250145 VO Algebraic Geometry III (2023S)

The course will introduce fundamental techniques and results in the theory of schemes. Even though being a continuation of Alg Geom II, newcomers are welcome (and can get the notes of the former course).

Prospective topics are as follows (there will be some flexibility in our choice):

Graded rings and projective schemes, with the example of elliptic curves and blowups.

Gluing of affine schemes to construct general schemes.

Properties of morphisms: Semicontinuity of fiber dimension, Chevalley's theorem about the image of constructible sets, flat morphisms, isomorphisms and automorphism groups. Group schemes and moduli spaces.

Singularities: Normalization and integral closure, blowups, resolution of singularities.

Dimension theory: Hilbert functions and Hilbert polynomials.