Graphics Reference
In-Depth Information
Consider the curve V(Y - X
2
) in
R
2
. Using only the definition, find the multiplicity of
the curve at (-1,1) and its tangent line.
10.6.2.
10.6.3.
Draw the plane curves below and analyze them at the point
p
like we did in Examples
10.6.6-10.6.10.
V (X
2
+ Y
2
(a)
- 2Y),
p
= (1,1)
V (X
3
- Y
4
),
p
= (0,0)
(b)
V ((X
2
+ Y
2
)
2
+ 3X
2
Y - Y
3
),
p
= (0,0)
(c)
Show that any curve in
C
2
of degree n that has a point
p
of multiplicity n consists of
n (not necessarily distinct) lines through
p
.
10.6.4.
Section 10.7
Consider the curves
C
1
= V(YZ - X
2
) and
C
2
= V(Y
2
- XZ) in
P
2
(
C
). Compute the inter-
section multiplicities m
p
(
C
1
,
C
2
) at the intersection points
p
and verify the validity of
Bézout's theorem
10.7.1.
Section 10.8
Prove that the graph of any polynomial function f(x,y) in
R
3
is an irreducible variety.
10.8.1.
10.8.2.
Show that
2
2
2
<
XXY
,
>=<
X
>«<
XY
,
>=<
X
>«<
XX Y
,
+
>
are two irredundant intersections of irreducible ideals. Therefore the direct analog of
Theorem 10.8.16 for ideals fails. The closest we come is Theorem B.6.8. By this
theorem and Lemma B.6.7 only the associated intersection into prime ideals is unique.
10.8.3.
Prove that the radical of a primary ideal in a commutative ring is prime.
10.8.4.
Prove that a prime ideal in a commutative ring is irreducible.
10.8.5.
This exercise describes another approach to projective varieties. Assume that the field
k is infinite in this exercise.
Definition.
We say that a polynomial f Πk[X
1
,X
2
,...,X
n+1
]
vanishes
at a point
p
Œ
P
n
(k) if
(
)
=
(
)
=
[
]
f c
,
c
,...,
c
0
for all
c
,
c
,...
c
with
p
c
,
c
,...,
c
.
12
n
+
1
12
n
+
1
12
n
+
1
Given a set S of polynomials in k[X
1
,X
2
,...,X
n+1
], define
{
}
n
()
=Œ
()
VS
pP
k
every f
Œ
S vanishes at
p
.
Any such subset of
P
n
(k) is called a
projective variety
.
(a)
Let f Πk[X
1
,X
2
,...,X
n+1
]. Show that if
ff f
=++ +
01
...
d
,
