Graphics Reference
In-Depth Information
(
)
-
2
(
)
(
)
=+++-
2
2
2
2
2
22
2
f XYZ XYZrs
,,
4
rXY
+
.
The torus
T
intersects the plane
P
defined by Z = s in a circle defined by
2
2
2
XYr
+-=.
0
Note that
P
= V(Z - s). Note also that
T
and
X
intersect properly but not transversally.
Now consider everything as defined over the complex numbers. The correspond-
ing
complex
varieties V(f) and V(Z - s) have degree 4 and 2, respectively. The complex
variety
()
«-
(
)
S
=
Vf
VZ s
has degree 2. It is also easy to see that
(
()
(
)
)
=
iVf VZ s
,
-
;
p
2
for all points
p
in
S
, and so
(
()
(
)
)
=
iVf VZ s
,
-
;
S
2
.
It follows that
()
(
)
=
Vf VZ s
•
-
2
S
,
and
(
()
(
)
)
=
(
()
«-
(
)
)
=
deg
Vf VZ s
•
-
2
deg V
f
VZ s
4
.
We can at last state Bèzout's theorem.
10.18.16. Theorem.
(Bézout's Theorem) If two pure dimensional varieties
V
and
W
in
P
n
(
C
) intersect properly, then
(
)
=
(
)(
)
deg
VW
•
deg
V
deg
W
.
If they intersect transversally, then
(
)
=
(
)(
)
deg
VW
«
deg
V
deg
W
.
Proof.
See [Kend77].
10.18.17. Corollary.
If two plane curves in
P
2
(
C
) of degree m and n intersect in
more than nm points counted with their multiplicity, then they must have at least one
irreducible component in common.
There is an alternate approach to the degree of a variety and Bèzout's theorem
that connects these ideas to homology theory and topology. Suppose that
V
is a