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German original

Algebraic Geometry: Advanced Methods
(dt. Algebraische Geometrie: Weiterführende Methoden)

Level, degree of commitment in original study programme Advanced 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: 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 study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency
One semester,
irregular
Person in charge of the module's outline Prof. Dr. Sönke Rollenske

Contents

Basic characteristics of algebraic varieties and morphisms are studied, including Zariski topology, dimension and regularity. The general techniques are illustrated on a representative class of examples, e.g. curves.

This module builds on the techniques learned in the Commutative Algebra course to provide a deeper insight into algebraic geometry.


Qualification Goals

Students can

  • to capture the basic characteristics of affine algebraic and projective varieties,
  • learn about the interaction of abstract methods and results of commutative algebra and geometric intuition.

They deepen

  • the practice of mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • in the exercises, their oral communication skills through discussion and free speech in front of an audience.

Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Algebra, Commutative Algebra (Large Specialization Module). Previous knowledge of differential geometry, number theory or topology is helpful.


Recommended Reading

  • - Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I., Vieweg + Teubner, Wiesbaden, 2010.
  • - Liu, Qing Algebraic geometry and arithmetic curves, Oxford University Press, Oxford, 2002.
  • - Perrin, Daniel Algebraic geometry. An introduction., Universitext. Springer-Verlag London, 2008.



Please note:

This page describes a module according to the latest valid module guide in Sommersemester 2021. 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:

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.