Elements of Algebraic Geometry I

Timetable

Module aim (aims)

This will be an elementary course of modern algebraic geometry which shall introduce students to the backbone of the subject in a friendly and not too technical way. We will discuss basics of commutative algebra along the way and plenty of geometric examples. After introducing classical varieties and their morphisms we will focus attention at sheaves and geometric properties of abstract algebraic varieties.

We will start with setting up foundations such as affine and projective algebraic varieties and explain morphisms, local rings, function fields, dimension and nonsingular varieties. We will conclude with a piece of more abstract story concerning schemes and sheaves. This will allow us to discuss Riemann-Roch theorem over an algebraically closed field for curves.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Knowledge of abstract algebra at the level of an undergraduate level: Groups, Rings an Fields will be very helpful. 

Syllabus

Week 1, 2	Affine varieties (Zariski topology on A^n, Hilbert’s Nullstellensatz, decomposition into components, Commutative Algebra interlude: finiteness conditions)
Week 3	Regular functions and morphisms (regular functions, morphisms between quasi-affine varieties, products of quasi-affine varieties, linear algebraic groups, CA interlude: localization)
Week 4	Projective varieties I (Zariski topology on P^n, standard affine open covering, some classical projective geometry)
Week 5	Projective varieties II
Week 6	Sheaves (sheaves of abelian groups, operations on sheaves, other types of sheaves)
Week 7	Algebraic varieties (pre-varieties, algebraic varieties, birational geometry, Grassmannians)
Week 8	Dimensions, tangent spaces, and singularities I (dimension of algebraic varieties, tangent space at a point, blowing, CA interlude: some dimension theory)
Week 9	Dimensions, tangent spaces, and singularities II
Week 10	Proper morphisms and complete varieties I (definition and basic properties of proper morphisms, completeness versus projectivity, some classical constructions in projective geometry)
Week 11	Proper morphisms and complete varieties II
Week 12	Normalization I (applications to curves, finite morphisms, normalization of a variety, complete non-singular curves, CA interlude: integral dependence)
Week 13	Normalization II
Week 14	Vector bundles and locally free sheaves I
Week 15	Vector bundles and locally free sheaves II

Reading list

[1] R. Hartshorne, Algebraic Geometry. GTM 52, Springer, 1977.

[2] D. Mumford, The red book of varieties and schemes. Springer, 1988.

[3] N. Bourbaki, Algebre commutative. Masson, 1985. 

[4] D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry. SV, 1995.

[5] R. Godement, Topologie algebrique et theorie des faisceaux. Hermann, 1958.

[6] J. Harris, Algebraic Geometry, a first course. GTM 133, Springer, 1992.

[7] T.A. Springer, Linear Algebraic Groups, Birkhauser 1998.