Instructor:  Dr. Tamás SZAMUELY

Text: Igor R. Shafarevich: Basic Algebraic Geometry I, Springer-Verlag, 1994. Chapters I, II/1,5,6, III/1.1, 2, 3 + as time permits.

Prerequisite: Introductory Abstract Algebra (topics on group actions not needed)

Course description:
The course provides an introduction to some basic notions in algebraic geometry.
The methods correspond to an introductory level.

Topics:
Introduction: algebraic curves in the affine and projective plane, their maps, singularities and intersections.

Elementary global theory of algebraic varieties: affine and projective varieties, regular and rational maps, function fields, products, dimension. Dimension of intersections. Applications: lines on surfaces, Tsen's theorem.

Local theory: Regular (smooth) and singular points, tangent spaces. Normal varieties, normalisation. Normalisation of curves, the correspondence between smooth curves and one-dimensional function fields. Singularities of maps, Bertini theorems.

Divisors on curves: divisors and the Picard grup, Bezout's theorem for plane curves. Group law on a smooth cubic curve.

Additional topics: to be determined.