## M953 Homework 7 (Click
here for solutions)

Due Monday, April 15, 2002

[1] Let X and Y be nonempty subsets of projective space, **P**^{n}.
- (a) If C(X) = C(Y), then show that X = Y.
- (b) Show that X is a projective algebraic set if and only if C(X) is an affine
algebraic set.
- (c) Say X is a projective algebraic set. Show that
X is irreducible if and only if C(X) is. (You may assume that X is irreducible if
and only if I
_{p}(X) is prime.)

[2] Let X = V_{p}(x^{2}y - z^{3}) in **P**^{2}.
Show that k(X) is isomorphic to k(**A**^{1}).

[3] Let X be an algebraic subset of **A**^{n}, which
we regard as the subset U_{0} of **P**^{n}, and let
Y be a projective algebraic subset of **P**^{n}.
Take k[**P**^{n}] to be k[x_{0}, ..., x_{n}].
- (a) Show that the union of X and V
_{p}(x_{0}) is a projective algebraic set.
- (b) Show that X is contained in X
^{*}, which is contained in the union
of X and V_{p}(x_{0}) which equals the union
of X^{*} and V_{p}(x_{0}).
- (c) Show that Y
_{*}^{*} is contained in Y.