# Algebraic geometry (WS 2020)

This webpage contains lecture notes, homeworks for algebraic geometry.

## Lecture notes:

Lecture 1 (02.10.2020) Systems of equations, rings, ideals, quotient ring. Functor of points, category theory, Yoneda lemma.

Homework 1 (due 16.10.2020)

Lecture 2 (09.10.2020) Noetherian condition, prime ideals, maximal ideals, radical, Spec, Zariski topology.

Lecture 3 (16.10.2020) Points, maximal ideals, closed points, geometric points, Specm, Nullstellensatz, ringed spaces, schemes.

Homework 2 (due 23.10.2020)

Lecture 4 (23.10.2020) Irreducibility, minimal primes, height, dimension, more on construction of Spec as a ringed space.

Homework 3 (due 30.10.2020)

Lecture 5 (30.10.2020) Proof of the sheaf property for the structure sheaf on Spec®. Heights of prime ideals and numbers of generators, localization.

Homework 4 (due 06.11.2020)

Lecture 6 (06.11.2020) Local rings, Nakayama lemma, artinian rings, length. Proof of the Krull Hauptidealsatz. Dimension of the polynomial ring.

Homework 5 (due 13.11.2020)

Lecture 7 (13.11.2020) Exercises. Schemes.

Homework 6 (due 20.11.2020)

Lecture 8 (20.11.2020) Morphisms of schemes, affinization, gluing.

Homework 7 (due 27.11.2020)

Lecture 9 (27.11.2020) More gluing. Proj.

Homework 8 (due 04.12.2020)

Lecture 10 (04.12.2020) Prime ideals and points with coefficients in fields. Locally ringed spaces. More Proj.

Homework 9 (due 11.12.2020)

Lecture 11 (11.12.2020) Proof that Proj is a scheme. Examples.

Lecture 12 (08.01.2021) More examples. Weighted projective spaces. Blowup. Curves on $\mathbb{P}^2$.

Homework 10 (due 15.01.2021)

Lecture 13 (15.01.2021) Parametrization of conics. Functor of points. Fiber products.

Homework 11 (due 22.01.2021)

Lecture 14 (22.01.2021)

Lecture 15 (29.01.2021)