To give an introduction to the modern Algebraic Geometry.

To develop and study the basic concepts of affine and projective geometry, such as Zariski topology, irreducibility, projections, blow ups, intersection multiplicities, regular and rational maps.

To explain in detail the techniques of differential forms, with applications to birational geometry of higher-dimensional algebraic varieties.

(LO1) To know:basic concepts of smooth geometry and algebraic geometry.

(LO2) To understand: the interplay between local and global geometry, the duality between differential forms and submanifolds, between curves and divisors, between algebraic and geometric data.

(LO3) To be able to: perform elementary computations with differential forms, patch together local objects into global ones, compute elementary intersection indices.

(S1) Problem solving skills

Affine varieties: hypersurfaces; non-singular and singular points; formal power series; Zariski topology; irreducibility; regular and rational functions. Linear projections and Hilbert’s Nullstellensatz.

Projective varieties: the projective space; Euler’s identity; projective sets; direct products; Segre’s embedding; blow ups; intersection multiplicities and branches of plane curves.

The Grassmann algebra: the dual space; multi-linear maps; skew-symmetric functions; the tensor and wedge products; Grassmanians.

Differential forms: the tangent and cotangent spaces; differential forms on affine and projective varieties; the restriction operation for differential forms; Poincaré’s construction.

Birational geometry: regular, rational and birational maps; rational varieties; the pull back operation for differential forms; non-rational projective hypersurfaces.