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Geometry of Quadratic Equations

Module MA5221

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

MA5221 is a semester module in English language at Master’s level which is offered once.

This module description is valid from SS 2016 to WS 2016/7.

Total workloadContact hoursCredits (ECTS)
90 h 30 h 3 CP

Content, Learning Outcome and Preconditions

Content

Linear systems of quadrics,
base loci, discriminant loci,
Jacobian and Reye varieties,
complete intersections of quadrics,
examples from classical algebraic geometry,
symmetric determinantal representations of hypersurfaces,
conic fibrations and rationality questions

Learning Outcome

At the end of the lecture, the students are familiar with the theory of linear systems of quadrics and their attributes, such as base loci and discriminant loci. They will know the basic examples from classical algebraic geometry, and will be able to perform simple computations with them. The will understand the connection to complete intersection and other classical constructions. Moreover, they will be able to apply the theory to the symmetric determinantal representations of hypersurfaces. Finally, they will be able to understand the connection to conic fibrations and rationality questions.

Preconditions

MA1101 Linear Algebra 1
MA1102 Linear Algebra 2
MA2101 Algebra
useful, but not necessary:
MA5120 Algebra 2
MA5107 Algebraic Geometry
MA3203 Projective Geometry

Courses, Learning and Teaching Methods and Literature

Learning and Teaching Methods

The module consists of a series of lectures. In the lectures, theoretical principles and examples are presented.

Media

blackboard

Literature

Dolgachev: Classical Algebraic Geometry

Module Exam

Description of exams and course work

The examination will be oral (30 minutes). Students are supposed to know the basic objects and notions of the lecture, as well as the main theorems connected these. They are supposed to know the basic examples of linear systems of quadrics from classical algebraic geometry. They are supposed to perform elementary computations, as well as to solve simple exercises with them, such as computing the basic attributes of linear systems, e.g., base and discriminant loci. They are able to name and explain applications of the theorems to to algebraic surfaces, hypersurfaces, and rationality questions.

Exam Repetition

The exam may be repeated at the end of the semester.

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