Fundamental part:

Affine varities. Hilbert Nullstellensatz.

Ring of polynomial functios. Rational functions.

Local properties of plane curves.

Projective varieties. Regular and rational functions.

Projective plane curves. Bezout's Theorem.

Max Noether Theorem.

Affine and rational maps. Resolutions of singularities.

Hilbert polynomial and Hilbert function.

Divisors on varieties.

Curves. Plane cubic curves. Linear systems on curves. Projective embeddings of curves.

Elective topics:

Riemann-Roch Theorem.

# Introduction to algebraic geometry

B. Hassett. Introduction to algebaric geometry. Cambridge Univ. Press, 2007.

M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Surfaces and Projective Varieties. A Classical View of Algebraic Geometry, EMS Text-books in Mathematics, 2009.

I. Shafarevich: Basic Algebraic Geometry I : Varieties in Projective Space, 2nd edition, Springer, Berlin, 1994.

K. Hulek: Elementary Algebraic Geometry, AMS, Providence, 2003.

W. Fulton: Algebraic Curves, Addison-Wesley, Redwood City, 1989.

J. Harris: Algebraic Geometry : A First Course, Springer, New York, 1995.

Student masters basic concepts and tools of algebraic geometry.

Knowledge and understanding: Understanding of basic concepts and theorems of algebraic geometry, and their role in some other areas.

Application: In the areas of mathematics that deal with geometric objects, in theoretical physics, and elsewhere.

Reflection: Understanding the theory on the basis of examples and applications.

Transferable skills: Formulation and solution of problems in an appropriate setup, solution and analysis of the results in examples, recognizing algebraic structue in geometric objects.

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Tomaž Košir:

GRUNENFELDER, Luzius, KOŠIR, Tomaž. Geometric aspect of multiparameter spectral theory. Transactions of the American Mathematical Society, ISSN 0002-9947, 1998, let. 350, št. 6, str. 2525-2546. [COBISS-SI-ID 8449113]

KOŠIR, Tomaž, SETHURAMAN, B. A. Determinantal varieties over truncated polynomial rings. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2005, vol. 195, no. 1, str. 75-95. [COBISS-SI-ID 13266265]

BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]