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School of Mathematics

# MATH31442 - 2006/2007

General Information
• Title: Introduction to Algebraic Geometry
• Unit code: MATH31442
• Credits: 10
• Prerequisites: 212, 252 (ex-UMIST), MT2212, MT2262 (ex-VUM)
• Co-requisite units:
• School responsible: Mathematics
• Member of staff responsible: Dr Gabor Megyesi (Mathematics and Social Sciences P.7, Tel: 63644)
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## Specification

### Brief Description of the unit

Algebraic geometry studies objects called varieties defined by polynomial equations. A very simple example is the hyperbola defined by the equation xy = 1 in the plane. There is a way of associating rings to varieties, and then the geometric properties can be studied using algebra, for example points correspond to maximal ideals, or the geometry of the variety can give information about certain algebraic properties of the ring. Algebraic geometry originated in nineteenth century Italy, but it is still a very active area of research. It has close connections with algebra, number theory, topology, differential geometry and complex analysis.

### Syllabus

1. Affine and projective spaces.
2. Affine and projective varieties.
3. Co-ordinate rings.
4. Function fields.
5. Morphisms and rational maps.
6. Hilbert's Nullstellensatz.
7. The classification of curves.
8. The group law on the points of an elliptic curve.

### Teaching and learning methods

Two lectures per week plus one weekly examples class.

Assessment
End of semester examination (2 hours) 100%.

## Arrangements

Online course materials are available for this unit.