Introduction to algebraic geometry

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M3
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

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.

Readings

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.

Objectives and competences

Student masters basic concepts and tools of algebraic geometry.

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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]