This class provides a systematic introduction into the modern language of algebraic geometry based on Grothendieck's notion of schemes. Schemes are a generalization of algebraic varieties defined over an algebraically closed field to situations locally modelled on polynomial equations with coefficients in an arbitrary commutative ring. The theory of schemes thus provides a framework also for arithmetic geometry and for dealing with families of algebraic varieties.

The following topics will be discussed. Basic concepts of scheme theory, including cohomological methods: General theory of sheaves, affine schemes, projective schemes, types of morphisms, coherent sheaves, divisors, differentials, cohomology of sheaves, Serre duality, higher direct images, flat and smooth morphisms, formal schemes, base change.

We mostly follow Chapter 2 and 3 in Hartshorne's "Algebraic geometry", including many of the exercises. Illustrations of the abstract concepts will come from the theory of toric varieties.

Lectures and exercise classes are to be conducted in English, as it is stated on STiNE.