Algebraic Geometry - NMAG401

• Title: Algebraická geometrie
• Guaranteed by: Department of Algebra (32-KA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2020
• Semester: winter
• E-Credits: 5
• Hours per week, examination: winter s.:2/2, C+Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: taught
• Language: English, Czech
• Teaching methods: full-time
• Teaching methods: full-time
• Guarantor: Liran Shaul, Ph.D.
• Teacher(s): Liran Shaul, Ph.D.
• Class: M Mgr. MMIB; M Mgr. MMIB > Povinně volitelné; M Mgr. MSTR; M Mgr. MSTR > Povinné
• Classification: Mathematics > Algebra

Annotation

English

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).

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (12.09.2013)

Czech

Přednáška slouží jako úvod do základních aspektů algebraické geometrie. Probíraná látka zahrnuje Zariského spektrum komutativního okruhu a jeho vztah k algebraickým varietám, geometrický význam lokalizace okruhů, zobrazení mezi varietami, některé vlastnosti abstraktních a projektivních variet a lokální vlastnosti variet (především pojem Krullovy dimenze a jeho vlastnosti).

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (12.09.2013)

Course completion requirements

In order to complete the course, the students must submit all homework, and to pass the final exam.

Last update: Shaul Liran, Ph.D. (25.09.2020)

Literature

English

[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.

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

Requirements to the exam

The course is completed with a written 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.

Last update: Shaul Liran, Ph.D. (25.09.2020)

Syllabus

English

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.

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

Czech

1. spektrum komutativního okruhu a jeho vztah k algebraickým varietám,

2. geometrický význam lokalizace okruhů,

3. zobrazení mezi varietami,

4. abstraktní variety,

5. projektivních variety a jejich vlastnosti,

6. Krullova dimenze.

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

Entry requirements

English

Basics of commutative algebra on level of the course Introduction to commutative algebra.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (28.06.2022)

Czech

Základy komutativní algebry na úrovni kurzu Úvod do komutativní algebry.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (28.06.2022)