discipline

Pure Mathematics

subject

Algebraic Geometry

lecturers

József Pelikán, Gyula Károlyi, Endre Szabó

credits

4

period

 

curriculum

Affine varieties, coordinate ring, local rings. Function fields. Zariski topology. The notion of dimension. Krull’s principal ideal theorem. Projective varieties. Morphisms, direct product of varieties, rational maps, blowing up. Smooth varieties, smooth curves. The Hilbert polynomial, degree, intersection multiplicity, arithmetical genus. Complex manifolds, bundles, differential forms. De Rham cohomology, Dolbeault cohomology. Sobolev spaces. The Fourier transform. Differential operators, pseudo-differential operators, elliptic operators. Hodge decomposition. Exact sequences, cohomology of line bundles over the projective space. Kodaira’s Vanishing Theorem. Rich bundles, Kodaira’s embedding theorem. Mumford regularity. Representable functors. Definition and existence of Hilbert scheme. Picard variety, Cow variety. The moduli space of a curve.

literature

Hartshorne: Algebraic Geometry

form of tuition

Lecture

mode of assessment

written/oral exam