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\begin{document}
\title{SYMMETRIC PLANE CURVES OF DEGREE  $7$~: PSEUDO-HOLOMORPHIC AND ALGEBRAIC CLASSIFICATIONS}
\author{Erwan Brugall\'e}
\date{}
\maketitle
\begin{abstract}
Almost all known restrictions on the topology of nonsingular real algebraic curves in the projective plane  are also valid for a wider class of objects: real pseudo-holomorphic curves. It is still unknown if there exists a nonsingular real pseudo-holomorphic curve not isotopic in the projective plane to a real algebraic curve of the same degree.
In this article, we focus our study on symmetric real curves in the projective plane. We give a classification of real schemes (resp.  complex schemes) realizable by symmetric real curves  of degree $7$ with respect to the type of the curve (resp. $M$-symmetric real curves  of degree $7$).
In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudo-holomorphic curves of degree $7$ in the projective plane but not by algebraic ones. The algebraic prohibitions are proved using tools introduced by S. Orevkov: the comb theoretical method and the cubic resolvent.
\end{abstract}
\tableofcontents

\section{Introduction and statements of results}


\subsection{Brief History}
The origin of topology of real algebraic curves can be traced back to
the paper of A. Harnack which was published in 1876.
D. Hilbert formulated the following question in the 16-th problem of his famous list (see \cite{Hil}):
up to ambient isotopy of
$\mathbb RP^2$, how can be arranged the connected components of the real point set of a nonsingular real algebraic curve of a given degree $m$
in $\mathbb RP^2$?


In 1900, the answer was known only for $m\le5$ and nowadays the problem is also solved
for $m=6$ (D. A. Gudkov 1969, \cite{Gud}) and $m=7$ (O. Ya. Viro 1979).
\begin{thm}[Viro, \cite{V1}, \cite{V3}]\label{class7}
 Any nonsingular real algebraic curve of degree $7$ in $\mathbb RP^2$ has one of the following real schemes~:
\begin{itemize}
\item $\langle J\amalg \alpha\amalg 1\langle\beta\rangle\rangle$
 with $\alpha+\beta\le 14$, $0\le\alpha\le 13$, $1\le\beta\le 13$,
\item $\langle J\amalg \alpha\rangle$ with $0\le\alpha\le 15$,
\item $\langle J\amalg 1\langle 1 \langle 1\rangle\rangle\rangle.$
\end{itemize}
Moreover, any of these 121 real schemes is realizable by nonsingular real algebraic  curves of degree $7$ in $\mathbb RP^2$.
\end{thm}
\textbf{Remark. }The notations used to encode real schemes are the usual ones introduced in \cite{V3}\footnote{for example,
$\langle J\amalg \alpha\amalg 1\langle\beta\rangle\rangle$ means a non-contractible (in $\mathbb RP^2$) component,
an oval with $\beta$ ovals in its interior and $\alpha$ ovals in its exterior. All the ovals $\alpha$ and $\beta$ lie outside each other. An oval is a contractible (in $\mathbb RP^2$) component of the real curve}.

In the study of the topology of real algebraic varieties, one can distinguish two directions~: \textit{constructions} and \textit{prohibitions}.
For example, the historically first prohibition result, the Harnack
theorem,
asserts that the real part of a real algebraic curve of genus $g$ cannot have more than
$g+1$ connected components (curves with the maximal number of connected components are called \textit{$M$-curves}).
In \cite{Arn}, V. I. Arnold paved the way to the use of powerful
topological methods in the study of this problem.
Almost all known prohibitions
can be obtained via a topological study of the double covering of $\mathbb CP^2$ ramified along the complex point set of  a curve of even degree
or looking at the braid defined by the intersection of a curve and a certain sphere $S^3$ in $\mathbb CP^2$ (the interested reader can see the surveys \cite{DK}, \cite{V3} and \cite{Wil}).
In particular, almost all known restrictions on the topology of real algebraic curves are also valid for a
category of  more flexible objects~: \textit{pseudo-holomorphic curves}. These objects, introduced by M. Gromov to study symplectic $4-$manifolds
in \cite{Gro}, share a lot of properties with the algebraic curves (for example,
the Harnack theorem is still valid in the real
pseudo-holomorphic case) and are much easier to deal with. Indeed, there are methods to construct pseudo-holomorphic curves
which are not necessarily algebraic~: S. Yu. Orevkov (see \cite{O1}) proved that it is sufficient to show the quasipositivity of a braid,
and I. Itenberg and E. Shustin (see \cite{IS}) proved that any T-construction gives a pseudo-holomorphic curve
 even if
the convexity condition on the triangulation is not fulfilled.
For example, the classification up to isotopy of real pseudo-holomorphic $M$-curves
of degree $8$ in $\mathbb RP^2$ is achieved by Orevkov \cite{O2}. However,
it remains $6$ real schemes for which it is unknown whether they are
realizable by algebraic $M$-curves of degree $8$ in $\mathbb RP^2$.

Then a natural question arises~: does there exist a real scheme in  $\mathbb RP^2$ which would be realizable pseudo-holomorphically but not algebraically?
Strictly speaking (i.e., dealing with nonsingular
curves in the projective plane), this interesting problem is still open.
However,
dealing with
singular curves or curves in other algebraic surfaces, the answer is \textit{no}~: S. Fiedler Le Touz\'e and S. Orevkov
(see \cite{O2}, \cite{FTO}) exhibit mutual arrangements of two
nonsingular curves in $\mathbb RP^2$ which are realizable pseudo-holomorphically but not
algebraically (examples with many irreducible components were known for a long time, see the ``pseudo-holomorphic Pappus
theorem'' in
\cite{FTO}), and
Orevkov and Shustin (see \cite{OS1} and \cite{OS2}) exhibit nonsingular \textit{$\mathcal L$-schemes} in the second rational geometrically ruled surface $\Sigma_2$ which are realizable
pseudo-holomorphically but not algebraically. One can note that  pseudo-holomorphic curves are a special case of \textit{symplectic curves} and that J. Y. Welschinger constructed
in \cite{W} examples of symplectic curves realizing algebraically unrealizable real schemes in $\Sigma_n$ for $n\ge2$.

However in the case of
nonsingular curves in $\mathbb RP^2$, for each known algebraic classification, the pseudo-holomorphic classification is the same
(and even the proofs for both classifications are the same!).
Thus, Theorem \ref{class7} is still true replacing "algebraic" by "pseudo-holomorphic".
The existence of a real scheme in $\mathbb RP^2$ pseudo-holomorphically but not algebraically realizable  turns out to be a difficult problem. So, one can tackle a simpler question, looking for example at complex curves
which admit more symmetries
than the action of $\mathbb Z/2\mathbb Z$ given by the complex conjugation.
The first natural action is an additional holomorphic
action of $\mathbb Z/2\mathbb Z$, which can be given by a symmetry of the projective plane.
Such a real plane curve, invariant
under
a symmetry, is called a \textit{symmetric curve}. The systematic study of symmetric curves was started by T. Fiedler \cite{Fie} and continued
by S. Trille \cite{Tr}.
The \textit{rigid isotopy classes} (two nonsingular curves of degree $m$ in $\mathbb RP^2$ are said to be rigidly isotopic if
they belong to the same connected component
of the complement of the discriminant hypersurface in the space of curves of degree $m$) of nonsingular sextic
in $\mathbb RP^2$ which contain a symmetric curve can be obtain from \cite{It1}.
Recently (see \cite{It}), using auxiliary conics, I. Itenberg and V. Itenberg found an elementary proof of this classification.
Once again, algebraic and pseudo-holomorphic classifications coincide.
On the other hand, in \cite{OS2}, Orevkov and Shustin
showed that there exists a  real scheme which is realizable by nonsingular symmetric pseudo-holomorphic $M$-curves of
degree $8$, but which is not realizable by real symmetric algebraic curves of degree $8$.

Hence, it is natural to wonder about the degree $7$ and this is the subject of this paper. It turns out that  the classification of real schemes which are realizable by
 nonsingular symmetric curves of degree $7$ in $\mathbb RP^2$
are the same in both algebraic and pseudo-holomorphic cases, as well as  the classification of \textit{complex schemes} which are realizable by
 nonsingular symmetric $M$-curves of degree $7$ in $\mathbb RP^2$  (Theorems
\ref{Main thm1} and \ref{Main thm2}). However, if we look at real schemes which are realizable by \textit{nonsingular dividing
symmetric curves} of degree $7$ in $\mathbb RP^2$, the answers are different. In Theorems \ref{Main thm3} and \ref{Main thm4}, 
we exhibit two real schemes which are realizable by symmetric dividing pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$ but not
by algebraic ones.

We state our classification results in the next subsection. In section \ref{prelim}, we give some definitions and properties of the objects  used in this paper (\textit{rational geometrically rules surfaces, braid and comb associated to an $\mathcal L$-scheme}). We also give an efficient algorithm to deal with combs.
In section \ref{pseudo state}, we give results related to the pseudo-holomorphic category. Algebraic results are given in section \ref{algebraic state}.
In the pseudo-holomorphic case we mostly use the braid theoretical approach.
The algebraic results are obtained using various methods.

In section \ref{recap}, we recapitulate the proofs of the four main
theorems stated in section \ref{statement}.

\textbf{Aknowledgements : }I am very grateful to I. Itenberg for advices and encouragements, and to S. Orevkov for useful discussions.

\subsection{Classification results}\label{statement}
First we state two simple lemmas.
Surprisingly, we did not succeed to find these statements in the literature.
The proof of these prohibitions is a straightforward application of the Bezout
theorem, the Fiedler orientation alternating rule (see \cite{V3}) and the Rokhlin-Mischachev orientation formula (see \cite{Rok} or section \ref{prelim}),
and all the constructions are performed in \cite{Soum} and \cite{FLT2}.
\begin{lemma}\label{class7 div}
 Any nonsingular dividing real pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ has one of the following real schemes~:
\begin{itemize}
\item $\langle J\amalg \alpha\amalg 1\langle\beta\rangle\rangle$
 with $\alpha+\beta\le 14$, $\alpha+\beta=0$ mod $2$, $0\le\alpha\le 13$, $1\le\beta\le 13$, if $\alpha=0$ then $\beta\ne 2, 6, 8$ and if $\alpha=1$ then $\beta\ge 5$,
\item $\langle J\amalg \alpha\rangle$ with $6\le\alpha\le 15$, $\alpha=1$ mod $2$,
\item $\langle J\amalg 1\amalg 1\langle 1 \langle 1\rangle\rangle\rangle.$
\end{itemize}
Moreover, any of these real schemes is realizable by nonsingular dividing real algebraic  curves of degree $7$ in $\mathbb RP^2$.
\end{lemma}
\begin{lemma}\label{class7 nondiv}
 Any nonsingular non-dividing real pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ has one of the following real schemes~:
\begin{itemize}
\item $\langle J\amalg \alpha\amalg 1\langle\beta\rangle\rangle$
 with $\alpha+\beta\le 13$, $0\le\alpha\le 12$, $1\le\beta\le 13$,
\item $\langle J\amalg \alpha\rangle$ with $0\le\alpha\le 14.$
\end{itemize}
Moreover, any of these real schemes is realizable by nonsingular non-dividing real algebraic  curves of degree $7$ in $\mathbb RP^2$.
\end{lemma}
Now, we introduce \textit{symmetric curves}. Denote by $s$ the holomorphic involution of $\mathbb CP^2$ given by
$[x:y:z]\mapsto [x:-y:z]$.
A real curve in $\mathbb RP^2$ is called \textit{symmetric} if $s(A)=A$.
\\Here we state the four main theorems of this article.
\begin{thm}\label{Main thm1}
The following real schemes are not realizable by symmetric real pseudo-holomorphic curves of degree $7$  in $\mathbb RP^2$~:
\begin{itemize}
\item $\langle J\amalg (14-\alpha)\amalg 1\langle \alpha\rangle \rangle\textrm{  with } \alpha=6,7,8,9$,
\item $\langle J\amalg (13-\alpha)\amalg 1\langle \alpha\rangle \rangle \textrm{  with } \alpha=6,7,9$.
\end{itemize}
Moreover, any real scheme realizable by real algebraic curves of degree $7$ in $\mathbb RP^2$
and not mentioned in the list above
is realizable by symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{thm}
\begin{thm}\label{Main thm2}
A complex scheme is realizable by  symmetric real algebraic (or pseudo-holomorphic) $M$-curves of degree $7$
in $\mathbb RP^2$  if
and only if it is contained in the following list~:
\begin{itemize}
\item $\langle J\amalg \left (\frac{12-\alpha}{2} \right)_+ \amalg\left ( \frac{16-\alpha}{2} \right)_-
\amalg 1_-\langle \left ( \frac{\alpha+2}{2} \right)_+\amalg\left ( \frac{\alpha-2}{2} \right)_-\rangle \rangle_I$  with $\alpha=2,4,10,12$,
\item $\langle J\amalg \left (\frac{15-\alpha}{2} \right)_+ \amalg\left ( \frac{13-\alpha}{2} \right)_-
\amalg 1_-\langle \left ( \frac{\alpha+1}{2} \right)_+\amalg\left ( \frac{\alpha-1}{2} \right)_-\rangle \rangle_I$  with $\alpha=1,5,13$,
\item $\langle J\amalg \left (\frac{15-\alpha}{2} \right)_+ \amalg\left ( \frac{13-\alpha}{2} \right)_-
\amalg 1_+\langle \left ( \frac{\alpha-1}{2} \right)_+\amalg\left ( \frac{\alpha+1}{2} \right)_-\rangle \rangle_I$  with $\alpha=3,11$,
\item $\langle J\amalg 9_+\amalg 6_-\rangle_I$.
\end{itemize}
\end{thm}
\textbf{Remark. }The notations used to encode complex schemes are the usual ones proposed in \cite{V3}.
\begin{thm}[Pseudo-holomorphic classification]\label{Main thm3}
The following real schemes are not realizable by dividing symmetric real pseudo-holomorphic curves of degree $7$
in $\mathbb RP^2$~:
 \begin{center}
$\langle J\amalg \alpha\amalg 1\langle \beta\rangle \rangle$  with
 $\alpha=2,6$ and $\alpha+\beta=12$,
$\alpha=\beta=4$,
\end{center}
Moreover, any real scheme  mentioned in  Lemma \ref{class7 div} and   not forbidden by Theorem \ref{Main thm1} and  the above list
 is realizable by dividing symmetric real pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$; any
real scheme  mentioned in Lemma \ref{class7 nondiv} which is  not forbidden by Theorem \ref{Main thm1}
 is realizable by non-dividing symmetric real pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$.
\end{thm}

\begin{thm}[Algebraic classification]\label{Main thm4}
The real schemes
$$\langle J\amalg 8\amalg 1\langle 4\rangle \rangle\textrm{ and }\langle J\amalg 4\amalg 1\langle 8\rangle \rangle$$
are not realizable by
a dividing symmetric real algebraic curve of degree $7$ in $\mathbb RP^2$.
Any other real scheme which is realizable by dividing symmetric real pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$ is realizable by
dividing symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.

