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Classification of flexible \(M\)-curves of degree 8 up to isotopy. (English) Zbl 1105.14080
From 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.

14P25 Topology of real algebraic varieties
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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