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Next: MA4141 Representations of algebras
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MA4121 Projective Curves
Credits: 20 |
Convenor: Dr. D. Notbohm |
Semester: 1 |
Prerequisites: |
essential: MA2102(=MC241), MA2111(=MC254), MA2001(=MC224) |
desirable: MA1271(=MC149) |
Assessment: |
Regular course work: 15% |
Three hour examination in January: 85% |
Lectures: |
36 |
Problem Classes: |
10 |
Tutorials: |
none |
Private Study: |
104 |
Labs: |
none |
Seminars: |
none |
Project: |
none |
Other: |
none |
Surgeries: |
none |
Total: |
150 |
Subject Knowledge
Aims
This module aims to introduce
basic ideas of Algebraic Geometry, to show how basic ideas
from pure mathematics could be brought together in one of the
very beautiful subjects of mathematics and to demonstrate
the power of the interaction between algebra and geometry.
Learning Outcomes
To know the definitions of and understand the key concepts introduced in this
module.
To be able to understand, reproduce and apply the main results and proofs in
this module.
To be able to solve routine problems in the subject matter covered.
Subject Skills
Aims
To know the definitions of and understand the key concepts introduced in this
module.
To be able to understand, reproduce and apply the main results and proofs in
this module.
To be able to solve routine problems in the subject matter covered.
Learning Outcomes
The ability to present arguments and solutions in a coherent and logical form.
The ability to use the techniques taught within the module to solve problems.
The ability to apply taught principles and concepts to new situations
Explanation of Pre-requisites
This module discusses relations between algebra, geometry and analysis.
MA2102 Linear Algebra, MA2111 Algebra 1 and MA2001 Vector Calculus
are essential as well as basic
knowledge about complex numbers
MA1271 Geometry of the plane is desirable.
Course Description
Curves in the plane like lines, circles, ellipse, parabola and hyperbola
can be described by polynomial equations, the first by a linear equation
the other by quadratic equations. Curves in the plane of this type, i.e. sets
of solutions of a polynomial equation in two variables, are the subject of this
module. In the study of these objects, algebraic and geometric considerations
come together. It is a first step into the field of Algebraic Geometry
which is nowadays one of the most fashionable subjects of mathematics.
This is an introductory course and we will concentrate
on presenting some basic ideas of the powerful interaction
between geometry and algebra. This intrplay will be used
to study and analyze the
geometry of plane algebraic curves.
We can think of algebraic curves as objects defined over the real
or over the complex numbers;
i.e. we look at polynomials with real coefficients or complex coefficients.
It turns out that the theory is much simpler
and more beautiful over the complex numbers. And
therefore we will mainly work with complex numbers.
Part of the module is the discussion of examples which we will use to
explain and illustarte main results and concepts.
Syllabus
Affine curves, homogeneous coordinates, projective plane, varieties,
irreducible components, coordinate transformations,
minimal polynomials,
projective and
plane algebraic curves, intersection of curves, singular and simple points,
the degree
of a curve, Bezout's Theorem, points of inflection and cubics.
Reading list
E. Brieskorn and H. Knörrer,
Plane Algebraic Curves,
Birkhäuser.
Gerd Fischer,
Plane Algebraic Curves,
American Mathematical Society.
Frances Kirwan,
Complex Algebraic Curves,
Cambridge University Press.
E.J.F. Pimrose,
Plane Algebraic Curves,
Macmillan& Co LTD.
Miles Reid,
Undergraduate Algebraic Geometry,
Cambridge University Press.
A. Seidenberg,,
Elements of the Theory of Algebraic Curves,
Addison-Wesley.
Module Evaluation
Module questionnaires, module review, year review.
Next: MA4141 Representations of algebras
Up: ModuleGuide03-04
Previous: MA4111 Differential Geometry
Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.