Any real scheme which is realizable by non-dividing symmetric real pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$  is  realizable
by non-dividing symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{thm}



\section{Preliminaries}\label{prelim}

\subsection{Rational geometrically ruled surfaces}\label{ruled}
Let us define \textit{the $n^{th}$ rational geometrically ruled surface}, denoted by $\Sigma_n$, the surface obtained by taking four copies of $\mathbb C^2$ with coordinates $(x,y)$, $(x_2,y_2)$, $(x_3,y_3)$
and $(x_4,y_4)$, and by gluing it along $(\mathbb C^*)^2$ with the identifications $(x_2,y_2)=(1/x,y/x^n)$, $(x_3,y_3)=(x,1/y)$
and $(x_4,y_4)=(1/x,x^n/y)$. Define in $\Sigma_n$ the algebraic curve $E$ (resp. $B$ and $F$) defined by the equation $\{y_3=0\}$ (resp. $\{y=0\}$ and $\{x=0\}$). The coordinate system $(x,y)$
is called \textit{standard}.
The projection $\pi$ : $(x,y)\mapsto x$ on $\Sigma_n$ defines a $\mathbb CP^1$-bundle over $\mathbb CP^1$.
We have $B\circ B=n$, $F\circ F=0$ and $B\circ F=1$. The surface $\Sigma_n$ has a natural real structure induced by the complex conjugation in $\mathbb C^2$, and
the real part of $\Sigma_n$ is a torus if $n$ is even and a Klein bottle if $n$ is odd. The restriction of $\pi$ on $\mathbb R\Sigma_n$ defines a pencil of lines denoted by $\mathcal L$.

 The group $H_2(\Sigma_n,\mathbb Z)$ is isomorphic to $\mathbb Z\times\mathbb Z$ and is generated by the classes of $B$ and $F$. Moreover, one has $E=B-nF$. An algebraic or pseudo-holomorphic curve on $\Sigma_n$ is
said to be of \textit{bidegree} $(k,l)$ if it realizes the homology class $kB+lF$ in $H_2(\Sigma_n,\mathbb Z)$. A curve of bidegree $(1,0)$ is called \textit{a base} of $\Sigma_n$.

In the rational geometrically ruled surfaces, we study real curves up to \textit{isotopy with respect to $\mathcal L$}. Two curves are said
isotopic with respect to the fibration $\mathcal L$ if there exists an isotopy of $\Sigma_n$ which brings the first curve to the second one, and
which maps each line of $\mathcal L$ in another line of $\mathcal L$.

The work of Gromov (see \cite{Gro}) shows that the  following proposition, trivial in the algebraic case, it is also true in the pseudo-holomorphic case.
\begin{prop}
There exists a unique base of $\Sigma_2$ which passes through $3$ generic points.
\end{prop}

In this paper, curves in a rational geometrically ruled surface are depicted up to isotopy with respect to $\mathcal L$.

\subsection{Real curves}
A \textit{real pseudo-holomorphic curve} $C$ in $\mathbb CP^2$ or $\Sigma_n$ is an immersed Riemann surface which is a $J$-holomorphic curve in some tame almost complex structure $J$ such that the exceptional section (in $\Sigma_n$ with $n\ge 1$) is $J$-holomorphic (see \cite{Gro},
$conj(C)=C$, and $conj_*\circ J_p=-J_p\circ conj_*$, where $conj$ is the complex conjugation. If the curve $C$ is immersed in $\mathbb CP^2$ (resp. $\Sigma_n$) and realizes the homology class $d[\mathbb CP^1]$ in $H_2(\mathbb CP^2,\mathbb Z)$ (resp $kB+lF$ in $H_2(\Sigma_n,\mathbb Z$), $C$ is said to be of degree $d$ (resp. of bidegree $(k,l)$).
For example, a real algebraic curve  of degree $d$ in $\mathbb CP^2$ is a real pseudo-holomorphic curve of degree $d$ for the standard complex structure. All intersections of two $J$-holomorphic curves are positive, so the Bezout theorem is still true
for two $J$-holomorphic curves.

Gromov pointed out that through $2$ (resp. 5) given generic points in $\mathbb CP^2$,
there exists a unique $J$-line (resp. $J$-conic) passing through these points.
As soon as the degree is greater than $3$, such statement is no more true. This is a direction to find
some difference between algebraic and pseudo-holomorphic curves (see \cite{FTO} where the authors use pencils of cubics).

From now on, we state results about "curves" if they are valid in both cases, algebraic and pseudo-holomorphic. A nonsingular real curve of genus $g$ is called an \textit{$(M-i)$-curve}
if its real part has $g+1-i$ connected components. If $i=0$, we simply speak about \textit{$M$-curve} or \textit{maximal curve}.
A connected component of the real part of a nonsingular real curve in a real surface is called an \textit{oval} if it is contractible in the surface.
In $\mathbb RP^2$ the complement of an oval is formed by two connected components, one of which is homeomorphic to a disk
(called the \textit{interior} of the oval) and the other to a M\"obius strip (called the \textit{exterior} of the oval).
Two ovals in $\mathbb RP^2$ are said to constitute an \textit{injective pair} if one of them is enclosed by the other. A set of ovals such that any two ovals of this set form an injective pair is called a nest.
For a nonsingular real curve of degree $7$ with a nest, the oval which contains some other ovals is called the
 \textit{non-empty oval} of the curve. The ovals lying
inside this oval are called the \textit{inner ovals} while those lying outside are called the
\textit{outer ovals}.

A nonsingular
real curve $C$ is a $2$-dimensional object and $C\setminus\mathbb RC$ is either connected or it has two connected components. In the former
case, we say that $\mathbb RC$ is a \textit{non dividing curve},
or of type $II$, and in the latter case, we say that $\mathbb RC$ is a \textit{dividing curve}, or of type $I$.

Let $C$ be a dividing curve.
The two halves of $A\setminus\mathbb RA$ induce two opposite orientations on $\mathbb RA$ which are called \textit{complex orientations}
of the curve. Fix such a complex orientation of the curve.
An injective pair of ovals in $\mathbb RP^2$ is called \textit{positive} if the orientations of the two ovals are induced by one of the orientations
of the annulus in $\mathbb RP^2$
bounded by the two ovals, and \textit{negative} otherwise. Now suppose that $C$ is of odd degree. If the integral homology classes realized by the odd component of the curve and an oval in the
M\"obius strip defined by the exterior of
this oval have the same sign, we say that this oval is \textit{negative}, and
\textit{positive} otherwise.
Denote by
$\Pi_+$ (resp. $\Pi_-$) the number of positive (resp. negative)  injective pairs of ovals and by
$\Lambda_+$ (resp. $\Lambda_-$) the number of positive (resp. negative) ovals.
\begin{prop}[Rokhlin-Mischachev's orientation formula]\label{orientation formula}
If $C$ is a dividing nonsingular curve of degree $2k+1$ in $\mathbb RP^2$ with $l$ ovals, then
$$\Lambda_+-\Lambda_-+2(\Pi_+-\Pi_-)=l-k(k+1)$$
\end{prop}

The following fact about curves in toric surfaces is well known.
\begin{prop}\label{lattice}
Let $C$ be a nonsingular real curve with Newton polygon $\Delta$ in a toric surface. Then the genus of $C$ is equal to the number of
integer points in the interior of $\Delta$.
\end{prop}

\begin{lemma}\label{lem bezout1}
Let $X$ be a curve  of bidegree $(3,1)$ on $\Sigma_2$, whose the real part is as depicted in Figure \ref{bezout1}a). We orient
$\mathcal L$, the pencil of lines of $\Sigma_2$, from the left to the right. Then, the lines of $\mathcal L$
 meet first all the ovals $\beta$ and then all the ovals $\alpha$ or conversely.

Moreover, if $\alpha+\beta\ge 3$, then, the lines of $\mathcal L$
 meet first all the ovals $\beta$ and then all the ovals $\alpha$.
\end{lemma}
\textit{Proof.  }Suppose there exists $3$ ovals contradicting the lemma. Then the base passing through these ovals intersects the curve in $9$ points what contradicts the Bezout theorem.~\findemo

\begin{figure}[h]
\centering
\begin{tabular}{ccc}
\includegraphics[height=2cm, angle=0]{Figures2/Bezout1_a.eps}&
\includegraphics[height=2.5cm, angle=0]{Figures2/Bezout2_a.eps}&
\includegraphics[height=2.5cm, angle=0]{Figures2/Bezout2_b.eps}
\\ a)&b)&c)
\end{tabular}
\caption{}
\label{bezout1}
\end{figure}
Given a curve of odd degree in $\mathbb RP^2$, one can speak about convexity~:
the segment defined by two points $a$ and $b$ is the
connected component of the line $(a,b)\setminus\{a,b\}$ which has an even number of intersection points
with the odd component of the curve.
\begin{lemma}\label{convexe}
 If a real pseudo-holomorphic curve of degree $7$ with at least $6$ ovals has a nest, then it is possible to choose a point in each inner oval such that these points are the vertices of a convex polygon in $\mathbb RP^2$
Moreover, if a line $L$ passes through two outer ovals $O_1$ and $O_2$ and separates the inner ovals, then $O_1$ and $O_2$ are not
on the same connected component of $L\setminus (Int(O)\cup\mathcal J)$ where $O$ is the non-empty oval.
\end{lemma}
\textit{Proof. }Suppose there exists four empty ovals contradicting the lemma as depicted in Figure \ref{bezout1}b) and c). Then the conic passing through these ovals and another  one intersects the curve in at least $16$ points what contradicts the Bezout theorem.~\findemo



\subsection{Braid theoretical methods}\label{braid}
Here we recall the method exposed by Orevkov in \cite{O1}. We call the \textit{group of braids with $m$-strings} the group
$B_m=\langle \sigma_1, ..., \sigma_{m-1}| \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_{i}\sigma_{i+1},
 \sigma_i\sigma_j=\sigma_j\sigma_i \textrm{ if }|i-j|>1\rangle$.
We call the \textit{exponent sum} of a braid $b=\prod_{j=1}^n \sigma_{i_j}^{k_j}$ the integer
$e(b)=\sum_{j=1}^n k_j. $
A braid $b$ is called quasipositive if it is a product of conjugated elements of $\sigma_1$ in $B_m$.

An arrangement of the real part of a real curve  in $\Sigma_n$ up to isotopy
which respect the pencil of lines $\mathcal L$ is called
an $\mathcal L-scheme$. From now on, we fix a base $B$ of  $\Sigma_n$,
a fiber $l_{\infty}$ of $\mathcal L$ and an $\mathcal L-scheme$ $A$ in $\Sigma_n$, which may be nodal. Suppose moreover that all the intersections of $A$
and $E$ are real
and that $A$ intersects each fiber in $m$ or $m-2$ real points (counted with multiplicities) and $l_{\infty}$  in $m$ distinct real points. Choose a standard coordinate system on $\Sigma_n$ such that $l_\infty$ has equation $\{x_2=0\}$ (see section \ref{ruled}).
a trivialization of the $\mathbb CP^1$-bundle over $B\setminus(B\cap l_\infty)$.
Then, examine the real part of the fibration from $x_1=-\infty$ to $x_1=+\infty$, and
encode the $\mathcal L-scheme$ $A$ in the following way~:
\begin{itemize}
\item  if the pencil of real lines has a tangency point with $A$,  write $\supset_k$ if $A$ intersects a fiber in $m$
real points before the tangency point, and $\subset_k$ otherwise,
\item if the pencil of real lines meets a double point of $A$, write $\times_k$ if the tangents are non-real and $\bullet_k$ otherwise,
\item if $A$ intersects the infinite section, write $/_k$ if the branch of $A$ passing through the infinity
lies in the region $\{y>0\}$ just before the infinite point, and  $\setminus_k $ otherwise.
\end{itemize}
Here, $k$ is the number of real intersection points of the fiber and $A$ below the ramification point. Replace all the occurences
$\subset_k$ $\supset_k $ and $\bullet_k$ by $o_k$. Thus, we have now
a coding $s_1\ldots s_r$ of the $\mathcal L-scheme$ $A$. In order to obtain a braid from this coding, perform the
 following substitutions~:
\begin{itemize}
\item replace each $\times_j$ which appears between  $\subset_s$ et $\supset_t$
 by $\sigma_j^{-1}$,
\item replace each $\setminus_1$  which appears between  $\subset_s$ et $\supset_t$ by $\sigma_1\sigma_2\ldots\sigma_{m-1}$,
\item replace each $/_{m}$  which appears between  $\subset_s$ et $\supset_t$ by $\sigma_{m-1}\sigma_{m-2}\ldots\sigma_{1}$,
\item replace each subword $\supset_s\times_{i_{1,1}}\ldots\times_{i_{r_1,1}} ?_1 \times_{i_{1,2}}\ldots\times_{i_{r_2,2}} ?_2
\ldots\times_{i_{1,p}}\ldots\times_{i_{r_p,p}} ?_1 \subset_t$
\\with $?_i=\setminus_1,\setminus_3,/_m$ or $/_{m-2}$
by $\sigma_s^{-1}u_{1,1}\ldots u_{r_1,1} v_1 u_{1,2}\ldots u_{r_2,2} v_2\ldots  v_p u_{1,p}\ldots u_{r_p,p} \tau_{s,t}$,
\\where
\begin{center}
\begin{tabular}{ll}
$u_{j,k}=\left\{\begin{array}{ll}
\sigma_{i_{j,k}}^{-1} & \textrm{ if }i_{j,k}<s-1,\\
\sigma_{i_{j,k}+2}^{-1} & \textrm{ if }i_{j,k}>s-1,\\
\tau_{s,s+1}\sigma_{s-1}^{-1}\tau_{s+1,s} & \textrm{ if }i_{j,k}=s-1,
\end{array}\right.$&
$v_l=\left\{\begin{array}{ll}
\sigma_1\sigma_2\ldots\sigma_{m-1} & \textrm{ if }?_l=\setminus_1,\\
\sigma_2^{-1}\sigma_1^2\sigma_2\ldots\sigma_{m-1} & \textrm{ if }?_l=\setminus_3,\\
\sigma_{m-1}\sigma_{m-2}\ldots\sigma_1 & \textrm{ if }?_l=/_m,\\
\sigma_{m-2}^{-1}\sigma_{m-1}^2\sigma_{m-2}\ldots\sigma_1 & \textrm{ if }?_l=/_{m-2},\\
\end{array}\right.$
\end{tabular}

