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.
- 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
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
- 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
MATH21800 Algebra 2
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.
Further exam information can be found on the Maths Intranet.