Graphics Reference
In-Depth Information
Proof.
See [Seid68]. The transformation (10.79) sets up the desired correspondence
between parameterizations.
The next lemma is another special case of Theorem 10.12.6.
10.12.14. Lemma.
An ordinary r-fold point
p
of
C
has exactly r parameterizations
centered at
p
that are linear and that have the same tangents as those of
C
.
Proof.
We may assume that the r-fold point, r > 0, is the origin and, since our base
field is the complex numbers,
r
'
1
(
)
=
(
)
+
fXY
,
Y mX
i
-
....
i
=
The transformed curve
C
n
will be defined by
r
'
1
(
)
=
(
)
+
(
)
fXY
,
Y m
-
X
... .
n
i
i
=
It follows that X = 0 intersects
C
n
precisely in the points (0,m
i
). Furthermore, the
multiplicity of the intersection points is 1 because all the m
i
are distinct. This
means that the points (0,m
i
) are simple points for
C
n
and that
C
n
has r parameteriza-
tions centered on X = 0. We conclude that
C
has r branches centered at the origin. From
Lemma 10.12.7 and Theorem 10.12.8, the parameterization of
C
n
at (0,m
i
) is given by
Xt
Ymt
i
=
=
...
and its tangent line is Y = m
i
X. The lemma is proved.
10.12.15. Lemma.
Any irreducible curve can be transformed into a curve with only
ordinary singularities with a finite sequence of quadratic transformations.
Proof.
See [Walk50].
After these preliminaries, we are ready to prove Theorem 10.12.6.
Sketch of proof for Theorem 10.12.6.
Lemma 10.12.7 and Lemma 10.12.14
already proved two special cases. Let
p
be an arbitrary r-fold point of
C
. We may
assume that
C
is irreducible,
p
=
0
, and X = 0 is not a tangent of
C
at
0
. Since our
base field is the complex numbers,
r
'
1
(
)
=
(
)
+
fXY
,
Y mX
-
...,
i
i
=
where the m
i
are not necessarily distinct. It follows that
