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# Algebraic Geometry: A First Geometric Look

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Дисциплина общефакультетского пула
Когда читается:
3, 4 модуль

### 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, that it provides an algebraic explanation to various geometric properties of the figures and it the same time it give geometric intuition to purely algebraic constructions. It plays an important role in many areas of mathematics and theoretical 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 examples and applications of the ideas of algebraic geometry, as well as a first discussion of its technical tools.

#### 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 or apply their knowledge to study the courses like algebraic groups, curves and surfaces.

#### Expected Learning Outcomes

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

#### Course Contents

• Lines, conics. Rational curves and Veronese curves. Plane cubic curves. Additive law on the points of cubic curve
• Grassmannians, Veronese’s, and Segre’s varieties. Examples of projective maps coming from tensor algebra
• Integer elements in ring extensions, finitely generated algebras over a field, transcendence generators, Hilbert’s theorems on basis and on the set of zeros
• Affine Algebraic Geometry from the viewpoint of Commutative Algebra. Maximal spectrum, pullback morphisms, Zariski topology, geometry of ring homomorphisms
• Algebraic manifolds, separateness. Irreducible decomposition. Projective manifolds, properness. Rational functions and maps.
• Dimension. Dimensions of subvarieties and fibers of regular maps. Dimensions of projective varieties.
• Linear spaces on quadrics. Lines on cubic surface. Chow varieties.
• Vector bundles and their sheaves of sections. Vector bundles on the projective line. Linear systems, invertible sheaves, and divisors. The Picard group
• Tangent and normal spaces and cones, smoothness, blowup. The Euler exact sequence on a projective space and Grassmannian

• Final exam

#### Interim Assessment

• Interim assessment (4 module)
0.667 * Final exam + 0.333 * Problem sheets grade

#### Recommended Core 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