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Regular version of the site
2016/2017

Algebraic Geometry: a Start-up Course

Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 5

Course Syllabus

Abstract

Algebraic geometry studies geometric loci looking locally as a solution set for a system of polynomial equations on an affine space. It gives an explicit algebraic explanation for various geometric properties of figures, and in the same time, brings up a geometric intuition underlying abstract 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 geometric 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

• 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 improve their presentation skills and prepare for participation in research projects in the subject area

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
• 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.
• Affine Algebraic Geometry from the viewpoint of Commutative Algebra. Maximal spectrum, pullback morphisms, Zariski topology, geometry of ring homomorphisms

Assessment Elements

• class written exam
• oral exam

Interim Assessment

• Interim assessment (4 module)
0.5 * class written exam + 0.5 * oral exam

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