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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 (individual examination) |
Language, Grading |
English,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
Students will
- master the use of algebraic methods to describe geometric objects (algebraic varieties),
- understand the geometry-algebra-geometry translation process and can apply it to posed problems,
- have experienced how geometric problems can be mastered by using abstract algebraic techniques,
- have been introduced to current developments and results by learning modern methods of algebraic geometry,
- have deepened mathematical ways of working (developing mathematical intuition and its formal justification, abstraction, proof),
- have improved their oral communication skills in exercises by practicing free speech in front of an audience and in 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
Module imported from M.Sc. Mathematics.
It can be attended at FB12 in study program(s)
- B.Sc. Mathematics
- M.Sc. Computer Science
- M.Sc. Mathematics
- LAaG Mathematics
When studying LAaG Mathematics, this module can be attended in the study area Advanced Modules.
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 2023/24. 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.