Graphics Reference
In-Depth Information
Definition.
The fixed number in Theorem 10.18.12 is called the
intersection multi-
plicity
of
V
1
and
V
2
at
p
and is denoted by i(
V
1
,
V
2
;
p
).
10.18.13. Theorem.
Let
V
1
and
V
2
be any two pure dimensional varieties in
C
n
or
P
n
(
C
) which intersect properly. If
C
is an irreducible component of
V
1
«
V
2
, then at
almost every point
p
in
C
, i(
V
1
,
V
2
;
p
) has a common fixed value.
Proof.
See [Kend77].
Definition.
The fixed number in Theorem 10.18.13 is called the
intersection multi-
plicity
of
V
1
and
V
2
along
C
and is denoted by i(
V
1
,
V
2
;
C
).
Finally,
Definition.
Let
V
1
and
V
2
be any two pure dimensional varieties in
C
n
or
P
n
(
C
) that
intersect properly. The formal sum
m
Â
(
)
i
VV CC
,
;
i
2
i
i
i
=
1
over all the distinct irreducible components
C
i
of
V
1
«
V
2
is called the
intersection
product
of
V
1
and
V
2
and is denoted by
V
1
•
V
2
.
Note the purely formal nature of the intersection product
V
1
•
V
2
and its simi-
larity between the formal sum that it is and the formal sums that are used to define
homology groups. We can now tie together the concepts of intersection degree and
intersection multiplicities, namely,
If
V
1
and
V
2
are any two pure dimensional varieties in
C
n
10.18.14. Theorem.
or
P
n
(
C
) that intersect properly, then
m
Â
i
(
)
=
(
)
deg
VV
•
VV C
,
,
deg
C
.
12
12
i
i
i
=
1
where the sum on the right-hand side of this equation is taken over all the distinct
irreducible components
C
i
of
V
1
«
V
2
.
Proof.
See [Kend77].
Note that the right hand side of the equation in Theorem 10.18.14 would be the
natural definition for the degree of the intersection product
V
1
•
V
2
although we do not
bother to make such a definition.
10.18.15. Example.
Let
T
be the torus in
R
3
obtained by rotating the circle in the
x-z plane with center (r,0,0) and radius s, where r > s > 0, about the z-axis.
Analysis.
As a surface of revolution, it is easy to see that
T
is the
real
hypersurface
V(f), where
