Università Roma Tre

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.