12
I. TH E NOTIO N O F "INVARIANT "
(2) Fo r ever y extensio n k of ko wit h k infinite , ever y G-torso r T ove r &; , an d
every nonempt y ope n subvariet y U of X, ther e exist s x G U(k) whos e
fiber Q
x
i s isomorphic t o T (i.e. , th e set of x G X(k) wit h Q ^ = T is
dense i n X).
EXERCISE 5.2 . Show : I f (P, X) i s a versal G-torso r a s above, th e field K is a regular
extension o f fco, i.e. , th e /co-variet y X i s absolutely irreducible .
5.3. V E R S A L G - T O R S O R S EXIST . Choos e a n embeddin g o f G i n G L J V fo r
some N. Le t X b e the homogeneous spac e GL^v/G . Pu t Q G L J V , viewe d a s a
G-torsor ove r X. The fiber P of Q at the generic point of X is versal. Indeed , if
T i s as in 5.1.2, i t follow s fro m [Se65 , 1.5.4, Cor . 1], combine d wit h th e fact tha t
every GLjy-torso r i s trivial, tha t T i s isomorphic t o Qx fo r some x G X(k), an d
that suc h a n x i s unique u p to the action o f GLjv(fc) o n X(k). Le t OT be th e
corresponding orbi t o f GLJV(/C). Sinc e k i s infinite, GLAT(/C ) i s dense i n G L J V fo r
the Zarisk i topology , an d OT is dense i n X. Henc e P i s versal.
Note tha t thi s give s more . I t shows tha t th e functo r
k^H\k,G)
is isomorphi c t o the functo r k i-» X(k)/GLiN(k).
E X A M P L E 5.4 . A s in 5.3, choos e a n embedding o f G in GL^v fo r someT V over
ko; writ e V fo r the affin e iV-spac e o n which GLA T (an d henc e G ) acts.
We als o mak e th e followin g assumption :
(F) The scheme-theoretic stabilizer in G of the generic point in V is trivial
Property (F ) implie s cf. [Th86, Prop . 4.7 ] that ther e i s an open dens e subse t
V o f V whic h i s G-stable, an d is a G-torsor . Le t X = V'/G, an d let K b e th e
function field o f X ove r fco; b y takin g th e generi c fiber o f V' » X, we get a G-torso r
P ove r K. The torsor P is versal. Indeed , becaus e o f the definitio n o f versal, we
may assum e tha t ko is infinite, henc e V ha s a rationa l point . On e ca n use this
point t o define a G-equivarian t ma p G L ^ » V, henc e a G-equivarian t dominan t
map P V, wher e P i s open dens e an d G-stable i n GL^y Th e fac t tha t GLA T
has th e versa l propert y 5.1.2 implie s th e sam e propert y fo r V'.
(Note tha t th e dimension o f X i n 5.4 isTV dimG whic h i s typically muc h
lower tha n dim(GL]v/G ) i n 5.3. )
E X A M P L E 5.5 . I f G is finite, the n th e actio n o f G on V ha s propert y (F) , and
one ma y tak e fo r V' th e spac e V wit h th e linea r varietie s ker( g 1) removed , fo r g
i n G - { l } .
5.6. E X A M P L E S .
(1) Choos e G = O
n
, so that A = Quad
n
, an d assume tha t char(/c ) ^ 2 .
Take K /co(£i^2 , ,tn) wher e th e ti ar e indeterminates. Th e torso r
corresponding t o the quadrati c for m ( t i , . . . , tn) i s versal.
The densit y propert y 5.1.2 hold s i n thi s cas e becaus e th e class of
the quadrati c for m stay s th e same whe n on e multiplie s th e £i, ... , tn by
arbitrary squares , an d suc h change s giv e ris e t o a dens e se t in the spac e
of th e V s . A similar remar k applie s t o practically al l of the versa l object s
that appea r i n these note s (e.g. , fo r Gi o r F
4
); i n each case , ther e are
trivial change s o f the parameter s whic h giv e a dense set .
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