1. komponenta

Lecture type	Total
Lectures	45

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

COURSE AIMS AND OBJECTIVES:
To familiarize students with the theory of plane algebraic curves combining analytic and algebraic approach. In this way, students will be introduced in the methodology and basic classification problems in algebraic geometry.

COURSE DESCRIPTION AND SYLLABUS:
1. Projective and affine plane curves. Irreducible components. Ring of regular functions and field of rational functions of an irreducible curve.
2. Morphism, isomorphism and biratinal equivalence of plane curves.
3. Singularity of plane curves. Definition of non-singular curves. Curves over R and the Implicit function theorem. The classification of singular points.
4. Dimension of a ring.
5. Notion of a local ring. The characterization of non-singularity of a curve in terms of local ring.
6. Valuation rings and their extensions.
7. Types of classification of curves. Difference between the affine and the projective case. The classification of curves into classes of birational equivalence. The relation to the field theory
8. Resolvent of a polynomial. Definition of the intersection multiplicty of two curves. Bezout theorem. Pascal's theorem.
9. Rational and irrational curves. Projective conis is isomorphic to a line. Elliptic curves.
10. Weierstrass normal form of an elliptic curves. Geometric definition of the structure of a group on an elliptic curve
11. Lattices in C. Periodic and double periodic functions on elliptic curves.
12. The field of meromorphic functions on a lattice and the field of rational functions on elliptic curves.
13. Elliptic curves and finite fields.

TEACHING AND ASSESSMENT METHODS:
Attending of at least 70% of lectures and examples classes, sloving at least 70% of homework assignments, and passing grade on two mid-term exams.

1. D. Cox, J. Little, D. O'Shea: Ideals, Varieties and Alghorithms. An Introduction to Computational Algebraic Geometry and Commutaive Algebra, 2nd edition, Undergraduate Texts in Mathematics
2. N. Koblitz: Introduction to Elliptic Curves and Modular Forms, 2nd edition, Graduate Texts in Mathematics
3. I. R. Shafarevich: Basic Algebraic Geometry 1: Varieties in Projective Space, 2nd edition

Enrollment :
Attended : Algebra 1

Examination :
Passed : Algebra 1

Mandatory course - Regular study - Theoretical Mathematics