Algebraic Geometry 1000-135GEA

Algebraic properties of rings and the field of rational functions. Algebraic subsets of affine and projective spaces. Regular mappings. Rational and bi-rational maps. Segre embeddings. Local rings of functions. Tangent and co-tangent spaces. Algebraic theory of local rings. Smooth point of algebraic sets. Integral extensions of rings. The dimension of an algebraic set. Normal points and sets.

Course coordinators

Term 2024L:

Katarzyna Ragus

Term 2023L:

Katarzyna Szeryńska

Type of course

elective courses

Requirements

Commutative algebra
Algebraic methods in geometry and topology

Prerequisites

Algebra I

Prerequisites (description)

(in Polish) Wymagane: Algebra przemienna, Metody algebraiczne geometrii i topologii, zalecane: Geometria różniczkowa, Algebra 2, Topologia 2, Funkcje analityczne. Inne przedmioty fakultatywne związane z wykładem: Topologia algebraiczna, Analiza zespolona, Geometria różniczkowa 2, Grupy i algebry Liego.

Learning outcomes

A student should be able to:
- formulate notions from the syllabus and explain them in examples
- formulate theorems from the syllabus and give some chosen proofs

Assessment criteria

The final mark will be given on basis of the results of exercises and the final exam. Detailed rules for completing the course are provided in the information on classes in the relevant academic year.

Bibliography

D. Eisenbud, J. Harris, The geometry of schemes, Graduate Texts in Mathematics 197, Springer-Verlag, 2000.
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977.
K. Hulek, Elementary algebraic geometry, Student Mathematical Library 20, American Mathematical Society, 2003.
D. Mumford, Algebraic geometry I: Complex projective varieties, Classics in Mathematics, Springer-Verlag, 1995.
M. Reid, Undergraduate algebraic geometry, London Mathematical Society Student Texts 12, Cambridge University Press, 1988.
I. R. Shafarevich, Basic algebraic geometry 1, 2, 2nd ed., Springer-Verlag, 1994.