3.6 Flexible Curves of Degrees $\le 5$

In this subsection, I show that for degrees $\le 5$ the prohibitions coming from topology allow the same set of complex schemes as all prohibitions. The set of complex schemes of algebraic curves of degrees $\le 5$ was described in 2.8. In fact the same is true for degree 6 too. For degree greater than 6, it is not known, but there is no reason to believe that it is the case.

Degrees $\le 3$. Theorems 3.3.A and the Harnack inequality 3.3.B prohibit all non realizable real schemes for degree $\le 3$. To obtain the complete set of prohibitions for complex schemes of degrees $\le 3$ one has to add the Klein congruence 3.4.B, 3.4.D and the complex orientation formula 3.4.C; cf. Section 2.8.

Degree 4. By the Arnold inequlity 3.3.K, a flexible curve of degree 4 cannot have a nest of depth 3. By the Arnold inequality 3.3.J, it has at most one nonempty positive oval, and if it has a nonempty oval then, by the extremal property 3.3.L of this inequality, the real scheme is $\langle1\langle1 \rangle\rangle$. Together with 3.3.A and the Harnack inequality 3.3.B, this forms the complete set of prohibitions for real schemes of degree 4.

From the Klein congruence 3.4.B, it follows that the real schemes $\langle 1\rangle$ and $\langle3 \rangle$ are of type II. The empty real scheme $\langle 0\rangle$ is of type II by 3.4.A. By the extremal property 3.4.D of the Harnack inequality, $\langle 4\rangle$ is of type I. The real scheme $\langle 2\rangle$ is of type II by the complex orientation formula 3.4.C, cf. Section 2.8. By 3.4.F, the scheme $\langle1\langle1 \rangle\rangle$ is of type I. By the complex orientation formula, it admits only the complex orientation $\langle1\langle1^-\rangle\rangle$.

Degree 5. By the Viro-Zvonilov inequality 3.3.O, a flexible curve of degree 5 can have at most one nonempty oval. By the extremal property of this inequality 3.3.P, if a flexible curve of degree 5 has a nonempty oval, then its real scheme is $\langle J\amalg1\langle1\rangle\rangle$. Together with 3.3.A and the Harnack inequality 3.3.B, this forms the complete set of prohibitions for real schemes of degree 5.

From the Klein congruence 3.4.B, it follows that the real schemes $\langle J\amalg 1\rangle$, $\langle J\amalg 3\rangle$, and $\langle J\amalg 5\rangle$ are of type II. From the complex orientation formula, one can deduce that the real schemes $\langle J\rangle$ and $\langle J\amalg 2\rangle$ are of type II, cf. 2.8. By the extremal property 3.4.D of the Harnack inequality, $\langle J\amalg 6\rangle$ is of type I. The complex orientation formula allows only one complex semiorientation for this scheme, namely $\langle J\amalg 3^-\amalg3^+\rangle$. By the 3.4.H, the real scheme $\langle J\amalg1\langle1\rangle\rangle$ is of type I. The complex orientation formula allows only one complex semiorientation for this scheme, namely $\langle J\amalg1^-\langle1^-\rangle\rangle$, cf. 2.8. The real scheme $\langle J\amalg 4\rangle$ is of indefinite type (even for algebraic curves, see 2.8). In the case of type I, only one semiorientation is allowed by the the complex orientation formula. It is $\langle J\amalg 3^-\amalg1^+\rangle$.