Classification of flexible \(M\)-curves of degree 8 up to isotopy.

*(English)*Zbl 1105.14080From the introduction: In the first part of his 16-th problem, Hilbert asked how the connected components of a plane real algebraic curve of degree \(m\) can be arranged on \(\mathbb{R}\mathbb{P}^2\) up to isotopy. At that time, the answer was known only for \(m\leq 5\) solved this problem for \(m= 6\) and Viro did it for \(m= 7\). For \(m= 8\), the complete answer is still unknown. We shall call the real scheme of a curve the arrangement of its connected components on \(\mathbb{R}\mathbb{P}^2\) up to isotopy.

It is reasonable to start the classification with \(M\)-curves (a curve is called an \(M\)-curve if it has the maximal possible number \((m-1)(m- 2)/2+ 1\) of connected components). After the studies of Fiedler, Viro, Shustin, Korchagin, and Chevallier, there remained only 9 real schemes whose realizability was open. Here we exclude two of them:

Theorem 1.1. There do not exist real algebraic curves of degree 8 with real schemes

\[ \langle 1\sqcup 1(3)\langle 3\rangle\sqcup 1\langle 6\rangle\rangle, \quad \langle 1\sqcup 1\langle 6\rangle\sqcup 1\langle 13\rangle\rangle. \]

This is an immediate consequence of Theorem 1.2 (a) below.

We say that a real scheme in \(\mathbb{R}\mathbb{P}^2\) is realized by a real pseudo-holomorphic curve \(A\) if it is isotopic in \(\mathbb{R}\mathbb{P}^2\) to \(\mathbb{R} A= A\cap\mathbb{R}\mathbb{P}^2\), the set of real points of \(A\). Here we complete the classification of up to isotopy of the arrangements of 22 ovals on \(\mathbb{R}\mathbb{P}^2\) realizable by real pseudo-holomorphic \(M\)-curves of degree 8:

Theorem 1.2. (a) The two real schemes in Theorem 1.1. are not realizable by real pseudo-holomorphic curves of degree 8. (b) The other seven real schemes (1) are realizable by real pseudoholomorphic curves of degree 8.

It is reasonable to start the classification with \(M\)-curves (a curve is called an \(M\)-curve if it has the maximal possible number \((m-1)(m- 2)/2+ 1\) of connected components). After the studies of Fiedler, Viro, Shustin, Korchagin, and Chevallier, there remained only 9 real schemes whose realizability was open. Here we exclude two of them:

Theorem 1.1. There do not exist real algebraic curves of degree 8 with real schemes

\[ \langle 1\sqcup 1(3)\langle 3\rangle\sqcup 1\langle 6\rangle\rangle, \quad \langle 1\sqcup 1\langle 6\rangle\sqcup 1\langle 13\rangle\rangle. \]

This is an immediate consequence of Theorem 1.2 (a) below.

We say that a real scheme in \(\mathbb{R}\mathbb{P}^2\) is realized by a real pseudo-holomorphic curve \(A\) if it is isotopic in \(\mathbb{R}\mathbb{P}^2\) to \(\mathbb{R} A= A\cap\mathbb{R}\mathbb{P}^2\), the set of real points of \(A\). Here we complete the classification of up to isotopy of the arrangements of 22 ovals on \(\mathbb{R}\mathbb{P}^2\) realizable by real pseudo-holomorphic \(M\)-curves of degree 8:

Theorem 1.2. (a) The two real schemes in Theorem 1.1. are not realizable by real pseudo-holomorphic curves of degree 8. (b) The other seven real schemes (1) are realizable by real pseudoholomorphic curves of degree 8.

##### MSC:

14P25 | Topology of real algebraic varieties |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |