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10.17.1. Lemma.
The map m is an imbedding.
Proof.
The lemma follows from the fact that if m( V ) = [w], then
{
}
nk
VvR
v
Ÿ=
w
0.
10.17.2. Theorem. The set m(G n ( R n+k )) is a smooth submanifold of P N-1 and is the
set of zeros of a finite set of quadratic homogeneous polynomials called the Plücker
relations .
Proof.
See the algebraic geometry references listed above.
Finally, one can generalize the definition of the Grassmann varieties both by allow-
ing a more general field and by passing from an affine to a projective version. In par-
ticular, G n ( C n+k ) denotes the space of complex n-dimensional linear subspaces of C n+k
and G n ( P n+k ) and G n ( P n+k ( C )) denote the spaces of n-dimensional linear subspaces of
the projective spaces P n+k and P n+k ( C ), respectively. In all these cases one gets nice
varieties that are also manifolds.
10.18
N-Dimensional Varieties
This section gives a very brief overview of algebraic geometry in higher dimensions.
We continue the approach to higher-dimensional varieties taken in Section 10.16. The
results will divide up into two types, those that deal with properties of varieties that
are local in nature and those that are global . For example, some theorems deal with
what neighborhoods of points look like (see Theorem 10.18.9). This is a local prop-
erty. Others deal with whether or not the space is connected, whether it is orientable
(in the case where we are dealing with a manifold), or what its homology groups or
derived invariants, such as the Euler characteristic, are. These are global properties.
In addition to considering intrinsic properties of varieties by themselves, we also want
to generalize how intersections behave. We shall see that what we learned about plane
curves will generalize if we use the concept of codimension.
Let V = V(f 1 ,f 2 ,...,f m ), f i ΠC [X 1 ,X 2 ,...,X n ], be a variety in C n . Define
n m
: CC
F
Æ
by
() =
(
() ()
()
)
F
x
f
x
,
f
x
,...
f m
x
.
1
2
Here is a local criterion for smoothness.
10.18.1. Theorem.
If V is irreducible, then it is smooth at a point p if and
only if
codim
V
=
rank of Jacobian matrix F at
¢
p
.
(10.93)
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