In the following problems k is an algebraically closed field of characteristic zero.

1. Let X be a noetherian topological space. Prove that there exists a decomposition X = U7.1 Ci such that (a) Ci is a closed irreducible subset of X. (b) If F C X is a closed irreducible subset of X then F C Ci for some i. (c) The Ci’s are unique with properties (a) and (b)

Let X,Y be affine algebraic varieties and let f : X Y be a morphism (i.e. hof E k[X] for all h E k[Y]). (a) Prove that X is a topological space with the Zariski topology. (b) Prove that f is continuous in the Zariski topology. (c) Prove that f(X) C Y is dense if and only if f* is injective.

3. Let X be an irreducible algebraic variety and let y E k(X). Prove that {x E I-9 for some f, g E k[X] with g(x) O} is an open subset of X . 4. Let X be an irreducible algebraic variety and let K = k(X) be the field of rational functions on X. Let L be a subfield of K containing k. Prove that there exists a finite subset fm} of L such that L fin)

y%=

.5. Let C x X > X be an action and let x E X. Prove that Gx is an open subset of Gx.

6. Let C x X X be an action and let V C X be a closed subset of X. Prove that {g E gV C V} = {g E G I gV = V}, and this is a closed subgroup of G.

– Let G = Gln(k) and let X = 111,.„(k). Define G x X X by (g, x) gxg-1.

(a) Determine the set of closed orbits. (b) Determine the set of orbits of maximal dimension. What is this maximum?

(c) For each orbit Cl(x) C X there is a unique closed orbit Cl(s) C Cl(x). Charac-terize s in terms of x.

1. Let X be a noetherian topological space. Prove that there exists a decomposition X = U7.1 Ci such that (a) Ci is a closed irreducible subset of X. (b) If F C X is a closed irreducible subset of X then F C Ci for some i. (c) The Ci’s are unique with properties (a) and (b)

Let X,Y be affine algebraic varieties and let f : X Y be a morphism (i.e. hof E k[X] for all h E k[Y]). (a) Prove that X is a topological space with the Zariski topology. (b) Prove that f is continuous in the Zariski topology. (c) Prove that f(X) C Y is dense if and only if f* is injective.

3. Let X be an irreducible algebraic variety and let y E k(X). Prove that {x E I-9 for some f, g E k[X] with g(x) O} is an open subset of X . 4. Let X be an irreducible algebraic variety and let K = k(X) be the field of rational functions on X. Let L be a subfield of K containing k. Prove that there exists a finite subset fm} of L such that L fin)

y%=

.5. Let C x X > X be an action and let x E X. Prove that Gx is an open subset of Gx.

6. Let C x X X be an action and let V C X be a closed subset of X. Prove that {g E gV C V} = {g E G I gV = V}, and this is a closed subgroup of G.

– Let G = Gln(k) and let X = 111,.„(k). Define G x X X by (g, x) gxg-1.

(a) Determine the set of closed orbits. (b) Determine the set of orbits of maximal dimension. What is this maximum?

(c) For each orbit Cl(x) C X there is a unique closed orbit Cl(s) C Cl(x). Charac-terize s in terms of x.

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