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Let
V
1
and
V
2
be two irreducible varieties in
C
n
10.18.5. Theorem.
that have a
nonempty
intersection. Then
(
)
£
codim
VV
«
codim
V
+
codim
V
.
1
2
1
2
Proof.
See [Kend77].
If
V
1
and
V
2
are irreducible varieties in
P
n
(
C
), then
10.18.6. Corollary.
(
)
£
codim
VV
«
codim
V
+
codim
V
.
1
2
1
2
Proof.
This corollary is an immediate consequence of Theorem 10.18.5 because
any
two varieties in
P
n
(
C
) intersect.
We say that two irreducible varieties
V
1
and
V
2
in
C
n
or
P
n
(
C
)
intersect
Definition.
properly
if
(
)
£
codim
VV
«
codim
V
+
codim
V
.
1
2
1
2
Two arbitrary varieties
V
1
and
V
2
are said to
intersect properly
if each irreducible com-
ponent of
V
1
intersects properly with each irreducible component of
V
2
.
The idea of intersecting properly tries to capture the idea that two varieties overlap
as little as possible. Too much overlap corresponds to degenerate cases about which
not much can be said. The condition on codimension in the definition of intersecting
properly is equivalent to saying that the dimension of the intersection is as small as
possible, namely, dim
V
1
+ dim
V
2
- n. The ideal case is where varieties intersect trans-
versally but the weaker condition of intersecting properly is adequate. To see that to
intersect properly is not the same as intersecting transversally consider the varieties
V(X
2
+ Y
2
- 1) and V(Z - 1), which intersect properly but not transversally. If varieties
intersect transversally, then they will also intersect properly, so that the former con-
dition is stronger than the latter.
In order to state the generalized version of Bèzout's theorem we need to define
the degree of an arbitrary variety. There are a number of ways to do this, but first of
all, just like in the case of dimension, one has to agree on what the degree should be
in simple cases. We already agreed earlier in Section 10.5 that a hypersurface defined
by an irreducible polynomial should have degree equal to the degree of that polyno-
mial. A natural way to deal with the general case would be to divide it into two steps:
Step 1:
Define the degree of an arbitrary irreducible variety (which may not be
defined by a single irreducible polynomial since the polynomial ring in more than
one variable is not a principal ideal domain).
Step 2:
Define the degree of an arbitrary variety to be the sum of the degrees of
its irreducible components.
Step 2 is plausible given the relationship between the degree of a plane curve and the
number of intersections it has with a line discussed at the beginning of Section 10.6.
Step 1 is clearly the hard part but we shall deal with it in a similar way, in terms of
intersections with linear subspaces.
