Graphics Reference
In-Depth Information
10.19
E
XERCISES
Section 10.1
10.1.1.
Show that
C
- {
0
} cannot be a variety in
C
.
10.1.2.
Show that a proper nonempty variety in
C
consists of a finite set of points.
Section 10.2
Show that the closure in
P
2
(
C
) of the variety in Example 10.2.3 really is a sphere.
10.2.1.
Section 10.3
10.3.1.
Prove Proposition 10.3.1.
Find coordinate neighborhoods (
U
,j) for
P
2
so that the lines defined by the equations
below becomes their lines at infinity:
(a)
10.3.2.
Y = 0
(b)
X - 3Y = 0
Find coordinate neighborhoods (
U
,j) for
P
3
10.3.3.
so that the planes defined by the equa-
tions below becomes their planes at infinity:
(a)
W = 0
(b)
X + Y + Z + W = 0
10.3.4.
Consider the polynomial
(
)
=
2
gXY
,
YX X
+
-
1
and let
W
= V(g) Õ
R
2
. Show that the topological closure of
W
in
P
2
is
W
» {[0,1,0]}.
10.3.5.
Consider the polynomial
(
)
=-
2
2
(
)
fXY
,
Y
X X
-
1
and let
X
be the projective variety in
P
2
(
C
) defined by H(f). Show that topologically
X
is a pinched sphere. (
Hint:
We can relate this problem to the one in Example 10.2.4,
which considered the variety
(
)
2
(
)
(
)
VY
-+
X
e
XX
-
1
for e=1. If we let e go to zero, then we will get the variety
V
= V(f) Õ
R
2
. In Figure
10.4 (a) the circle will shrink to a point.)
Find a coordinate system for
P
2
that shows that the projective completion of the curve
XY = 1 in
R
2
is an ellipse.
10.3.6.
Show that for any finite set of points in
P
2
, there is a line in
P
2
that does not pass
through these points.
10.3.7.
(a)