$\tau_{s,t}=\left\{\begin{array}{ll}
(\sigma_{s+1}^{-1}\sigma_s)(\sigma_{s+2}^{-1}\sigma_{s+1})
\ldots(\sigma_t^{-1}\sigma_{t-1}) & \textrm{if }t>s,\\
(\sigma_{s-1}^{-1}\sigma_s)(\sigma_{s-2}^{-1}\sigma_{s-1})
\ldots(\sigma_t^{-1}\sigma_{t+1}) & \textrm{if }t<s,\\
1 & \textrm{if }t=s.
\end{array}\right.$
\end{center}
\end{itemize}
Then we obtain a braid $b_\mathbb R$. We define the braid associated to the $\mathcal L-scheme$ $A$, denoted $b_A$,
 as the braid
$b_\mathbb R\Delta_m^n$, where $\Delta_m$ is the Garside element of $B_m$ and which is given by
$$\Delta_m=(\sigma_1\ldots\sigma_{m-1})(\sigma_1\ldots\sigma_{m-2}\sigma_1)\ldots(\sigma_1\sigma_2)\sigma_1.$$
For example, the coding and the braid corresponding to the real $\mathcal L-scheme$ in $\Sigma_2$ depicted in Figure \ref{ex braid}a) are~:
\begin{center}
\begin{tabular}{lll}
$\supset_3 o_3^2\times_1 o_2^2\times_1^4 /_2 \subset_3\times_2^2\supset_3\subset_3$& and
$\sigma_3^{-3}\sigma_1^{-1}\sigma_2^{-1}\sigma_3\sigma_2^{-2}\sigma_3^{-1}\sigma_2\sigma_1^{-4}
\sigma_2^{-1}\sigma_3^{2}\sigma_2\sigma_1\sigma_2^{-2}\sigma_3^{-1}\Delta_4^2$
\end{tabular}
\end{center}
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2cm, angle=0]{Figures0/Ex_braid.eps}&
\includegraphics[height=3cm, angle=0]{Figures-1/oval.eps}\\
a)&b)
\end{tabular}
\end{center}
\caption{}
\label{ex braid}
\end{figure}
In \cite{O1}, the following result is proved, where $r$ is the number of (real) intersection points of
$A$ with the exceptionnal section.
\begin{thm}
The  $\mathcal L-scheme$ $A$ is realizable by a pseudo-holomorphic curve of bidegree $(m,r)$ if and only if the braid $b_A$ is
quasipositive.
\end{thm}
\begin{prop}\label{operations}
Let $A$ an $\mathcal L-scheme$, and $A'$ the $\mathcal L-scheme$ obtained
from $A$ by one of the following elementary operations~:
$$\begin{array}{c c c c c c cc}
\times_j\supset_{j\pm 1}\to\times_{j\pm 1}\supset_j &, & &
\subset_{j\pm 1}\times_j\to\subset_j\times_{j\pm 1} &, & &
\times_j u_k\to u_k\times_j &,
\end{array}$$
$$\begin{array}{c c c c c c}
o_k\subset_k\supset_{k- 1}&\longleftrightarrow&\subset_k\times_{k-1}\supset_{k}&\longleftrightarrow&
\subset_{k-1}\supset_{k}o_k&,
\end{array}$$
$$\begin{array}{c c c c c c cc}
\subset_j\supset_{j\pm 1}\to\emptyset &, & &
\subset_j\supset_{k}\to\supset_k\subset_j &, & &
o_j\to \emptyset &,
\end{array}$$
where $|k-j|>1$ and $u$ stands for  $\times$, $\subset$, or $\supset$.

Then if $A$ is realizable by a pseudo-holomorphic curve of bidegree $(m,r)$, so is  $A'$.

Moreover, if $A$ is realizable by a dividing pseudo-holomorphic curve of bidegree $(m,r)$ and if $A'$ is obtained from $A$ by one
of the previous elementary operations but the last one, so is $A'$.
\end{prop}
According to Proposition \ref{operations}, if an  $\mathcal L$-scheme $A$ is realizable by a pseudo-holomorphic curve, so is any $\mathcal L$-scheme obtained from $A$ by some $\subset_j\supset_{j\pm 1}\to\emptyset $ operations. Unfortunately, as Orevkov showed (\cite{O5}), this is
not the case for algebraic curves.

In this paper, we use the following tests to show that a given braid is not quasipositive.
Propositions \ref{alex} and \ref{double alex} are
corollary of the Murasugi-Tristram inequality (see \cite{O1}) and Proposition \ref{square} is a corollary of the generalized Fox-Milnor theorem (see \cite{O2}).
\begin{prop}\label{alex}
If a braid $b$ in $B_m$ is quasipositive and $e(b)<m-1$, then the Alexander polynomial of $b$ is identically zero.
\end{prop}
\begin{prop}\label{double alex}
If a braid $b$ in $B_m$ is quasipositive and $e(b)=m-1$, then all the roots of the Alexander polynomial of $b$ situated on the unit circle are of order at least two.
\end{prop}
\begin{prop}\label{square}
If a braid $b$ in $B_m$ is quasipositive and $e(b)=m-1$, then $det(b)$ is a square in $\mathbb Z$.
\end{prop}
One can refer to \cite{Lic} or \cite{O1} for the definitons of the Alexander polynomial and the determinant of a braid.


\subsection{Comb theoretical methods}
In \cite{O3}, Orevkov reformulates the existence of real algebraic curves of bidegree $(3,0)$ realizing a given $\mathcal L$-scheme in $\Sigma_n$ in terms of some colored graphs on $\mathbb CP^1$. Guided by this article, we give in this section an efficient algorithm to check whether an $\mathcal L$-scheme is realizable by  a real algebraic curve of bidegree $(3,0)$ in $\Sigma_n$.

Using these \textit{Trigonal graphs on $\mathbb CP^1$}, Orevkov (\cite{O5}) obtained a classification of curves of bidegree $(3,0)$ in $\Sigma_n$ up to isotopy which respects the pencil of lines, in terms of gluing of cubics.

\subsubsection*{Trigonal graphs on $\mathbb CP^1$}
In what follows, $n$ is a positive integer and $P(X,Y)=Y^3+b_2(X)Y+b_3(X)$ is a real polynomial, where $b_i(X)$ is a real polynomial of degree $in$ in $X$. By a suitable change of coordinates in $\Sigma_n$, any curve of bidegree $(3,0)$ in $\Sigma_n$ can be put into this form.
 \begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=0.7cm, angle=0]{Figures3/RP1.eps}&
\includegraphics[height=1cm, angle=0]{Figures2/Vertex1.eps}    &
\includegraphics[height=1cm, angle=0]{Figures2/Vertex2.eps}    &
\includegraphics[height=1.3cm, angle=0]{Figures2/Vertex3.eps}
\\ a)&b) &c)&d)
\end{tabular}
\caption{}
 \label{pregraph}
\end{figure}
Denote by $D=-4b_2^3-27b_3^2$ the discriminant of $P$ with respect to the variable $Y$ and consider the rational function $f(X)=\frac{D}{b_3^2}=-\frac{4b_2^3}{b_3^2}-27$. Color and orient $\mathbb RP^1$ as depicted in Figure \ref{pregraph}a). Let $\Gamma$ be the coloured and oriented graph $f^{-1}(\mathbb RP^1)$ with the  colouring and the orientation induced by those chosen on $\mathbb RP^1$.  Then, $\Gamma$ is a coloured and oriented graph on $\mathbb CP^1$, invariant under the action of the complex conjugation. Moreover, $\Gamma$ has exactly $6n$ vertices depicted in Figure \ref{pregraph}a), $3n$ vertices depicted on \ref{pregraph}b) and $2n$ vertices depicted on \ref{pregraph}c). The intersection of $\Gamma$ and $\mathbb RP^1$ can be extracted from the real part of the curve $P$ as shown on Figure \ref{real graph} (here we also depicted a small neighborhood of $f^{-1}(\mathbb RP^1)\cap\mathbb RP^1$ and we omitted the arrows).
 \begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures2/Real_graph1.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Real_graph2.eps}    &
\includegraphics[height=2cm, angle=0]{Figures2/Real_graph3.eps}    &
\includegraphics[height=2cm, angle=0]{Figures2/Real_graph4.eps}
\\ a)&b) &c)&d)
\end{tabular}
\caption{}
 \label{real graph}
\end{figure}
The correspondances on Figures \ref{real graph}a) and b) are obvious, and those on Figures \ref{real graph}c) and d) are given by the two following propositions proved in \cite{O3}. Let $x_1$ and $x_2$ be two real numbers such that $x_1<x_2$.
\begin{prop}\label{alt int}
Suppose the interval $[x_1,x_2]$ verifies the following conditions :
\begin{itemize}
\item each of the polynomials $P(x_1,Y)$ and $P(x_2,Y)$ has one simple root and one double root,
\item the double root is greater than the simple root for one of the polynomials  $P(x_1,Y)$ and $P(x_2,Y)$ and
vice-versa for the other polynomial,
\item the polynomial $P(x_0,Y)$ has exactly one real root for $x_1<x_0<x_2$.
\end{itemize}
 Then, $[x_1,x_2]$ contains at least two
real roots of $b_2$ and one root of $b_3$ which is between two real roots of $b_2$.
\end{prop}
\begin{prop}
Suppose the interval $[x_1,x_2]$  verifies  the first two conditions in proposition \ref{alt int} and the polynomial $P(x_0,y)$ has exactly three real roots for $x_1<x_0<x_2$. Then, $[x_1,x_2]$ contains at least one root of $b_3$.
\end{prop}
In the same way, one can associate  a colored and oriented graph on $\mathbb RP^1$ to an $\mathcal L$-scheme $A$ of bidegree $(3,0)$ as indicated on Figure \ref{real graph}.
\begin{defi}
The colored and oriented graph on $\mathbb RP^1$ constructed above is called the real graph associated to $A$.
\end{defi}
 \begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=2.5cm, angle=0]{Figures2/Operation1.eps}&
\includegraphics[height=2.5cm, angle=0]{Figures2/Operation2.eps}    &
\includegraphics[height=2.5cm, angle=0]{Figures2/Operation3.eps}\\
 a)&b) &c)
\end{tabular}
\caption{}
 \label{oper}
\end{figure}

\begin{defi}
Let $\Gamma$ be a subset of $\mathbb CP^1$ and  $\pi:\Gamma\to\mathbb RP^1$ a topological ramified covering. Then the coloring and oriention of $\mathbb RP^1$ shown on Figure \ref{pregraph}a) defines a coloring and an orientation of $\Gamma$ via $\pi$. We say that $\Gamma$ is a trigonal graph of degree $n$ if
\begin{itemize}
\item $\Gamma$ has exactly $6n$ vertices depicted in Figure \ref{pregraph}a), $3n$ vertices depicted on \ref{pregraph}b) and $2n$ vertices depicted on \ref{pregraph}c), and no other non-real multiple points,
\item The set $\pi^{-1}([\infty;-27])$ is connected.
\end{itemize}
\end{defi}
\begin{thm}\label{constr trig}
Let $A$ be an $\mathcal L$-scheme of bidegree $(3,0)$ in $\Sigma_n$ and $G$ its real graph. Then $A$ is realizable by  real algebraic curves of bidegree $(3,0)$ in $\Sigma_n$ if and only if there exists a trigonal graph $\Gamma$ of degree $n$ such that $\Gamma\cap \mathbb RP^1=G$.
\end{thm}
\textit{Proof. }If there exists such a trigonal graph of degree $n$, the existence of the desired real algebraic curve is proved in \cite{O3}.

Suppose there exists a real algebraic curve $P$ realizing $A$. Define $f$ and $\Gamma$ as in the begining of this section. Made all the following operations on one of the halves of $\mathbb CP^1\setminus\mathbb RP^1$. The final picture will be obtained by gluing the obtained graph with its image under the complex conjugation.

If $\Gamma$ has non-real double points, smooth them as depicted on Figure \ref{oper}a).

