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Unformatted text preview: 3 Hilbert-Mumford type theorems. 3.1 Basics on group actions. 3.1.1 Algebraic group actions. Let X be an algebraic variety and let G be an algebraic group both over C . Then an (algebraic group) action of G on X is a morphism & : G & X ! X satisfying: 1. &(1 ;x ) = x ( 1 denoting the identity element of G ) for all x 2 X . 2, &( gh;x ) = &( g; &( h;x )) for all g;h 2 G and x 2 X . We will denote such an action by gx . The set Gx is called the orbit of x . Our main example is G a Z-closed subgroup of GL ( n;F ) , X = F n and gx is the matrix action of G on F n . More generally, we de&ne a regular representation of an algebraic group, G , to be a group morphism & : G ! GL ( V ) where V is a &nite dimensional vector space over C . We denote it ( &;V ) . One more bit of notation the isotropy group of x 2 X is the set f g 2 G j gx = x g and it will be denoted G x . We will con&ne our attention to the case when G is irreducible (that is connected in the Z-topology). Lemma 1 Let G be irreducible and act on an algebraic variety, X . Let x 2 X and let Y be the Z-closure of Gx . Then 1. Y is irreducible. 2. Gx is Z-open in Y: 3. There is a Z-closed G orbit in Y . 4. Y is the S-closure of Gx . Proof. We have seen in Theorem 26 of 1 : 3 : 4 that Gx has interior in Y . Since y ! gy de&nes an automorphism of Y we see that Gx is a union of open subsets of Y . This proves 2. Since Gx is the image under a morphism of an irreducible variety it is irreducible. As it is dense in Y , Y is irreducible. Let Z = Y Gx . Then Z is closed and G-invariant. If Z = ; then Y is closed. If not let V = Gz be an orbit of minimal dimension in Z: If W is the closure of V then W V is closed in W and since W is irreducible dim( W V ) < dim W . Thus the dimension of any orbit in W V would be lower than the minimum possible. Thus we must have V is closed proving 3. We note that 4 is an immediate consequence of 2 and Theorem 20 of 126.96.36.199. 1 Proposition 2 Let G be an a ne algebraic group acting on an irreducible a ne variety X . Then there exists an imbedding of & : X ! C n as a Z-closed subset of C n for some n and an algebraic group homomorphism, : G ! GL ( n; C ) such that & ( gx ) = ( g ) & ( x ) for all x 2 X and g 2 G . Proof. We may assume that X & C m is Z-closed, Let f i = x i j X for i = 1 ;::;m ( x i the standard coordinates on X ). Let the action of G on X be given by F: Then F & O ( X ) is a subalgebra of O ( G X ) = O ( G ) C O ( X ) . We also note that F & ( f )(1 ;x ) = f ( x ) also if we set ( g ) f ( x ) = f ( g 1 x ) then F & ( g ) f ( h;x ) = F & f ( hg 1 ;x ) . This implies that the linear span of f F & ( g ) f j g 2 G g is &nite dimensional for each f 2 O ( X ) . Thus we see that the linear span, W , of f ( g ) f i j g 2 G;i = 1 ;::;m g is &nite dimensional. Let u 1 ;:::;u d be a basis of W . Then the map & ( x ) = ( u 1 ( x ) ;:::;u d ( x )) de&nes an isomorphism of X into C d . We assert that & ( X ) is S-closed in C n...
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