This entry is from Winter semester 2018/19 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): Written or oral examination Examination type: Successful completion of at least 50 percent of the points from the weekly exercises. 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

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Basismodulen sowie im Aufbaumodul Elementare Algebraische Geometrie oder im Aufbaumodul Algebra vermittelt werden.

## 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.

Die Wahlmöglichkeit des Moduls ist dadurch beschränkt, dass es der Reinen Mathematics zugeordnet ist.

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