Main content

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

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 Compulsory Elective 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 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:

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.