Algebraic Geometry - MAT00001M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Emilie Dufresne
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2019-20

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

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

[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.]

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).

Assessment

Task Length % of module mark
University - closed examination
Algebraic Geometry
2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
University - closed examination
Algebraic Geometry
2 hours 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.



The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.