If $f^{-1}([\infty;-27])$ is not connected, choose $p$ in $f^{-1}(-27)$ and $q$ in $f^{-1}(\infty)$ which belong to different connected components of $f^{-1}([\infty;-27])$. If $p$ and $q$ belong to the same connected components of $\Gamma$, choose $p$ and $q$ such that they are connected in $\Gamma$ by an arc of $f^{-1}(]-27;\infty[)$ and apply the operation depicted on Figure \ref{oper}b) to $\Gamma$. Otherwise, choose $p$ and $q$ such that they lie on the boundary of one of the connected components of $\mathbb CP^1\setminus\Gamma$ and apply the operation depicted on Figure \ref{oper}c) to $\Gamma$. Apply these operations until $f^{-1}([\infty;-27])$ is connected.\findemo

\subsubsection*{The algorithm}
 \begin{figure}[h]
      \centering
 \begin{tabular}{ccccccc}
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g1.eps}    &
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g2.eps}    &
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g3.eps}     &
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g4.eps}     &
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g5.eps}&
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g6.eps} &
\includegraphics[height=0.7cm, angle=0]{Figures2/gene_g1g4.eps}
\\ $g_1$&$g_2$ &$g_3$&$g_4$&$g_5$&$g_6$&$g_1g_4$
\end{tabular}
\caption{}
 \label{gener semi group}
\end{figure}
Denote by $CB_n$ the semigroup generated by the elements $g_1,\ldots g_6$ in $\mathbb R^2$ depicted in Figure \ref{gener semi group}. The multiplication of two elements $cb_1$ and $cb_2$ in $CB$ is the attachement of the right endpoint of $cb_1$ to the left endpoint of $cb_2$. \begin{defi}
The elements of $CB$ are called combs.
\end{defi}
For example, the comb $g_1g_4$ is depicted on figure \ref{gener semi group}. The unit element of $CB$ is denoted by $1$.
\begin{defi}
A weighted comb is a quadruplet $(cb,\alpha,\beta,\gamma)$ in $CB\times\mathbb Z^3$.
\end{defi}

Consider  an $\mathcal L$-scheme $A$ of bidegree $(3,0)$ in $\Sigma_n$ satisfying the hypothesis of section \ref{braid}, and its coding $s_1 \ldots s_r$  defined in section \ref{braid}, using the symbols $\subset$, $\supset $, $o$ and $\times$. In this coding, replace all the occurences $\times_k$ (resp. $o_k$) by $\supset_k$ $\subset_k$ (resp. $\subset_k$ $\supset_k$).
This coding is denoted by $r_1\ldots r_q$. Define the weigted combs $(cb_i,\alpha_i,\beta_i,\gamma_i)$ as follows~:
\begin{itemize}
\item $(cb_0,\alpha_0,\beta_0,\gamma_0)=(1,6n,3n,2n)$
\item $(c_1,\alpha_1,\beta_1,\gamma_1)=\left\{\begin{array}{ll}
(g_3,6n-1,3n,2n)& \textrm{ if }n\textrm{ is even and }r_1=\supset_k \textrm{ and }r_q=\subset_k, \\ &
\textrm{ or }n\textrm{ is odd and }r_1=\supset_k\textrm{ and }r_q=\subset_{k\pm 1},\\
(g_5,6n-1,3n-1,2n) & \textrm{ otherwise },\end{array}\right.$
\item for $i>1$, \\$(c_i,\alpha_i,\beta_i,\gamma_i)=\left\{\begin{array}{ll}
(c_{i-1}g_2,\alpha_{i-1}-1,\beta_{i-1},\gamma_{i-1})& \textrm{ if }r_i=\subset_k\textrm{ and }r_{i-1}=\supset_{k},\\
(c_{i-1}g_3,\alpha_{i-1}-1,\beta_{i-1},\gamma_{i-1})& \textrm{ if }r_i=\supset_k\textrm{ and }r_{i-1}=\subset_{k},\\
(c_{i-1}g_6g_1g_4g_1g_6,\alpha_{i-1}-1,\beta_{i-1}-1,\gamma_{i-1}-2)& \textrm{ if }r_i=\supset_k\textrm{ and }r_{i-1}=\subset_{k\pm 1},\\
(c_{i-1}g_5,\alpha_{i-1}-1,\beta_{i-1}-1,\gamma_{i-1}) & \textrm{ if }r_i=\subset_k\textrm{ and }r_{i-1}=\supset_{k\pm1}.\end{array}\right.$
\end{itemize}
\begin{defi}
One says that the weighted comb $(cb_q,\frac{\alpha_q}{2},\frac{\beta_q}{2},\frac{\gamma_q}{2})$  is associated to the $\mathcal L$-scheme $A$.
\end{defi}
\begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
 \includegraphics[height=2cm, angle=0]{Figures2/Ex_pregraph_a.eps} &
\includegraphics[height=2cm, angle=0]{Figures2/Ex_pregraph_b.eps} &
\includegraphics[height=2cm, angle=0]{Figures2/Ex_pregraph_c.eps}

\\ a)&b)&c)
 \end{tabular}
\caption{}
 \label{ex comb}
\end{figure}
\begin{defi}
Let $cb$ be a comb. A closure or $cb$ is a subset of $\mathbb R^2$ obtained by joining each generator $g_1$ (resp. $g_3$, $g_5$) in $cb$ to a generator $g_2$ (resp. $g_4$, $g_6$) in $cb$ by a path in $\mathbb R^2$ without double points.

If there exists a closure of $cb$, one says that cb is closed.
\end{defi}
For example, the weighted comb associated to the $\mathcal L$-scheme in $\Sigma_1$ depicted in Figure \ref{ex comb}a) is \\$(g_5g_6g_1g_4g_1g_6g_5g_2g_3g_2,0,0,0)$. A closure of this comb is shown on Figure \ref{ex comb}b). The comb depicted on Figure \ref{ex comb}c) is not closed.
\begin{defi}\label{chain}
A chain of weighted combs is a sequence $(w_i)_{1\le i\le k}$ of weighted combs such that :
\begin{quote}
$w_k=(cb,0,0,0)$ where $cb$ is a closed comb,\\
$\forall i\in \{1\ldots k-1\}$, the weighted comb $w_{i+1}$ is obtained from $w_i$ by one of the following operations :

 \begin{tabular}{llll}
(1) &if $\gamma>0$ : &&$g_2\to (g_6g_1)^2g_6$, $\alpha_{i+1}=\alpha_i$, $\beta_{i+1}=\beta_i$, $\gamma_{i+1}=\gamma_i-1$\\
&& or & $g_5\mapsto (g_3g_6)^2g_3$, $\alpha_{i+1}=\alpha_i-3$, $\beta_{i+1}=\beta_i$ , $\gamma_{i+1}=\gamma_i-1$
\\ (2) &if $\alpha>0$ : & & $g_1\to g_3$, $\alpha_{i+1}=\alpha_i-1$, $\beta_{i+1}=\beta_i$, $\gamma_{i+1}=0$
\\  (3) & else : && $g_5\to g_4g_5g_4$, $\alpha_{i+1}=0$, $\beta_{i+1}=\beta_i-1$, $\gamma_{i+1}=0$.
\end{tabular}
\end{quote}
One says that the chain $(w_i)_{1\le i\le k}$ starts at $w_1$.
\end{defi}
\begin{defi}
Let $w$ be a weighted comb. The multiplicity of $w$, denoted by $\mu(w)$, is defined as the number of chains of weighted combs which start at $w$.
\end{defi}
\begin{thm}\label{prohib comb}
Let $A$ be an $\mathcal L$-scheme of bidegree $(3,0)$ in $\Sigma_n$ and $w$ its associated weighted comb. Then $A$ is realizable by  real algebraic curves of bidegree $(3,0)$ in $\Sigma_n$ if and only if $\mu(w)>0$ or  $w=(1,6n,3n,2n)$.
\end{thm}
\textit{Proof. }Let $w=(cb,\alpha,\beta,\gamma)$ be the comb and $G$ be the real graph associated to the $\mathcal L$-scheme $A$. If $cb=1$, it is well known that $A$ is realizable by a real algebraic curve of bidegree $(3,0)$ in $\Sigma_n$. Otherwise, a chain of weighted combs starting at $w$ is a reformulation of the statement~:

"choose $D$ one of the halves of $\mathbb CP^1\setminus\mathbb RP^1$. There exists a finite sequence $(G_i)_{1\le i\le k}$ of subset of $\mathbb CP^1$ such that :
\begin{itemize}
\item $G_0=G$,
\item $G_k$ is a trigonal graph such that $G_k\cap\mathbb RP^1=G$,
\item for $i$ in $\{1,\ldots,\alpha\}$, $G_i$ is obtained from $G_{i-1}$ by one of the gluing in $D$ depicted in Figures \ref{gluing}a) and b).
 Denote by $c$ the number of gluing depicted in Figure \ref{gluing}b).
\item for $i$ in $\{\alpha+1,\ldots,\alpha+\beta-3c\}$, $G_i$ is obtained from $G_{i-1}$ by the gluing in $D$ depicted in Figure \ref{gluing}c).
\item for $i$ in $\{\alpha+\beta-3c+1,\ldots,k-2\}$, $G_i$ is obtained from $G_{i-1}$ by the gluing in $D$ depicted in Figure \ref{gluing}d).
\item $G_{k-1}$ has no boundary and contains $G_{k-2}$.
\item $k=\alpha+\beta+\gamma-3c+2$,
\item $G_k$ is the gluing of $G_{k-1}$ and its image under the complex conjugation".
\end{itemize}
So, according to Theorem \ref{constr trig}, there exists  a chain of weighted combs starting at $w$ if and only if $A$ is realizable by a real algebraic curve of bidegree $(3,0)$ in $\Sigma_n$.
\findemo
 \begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=1.5cm, angle=0]{Figures2/Gluing1.eps}&
\includegraphics[height=1.5cm, angle=0]{Figures2/Gluing2.eps}    &
\includegraphics[height=1.5cm, angle=0]{Figures2/Gluing3.eps}    &
\includegraphics[height=1.5cm, angle=0]{Figures2/Gluing4.eps}
\\ a)&b) &c)&d)
\end{tabular}
\caption{}
 \label{gluing}
\end{figure}

Theorem \ref{prohib comb} provides an algorithm to check wether an $\mathcal L$-scheme is realizable by  a real algebraic curve of bidegree $(3,0)$ in $\Sigma_n$. In order to reduce computations, one can use the following observations.
\begin{lemma}\label{l1}
Let $cb$ be a closed comb and $Cl$ one of its closures. Suppose that $cb=c_1g_{i_1}c_2g_{i_2}c_3$ and that $g_{i_1}$ and $g_{i_2}$ are joined in $Cl$. Then the combs $c_1c_3$ and $c_2$ contain the same number of $g_1$ (resp. $g_3$, $g_5$) and $g_2$ (resp. $g_4$, $g_6$).
\end{lemma}
\textit{Proof. }Straightforward.
\begin{lemma}\label{l2}
Let $(cb,0,\beta,0)$ be an element of a chain of weighted combs. Then it is possible to join each $g_1$ in $cb$ to a $g_2$ in $cb$ by a path in $\mathbb R^2$ without double points, such that  if $cb=c_1g_{i_1}c_2g_{i_2}c_3$ with $g_{i_1}$ and $g_{i_2}$ joined, then the combs $c_1c_3$ and $c_2$ contain the same number of $g_1$ and $g_2$.
\end{lemma}
\textit{Proof. }Straightforward.\findemo
\begin{lemma}\label{l3}
Let $(cb,\alpha,\beta,0)$ be an element of a chain of weighted combs, where $cb=\prod_{j=1}^k g_{i_j}$. Define the equivalence relation $\sim$ on $\{j|i_j=1$ or $2\}$ as follows :
\begin{center}
$r\sim s$ if  the number of $g_1$, $g_2$, $g_3$, and $g_4$ in $\prod_{j=r}^s g_{i_j}$ is odd.
\end{center}
Denote by $E_1^{cb}$ and $E_2^{cb}$ the two equivalence classes of $\sim$. Then
$$|Card(E^{cb}_1)-Card(E_2^{cb})|\le \alpha.  $$
\end{lemma}
\textit{Proof. }Choose a chain of weighted combs $(w_i)_{1\le i\le k}$ which contains $(cb,\alpha,\beta,0)$ and let $(\widetilde{cb},0,\beta,0)$ be an element of this chain. Then there exists $l\in\{1\ldots \alpha\}$ such that
\begin{center} $Card(E_1^{\widetilde{cb}})=Card(E_1^{cb})-l$ and $Card(E_2^{\widetilde{cb}})=Card(E_2^{cb})-\alpha +l$.
\end{center}
 It is obvious that in a closure of $\widetilde{cb}$, an element of $E_1^{\widetilde{cb}}$ has to be joined to an element of $E_{2}^{\widetilde{cb}}$ so the cardinal of these two sets are equal.\findemo

The algorithm given by Theorem \ref{prohib comb} improved by Lemmas \ref{l1}, \ref{l2}, and \ref{l3},  is quite efficient~: the computations needed in this article have been done in less than 5 seconds on a computer.



\subsection{General facts about symmetric curves in the real plane}\label{general}
 We denote by $B_0$ the line $\{y=0\}$. The involutions $c$ and $conj$ commute, so $s\circ conj$ is an anti-holomorphic involution on $\mathbb CP^2$. The real part of this real structure
 is a real projective plane $\widetilde{\mathbb RP^2}=\{[x_0:ix_1:x_2]\in  \mathbb CP^2|
(x_0,x_1,x_2)\in \mathbb R^3\setminus\{0\}\}$, and it is clear that
$\widetilde{\mathbb RP^2}\cap \mathbb RP^2=\mathbb R\Delta\cup\{[0:1:0]\}$.
A symmetric curve $C$ is real for the structures defined by $conj$ and  $s\circ conj$. Denote $\widetilde{\mathbb RC}$ its real part for $s\circ conj$ and call it the mirror curve of $\mathbb RC$.

For a maximal symmetric curve, the real scheme realized by the mirror curve is uniquely determined.
\begin{thm}[Fiedler]\label{max sym}
The mirror curve of a maximal symmetric curve of degree $2k+1$ is a nest of depth $k$ with a pseudo-line.
\end{thm}
Denote by $\mathcal L_p$ the pencil of lines through the point $[0:1:0]$ in $\mathbb CP^2$.
If $C$ is a real symmetric curve of degree $7$ in $\mathbb RP^2$, the curve $X=C/s$ is a curve of bidegree $(3,1)$ in $\Sigma_2$ and is called the \textit{quotient curve} of $C$. The $\mathcal L$-scheme realized by $\mathbb RX$
is obtained by gluing the $\mathcal L_p$-schemes realized by $\mathbb RC$ and $\widetilde{\mathbb RC}$ along $B_0$.

Conversely, a symmetric curve is naturally associated  to an arrangement of a curve $X$ of bidegree $(3,1)$ and a base in $\Sigma_2$.
\begin{prop}[Fiedler]\label{inter max}
If $C$ is a dividing symmetric curve of degree $d$ in $\mathbb RP^2$, then
\begin{displaymath}
Card(\mathbb RC\cap\Delta)=d\textrm{ or }Card(\mathbb RC\cap\Delta)=0
\end{displaymath}
\end{prop}
Thus, in the case of a dividing symmetric curve of odd degree, all the common points between the curve and $B_0$ are real.

\begin{prop}\label{cplx orient}
Let $C$ be a nonsingular real pseudo-holomorphic symmetric curve in $\mathbb RP^2$. Then $C$ is smoothly isotopic to a
nonsingular real pseudo-holomorphic symmetric curve $C'$ in $\mathbb RP^2$ whose  invariant components
of $\widetilde{\mathbb RC'}$ have all their tangency points with a line of $\mathcal L_p$ on $B_0$.
\end{prop}
\textit{Proof. }Suppose that $\widetilde{\mathbb RC'}$ have tangency points with a line of $\mathcal L_p$ not on $B_0$. Then push all the corresponding tangent points of the quotient curve above $B_0$
 applying the first elementary operation of Proposition \ref{operations}.
The resulting symmetric curve satisfies the conditions of the proposition.~\findemo

For example, the symmetric curves of degree $4$ depicted in Figures \ref{sym isotopy}a) and d) are isotopic in $\mathbb CP^2$.
The dashed curve represent their mirror curves. The corresponding quotient curves
 are depicted in Figures \ref{sym isotopy}b) and c).

