Algebraic Curves - 2019 entry

This module introduces you to the basic concepts of algebraic geometry and algebraic curves. This includes, affine and projective varieties, affine and projective curves, intersection theory in projective space and Bezout's Theorem. It also includes desingularisation of algebraic curves, curves and function fields in one variable, and the Riemann-Roch Theorem.

Pre-requisite Module: MTH2002 Algebra, or equivalent

The module aims to introduce you to some of the central concepts of modern algebraic geometry in an accessible form. The treatment will be in the language of varieties, and will cover the standard properties of affine and projective curves over an algebraically closed field.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Demonstrate a good understanding of the basic concepts of algebraic geometry in the context of affine and projective curves;

Discipline Specific Skills and Knowledge:

2 Reveal an enhanced understanding of the role of algebraic techniques in the formulation and solution of problems in geometry;

Personal and Key Transferable/ Employment Skills and  Knowledge:

3 Show enhanced problem-solving skills and ability to apply rigorous mathematical argument to the systematic study of geometric questions.

- Affine varieties: The Coordinate Ring; Hilbert's Nullstellensatz; irreducible components; multiple points and tangents;

- Projective varieties: projective space; projective plane curves; Bezout’s Theorem; morphisms and rational maps;

- Resolution of singularities;

- Riemann-Roch Theorem and applications.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

ELE – http://vle.exeter.ac.uk

Reading list for this module: