and the method of their study.

Course Contents

• Projective spaces. Geometry of projective quadrics. Spaces of quadrics.
• Lines, conics, and PGL(2). Rational curves and Veronese curves. Plane cubic curves.
• Grassmannians, Veronese's, and Segre's varieties. Examples of projective maps coming from tensor algebra.
• Elements of commutative algebra: Integer elements in ring extensions, finitely generated algebras over a field, transcendence generators, Hilbert's theorems.
• Affine Algebraic Geometry - Commutative Algebra dictionary. Maximal spectrum, pullback morphisms, Zariski topology, geometry of ring homomorphisms.
• Agebraic manifolds, separateness. Irreducible decomposition. Projective manifolds, properness. Rational functions and maps.
• Dimension. Dimensions of subvarieties and fibers of regular maps. Dimensions of projective varieties.
• Vector bundles and their sheaves of sections. Vector bundles on the projective line. Linear systems, invertible sheaves, and divisors. The Picard group.

Assessment Elements

• Solutions of problems from home task sheets
  Written solutions
• Final exam
  Written exam

Interim Assessment

• Interim assessment (2 module)
  0.5 * Final exam + 0.5 * Solutions of problems from home task sheets

Bibliography

Recommended Core Bibliography

• Alexey L. Gorodentsev. Algebra II: Textbook for Students of Mathematics. Springer, 2017. ISBN: 9783319508535,3319508539
• Igor R. Shafarevich Basic Algebraic Geometry 1 Varieties in Projective Space, Springer, 2013.

Recommended Additional Bibliography

• Igor R. Shafarevich. Basic Algebraic Geometry 2 Schemes and Complex Manifolds, Springer, 2013