 \begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures1/Symmetric_isotopy_a.eps}    &
\includegraphics[height=2cm, angle=0]{Figures1/Symmetric_isotopy_b.eps}    &
\includegraphics[height=2cm, angle=0]{Figures1/Symmetric_isotopy_c.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures1/Symmetric_isotopy_d.eps}
\\ a)&b) &c)&d)
 \end{tabular}
\caption{}
 \label{sym isotopy}
\end{figure}
The curves $\mathbb RC$ and $\mathbb RC'$ have the same complex orientations. Using  the invariant components of $\widetilde{\mathbb RC'}$, one can apply the Fiedler orientations alternating rule to $\mathbb RC'$.

\begin{prop}\label{oval quotient}
If $C$ is a dividing symmetric curve in $\mathbb RP^2$, then an oval of $\mathbb RC$ and an oval of $\widetilde{\mathbb RC}$
cannot intersect in more than $1$ point.
\end{prop}
\textit{Proof. }
Suppose that a curve $C$ contradicts the lemma. Then according to Proposition \ref{cplx orient}, $C$ is isotopic to a dividing pseudo-holomorphic symmetric curve $C'$
with an invariant oval  $O$ of $\mathbb RC$,  and an oval of $\widetilde{\mathbb RC}$ as shown in Figure \ref{ex braid}b), which is impossible according to the Fiedler alternating rule.~\findemo

In general, there is no link between the type of the symmetric curve and the type of its quotient curve. However, if
both the symmetric curve and its mirror curve are of type I, there is no ambiguity.
\begin{prop}[Trille]\label{quotient dividing}
If a symmetric curve in $\mathbb RP^2$ and its mirror curve are of type $I$, so is the quotient curve.
\end{prop}


\section{Pseudo-holomorphic statements}\label{pseudo state}

\subsection{Prohibitions for curves of bidegree $(3,1)$ in $\Sigma_2$}
\begin{prop}\label{first M}
The following real schemes cannot be realized by symmetric pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$:
\begin{center}
$\langle J\amalg \beta\amalg 1\langle \alpha\rangle \rangle$ with $\alpha=6,7,8,9$ and $\beta=14-\alpha$,  $\alpha=7,9$ and $\beta=13-\alpha$.
\end{center}
\end{prop}
\textit{Proof. }According to Proposition \ref{max sym}, \ref{inter max}, the Bezout theorem, Lemmas \ref{lem bezout1} and \ref{convexe},
 the only possibilities for the $\mathcal L$-schemes of the quotient curves
  are depicted in Figure \ref{prohib even} for the four $M$-curves, and
 in Figures \ref{prohib odd2}a), b) and c), for the two $(M-1)$-curves.
\begin{figure}[h]
\centering
\begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_even_a.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_even_a.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_oddM_a.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_oddM_b.eps}
\\ $(\alpha,\beta)=(3,3)$ or $(4,2)$&$(\alpha,\beta)=(3,3)$ or $(2,4)$&
$(\alpha,\beta)=(3,3)$ or $(4,2)$&$(\alpha,\beta)=(3,3)$ or $(4,2)$
\end{tabular}
\caption{}
\label{prohib even}
\end{figure}

\begin{figure}[h]
\centering
\begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_oddM-1_a.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_oddM-1_b.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_oddM-1_c.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/M-Complexa.eps}
\end{tabular}

$(\alpha,\beta)=(3,3)$ or $(4,2)$, $\gamma+\delta=6$
\caption{}
\label{prohib odd2}
\end{figure}

The curves in Figures  \ref{prohib odd2}a) (resp. c)) and \ref{prohib even}c) (resp. d))  have the same position with respect
to $\mathcal L$, so they give the same
braid. Moreover, the braid corresponding to the curves in Figures \ref{prohib even}d) and b) are the same.
Thus, according to Lemma \ref{lem bezout1},
the  corresponding braids are~:
\begin{quote}
$b^1_{\alpha,\beta}=\sigma_2\sigma_1\sigma_2^{-(1+\beta)}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\sigma_2^{-1}
\sigma_1\Delta_3^2  \textrm{ with   }
(\alpha,\beta)=(3,3),(4,2)$,
\\$b^2_{\alpha,\beta}= \sigma_1\sigma_2^{-\beta}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\Delta_3^2  \textrm{ with  }
(\alpha,\beta)=(3,3),(2,4),(4,2)$,
\\$b^3_{\alpha,\beta}= \sigma_2^{-1}\sigma_1^{-1}\sigma_2^{2}\sigma_1\sigma_2^{-\beta}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\Delta_3^2
  \textrm{ with  }
(\alpha,\beta)=(3,3),(4,2)$,
\\$b^4= \sigma_2^{-7}\sigma_1\sigma_2\Delta_3^2$.
\end{quote}
The Alexander polynomials of those braids are
\begin{center}
\begin{tabular}{lll}
$p^1_{3,3}=p^2_{3,3}=p^3_{4,2}=
(-1+t)^3$,&
$p^1_{4,2}=p^2_{2,4}=p^2_{4,2}=p^3_{3,3}=(-1+t)(t^2-t+1)$,\\
$p^4=(-1+t)(t^4-t^3+t^2-t+1)$.
\end{tabular}
\end{center}
In each case,  $e(b)=1$ and the Alexander polynomial is not identically zero. So, according to
 Proposition \ref{alex},
 none of these braids is
quasipositive.~\findemo

\begin{prop}\label{M scheme 1}
The topology of the quotient curve of a maximal nonsingular symmetric pseudo-holomorphic curve in $\mathbb RP^2$ of degree $7$
with a nest containing at least $2$ ovals, with respect
to the base $\{y=0\}$ of $\Sigma_2$ is uniquely determined by the real scheme of the symmetric curve.
\end{prop}
\textit{Proof. }Let $C$ be a symmetric $M$-curve of degree $7$ with a nest containing at least $2$ ovals.
 Then the possible  $\mathcal L$-schemes for the quotient curves of $C$ are depicted in Figures \ref{prohib even}a) and b) with $\alpha=1,2,3,4,5,6$ if
the number of inner ovals is even and in Figures \ref{prohib even}c) and d) with $\alpha=0,1,2,3,4,5,6$ if the number of inner ovals is odd.
So, according to Lemma \ref{lem bezout1},
the  corresponding braids are~:
\begin{quote}
$b^1_{\alpha,\beta}=\sigma_2\sigma_1\sigma_2^{-(1+\beta)}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\sigma_2^{-1}
\sigma_1\Delta_3^2 $ with $\alpha+\beta=6$ and $\alpha=1,2,3,4,5,6$,
\\$b^2_{\alpha,\beta}= \sigma_1\sigma_2^{-\beta}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\Delta_3^2 $ with $\alpha+\beta=6$ and $\alpha=0,1,2,3,4,5,6$,
\\$b^3_{\alpha,\beta}= \sigma_2^{-1}\sigma_1^{-1}\sigma_2^{2}\sigma_1\sigma_2^{-\beta}\sigma_1^{-1}\sigma_2\sigma_1^{-\alpha}\Delta_3^2$ with $\alpha+\beta=6$ and $\alpha=0,1,2,3,4,5,6$.
\end{quote}
The braids $b^1_{3,3}$, $b^1_{4,2}$, $b^2_{2,4}$, $b^2_{3,3}$, $b^2_{4,2}$, $b^3_{3,3}$ and $b^3_{4,2}$ were already shown to be not
quasipositive in the proof of Proposition \ref{first M}. The Alexander polynomials of the remaining braids are~:
\begin{center}
\begin{tabular}{ll}
$p^1_{2,4}=(1-t+t^2)$,&
$p^1_{6,0}=p^2_{0,6}=p^2_{6,0}=p^3_{1,5}=(-1+t)(t^4-t^3+t^2-t+1)$, \\
$p^3_{0,6}=(t^2+t+1)(t^2-t+1)(-1+t)^3$,  &
$p^3_{5,1}=(-1+t)(t^2-t+1)$,
\\ $p=0$ for the other braids.
\end{tabular}
\end{center}
In each case, $e(b)=1$, so if the Alexander polynomial of the braid is not identically null, according to  Proposition \ref{alex},
the braid is not quasipositive. Thus, $b^1_{\alpha,\beta}$ is not quasipositive for $\alpha=2,3,4,6$,
$b^2_{\alpha,\beta}$ is not quasipositive for $\alpha=0,2,3,4,6$, and $b^3_{\alpha,\beta}$ is not
quasipositive for $\alpha=0,1,3,4,5$. The construction in the next section of symmetric $M$-curves of degree $7$ which have not
been forbidden in Proposition \ref{first M}
 shows that the other braids are quasipositive (however, it is not difficult to check this directly on
the braids). Hence, the topology of the quotient curve with respect to the base $\{y=0\}$ of $\Sigma_2$ is uniquely
determined by the topology of the symmetric curve.
~\findemo

\begin{prop}\label{M scheme 2}
If $C$ is a maximal nonsingular symmetric pseudo-holomorphic curve in $\mathbb RP^2$ of degree $7$
with a nest containing only one oval, then the only possibilities for the arrangement of its quotient curve and the base $\{y=0\}$
of $\Sigma_2$ are depicted in Figures \ref{prohib odd2}d) with
$(\gamma,\delta)=(4,2)$ or $(0,6)$.
\end{prop}
\textit{Proof. } If $C$ has exactly one inner oval, the  $\mathcal L$-schemes of the arrangement of its
quotient curve and the base $\{y=0\}$ of $\Sigma_2$ can only be as  depicted in \ref{prohib odd2}d).
The braids corresponding to these curves are
\begin{quote}
$b^5_{\gamma,\delta}= \sigma_2^{-(1+\delta)}\sigma_1^{-1}\sigma_2^{2}\sigma_1\sigma_2^{-\gamma}\sigma_1^{-1}\sigma_2\Delta_3^2$ with $\gamma+\delta=6$ and $\gamma=0,1,2,3,4,5,6$.
\end{quote}
The corresponding Alexander polynomials are~:
\begin{center}
\begin{tabular}{ll}
$p^5_{1,5}=p^5_{3,3}=(-1+t)(t^2-t+1)$,&
$p^5_{2,4}=(-1+t)^3$,\\
$p^5_{5,1}=(-1+t)(t^4-t^3+t^2-t+1)$, &
$p^5_{6,0}=(t^2-t+1)(t^2+t+1)(-1+t)^3$,
\\$p^5_{0,6}=p^5_{4,2}=0$.
\end{tabular}
\end{center}
In each case, $e(b)=1$, so according to  Proposition \ref{alex},
only the braids $b^5_{0,6}$ and $b^5_{4,2}$ can be quasipositive.
~\findemo

\subsection{Prohibitions for reducible curves of bidegree $(4,1)$ in $\Sigma_2$}\label{prohib1}
\begin{prop}\label{prohib thm 1}
The real scheme $\langle J\amalg 7\amalg 1\langle 6\rangle \rangle$ is not realizable by a nonsingular symmetric pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Here, it is not sufficient to look at the
$\mathcal L$-schemes of the quotient curves in $\Sigma_2$ only,  we look at the mutual arrangement of the quotient curves and a base of $\Sigma_2$. The possible  $\mathcal L$-scheme
of the quotient curves for the symmetric curve announced above with respect to the base $\{y=0\}$ are depicted in Figure \ref{base1}.
\begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=2cm, angle=0]{Figures2/Quotient1a.eps}  &
\includegraphics[height=2cm, angle=0]{Figures2/Quotient1b.eps}
 &\includegraphics[height=2cm,angle=0]{Figures2/Quotient1c.eps}   \\
a)&b)&c)\\
 \end{tabular}
\caption{}
 \label{base1}
\end{figure}

The $\mathcal L$-scheme   in Figures \ref{base1}b) cannot be realized pseudo-holomorphically~: the braid corresponding to this
$\mathcal L$-scheme and the Alexander polynomial are~:
\begin{center}
\begin{tabular}{ll}
$ \sigma_1^{-4}\sigma_2^{2}\sigma_1^{-3}\sigma_2^{-1}\sigma_1\Delta_3^2$&
 and $(-1+t)^3 $
\end{tabular}
\end{center}
The Alexander polynomial is not null although $e(b)=1$, so according to Proposition
\ref{alex}, the braid is  not
quasipositive.

