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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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Size 3.4Mb Date Oct 28, 2004 |
groups special.* Among the special groups are:
1) the additive group of a field;
2) the multiplicative group of a field;
3) the group SLn;
4) the group Spn...
Invariant Theory 161
When N = {e}, a relative section is a section in the sense of 2.5, but the
converse, in general, is false, since points in general position in a section can
have nontrivial stabilizers...
If, in addition, G° has no
nontrivial characters (which in case (a) means it is unipotent), then k(X)G =
< We will assume G is connected...
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)...
We will call the G -module M algebraic if any element of M is con-
contained in some algebraic finite-dimensional G-submodule...
Consequently, all of the invariants C.16), hence all invariants of G, can be ex-
expressed in terms of invariants C.16) with n < N...
Here,
however, we consider a different approach to the construction of Chevalley
sections due to Seshadri [1962]...
// X is a smooth affine variety and the group G is reductive,
then fc[Z]G is a Cohen-Macaulay algebra (i.e...
Suppose A is a graded Cohen-Macaulay algebra, el, ..., ed are the degrees
of any parameters t1,...,ti, and p1,...,pr are the degrees of basis elements when
it is viewed as a k[t1,..., f^-module (it is easy to see that such elements can
be chosen to be homogeneous)...
If we trace the development of the isomorphisms
C.41) and C.23), we can see, for example, that to the identity covariant id:
Vd->Vd corresponds the invariant u(?) of the pair (u, ?)...
A pair (Y,nY), where Y is an algebraic variety and nY a
morphism of X into Y, is called a geometric quotient for the action G:X if
conditions (F1)-(F4) are satisfied...
X -* Z that is constant on the orbits of
G there exists a unique morphism a: Y-* Z such that nz = a...
Let Y be the space of symmetric (resp., skew-
symmetric) matrices of order m, and let 7t: X -> Y be the morphism that assigns
to x = (vl,..., vm) e X its Gram matrix (B(v(, Vj))...
Assume X is a projective variety embedded in a projective space PV so that
the action of G is induced by a linear representation in V...
Mumford [1977]
proved that for such curves there exists a coarse variety of moduli Mg that is an
irreducible projective variety and contains the variety Mg of Example 2 as an
open subset...
Since there exists a local section over any point of G/H, it
follows that G * F is an analytic variety...
Restriction of functions to F defines an algebra isomorphism k[G * F]G ^
k[F]H and therefore an isomorphism of varieties F/H 2^.(G* F)/G, by means of
which these varieties are usually identified...
¦4 Suppose some fiber of the quotient morphism different from the null-cone
contains an invariant subvariety M in which the codimension of an orbit in
general position is equal to m...
It follows from the preceding discussion that this morphism is injective
on the open subset G * V+{h)° a G * V+(h) and maps it onto <Sl(h)...
This theorem has many applications to questions on the structure of stabi-
stabilizers, orbit types, and others (Bredon [1972])...
The variety a'1(X) is equivariantly isomorphic to
the homogeneous fiber space G * (S nX) (see Proposition 4.21), hence S n X is
Gx
an etale slice at x for the action of G on X...
Then Nx can be canonically identified with some linear subspace of
V, which enables us to consider the sum x + 9lx...
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