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Find a parameterization of the curve using the method described in Example 10.13.1
and the point (3/2,1/2) on the curve.
10.13.2.
Prove Theorem 10.13.2(2).
10.13.3.
(a)
If f Œ k[ V ] is a polynomial function on an affine variety V , then f : V Æ k is a con-
tinuous function with respect to the Zariski topology.
(b)
Generalize (a) and prove that any polynomial function between affine varieties
is continuous.
10.13.4.
Prove or disprove that the following maps define isomorphisms:
(a)
f:V(XY - 1) Æ R , f(x,y) = x
g : R Æ V (Y 3
- X 4 ), g(t) = (t 3 ,t 4 )
(b)
h : R Æ V (Y - X k ), h(t) = (t,t k )
(c)
Let X and Y be varieties in k n . Let D = { ( v , v ) | v Πk n } be the diagonal in k 2n
10.13.5.
=
k n
¥ k n .
Show that X ¥ Y and D are varieties in k 2n .
(a)
(b)
Define
2
() = (
)
j
:
XY
«Æ
k y
j
v vv
,
.
Show that j defines an isomorphism between X « Y and ( X ¥ Y ) « D. In other
words, one can replace an intersection between varieties with the intersection
of another variety and a linear variety.
10.13.6.
Show that a rational function u : V Æ W between varieties V and W is dominant if
and only if W is the smallest variety in W containing u( V ).
Let f(X,Y) = X 3
- X 2
+ Y 2 and g(X,Y) = X 2
+ Y 2
10.13.7.
+ X. Show that the map
2
Ê
Á
X
XY
XY
XY
ˆ
˜
j XY
(
,
) =
,
2
2
2
2
+
+
sends the variety V(f) to the variety V(g). Show also that the two places of f with
center 0 are mapped to places of g with distinct centers. See Figure 10.22.
Y
X 3 - X 2 + Y 2 = 0
X
X 2 + Y 2 + X = 0
Figure 10.22.
The curves in Exercise
10.13.7.
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