Graphics Reference
In-Depth Information
(a)
(b)
Figure 10.8.
Reducible and irreducible varieties.
(
)
=+-
2
2
fXYZ
,,
X
Y
1
,
1
(
)
=+-
2
2
f
X Y Z
,,
X
Z
1
,
and
2
(
)
=+-
2
2
fXYZ X
,,
Z
4
.
3
V
is the intersection of two cylinders V(f
1
) and V(f
2
) of radius 1 centered on the z-
and y-axis, respectively. See Figure 10.8(a). It is easy to check that
V
is the union of
two ellipses that are the intersection of the cylinder V(f
1
) and the planes V(X - Z) and
V(X + Z), respectively. In other words,
(
)
»
(
)
V
=
Vf X Z
,
-
Vf X Z
,
+
,
1
1
and so
V
is reducible. The variety
W
, on the other hand, is the intersection of the
cylinder V(f
1
) and the cylinder V(f
3
) of radius 2, which is centered on the y-axis. See
Figure 10.8(b). It is irreducible because this time the two connected pieces of
W
cannot be separated by varieties. Although we do not yet have the tools to prove this,
we sketch the steps in the argument. The reader should return to this example after
reading Section 10.18. First of all, one needs to look at this as a problem in
C
3
. One
also needs to know about pure dimensional varieties, their degree, and Bézout's
Theorem for them. The degree of an n-dimensional variety in
C
m
has to do with the
number of points in the intersection of (m - n)-dimensional planes with the variety.
The complex varieties defined by f
1
and f
3
are pure two-dimensional varieties of degree
2 (seen in
R
3
by the fact that “most” lines intersect V(f
1
) and V(f
3
) in 2 points). It
follows from Bézout's Theorem (Theorem 10.18.16) that the degree of their intersec-
tion is the product of the two degrees, namely, 4. Assume that the intersection were
reducible and a union of two varieties of degree r and s, respectively. Then each would
contain precisely one of the two components of
W
. Such a union would have degree
r + s. It would follow that r + s = 4. But this leads to a contradiction because neither
degree 1 or 2 is possible (again seen in
R
3
by the fact that there are too many hori-
zontal and almost horizontal planes that meet the curves in
W
in 4 points, something
that would violate Theorem 10.18.7 and the definition of degree).
Example 10.5.12 shows that determining whether or not a general variety is irre-
ducible can be tricky. Making this determination in the case of hypersurfaces is easier.
