**3/4, 12.30-14.30: **Discussion of the exercises assigned in the previous lectures.

**4/4, 12.30-14.30: **Products of quasi-projective varieties. The case of P^{1}xP^{1}. Segre maps and Segre varieties in general. A subset of P^{n}xP^{m} is closed if and only if it is the zero locus of some bihomogeneous polynomials. The Zariski topology on P^{n}xP^{m} is finer then the product topology. The product of two affine charts of P^{n}xP^{m }is isomorphic to A^{n+m}. The projections are regular, a product of two regular maps is regular. A product of affine varieties is affine. The diagonal is closed. The intersection of two affine open subsets is an affine open subset. The graph of a regular map is closed. Example: the Veronese map of degree 2.

Exercises: (1) Show that a twisted cubic cannot be contained in a plane. (2) Show that a subset of A^{n}xP^{m} is closed if and only if it is the zero locus of some polynomials in x_{1},...,x_{n},y_{0},...,y_{m}, homogeneous in the y's. (3) Show that the image of P^{n}x{pt} is a linear subspace of dimension n contained in the Segre variety. (4) Show that the irreducible components of a product are given by the products of the irreducible components of the two factors.

**7/4, 10.30-12.30: **Rational functions on quasi-projective varieties: definition, examples, basic properties. If X is irreducible, the set k(X) of rational functions on X is a field. If X is irreducible and affine, then k(X) is isomorphic to the quotient field of k[X]. Examples: rational functions on A^{n} and on P^{n}. Rational maps between irreducible quasi-projective varieties. Examples. The Cremona map P^{2}-->P^{2}. Every rational map P^{1}-->P^{n} is regular. Dominant rational maps. Criterium for the definition of the composition of two rational maps. If either g is regular, or f is dominant, then g.f is defined. Birational maps, birational equivalence. Example: the Cremona map is birational.

Exercises: (1) Show that the composition of dominant rational maps is dominant. (2) Show that birational equivalence is an equivalence relation on the set of irreducible q.p. varieties. (3) S. ex. 6 and 7 p. 40 (Ch. I sec. 3)