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German original

Algebraic Geometry: Projective Varieties
(dt. Algebraische Geometrie: Projektive Varietäten)

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,
Regelmäßig im Wechsel mit anderen advanced moduleen 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.


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 Wintersemester 2019/20. 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.