Consider the base $H$ passing through the points
$a$, $b$ and $c$ in Figures \ref{base1}a) and c), where $c$ is a point of the fiber $L$. If the point $c$ varies on $L$ from $0$
to $\infty$, then, because of the choice of $L$, for some $c$, the base $H$ passes through an oval. Now, we want $H$ to pass through the first oval
we meet as $c$ varies from $0$
to $\infty$. The only possible mutual arrangements for $H$
and the quotient curves which do not contradict the Bezout theorem are shown in Figure \ref{base2}.
\begin{figure}[h]
      \centering
 \begin{tabular}{cc cc}
\includegraphics[height=2cm, angle=0]{Figures2/Quotient2b.eps}
 &\includegraphics[height=2cm,angle=0]{Figures2/Quotient2d.eps}   &
\includegraphics[height=2cm, angle=0]{Figures2/Quotient3b.eps} &
\includegraphics[height=2cm,angle=0]{Figures2/Quotient3d.eps}   \\
a)&b)&c)&d)\\
 \end{tabular}
\caption{}
 \label{base2}
\end{figure}
The corresponding braids
are~:
\begin{quote}
$b^6= \sigma_3^{-2}\sigma_2^{-2}\sigma_3^{-1}\sigma_1\sigma_2^{2}\sigma_1^{-4}\sigma_2^{-1}\sigma_3
\sigma_2^{-3}\sigma_3^{-1}\sigma_2\sigma_1^{-1}\Delta_4^2$,
\\$b^7= \sigma_3^{-3}\sigma_1\sigma_2^{2}\sigma_1^{-4}\sigma_2^{-1}\sigma_3\sigma_1^{-2}
\sigma_2^{-3}\sigma_3^{-1}\sigma_2\sigma_1^{-1}\Delta_4^2$,
\\$b^8=  \sigma_2^{-2}\sigma_3^{-3}\sigma_1^{-3}\sigma_2^{-1}\sigma_3\sigma_2^{-3}\sigma_3^{-1}\sigma_2\sigma_1^{-2}
\sigma_2^{-1}\sigma_3^{2}\sigma_2\sigma_1\Delta_4^2$,
\\$b^{9}= \sigma_3^{-3}\sigma_1^{-3}\sigma_2^{-1}\sigma_3\sigma_2^{-2}\sigma_1^{-2}
\sigma_2^{-1}\sigma_3^{-1}\sigma_2\sigma_1^{-2}\sigma_2^{-1}\sigma_3^2\sigma_2\sigma_1\Delta_4^2$.
\end{quote}
The computation of the corresponding Alexander polynomials gives~:
\begin{center}
\begin{tabular}{ll}
$p^6=(t^2-t+1)(t^6-3t^5+6t^4-5t^3+6t^2-3t+1)(-1+t)^3 $,&
$p^8=(-1+t)^7  $,\\
$p^7=(2t^4-2t^3+3t^2-2t+2)(t^2-t+1)^2(-1+t)^3 $,&
$p^{9}=(t^2-t+1)(-1+t)^3 $.
\end{tabular}
\end{center}
In each case, $e(b)=2$, so according to Proposition
\ref{alex}, none of these braids is quasipositive.
~\findemo



\subsection{Utilization of the Rokhlin-Mischachev orientation formula}
 \begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ia.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ib.eps} &
    \includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ic.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Id.eps}
\\ a)&b)&c)&d)\\
 \includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ie.eps} &
    \includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_If.eps}&
\includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ij.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures2/Quotient_Type_Ik.eps}
\\ e)&f)&g)&h)
 \end{tabular}
\caption{}
 \label{quotient type I}
\end{figure}
\begin{lemma}\label{lem1}
There does not exist a symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ with a quotient curve
realizing the $\mathcal L$-scheme depicted in Figures \ref{quotient type I}a) and d) with $\alpha+\beta$ odd.
\end{lemma}
\textit{Proof. }Such a quotient curve is of type $II$ because it is an $(M-1)$-curve. The mirror curve of the
initial symmetric curve is a nest of depth $3$ with an odd component, and so is of type $I$. Thaus, according to Proposition
\ref{quotient dividing}, the initial symmetric curve cannot be of type $I$.
~\findemo

\begin{lemma}\label{lem2}
There does not exist a symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ with a quotient curve
realizing the $\mathcal L$-scheme depicted in Figure \ref{quotient type I}b) with $\alpha+\beta$ odd.
\end{lemma}
\textit{Proof. }According to the Fiedler orientation alternating rule on symmetric curves corresponding to these quotient curves,
 the symmetric curves
cannot be of type $I$ if $\alpha+\beta$ is odd, as the two invariant empty ovals have opposite orientations.
~\findemo

\begin{lemma}\label{lem3}
There does not exist a symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ with a quotient curve
realizing the $\mathcal L$-scheme depicted in Figure \ref{quotient type I}c) with
$(\alpha,\beta)=(4,1)$, $(3,2)$, $(2,3)$, and $(1,2).$
\end{lemma}
\textit{Proof. }According to the Fiedler orientation alternating rule on symmetric curves corresponding to these quotient curves,
the three invariant ovals are positive and that we have
$\Lambda_+-\Lambda_-=1$,
$\Pi_+-\Pi_-=0$ if $\alpha$ is odd, and
$\Pi_+-\Pi_-=-2$ if $\alpha$ is even.
Thus, the Rokhlin-Mischachev orientation formula is fulfilled only for $(\alpha,\beta)=(3,2)$. Choose $L$ as the starting line for the pencil $\mathcal L$,
 and the corresponding braid and its Alexander polynomial are~:
\begin{center}
\begin{tabular}{ll}
$\sigma_1^{-4}\sigma_2^{-1}\sigma_1\sigma_2^{-2}\sigma_1\Delta_3^2$&
and $(-1+t)^3 $.
\end{tabular}
\end{center}
Since $e(b)=1$, according to Proposition \ref{alex}, the braid is not quasipositive.
~\findemo
\begin{lemma}\label{lem4}
There does not exist a symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ with a quotient curve
realizing the $\mathcal L$-scheme depicted in Figure \ref{quotient type I}e) with
$(\alpha,\beta)=(5,0)$, $(4,1)$, $(3,2)$, $(2,3)$, and $(2,1).$
\end{lemma}
\textit{Proof. }According to the Fiedler orientation alternating rule on symmetric curves corresponding to these quotient curves,
 the two invariant empty ovals have opposite orientations, the nom-empty oval is negative, and we have
$\Lambda_+-\Lambda_-=1$,
$\Pi_+-\Pi_-=0$ if $\alpha$ is even, and
$\Pi_+-\Pi_-=-2$ if $\alpha$ is odd.
Hence, the Rokhlin-Mischachev orientation formula is fulfilled only for $(\alpha,\beta)=(4,1)$ or $(2,3)$. Choose $L$ as the starting line for the pencil $\mathcal L$,
 and the corresponding braids are~:
\begin{quote}
$b^{10}_{\alpha,\beta}=\sigma_3^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_3^{-(1+\beta)}\sigma_2^{-1}\sigma_3
\sigma_2^{-\alpha}\sigma_3^{-1}
\sigma_2\sigma_1^{-3}\sigma_3\sigma_2\sigma_1\sigma_3^{-1}\Delta_4^2$.
\end{quote}
The computation of the corresponding Alexander polynomials gives
\begin{center}
\begin{tabular}{ll}
$p^{10}_{4,1}=(t^2+1)(t^2-t+1)(-1+t)^3$,
&$p^{10}_{2,3}=(t^4-2t^3+4t^2-2t+1)(-1+t)^3$
\end{tabular}
\end{center}
In each case we have $e(b)=2$, so according to Proposition \ref{alex}, both braids are not quasipositive.~\findemo

\begin{lemma}\label{lem5}
There does not exist a symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ with a quotient curve
realizing the $\mathcal L$-scheme depicted in Figure \ref{quotient type I}f) with $(\alpha,\beta)=(5,0)$, $(3,2)$, and  $(2,3)$.
\end{lemma}
\textit{Proof. }Choose $L$ as the starting line for the pencil $\mathcal L$,
 and the corresponding braids are~:
\begin{quote}
$b^{11}_{\alpha,\beta}=\sigma_2^{-\alpha}\sigma_3^{-1}\sigma_2\sigma_3^{-\beta}\sigma_1^{-3}
\sigma_1\sigma_2^{2}\sigma_1^{-4}
\sigma_2^{-1}\sigma_3\Delta_4^2$.
\end{quote}
The computation of the determinant gives
$ 976$ for $b^{11}_{5,0}$
and
$592$ for $b^{11}_{3,2}$ which
are not squares in $\mathbb Z$ although $e(b)=3$. So according to Proposition \ref{double alex}, these two braids are not quasipositive.

The computation of the Alexander polynomial of $b^{11}_{2,3}$ gives
\begin{quote}
$(t^2+1)(t^6-5t^5+12t^4-14t^3+12t^2-5t+1)(-1+t)^2$.
\end{quote}
The number $i$ is a simple root of this polynomial and $
e(b)=3$. Thus, according to Proposition \ref{square}, this braid is not quasipositive.
~\findemo

\begin{prop}\label{restr thm 4 3}
The real schemes $\langle J\amalg 2\amalg 1\langle 10\rangle \rangle$ and  $\langle J\amalg 6\amalg 1\langle 6\rangle \rangle$ are not realizable by nonsingular
 symmetric dividing pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }According to the Bezout theorem, Proposition \ref{oval quotient}, Lemma \ref{bezout1} and Lemma \ref{convexe},
the only possibilities for the  $\mathcal L$-scheme of
the quotient curve of such a dividing symmetric curve of degree $7$ in $\mathbb RP^2$ are depicted
in Figures \ref{quotient type I}a), g) and h) with $(\alpha,\beta+\gamma)=(4,1)\textrm{ and }(2,3)$
and in Figures  \ref{quotient type I}d), e) and f) with
$(\alpha,\beta)=(5,0),\textrm{ and }(3,2)$.
If a curve of bidegree $(3,0)$ realizes one of the two $\mathcal L$-schemes depicted in Figures \ref{quotient type I}g) and h) then $\gamma=0$. Otherwise, the base passing through
the points $a$ and $b$ and through an oval $\gamma$ intersects
the quotient curve in more than $7$ points, what contradicts the Bezout theorem.

The remaining quotient curves have been prohibited in Lemmas \ref{lem1}, \ref{lem2}, \ref{lem3}, \ref{lem4}, and \ref{lem5}~\findemo

\begin{prop}\label{only possibility}
If  a nonsingular symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ realizes
the real scheme $\langle J\amalg 8\amalg 1\langle 4\rangle \rangle$, then the  $\mathcal L$-scheme of
its quotient curve is as depicted in Figure \ref{quotient type I}c)
with $(\alpha,\beta)=(1,4)$.

If  a nonsingular symmetric dividing pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ realizes
the real scheme $\langle J\amalg 2\beta+2\amalg 1\langle 2\alpha\rangle \rangle$ with $(\alpha,\beta)=(4,1)$ or $(2,1)$,
 then the  $\mathcal L$-scheme of
its quotient curve is as depicted in Figure \ref{quotient type I}f).
\end{prop}
\textit{Proof. }The proof is the same as for the previous proposition.
~\findemo

\begin{prop}\label{non 4 4 I}
The real scheme $\langle J\amalg 4\amalg 1\langle 4\rangle \rangle$ is not realizable by
 nonsingular symmetric dividing pseudo-holomorphic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Suppose the contrary. Then, according to Proposition \ref{only possibility}, the quotient curve $X$ of such a symmetric curve is as depicted in Figure \ref{quotient type I}f) with $(\alpha,\beta)=(2,1)$. Using the Fiedler orientation alternating rule, and denoting by $\epsilon$ the sign of the two non-invariant outer ovals of the symmetric curve, we have
$\Lambda_+-\Lambda_-=-1+2\epsilon$ and
$\Pi_+-\Pi_-=0$.
Thus, the Rokhlin-Mischachev orientation formula is fulfilled only if $\epsilon=-1$. Hence, one of the two complex orientations of the curve is as depicted in Figure \ref{orientations 4 4}a). Using once again the Fiedler orientation alternating rule, we see that the pencil of lines through the point $p$ induces an  order on the $6$ non-invariant ovals of the symmetric curve as depicted in Figure \ref{orientations 4 4}a). So the ovals $4$ and $1$ are not on the same connected component of $\mathbb RP^2\setminus (L_1\cup \L_2)$. The mutual arrangement of the curve $X$ and the quotient curve of $(L_1\cup \L_2)$ (in bold line) is as depicted in Figure \ref{orientations 4 4}b). Now, consider the base of $\Sigma_2$ which passes through the three ovals of $X$. It has $9$ intersection points with $X$ which contradicts the Bezout theorem.~\findemo
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2.5cm, angle=0]{Figures2/Orientation_4_4.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures2/Contradiction_4_4_I.eps} &
\includegraphics[height=2cm, angle=0]{Figures3/Pseudo_constr_a.eps}    &
\includegraphics[height=2cm, angle=0]{Figures3/Pseudo_constr_b.eps}
\\ a)&b)&c)&d)
 \end{tabular}
\caption{}
 \label{orientations 4 4}
\end{figure}

\subsection{Constructions}
\begin{prop}\label{pseudo}
There exists nonsingular real pseudo-holomorphic curves of bidegree $(3,1)$ in $\Sigma_2$ such that the $\mathcal L$-scheme realized by the union of this curve and a base is
as shown in Figures \ref{orientations 4 4}c) and d). In particular, all the real tangency points of the curve with the pencil $\mathcal L$ are above
the base $\{y=0\}$.
\end{prop}
\textit{Proof. }The braids associated to these $\mathcal L$-schemes are
\begin{quote}
$b^{12}=\sigma_2^{-1}\sigma_3^{-1}\sigma_2 \sigma_3^{-1}\sigma_2^{-1}\sigma_3^{-3} \sigma_2^{-1}\sigma_3
 \sigma_1^{-1}\sigma_2^{-2}\sigma_3^{-1}\sigma_1\sigma_2^{2}\sigma_1^{-1}\sigma_2^{-2}\sigma_1^{-1}
\sigma_2^{-1}\Delta_4^2$,
\\$b^{13}=\sigma_2^{-3}\sigma_3^{-1}\sigma_2^{-1} \sigma_3\sigma_2^{-1}\sigma_3^{-1} \sigma_2\sigma_1^{-2} \sigma_2^{-1}\sigma_1^{-1}\sigma_3^{-1}\sigma_1\sigma_2^{2}\sigma_1^{-2}\sigma_2^{-1}\sigma_1^{-2}\sigma_2^{-1}\sigma_3
\Delta_4^2$.
\end{quote}
Using the Garside normal form (see \cite{Gar} or \cite{Jaq}), we see that these braids are trivial, so quasipositive.~\findemo

\begin{figure}[h]
      \centering
 \begin{tabular}{cc}
\includegraphics[height=4cm, angle=0]{Figures3/S_A.eps}    &
\includegraphics[height=4cm, angle=0]{Figures3/S_B.eps}
\\ a)&b)
 \end{tabular}
\caption{}
 \label{pseudo constr}
\end{figure}
\begin{figure}[h]
      \centering
\begin{tabular}{cc}
\includegraphics[height=3cm, angle=0]{Figures3/path.eps}  &
\includegraphics[height=3cm, angle=0]{Figures3/Perturb_p.eps}
\\a)&b)
\end{tabular}
\caption{}
 \label{perturb p}
\end{figure}
Denote by $C$ (resp. $C'$) the strict transform by the blow up of $\mathbb CP^2$ at $[0:1:0]$ of
a symmetric nonsingular pseudo-holomorphic curve of degree $7$ in $\mathbb RP^2$ corresponding to the
quotient curve depicted in Figure \ref{orientations 4 4}c) (resp. d)). If we note $p$ the intersection point of $C$ with the exceptionnal section $E$
and $F_p$ the fiber of $\Sigma_1$ through $p$, the curve $C$ has
a tangency point of order $2$ with $F_p$ at $p$.

In order to prove Propositon \ref{difference 1}, we look closer
at the construction of the braid associated to a pseudo-holomorphic curve.

