Algebraic Geometry

Unit aims

Algebraic geometry is the study of systems of polynomial equations. The solution set of a system of polynomial equations forms a geometric object called an algebraic variety. The aim of this course is to develop basic algebraic tools to explore the geometry of these varieties. We will build up a dictionary between geometric properties of varieties and numericalinvariants of equations.

Unit description

  • Affine, projective varieties and algebraic groups
  • Regular functions and maps
  • Morphisms and rational maps
  • Ideals of varieties, irreducible decomposition, and Hilbert's Nullstellensatz
  • Definitions of dimension and degree
  • Hilbert polynomials
  • Smoothness and tangent spaces
  • Examples of varieties: Grassmannians and flag varieties
  • Toric varieties

Learning objectives

Students who successfully complete the unit will be able to:

  • define affine, projective varieties and algebraic groups, morphisms and rational maps between them, the Zariski topology, irreducible components of a variety and the concepts of dimension and smoothness for algebraic varieties
  • understand the correspondences between varieties and their coordinate rings both in the affine and projective cases
  • understand proofs of some classic results in algebraic geometry, e.g. Hilbert's Nullstellensatz
  • compute the algebraic invariants of varieties such as degree, dimension and Hilbert polynomial
  • carry out calculations on classical families of varieties such as Grassmannians, and toric varieties
  • compute the singular points of algebraic varieties

Reading and references

Recommended

  • Joe Harris, Algebraic Geometry: A First Course, Springer, 1992
  • Robin Hartshorne, Algebraic Geometry (First chapter), Springer-Verlag, 1977
  • Miles Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 2001
  • Karen E. Smith et al., An Invitation to Algebraic Geometry, Springer, 2000

Unit code: MATHM0036
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Fatemeh Mohammadi and Dr Farhad Babaee

Pre-requisites

MATH21800 Algebra 2

Co-requisites

None

Methods of teaching

There are 3 lecture per week and every other week one session is designed as a problem session. The course is based on the lectures and exercises. The basic lecture notes will be posted and solutions to most of the exercises will be distributed. The last 2 weeks of the course will be devoted to review and revision, and in this time exercises (both assigned and not assigned) will be addressed. Besides the problems classes, there is also a weekly office hour during which students can ask questions about lectures and exercises.

Methods of assessment 

The pass mark for this unit is 50.

The final mark is calculated as follows:

  • 80% Exam
  • 20% Assessed Coursework

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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