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Parshin A.N., Shafarevich I.R., (eds.) Springer, Popov, Vinberg. Algebraic geometry IV. Linear algebraic groups. Invariant theory (Enc.Math.55, Springer)(T)(287s).djvu |
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is a rational mapping that is constant on the orbits of points of X in general
position, then there exists a rational mapping <p: Y -*¦ Z such that p = <pn (see
Proposition 1.9)...
Consequently, an action a possesses a section if
and only if c(a, §) = 0 for a suitable choice of subgroup § in a class of conjugate
subgroups...
A finitely generated extension of a field k is called rational if it is isomorphic to the field of rational
functions of independent variables...
This is not entirely the same as a complete system of invariants in the sense of 0.1 (see, however,
Proposition 3.4 and Theorem 3.3)...
If G acts on an
algebraic variety X, then, by Proposition 1, /c[X] is an algebraic G-algebra (but
k(X), in general, is not)...
As regards the question of finite generation of the algebra of invariants, this
feature is very significant, and the answer to the question is largely dependent
on properties of the pair (G, H)...
By Chevalley's theorem (see 1.4), any unipotent subgroup or, in
general, any subgroup having no nontrivial characters is observable...
Example 3 can be trivially generalized by replacing the homogeneous variety
G/H by any normal variety Z containing G/H as an open orbit whose complement
has codimension at least 2; indeed, in this situation k[Z] = /c[G/Jf]...
// fc[X] is factorial and the group G is connected and has no
nontrivial characters, then k[X~\G is factorial...
It is easy to see, though, that f2, f3 are indeed
algebraically independent, for if they were not, they would satisfy a relation
f2 = cf3(cek) contradicting the factorial nature of fc[F4]SL2...
Then Wk* = Wk,, where X* is obtained from X by applying
the automorphism of the lattice of weights induced by the canonical involution
of the Dynkin diagram (Vinberg and Onishchik [1988])...
Morphisms of algebraic varieties are the mappings that are
morphisms of topological spaces with sheaves of functions...
A more visual necessary
condition is that all of the orbits must be closed and, if X is irreducible, have the
same dimension (cf...
Let M = /c[Z] and take JVa to be the invariant ideal Ix of/c[X] corresponding
to the subset Ya c X...
Consequently, its image under
the quotient morphism
U{ x Uj^(Ui x Uj)/(G xG) = (UJG) x (UJG)
is closed in (UJG) x (Uj/G) (see 4.2), but this image is just AXIG n
((Ut/G) x (Uj/G)).+
One should not think that the condition in the hypothesis of this theorem is
always satisfied: Nagata [1956] constructed an example of an action of the
group Z/B) for which there is no geometric quotient...
Invariant Theory 193
polynomial h(x) can be parametrized by the points of some projective variety -
the Hilbert scheme Hilb^P")...
Other examples and references to the literature can be found in
Gieseker [1983] and Mumford and Fogarty [1982]...
In other
words, the quasiparabolic subgroups are the stabilizers of the highest vectors of
linear representations of G...
< Suppose Z is an unramified Galois covering of an open subset U c G/H
such that the fibering n: G -> G/H is trivialized over Z...
Indeed, since the coefficients
of the characteristic polynomial of the adjoint operator are homogeneous in-
invariants, each element of the null-cone is nilpotent...
It follows from the preceding discussion that h is a characteristic of any
element of g2(fc) with nonzero coordinates...
Secondly, even if the stabilizer Gx is reductive, it can happen that in any
neighborhood of x there is a point y such that Gy is not conjugate in G to any
subgroup of Gx (this is clearly impossible in a neighborhood equivariantly iso-
morphic to a homogeneous fiber space over G/Gx)...
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