Geometry of Quadratic Equations
Module MA5221
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 workload | Contact hours | Credits (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
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
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.