Let us introduce some notations. For $\epsilon>0$, denote by $\gamma_{1,\epsilon}$ and
 $\gamma_{2,\epsilon}$ the followings paths
\begin{center}
\begin{tabular}{cccc}
$\gamma_{1,\epsilon}~:$&$[-1,1]$&$\to$&$\mathbb C$\\
   &   $t$&$\mapsto$&$\frac{1}{\epsilon}t+i\epsilon$
\end{tabular} and
\begin{tabular}{cccc}
$\gamma_{2,\epsilon}~:$&$[0,1]$&$\to$&$\mathbb C$\\
  &  $ t$&$\mapsto$&$\frac{1}{\epsilon}e^{i\Pi t}+i\epsilon$
\end{tabular}.
\end{center}
Choose $\gamma_\epsilon$  a parametrisation of the union of the image of $\gamma_{1,\epsilon} $
and $\gamma_{2,\epsilon}$ (see Figure \ref{perturb p}a)). Denote also by $\pi$ the projection
$\Sigma_1\to\mathbb CP^1$ on the base $\{y=0\}$, $\pi'$ the restriction of $\pi$ to $\Sigma_1\setminus E$,
$S_A=\pi^{-1}(\mathbb RP^1)$, $D_\epsilon$ the compact region
of $\mathbb C$ bounded by $\gamma_\epsilon$,
$b_{1,\epsilon}=\pi'^{-1}(\gamma_{1,\epsilon}([-1:1])\cap A$, $b_{2,\epsilon}=\pi'^{-1}(\gamma_{2,\epsilon}([0:1])\cap  A$,
$b_{\epsilon}=b_{1,\epsilon}b_{2,\epsilon}$ and $N_\epsilon=\pi'^{-1}(D_\epsilon)\cap A$. Thus ,
$b_{1,\epsilon}$ and $b_{2,\epsilon}$ are braids and $b_{2,\epsilon}=\Delta_6$.

As $C$ is a real curve, $S_C$ is formed by a real part which is $\mathbb RC$, and by a non-real part.
This latter space has several connected components  which are globally invariant by
the complex conjugation. One can  deduce $S_C$ (resp. $S_{C'}$) from the quotient curve of $C$. The curve $S_C$ (resp. $S_{C'}$) is depicted
in Figure \ref{pseudo constr}a) (resp. c)), where the bold lines are used to draw $\mathbb RC$ ant the dash lines are used to draw
$S_C\setminus\mathbb RC$.
According to \cite{O1}, the braid $b_{1,\epsilon}$ for $\epsilon$ small enough can be viewed as a smoothing of $S_C$.
\begin{prop}\label{difference 1}
The real pseudo-holomorphic curve $C$ constructed above is a dividing curve.
\end{prop}
\textit{Proof. }Except for the point $p$, all singular points of $S_C$ are smoothed as explained in Section \ref{braid}.
There exist local coordinates $(z,w)$ in a neighbohrood of $p$ in $\Sigma_1$ such that in
this neighbohrood, a line of $\mathcal L$ has equation $z=const$ and $C$ has equation $z=\frac{1}{w^2}$.
Then, for $\epsilon$ small enough, the smoothing of $S_C$ at $p$ is given by the parametrisation
\begin{center}
\begin{tabular}{ccc}
$[0,1]$&$\to$&$ [0,1]\times\mathbb C$\\
      $t$&$\mapsto$&$(\epsilon e^{i\Pi (1-t)},\pm\frac{1}{\epsilon}e^{-\frac{i\Pi(1-t)}{2}})$.
\end{tabular}
\end{center}
So the smoothing of the connected component of $C$ which contain $p$
is depicted in Figure \ref{perturb p}b). The closure of the braid $b_\epsilon$ corresponding to $C$
has $6$ components denoted by
$L_1,\ldots,L_6$ as depicted in Figure \ref{gluing}.
By the Riemann-Hurwitz formula, we have
$$\mu(N_\epsilon)=g(N_\epsilon)+\frac{\mu(\hat b_\epsilon) + 6-e(b\epsilon)}{2} $$,
where $\mu(N_\epsilon)$ the number of
connected components of $N_\epsilon$ and $g(N_\epsilon)$ the sum of the genus of the connected components of $N_\epsilon$. We have   $\mu(\hat b_\epsilon)=6$,  $e(b_\epsilon)=0$ and $\mu(N_\epsilon)\le 6$, so $N_\epsilon$ is
composed by $6$ discs. Denote these discs $D_{1,\epsilon},\ldots,D_{6,\epsilon}$ in such a way  that
$\partial D_{i,\epsilon}=L_{i,\epsilon}$. Define also $\overline{D_{i,\epsilon}}=conj(D_{i,\epsilon})$. As $\epsilon\to 0$, these $12$ discs glue together along $S_C$, and $C$ is the
result of this gluing. Moreover, $C\setminus\mathbb RC$ is the result of the gluing of these $12$ discs along
$S_C\setminus\mathbb RC$. Hence, to find the type of $C$, we just have to study how the $12$ discs
glue along $S_C\setminus\mathbb RC$.
\begin{figure}[h]
      \centering
\begin{tabular}{cc}
\includegraphics[height=4.5cm, angle=0]{Figures3/Sym_gluing.eps}&
     \includegraphics[height=4.5cm, angle=0]{Figures3/Sym_gluing2.eps}
\\a)&b)
\end{tabular}
\caption{}
 \label{gluing}
\end{figure}

Denote by $D_{i,\epsilon}|| D_{j,epsilon}$ the relation ``$D_{i,\epsilon}$ glue with $D_{j,\epsilon}$ along a connected component of $S_C\setminus\mathbb RC$
as $\epsilon\to 0$''.
Using the fact that each connected component of $S_C\setminus\mathbb RC$ is globally invariant by the complex
conjugation, we have  (see Figure \ref{gluing}a)) :
\begin{center}
\begin{tabular}{lllll}
 $D_{1,\epsilon}|| \overline{D_{2,\epsilon}}$,&
$D_{1,\epsilon}|| \overline{D_{6,\epsilon}}$,&
$D_{1,\epsilon}|| \overline{D_{4,\epsilon}}$,&
$D_{2,\epsilon}|| \overline{D_{3,\epsilon}}$,&
$D_{2,\epsilon}|| \overline{D_{5,\epsilon}}$,\\
$D_{3,\epsilon}|| \overline{D_{6,\epsilon}}$,&
$D_{3,\epsilon}|| \overline{D_{4,\epsilon}}$,&
$D_{4,\epsilon}|| \overline{D_{5,\epsilon}}$,&
$D_{5,\epsilon}|| \overline{D_{6,\epsilon}}$,&
$D_{i,\epsilon}|| \overline{D_{j,\epsilon}}\implies D_{j,\epsilon}|| \overline{D_{i,\epsilon}}.$
\end{tabular}
\end{center}
The curve $C$ is a dividing curve
if and only if there exist two equivalence classes for $||$.
Here the equivalent classes are
$\{D_{1,\epsilon},D_{3,\epsilon},D_{5,\epsilon},\overline{D_{2,\epsilon}},
\overline{D_{4,\epsilon}},\overline{D_{6,\epsilon}} \} $
and
$\{D_{2,\epsilon},D_{4,\epsilon},D_{6,\epsilon},\overline{D_{1,\epsilon}},
\overline{D_{3,\epsilon}},\overline{D_{5,\epsilon}} \} ,$
Hence, $C$ is a dividing curve and the proposition is proved.~\findemo
\begin{prop}\label{difference 2}
The real pseudo-holomorphic curve $C'$ constructed above is a dividing curve.
\end{prop}
\textit{Proof. }We keep the same notations than in Proposition \ref{difference 1}. As in this propsition, the closure of the braid $b_\epsilon$ has $6$ components, the surface $N_\epsilon$ is
composed by $6$ discs and
the two  equivalence classes for the relation $||$ are
$\{D_{1,\epsilon},D_{2,\epsilon},D_{4,\epsilon},\overline{D_{3,\epsilon}},
\overline{D_{5,\epsilon}},\overline{D_{6,\epsilon}} \} $
and
$\{D_{3,\epsilon}, D_{5,\epsilon}, D_{6,\epsilon}, \overline{D_{1,\epsilon}},
\overline{D_{2,\epsilon}}, \overline{D_{4,\epsilon}} \}$  (see Figure \ref{gluing}b)).
Hence, $C$ is a dividing curve and the proposition is proved.
~\findemo


\begin{cor}\label{pseudo non alg 2}
The complex schemes $\langle J\amalg 4_+\amalg 4_-\amalg  1_+\langle 2_+\amalg 2_-\rangle\rangle_I$ and $\langle J\amalg 2_+\amalg 2_-\amalg  1_+\langle 4_+\amalg 4_-\rangle\rangle_I$ are realizable by
 nonsingular symmetric real pseudo-holomorphic
 curves of degree $7$ in $\mathbb RP^2$.
\end{cor}

\section{Algebraic statements}\label{algebraic state}

\subsection{Prohibitions}

\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures2/8_4_II_b.eps}    &
\includegraphics[height=2cm, angle=0]{Figures2/8_4_II_c.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures2/8_4_II_e.eps} &
 \includegraphics[height=2cm, angle=0]{Figures2/8_4_II_i.eps}
\\ a)&b) &c)&d)\\
 \end{tabular}
\caption{}
 \label{8 4 I alg}
\end{figure}
\begin{lemma}\label{lem I 1}
The $\mathcal L$-schemes depicted in Figures \ref{8 4 I alg}b), c) and d) are not realizable by   nonsingular real pseudo-holomorphic curves of bidegree $(3,0)$ in
$\Sigma_3$.
\end{lemma}
\textit{Proof. }Compute the braids associated to these $\mathcal L$-schemes~:
\begin{center}
\begin{tabular}{ll} $b^{14}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-2}\sigma_2^{-1}\sigma_1\sigma_2^{-4}\sigma_1^{-1}\Delta_3^3$,
& $b^{15}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2^{-5}\sigma_1^{-1}\Delta_3^3$,
\\$b^{16}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2^{-1}\sigma_1\sigma_2^{-4}\sigma_1^{-1}\sigma_2^{-1}\Delta_3^3$.
\end{tabular}
\end{center}
These braids verify $e(b)=0$, so they are quasipositive if and only if they are trivial. Computing
their Garside normal form (see \cite{Gar} or \cite{Jaq}), we find
\begin{center}
\begin{tabular}{lll}
$p^{14}\sigma_2^{3}\sigma_1^{2}\sigma_2^{2}\sigma_1^{2}\Delta_3^{-3}$,
&$p^{15}\sigma_1\sigma_2^{2}\sigma_1^2\sigma_2^{2}\sigma_1^{2}\Delta_3^{-3}$,
&$p^{16}\sigma_1\sigma_2^{3}\sigma_1^{2}\sigma_2^{2}\sigma_1\Delta_3^{-3}$.
\end{tabular}
\end{center}
Thus, no one of these braids is quasipositive.
~\findemo

\begin{lemma}\label{lem I 3}
The $\mathcal L$-scheme depicted in Figure \ref{8 4 I alg}a) is not realizable by
nonsingular real algebraic curves of bidegree $(3,0)$ in
$\Sigma_3$.
\end{lemma}
\textit{Proof. }The weighted comb associated to this $\mathcal L$-scheme is
$$w_1=(g_3g_6g_1g_4g_1g_6g_5g_2g_3g_6g_1g_4g_1g_6(g_3g_2)^3g_3g_6g_1g_4g_1g_6g_5g_2,1,2,0).$$ We have $\mu(w_1)=0$ so, according to Proposition \ref{prohib comb} the lemma is proved.
\begin{prop}\label{prop 84}
The real scheme $\langle J\amalg 8 \amalg 1\langle 4 \rangle\rangle$ is not realizable by nonsingular symmetric dividing real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Suppose that there exists a dividing symmetric curve which contradict the lemma. The real algebraic curve of bidegree $(3,0)$ in $\Sigma_3$ obtained under a blow up of $\Sigma_2$ at the intersection point of the quotient curve and the exceptionnal section and a blow down of the strict transform of the fiber has a double
point with non-real tangents. Smooth this double point in order to obtain an oval. Then,
according to Propositions \ref{only possibility} and \ref{prohib comb}, the only possible $\mathcal L$-schemes which can be obtained are  depicted in Figures
\ref{8 4 I alg}a), b), c) and d). However, according to Lemmas \ref{lem I 1} and
\ref{lem I 3}, these $\mathcal L$-schemes are not algebraically realizable so there is a contradiction.~\findemo


\begin{lemma}\label{lem 48 1}
The $\mathcal L$-schemes depicted in Figures \ref{4 8 I alg}a), ..., f) are not realizable by  nonsingular real pseudo-holomorphic curves of bidegree $(3,0)$ in
$\Sigma_5$.
\end{lemma}
\textit{Proof. }
Compute the braids associated to these $\mathcal L$-schemes~:
\begin{center}
\begin{tabular}{ll}
$b^{17}=\sigma_2^{-3}\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2\sigma_1^{-1}c$,
& $b^{18}=\sigma_2^{-2}\sigma_1^{-2}\sigma_2^{-2}\sigma_1^{-1}\sigma_2\sigma_1^{-1}c$,
\\
$b^{19}=\sigma_2^{-2}\sigma_1^{-1}\sigma_2^{-2}\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2^{-1}c$,
&
$b^{20}=\sigma_2^{-4}\sigma_1^{-2}\sigma_2\sigma_1^{-1}c$,
\\ $b^{21}=\sigma_2^{-1}\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2^{-2}\sigma_1^{-1}\sigma_2\sigma_1^{-1}c$,
&
$b^{22}=\sigma_2^{-2}\sigma_1^{-2}\sigma_2^{-1}\sigma_1\sigma_2^{-2}
\sigma_1^{-1}\sigma_2\sigma_1^{-1}c,$
\end{tabular}

where $c=\sigma_1^{-3}\sigma_2^{-1}\sigma_1\sigma_2^{-4}\sigma_1^{-1}\sigma_2\Delta_3^5.$
\end{center}
The braids $b^{17}$ and $b^{21}$ verify $e(b)=2$, and the computation of their determinant gives
$ 301\textrm{ and } 805$,
which are not squares in $\mathbb Z$. So according to Proposition \ref{double alex}, these two braids are not quasipositive.

