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(
) -
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
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