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Department of Mathematics & Computer Science



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MA4121 Projective Curves

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

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.

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.

Objectives

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.

Transferable Skills

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

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.

Details of Assessment

There will be a series of pieces of work set during the semester which together will count for 10% of the final mark.

There will be six questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to four questions. All questions will carry equal weight. The examination will be worth 90% of the final mark.

Next: MA4141 Representations of algebras Up: Level 4 Previous: MA4101 Algebraic Topology

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Last updated: 2002-10-25
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