arXiv subjects
Publications and source records attributed to Séverine Fiedler-Le Touzé.
In this note, we consider M-curves of odd degree with real scheme of the form $< \mathcal{J} \amalg α\amalg 1 < β> >$. With help of complex orientations, we prove that for $m = 9$, $α\geq 2$, and for $m = 11$, $α\geq 3$.
We consider the $M$-curves of degree nine with three nests $1 \langle α_i \rangle, i = 1, 2, 3$ in $\mathbb{R}P^2$. After systematic constructions, Korchagin conjectured that at least two of the $α_i$ must be odd. It was later proved that there is always one odd $α_i$. We say that the curve has a jump in a non-empty oval $O$ if there exist four ovals $A, B, C, D$, with $A$ interior to some other non-empty oval $O'$, $D$ exterior, $B, C$ interior to $O$, such that $B$ and $C$ are separated inside of $O$ by any line passing through $A$ and $D$. In this paper, we prove the conjecture for the curves without jump, and we find restrictions on the complex orientations and rigid isotopy types admissible for the curves even, even, odd with jump.
Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic a topological type (cubic, points), and combinatorial pencil the cyclic sequence of five combinatorial reducible cubics. Up to the action of the symmetric group $S_5$ on $\{2, ...6\}$, there are seven possible combinatorial pencils with node at 1. Consider now the set of six pencils obtained, making the node to be $1, ...6$. Up to the action of S_6 on ${1, ...6}$, there are four possible lists of six combinatorial pencils. Let seven points $1, ...7$ lie in general position in the plane. Up to the action of $S_7$ on $\{1, ...7\}$, there are fourteen possible lists of seven nodal combinatorial cubics passing through the seven points, with respective nodes at $1, ...7$.
To a generic configuration of eight points in convex position in the plane, we associate a list consisting of the following information: for all of the 56 conics determined by five of the points, we specify the position of each remaining point, inside or outside. We prove that the number of possible lists, up to the action of $D_8$ on the set of points, is 47, and we give two possible ways of encoding these lists. A generic compex pencil of cubics has twelve singular (nodal) cubics and nine distinct base points, any eight of them determine the ninth one, hence the pencil. If the base points are real, exactly eight of these singular cubics are distinguished, that is to say real with a loop containing some base points. We call combinatorial cubic a topological type (cubic, base points), and combinatorial pencil the sequence of eight successive combinatorial distinguished cubics. Let us choose representants of the 47 orbits. In most cases, but four exceptions, a list determines a unique combinatorial pencil. We give a complete classification of the combinatorial pencils of cubics with eight base points in convex position. Up to the action of $D_8$ on the set of points, there are $43$ such pencils.
A real algebraic plane curve $A$ is said to be dividing if its real part $\mathbb{R}A$ disconnects its complex part $\mathbb{C}A$. A pencil of curves is totally real with respect to $A$ if it has only real intersections with $\mathbb{C}A$. If there exists such a pencil, then $A$ is dividing, this is the case for the $M$-curves. Can conversely any dividing curve be endowed with a totally real pencil? We study here the case of $M-2$-sextics having 2 or 6 empty exterior ovals. Such sextics are always dividing. We prove that they may actually be endowed with a totally real pencil of cubics.
The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of a given degree $m$. For $m = 9$, the classification of the $M$-curves is still wide open. Let $C_9$ be an $M$-curve of degree 9 and $O$ be a non-empty oval of $C_9$. If $O$ contains in its interior $α$ ovals that are all empty, we say that $O$ together with these $α$ ovals forms a nest. The present paper deals with the $M$-curves with three nests. Let $α_i, i = 1, 2, 3$ be the numbers of empty ovals in each nest. We prove that at least one of the $α_i$ is odd. This is a step towards a conjecture of A. Korchagin, claiming that at least two of the $α_i$ should be odd.
The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of given degree $m$. For $m \geq 8$, one restricts the study to the case of the $M$-curves. For $m=9$, the classification is still wide open. We say that an $M$-curve of degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we prohibit 10 isotopy types with deep nests and no outer ovals.