Notes of lectures on graphics tablet

Affine and projective spaces

Affine algebraic sets and Zariski topology

Graded rings and homogeneous ideals. Projective varieties. Hypersurfaces.

Intersection of a hypersurface with a line. Product of affine spaces and comparison between topologies. The Segre map and the Segre quadric P¹xP^1

Embedding of the affine space in the projective space. Ideal of a subset of the affine or projective space and its properties.

The ideal of a point in the affine space is maximal.

Statement of the Hilbert Nullstellensatz. Algebraic independence, trascendence bases. Integral elements over a ring.

Proof of Nullestellensatz.

Projective Nullstellensatz.

Normalization Lemma

Projective closure. Twisted cubic.

The ideal of the projective twisted cubic

Irreducible topological spaces.

Irreducible components. Topological dimension.

Theorems on dimension

Dimension of K-algebras.

Regular functions

Regular maps.

No notes for Lecture 19.

Open covering with affine varieties. Language of categories.

Functors, equivalence of categories. Rational maps.

Rational and birational maps.

Projections, standard quadratic transformation, rational varieties.

Theorem on the dimension of an intersection

Intersection of projective varieties - Complete varieties

Completeness of projective varieties.

Tangent space and smoothness

Finite morphisms.

Examples of finite morphisms. Blow ups.

Examples of resolution of singularities.