3.8 Prohibitions not Proven for Flexible Curves

In conclusion of this section, let us come back to algebraic curves. We see that to a great extent the topology of their real point sets is determined by the properties which were included into the definition of flexible curves. In fact, it has not been proved that it is not determined by these properties completely. However some known prohibitions on topology of real algebraic curves have not been deduced from them.

As a rule, these prohibitions are hard to summarize, in the sense that it is difficult to state in full generality the results obtained by some particular method. To one extent or another, all of them are consequences of Bézout's theorem.

Consider first the restrictions which follow directly from the Bézout theorem. To state them, we introduce the following notations. Denote by $h_r$ the maximum number of ovals occurring in a union of $\le r$ nestings. Denote by $h'_r$ the maximum number of ovals in a set of ovals contained in a union of $\le r$ nests but not containing an oval which envelops all of the other ovals in the set. Under this notations Theorems 1.3.C and 1.3.D can be stated as follows:

3.8.A   $h_2\le m/2$; in particular, if $h_1=[m/2]$, then $l=[m/2]$.

3.8.B   $h'_5\le m$; in particular, if $h'_4=m$, then $l=m$.

These statements suggest a whole series of similar assertions. Denote by $c(q)$ the greatest number $c$ such that there is a connected curve of degree $q$ passing through any $c$ points of $\mathbb{R}P^2$ in general position. It is known that $c(1)=2$, $c(2)=5$, $c(3)=8$, $c(4)=13$

3.8.C. (Generalization of Theorem 3.8.A).   If $r\le c(q)$ with $q$ odd, then

$$h_r+\left[c(q)-\frac r2\right]\le \frac {qm}2.$$

In particular, if $h_{c(q)-1}=\left[\frac{qm}2\right]$, then $l=\left[\frac{qm}2\right]$.

3.8.D. (Generalization of Theorem 3.8.B).   If $r\le c(q)$ with $q$ even, then

$$h'_r+[(c(q)-r)/2]\le qm/2.$$

In particular, if $h'_{c(q)-1}=qm/2$, then $l=qm/2$.

The following two restrictions on complex schemes are similar to Theorems 3.8.A and 3.8.B. However, I do not know the corresponding analogues of 3.8.C and 3.8.D.

3.8.E   If $h_1=\left[\frac m2\right]$, then the curve is of type I.

3.8.F   If $h'_4=m$, then the curve is of type I.

Here I will not even try to discuss the most general prohibitions which do not come from topology. I will only give some statements of results which have been obtained for curves of small degree.

3.8.G   There is no curve of degree 7 with real scheme $\langle J\amalg 1\langle 14\rangle\rangle$.

3.8.H   If an M-curve of degree 8 has real scheme $\langle \alpha\amalg 1\langle \beta\rangle\amalg 1\langle \gamma\rangle\amalg 1 \langle \delta\rangle\rangle$ with nonzero $\beta,\gamma$ and $\delta$, then $\beta,\gamma$ and $\delta$ are odd.

3.8.I   If an $(M-2)$-curve of degree $8$ with $p-n\equiv 4\mod 8$ has real scheme $\langle \alpha\amalg 1\langle \beta\rangle\amalg 1\langle \gamma\rangle\amalg 1 \langle \delta\rangle\rangle$ with nonzero $\beta,\gamma$ and $\delta$, then two of the numbers $\beta,\gamma,\delta$ are odd and one is even.

Proofs of 3.8.G and 3.8.H are based on technique initiated by Fiedler [Fie-82]. It will be developed in the next Section.