Algebraic Geometry - MAT00001M

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Additional information

Pre requisite knowledge for MSc students: a first course in commutative algebra, covering basic properties of rings, ideals and quotients, factorisation, especially in polynomial rings.

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

To introduce Algebraic Geometry, which is an important research tool for several members of the department. The study of Algebraic Geometry mixes algebraic tools with geometric insight: algebraic varieties are defined by the vanishing of collections of polynomials, and the geometry of these sets is tightly controlled by the structure of the associated polynomial algebra. In the treatment of the subject in this course, the emphasis will be on the commutative algebra, following on from some of the tools developed in the third year (especially in Algebraic Number Theory). However, students interested in classical geometry and topology should also find plenty of interest here. This module is designed to lead on to the study of Algebraic Groups in the Spring Term.

Module learning outcomes

Topics covered:

• Affine n-space and the Zariski topology.
• Abstract affine varieties.
• Hilbert's Nullstellensatz.
• Rational curves.
• The general notion of a variety illustrated with projective n-space.

Module content

 

These topics will allow students to understand an important area of modern mathematics, giving a good preparation for research in Algebra (both in the final year project and beyond into postgraduate life). Algebraic Geometry is a subject at the interface of commutative algebra and geometry and its study involves a synthesis of techniques and ideas from many Pure Modules in previous stages.

Cognitive & Intellectual Skills: Analysis, Synthesis, Evaluation, Application, all developed through learning new techniques and applying them to complex problems.

Key/Transferable Skills: Management of information – this module will equip students for research in algebra and algebraic geometry; Autonomy and Problem Solving developed through regular coursework assignments and tested in the examination.

Technical Expertise: the skills and techniques developed in this module are widely applicable across mathematics (algebra generally, group theory, geometry).

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

Miles Reid, Undergraduate Algebraic Geometry, LMS student text 12.