SubjectsSubjects(version: 811)
Course, academic year 2017/2018
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Algebraic Geometry - NMAG401
Czech title: Algebraická geometrie
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Jan Šťovíček, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinné
Classification: Mathematics > Algebra
Annotation -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (12.09.2013)

The course serves as an introduction to basic aspects of algebraic geometry. The discussed material includes the Zariski spectrum of a commutative ring and its relation to algebraic varieties, geometric aspects of localization of rings, maps between varieties, certain properties of abstract and projective varieties, and local properties of varieties (especially the Krull dimension and its properties).
Terms of passing the course -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (09.10.2017)

The credit will be granted on the basis of handed in homework. The homework will consist of three sets of problems published on the web page of the lecturer. At least 50 % of points from solutions of the problems handed in within given deadlines are required. If the conditions are not met, it is still possible to have the credit granted, where the exact form of updated conditions (a new deadline for solving the problems and/or extending the homework sets) is decided by the lecturer.

Literature -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (12.09.2013)

[1] I. R. Shafarevich: Basic Algebraic Geometry I, Second edition, Springer-Verlag, Berlin, 1994.

[2] A. Gathmann, Algebraic Geometry, http://www.mathematik.uni-kl.de/~gathmann/alggeom.php

[3] D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, New York, 1997.

[4] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser Boston, Inc., Boston, MA, 1985.

[5] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., 1969.

[6] H. Matsumura, Commutative Ring Theory, Second edition, Cambridge University Press, 1989.

Requirements to the exam -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (09.10.2017)

The course is completed with an oral exam. The requirements for the exam correspond to the syllabus and will be applied to the extent to which the topic was presented in lectures. It will be also demanded that the student is able to work with particular examples and do computations to the extent exercised at problem sessions or in given homework.

Syllabus -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (12.09.2013)

1. the spectrum of a commutative ring and its relation to algebraic varieties,

2. geometric aspects of localization of rings,

3. maps between varieties,

4. abstract varieties,

5. projective varieties and their properties,

6. Krull dimension.

 
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