1. komponenta

Lecture type	Total
Lectures	45

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES:
The goal of the course is to introduce students with the basic notions of algebraic geometry of varieties in projective space over an algebraically closed field.

COURSE DESCRIPTION AND SYLLABUS:
1. Polynomial rings. Hilbert basis theorem. Proof of Hilbert theorem about zeros.
2. Closed subsets of affine space. Egular functions on closed sets. Regular maps.
3. Irreducible algebraic sets. Rational functions. Rational maps. Closed sets of projective space.
4. Quasiprojective varieties. Regular and rational functions and maps. Products Closedness of image of projective variety.
5. Finite maps. Normalization. Dimension.
6. Local ring of a point. Tangent spaces. Singular points. Tangent conus.
7. Local parameters at a point. Power series expansions. Varieties over real and complex fields. Nonsingular points and subvarieties.
8. Structure of birational maps. Exceptional submanifolds. Isomorphism and birational equivalence. Normal varieties. Normalization. Embedding into projective space.
9. Singularities of maps. Irreducibility, nonsingularity and ramification.
10. Divisors (overview).
11. Algebraic groups (overview). Differential forms (overview).

Enrollment :
Passed : Algebra 2
Passed : Algebraic curves

Izborni predmet 3, 4 - Regular study - Theoretical Mathematics

Izborni predmet 3, 4 - Regular study - Theoretical Mathematics