The first 48 hours of the course form an introduction to the language of schemes and coherent sheaves, and cover the material in Hartshorne, Algebraic Geometry, Section II.1-II.8: sheaves; schemes; open and closed subschemes, affine, finite, projective morphisms; properness and separatedness; quasicoherent and coherent sheaves, pushforward and pullback; divisors and invertible sheaves; Proj construction and blow-ups, sheaf of differentials. The remaining 12 hours will cover a quick introduction to infinitesimal deformation theory: Artinian local algebras, deformation functors, Schlessinger axioms, tangent and obstruction spaces. Prerequisites are the basics of point set topology and a solid background in basic commutative algebra.In particular, we will need rings, subrings, quotient rings, prime and maximal ideals, noetherianity, localization; modules over a ring, kernels and cokernels, localization, tensor product. Previous knowledge of complex manifolds of quasiprojective varieties, while not logically necessary, would be useful to support intuition.

## Algebraic Geometry

Lecturer:

Course Type:

PhD Course

Anno (LM):

Second Year

Academic Year:

2020-2021

Period:

October-January

Duration:

60 h

Description:

Research Group:

Location:

A-005