Università Roma Tre

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994