4.3.A
Let
![$ A$](img2.png)
be a non-singular dividing curve of
degree
![$ m$](img11.png)
. Let
![$ L_0$](img611.png)
,
![$ L_1$](img297.png)
be real lines,
![$ C$](img67.png)
be one of two
components of
![$ \mathbb{R}P^2\smallsetminus (\mathbb{R}L_0\cup\mathbb{R}L_1)$](img668.png)
. Let
![$ \mathbb{R}L_0$](img620.png)
and
![$ \mathbb{R}L_1$](img621.png)
be oriented so that the projection
![$ \mathbb{R}L_0\to\mathbb{R}L_1$](img669.png)
from
a point lying in
![$ \mathbb{R}P^2\smallsetminus (C\cup\mathbb{R}L_0\cup \mathbb{R}L_1)$](img670.png)
preserves the orientations. Let ovals
![$ u_0$](img671.png)
,
![$ u_1$](img672.png)
of
![$ A$](img2.png)
lie in
![$ \mathbb{R}P^2- C$](img673.png)
and
![$ u_i$](img674.png)
is
tangent to
![$ L_i$](img310.png)
at one point
![$ (i=1, 2)$](img675.png)
. If the intersection
![$ \mathbb{R}A\cap C$](img660.png)
consists of
![$ m-2$](img676.png)
components, each of which is an arc connecting
![$ \mathbb{R}L_0$](img620.png)
with
![$ \mathbb{R}L_1$](img621.png)
, then points of tangency of
![$ u_0$](img671.png)
with
![$ L_0$](img611.png)
and
![$ u_1$](img672.png)
with
![$ L_1$](img297.png)
are positive with respect to one of the complex
orientations of
![$ A$](img2.png)
.
Proof.
Assume the contrary: suppose that with respect to a
complex orientation of
![$ A$](img2.png)
the tangency of
![$ u_0$](img671.png)
with
![$ L_0$](img611.png)
is
positive and the tangency of
![$ u_1$](img672.png)
with
![$ L_1$](img297.png)
is negative. Rotate
![$ L_0$](img611.png)
and
![$ L_1$](img297.png)
around the point
![$ L_0\cap L_1$](img678.png)
in the directions
out of
![$ C$](img67.png)
by small angles in such a way that each of
the lines
![$ L'_0$](img630.png)
and
![$ L'_1$](img631.png)
obtained intersects transversally
![$ \mathbb{R}A$](img15.png)
in
![$ m$](img11.png)
points. Perturb the union
![$ A\cup L'_0$](img679.png)
and
![$ A\cup L'_1$](img680.png)
obeying the
orientations. By
2.3.A, the nonsingular curves
![$ B_0$](img681.png)
and
![$ B_1$](img682.png)
obtained are of type I. It is easy
to see that their complex schemes can be obtained one from another by
relocating the oval, appeared from
![$ u_1$](img672.png)
(see Figure
28). This
operation changes one of the numbers
![$ \Pi^+-\Pi^-$](img683.png)
and
![$ \Lambda ^+-\Lambda ^-$](img684.png)
by 1. Therefore the left hand side of the complex orientation formula
is changed. It means that the complex schemes both of
![$ B_0$](img681.png)
and
![$ B_1$](img682.png)
can not satisfy the complex orientation formula.
This proves that the assumption is not true.