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

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.

• Hulek, K.: Elementare Algebraische Geometrie, Vieweg
• Shafarevich, I.R.: Basic Algebraic Geometry, Springer
• Hartshorne, R.: Algebraic Geometry, Springer