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Introduction to algebraic geometry

Code: 92921
ECTS: 5.0
Lecturers in charge: prof. dr. sc. Goran Muić - Lectures
English level:

1,0,0

All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.
Load:

1. komponenta

Lecture typeTotal
Lectures 45
* Load is given in academic hour (1 academic hour = 45 minutes)
Description:
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).
Literature:
  1. I. R. Shafarevich: Basic algebraic geometry 1: Varieties in Projective Space, 2nd edition
  2. D. Eisenbud: Commutative algebra with a view towards algebraic geometry
Prerequisit for:
Enrollment :
Passed : Algebra 2
Passed : Algebraic curves
3. semester
Izborni predmet 3, 4 - Regular study - Theoretical Mathematics

4. semester
Izborni predmet 3, 4 - Regular study - Theoretical Mathematics
Consultations schedule: