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This entry is from Winter semester 2021/22 and might be obsolete. You can find a current equivalent here.
Algebraic Geometry: Projective Varieties
(dt. Algebraische Geometrie: Projektive Varietäten)
Level, degree of commitment | Specialization module, compulsory elective module |
Forms of teaching and learning, workload |
Lecture (4 SWS), recitation class (2 SWS), 270 hours (90 h attendance, 180 h private study) |
Credit points, formal requirements |
9 CP Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination |
Language, Grading |
German,The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics. |
Duration, frequency |
One semester, Regularly alternating with other specialization modules in Geometrie |
Person in charge of the module's outline | Prof. Dr. Thomas Bauer |
Contents
Algebraic varieties: Affine and projective varieties, Hilbert's Nullstellensatz, singularities, tangent spaces and dimensions
Morphisms of varieties: regular and rational functions and maps, blow-up and resolution of singularities
Geometric applications: Linear systems of plane curves, cubic surfaces in three-space
Advanced algebro-geometric techniques: Divisors, differential forms, Riemann-Roch theorem on curves
Qualification Goals
The students shall
- learn about the application of algebraic methods for the description of geometric objects (algebraic varieties),
- understand the geometry-algebra-geometry translation process and be able to apply it to presented problems,
- learn how geometric problems can be solved by using abstract algebraic techniques,
- to develop their capacity for abstraction,
- be introduced to current developments and results by learning modern methods of algebraic geometry,
- practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
- improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.
Prerequisites
None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, either Elementary Algebraic Geometry or Algebra.
Applicability
The module can be attended at FB12 in study program(s)
- B.Sc. Mathematics
- M.Sc. Computer Science
- M.Sc. Mathematics
- LAaG Mathematics
When studying M.Sc. Mathematics, this module can be attended in the study area Specialization Modules in Mathematics.
The module can also be used in other study programs (export module).
The module is assigned to Pure Mathematics. Further information on eligibility can be found in the description of the study area.
Recommended Reading
- Hulek, K.: Elementare Algebraische Geometrie, Vieweg
- Shafarevich, I.R.: Basic Algebraic Geometry, Springer
- Hartshorne, R.: Algebraic Geometry, Springer
Please note:
This page describes a module according to the latest valid module guide in Winter semester 2021/22. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:
- Winter 2016/17
- Summer 2018
- Winter 2018/19
- Winter 2019/20
- Winter 2020/21
- Summer 2021
- Winter 2021/22
- Winter 2022/23
- Winter 2023/24
The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.
The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.