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, pseudodifferential 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 