Graphics Reference
In-Depth Information
Definition.
If
p
is a point of a pure r-dimensional variety
V
in
C
n
or
P
n
(
C
), then the
number d in Theorem 10.18.9 is called the
multiplicity of
V
at
p
or the
order of
V
at
p
and is denoted by m
p
(
V
).
Note that for a hypersurface
V
the multiplicity of
V
at
p
is ord
p
(f), just like for
plane curves.
10.18.10. Theorem.
Let
V
1
and
V
2
be varieties in
C
n
or
P
n
(
C
) of pure dimension r
and s, respectively, and let
L
be any (2n - r - s)-dimensional plane in
P
n
(
C
). For almost
all transforms
V
1
¢,
V
2
¢, and
L
¢ of
V
1
,
V
2
, and
L
, respectively, (
V
1
¢«
V
2
¢) «
L
¢ consists
of a common fixed number d of points.
Proof.
See [Kend77].
The value 2n - r - s = (n - r) + (n - s) in the theorem comes from the fact that
this is the codimension of
V
1
«
V
2
if they intersect transversally.
Definition.
If
V
1
and
V
2
are any two pure dimensional varieties in
C
n
or
P
n
(
C
) that
intersect properly, then the number d in Theorem 10.18.10 is called the
degree of inter-
section
of
V
1
and
V
2
and is denoted by deg(
V
1
•
V
2
).
Note.
The degree of intersection, deg(
V
1
•
V
2
), is in general
not
equal to
deg(
V
1
«
V
2
), the degree of the intersection of the two varieties, as one will see in
Example 10.18.15 below. The two degrees
are
the same if the varieties intersect
transversally.
We have all the definitions needed to state Bèzout's theorem; however, it is worth-
while to show how they relate to a generalized concept of multiplicity of intersections.
In analogy with the plane curve case, this concept is introduced by considering inter-
sections of linear subspaces with varieties.
10.18.11. Theorem.
Let
V
1
and
V
2
be varieties in
C
n
or
P
n
(
C
) of pure dimension r
and s, respectively, and let
L
be any (2n - r - s)-dimensional linear subspace. If
V
1
,
V
2
, and
L
intersect properly at a point
p
, then for almost all transforms
V
1
¢ of
V
1
near
V
1
,
V
2
¢ of
V
2
near
V
2
, and
L
¢ of
L
near
L
respectively, there is a common fixed number
d of distinct points of
V
1
¢«
V
2
¢«
L
¢ near
p
.
Proof.
See [Kend77].
Definition.
The fixed number d in Theorem 10.18.11 is called the
intersection
multiplicity
of
V
1
,
V
2
, and
L
at
p
and is denoted by i(
V
1
,
V
2
,
L
;
p
).
10.18.12. Theorem.
Let
V
1
and
V
2
be varieties in
C
n
or
P
n
(
C
) of pure dimension
r and s, respectively, which intersect properly. Then for almost all transforms
L
¢ of a
linear subspace
L
of dimension 2n - r - s that contain a point
p
, the number
i(
V
1
,
V
2
,
L
¢;
p
) is defined and has a common fixed value.
Proof.
See [Kend77].
