Algebraic curves

2018/2019
Programme:
Mathematics, First Cycle
Year:
2 year
Semester:
second
Kind:
optional
ECTS:
5
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed course Algebra 1.

Content (Syllabus outline)

Affine algebraic curves. Irreducibility and connectedness.
Projectivization. Multiplicity of intersection between a line and a curve. Bezout lemma.
Tangents. Singularity.
Polars and Hess curves.
Dual curve. Plücker formula.
Rational curves , Conics.
Cubic curves.
Degree-genus formula for nonsingular curves.

Readings

G. Fisher: Plane Algebraic Curves, AMS, Providence, 2001.
C. G. Gibson: Elementary Geometry of Algebraic Curves, Cambridge Univ. Press, Cambridge, 1998.
M. Reid: Undergraduate Algebraic Geometry, Cambridge Univ. Press, Cambridge, 1988.
K. Hulek: Elementary Algebraic Geometry, AMS, Providence, 2003.
F. Kirwan: Complex Algebraic Curves, Cambridge Univ. Press, Cambridge, 1992.
C. H. Clemens: A Scrapbook of Complex Curve Theory, 2nd edition, AMS, Providence, 2003.

Objectives and competences

This is one of the three basic courses in which students learn to think geometrically. The basic goal is to understand the basic definitions and properties of algebraic curves.

Intended learning outcomes

Knowledge and understanding: Understanding the relation between the algebraic equations and the geometric objects. Ability of treating some geometric problems by means of tools, coming from the theory of polynomials. Knowledge and understanding of the fundamental concepts of the theory of algebraic curves and algebraic geometry.
Application: Algebraic description of objects, appearing in problems from other areas of mathematics and its applications. Application of algebro-geometric methods in the treatment of such problems.
Reflection: Ability of percieving mathematical object from different points of view. Development of the geometric approach to solving problems in applicative mathematics.
Transferable skills: Formulation of problems in suitable contexts, evaluation of developed tools in concrete examples. This course demands a firm knowledge of certain chapters from mathematical analysis and algebra. Therefore students learn how to use previously acquired knowledge in new situations. Students learn the use of study literature in foreign languages.

Learning and teaching methods

Lectures, exercises, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

GRUNENFELDER, Luzius, GURALNICK, Robert M., KOŠIR, Tomaž, RADJAVI, Heydar. Permutability of characters on algebras. Pacific journal of mathematics, ISSN 0030-8730, 1997, let. 178, št. 1, str. 63-70. [COBISS-SI-ID 7437145]
GRUNENFELDER, Luzius, KOŠIR, Tomaž. Coalgebras and spectral theory in one and several parameters. V: GOHBERG, I. (ur.), LANCASTER, P. (ur.), SHIVAKUMAR, P. N. (ur.). Recent developments in operator theory and its applications : International Conference in Winnipeg, October 2-6, 1994, (Operator theory, ISSN 0255-0156, vol. 87). Basel, Boston, Berlin: Birkhäuser, cop. 1996, str. 177-192. [COBISS-SI-ID 7436889]
GRUNENFELDER, Luzius, KOŠIR, Tomaž. Koszul cohomology for finite families of comudule maps end applications. Communications in algebra, ISSN 0092-7872, 1997, let. 25, št. 2, str. 459-479. [COBISS-SI-ID 7127641]
SAKSIDA, Pavle. Nahm's equations and generalizations Neumann system. Proceedings of the London Mathematical Society, ISSN 0024-6115, 1999, let. 78, št. 3, str. 701-720. [COBISS-SI-ID 8853849]
SAKSIDA, Pavle. Neumann system, spherical pendulum and magnetic fields. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2002, vol. 35, no. 25, str. 5237-5253. [COBISS-SI-ID 11920217]
SAKSIDA, Pavle. Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity, ISSN 0951-7715, 2001, vol. 14, no. 5, str. 977-994. [COBISS-SI-ID 10942809]