The braids $b^{18}$,  $b^{19}$, $b^{20}$, and $b^{22}$ verify $e(b)=1$, and the computation of their Alexander polynomials gives
\begin{quote}
$p^{18}=(-1+t)(t^2+1)(t^2-t+1)$,
\\$p^{19}=(-1+t)(t^2-t+1)(t^2+1)(t^6-t^5+2t^4-3t^3+2t^2-t+1)$,
\\$ p^{20}=(-1+t)(t^4-t^3+2t^2-2t+1)(t^4-2t^3+2t^2-t+1)(t^2+1)^2$,
\\$p^{22}=(-1+t)(t^4-t^3+t^2-t+1)(t^6-2t^5+4t^4-5t^3+4t^2-2t+1)(t^2-t+1)(t^2+1)$.
\end{quote}
According to Proposition \ref{alex}, these four braids are not quasipositive.
~\findemo
  \begin{figure}[h]
      \centering
\begin{tabular}{cccc}
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_a.eps}    &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_b.eps}    &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_c.eps}    &   \includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_d.eps}
\\ a)&b) &c)&d)\\
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_e.eps}    &
 \includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_f.eps} &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_g.eps} &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_h.eps}
\\ e) &f)&g)&h)\\
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_i.eps}    &
 \includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_j.eps} &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_k.eps} &
\includegraphics[height=1.8cm, angle=0]{Figures2/4_8_II_l.eps}
\\i)&j)&k)&l)
 \end{tabular}
\caption{}
 \label{4 8 I alg}
\end{figure}

\begin {lemma}\label{lem 48 3}
The $\mathcal L$-schemes depicted in Figures \ref{4 8 I alg}g), ..., l) are not relizable by
  nonsingular real algebraic curves of bidegree $(3,0)$ in
$\Sigma_5$.
\end{lemma}
\textit{Proof. }The weighted combs associated to these $\mathcal L$-schemes are
\begin{center}
\begin{tabular}{ll}
$w_2=(ag_3bg_5g_2ag_3bag_3bg_5g_2ag_3ba^3g_3b,1,4,0)$,&
$w_5=(g_3bg_5g_2a^2g_3b(a^3g_3b)^2,2,5,1)$,\\
$w_3=(ag_3bg_5g_2ag_3ba^3g_5bg_3ba^3g_3b,1,4,0)$,&
$w_6=(ag_3bg_5g_2ag_3b(a^3g_3b)^2,2,5,1)$,\\
$w_4=(g_3bg_5g_2a^2g_3bg_3bg_5g_2a^2g_3ba^3g_3b,1,4,0)$,&
$w_7=((a^3g_3b)^3,3,6,2)$.\\
\end{tabular}

where $a=g_3g_2$ and $b=g_6g_1g_4g_1g_6$.
\end{center}
The multiplicity of all these weighted comb is $0$.
Thus, according to Proposition \ref{prohib comb}, none of these $\mathcal L$-schemes is relizable by
  nonsingular real algebraic curves of bidegree $(3,0)$ in
$\Sigma_5$.
\findemo
\begin{defi}
A $\mathcal L$-scheme is minimal if no operation $\subset_j\supset_{j\pm 1}\to\emptyset$ is possible.
\end{defi}
\begin {prop}\label{prop 48}
The real scheme $\langle J\amalg 4 \amalg 1\langle 8 \rangle\rangle$ is not realizable by nonsingular symmetric dividing real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Suppose that there exists a dividing symmetric curve which contradict the lemma. Then,
according to Proposition \ref{only possibility}, the only possible  minimal $\mathcal L$-scheme
for its quotient curve with respect to a base is depicted in Figure \ref{quotient type I}f). Blowing up
$\Sigma_2$ at the intersection point of the quotient curve and the exceptionnal section and blowing down
the strict transform of the fiber, we obtain a curve of bidegree $(3,0)$ in $\Sigma_3$ with a double
point with non-real tangents at $p$ and arranged with respect to a base of $\Sigma_3$ as shown in Figure \ref{4 8 I alg 7}a).
Then the cubic resolvante (see \cite{OS2}) of the union of the base and the curve of bidegree $(3,0)$ is a real algebraic curve of bidegree $(3,0)$ on $\Sigma_6$, depicted in Figure  \ref{4 8 I alg 7}b). The point $p$ is a triple point and the other singular points are
double points with non-real tangents. Smoothing all the double points in order to obtain ovals,
blowing up $\Sigma_6$ at $p$ and blowing down the strict transform of
the fiber, we obtain a nonsingular algebraic curve $C$
of bidegree $(3,0)$ in $\Sigma_5$ realizing the minimal $\mathcal L$-scheme depicted in Figure \ref{4 8 I alg 7}c).
According to Proposition \ref{operations} and Lemma \ref{lem 48 1}, the only possibilities
for the $\mathcal L$-scheme realized by $C$ which do not contradict obviously Proposition \ref{prohib comb} are depicted in Figures
\ref{4 8 I alg}g), ..., l). It
has been proved in Lemma \ref{lem 48 3} that all these $\mathcal L$-schemes
 are not algebraically realizable, so there is a contradiction.~\findemo
\begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=2cm, angle=0]{Figures2/4_8_II_2.eps}    &
\includegraphics[height=2cm, angle=0]{Figures2/4_8_II_3.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures2/4_8_II_4.eps}
\\ a)&b) &c)
 \end{tabular}
\caption{}
 \label{4 8 I alg 7}
\end{figure}



\subsection{Perturbation of a reducible symmetric curve}\label{viro}
The standard method to construct a lot of different isotopy types of nonsingular algebraic curves is to perturb a singular curve in many ways.
 So the first idea to construct symmetric algebraic curves is to perturb in many symmetric ways a singular symmetric
algebraic curve.
\begin{prop}\label{main constr}
All the real schemes listed in table \ref{list scheme} are realizable by nonsingular symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$. Moreover, those marked with a $^*$ are realized
by a dividing symmetric curve and those marked with a $^\circ$ are realized by a non-dividing curve.
\begin{table}[h]\label{list scheme}
\centering
\begin{tabular}{lllll}
$\langle J\rangle^\circ$ &$\langle J\amalg 10 \amalg 1\langle 1 \rangle\rangle^\circ$ &
 $\langle J\amalg 8 \amalg 1\langle 3 \rangle\rangle^\circ$ &$\langle  J \amalg 1\langle 6 \rangle\rangle^\circ$ &
 $\langle  J \amalg 1\langle 9 \rangle\rangle^\circ$

\\

$\langle J\amalg 1\rangle^\circ$ &$\langle J\amalg 11 \amalg 1\langle 1 \rangle\rangle$$^{\circ,*}$&
  $\langle J\amalg 9 \amalg 1\langle 3 \rangle\rangle$$^{\circ,*}$ &  $\langle J\amalg 1 \amalg 1\langle 6 \rangle\rangle^\circ$ &
  $\langle J\amalg 1 \amalg 1\langle 9 \rangle\rangle$$^{\circ,*}$

\\

$\langle J\amalg 2\rangle^\circ$ &$\langle J\amalg 12 \amalg 1\langle 1 \rangle\rangle^\circ$&
  $\langle J\amalg 10 \amalg 1\langle 3 \rangle\rangle^\circ$ &$\langle J\amalg 2 \amalg 1\langle 6 \rangle\rangle^\circ$  &
    $\langle J\amalg 2 \amalg 1\langle 9 \rangle\rangle^\circ$

\\

$\langle J\amalg 3\rangle^\circ$ &$\langle J\amalg 13 \amalg 1\langle 1 \rangle\rangle$$^{*}$&
 $\langle J\amalg 11 \amalg 1\langle 3 \rangle\rangle$$^{*}$  & $\langle J\amalg 3 \amalg 1\langle 6 \rangle\rangle^\circ$ &
    $\langle J\amalg 3 \amalg 1\langle 9 \rangle\rangle$$^{\circ,*}$

\\

$\langle J\amalg 4\rangle^\circ$ & &
  &$\langle J\amalg 4 \amalg 1\langle 6 \rangle\rangle$$^{\circ,*}$  &


\\

$\langle J\amalg 5\rangle^\circ$ & $\langle  J \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J \amalg 4\langle 1\rangle\rangle^\circ$ &  &


\\

$\langle J\amalg 6\rangle^\circ$ &  $\langle J\amalg 1 \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J\amalg 1 \amalg 1\langle 4 \rangle\rangle^\circ$ &$\langle J\amalg 6 \amalg 1\langle 6 \rangle\rangle^\circ$  &


\\

$\langle J\amalg 7\rangle$$^{\circ,*}$ &  $\langle J\amalg 2 \amalg 1\langle 2 \rangle\rangle$$^{\circ,*}$ &
 $\langle J\amalg 2 \amalg 1\langle 4 \rangle\rangle$$^{\circ,*}$   &  &
  $\langle  J \amalg 1\langle 10 \rangle\rangle$$^{\circ,*}$

\\

$\langle J\amalg 8\rangle^\circ$ & $\langle J\amalg 3 \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J\amalg 3 \amalg 1\langle 4 \rangle\rangle^\circ$  &  &
  $\langle J\amalg 1 \amalg 1\langle 10 \rangle\rangle^\circ$

\\

$\langle J\amalg 9\rangle$$^{\circ,*}$ & $\langle J\amalg 4 \amalg 1\langle 2 \rangle\rangle$$^{\circ,*}$  &
 $\langle J\amalg 4 \amalg 1\langle 4 \rangle\rangle^\circ$   &  &
   $\langle J\amalg 2 \amalg 1\langle 10 \rangle\rangle^\circ$

\\

$\langle J\amalg 10\rangle^\circ$ & $\langle J\amalg 5 \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J\amalg 5 \amalg 1\langle 4 \rangle\rangle^\circ$   & $\langle J \amalg 1\langle 7 \rangle\rangle^\circ$  &
  $\langle J\amalg 3 \amalg 1\langle 10 \rangle\rangle^\circ$

\\

$\langle J\amalg 11\rangle$$^{\circ,*}$ & $\langle J\amalg 6 \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J\amalg 6 \amalg 1\langle 4 \rangle\rangle$$^{\circ,*}$  & $\langle J\amalg 1 \amalg 1\langle 7 \rangle\rangle$$^{\circ,*}$  &
  $\langle J\amalg 4 \amalg 1\langle 10 \rangle\rangle$$^{*}$

\\

$\langle J\amalg 12\rangle^\circ$ & $\langle J\amalg 7 \amalg 1\langle 2 \rangle\rangle^\circ$  &
  &$\langle J\amalg 2 \amalg 1\langle 7 \rangle\rangle^\circ$   &


\\

$\langle J\amalg 13\rangle$$^{\circ,*}$ & $\langle J\amalg 8 \amalg 1\langle 2 \rangle\rangle$$^{\circ,*}$  &
 $\langle J\amalg 8 \amalg 1\langle 4 \rangle\rangle^\circ$  &$\langle J\amalg 3 \amalg 1\langle 7 \rangle\rangle$$^{\circ,*}$   &
  $\langle  J \amalg 1\langle 11 \rangle\rangle^\circ$

\\

$\langle J\amalg 14\rangle^\circ$ & $\langle J\amalg 9 \amalg 1\langle 2 \rangle\rangle^\circ$  &
  &  &
 $\langle J\amalg 1 \amalg 1\langle 11 \rangle\rangle$$^{\circ,*}$

\\

$\langle J\amalg 15\rangle$$^{*}$ & $\langle J\amalg 10 \amalg 1\langle 2 \rangle\rangle^\circ$  &
 $\langle J\amalg 10 \amalg 1\langle 4 \rangle\rangle$$^{*}$  &$\langle J\amalg 5 \amalg 1\langle 7 \rangle\rangle$$^{*}$   &
  $\langle J\amalg 2 \amalg 1\langle 11 \rangle\rangle^\circ$

\\

 & $\langle J\amalg 11 \amalg 1\langle 2 \rangle\rangle^\circ$ &
   &  &
  $\langle J\amalg 3 \amalg 1\langle 11 \rangle\rangle$$^{*}$

\\

$\langle J \amalg 1\langle 1 \rangle\rangle^\circ$ & $\langle J\amalg 12 \amalg 1\langle 2 \rangle\rangle$$^{*}$ &
 $\langle J \amalg 1\langle 5 \rangle\rangle^\circ$  &  &


\\

$\langle J\amalg 1 \amalg 1\langle 1 \rangle\rangle^\circ$ &   &
 $\langle J\amalg 1 \amalg 1\langle 5 \rangle\rangle^\circ$ &  &
    $\langle  J \amalg 1\langle 12 \rangle\rangle^\circ$

\\

$\langle J\amalg 2 \amalg 1\langle 1 \rangle\rangle^\circ$ &  $\langle J \amalg 1\langle 3 \rangle\rangle^\circ$ &
  $\langle J\amalg 2 \amalg 1\langle 5 \rangle\rangle^\circ$ & $\langle J \amalg 1\langle 8 \rangle\rangle^\circ$ &


\\

$\langle J\amalg 3 \amalg 1\langle 1 \rangle\rangle$$^{\circ,*}$& $\langle J\amalg 1 \amalg 1\langle 3 \rangle\rangle^\circ$  &
  $\langle J\amalg 3 \amalg 1\langle 5 \rangle\rangle$$^{\circ,*}$   &  $\langle J\amalg 1 \amalg 1\langle 8 \rangle\rangle^\circ$ &
 $\langle J\amalg 2 \amalg 1\langle 12 \rangle\rangle$$^{*}$

\\

$\langle J\amalg 4 \amalg 1\langle 1 \rangle\rangle^\circ$ & $\langle J\amalg 2 \amalg 1\langle 3 \rangle\rangle^\circ$  &
   $\langle J\amalg 4 \amalg 1\langle 5 \rangle\rangle^\circ$  &$\langle J\amalg 2 \amalg 1\langle 8 \rangle\rangle$$^{\circ,*}$  &


\\

$\langle J\amalg 5 \amalg 1\langle 1 \rangle\rangle^\circ$ & $\langle J\amalg 3 \amalg 1\langle 3 \rangle\rangle$$^{\circ,*}$  &
  $\langle J\amalg 5 \amalg 1\langle 5 \rangle\rangle$$^{\circ,*}$   &$\langle J\amalg 3 \amalg 1\langle 8 \rangle\rangle^\circ$  &
  $\langle  J \amalg 1\langle 13 \rangle\rangle^\circ$

\\

$\langle J\amalg 6 \amalg 1\langle 1 \rangle\rangle^\circ$ &  $\langle J\amalg 4 \amalg 1\langle 3 \rangle\rangle^\circ$  &
  $\langle J\amalg 6 \amalg 1\langle 5 \rangle\rangle^\circ$   &$\langle J\amalg 4 \amalg 1\langle 8 \rangle\rangle^\circ$  &
  $\langle J\amalg 1 \amalg 1\langle 13 \rangle\rangle$$^{*}$

\\

$\langle J\amalg 7 \amalg 1\langle 1 \rangle\rangle$$^{\circ,*}$ &  $\langle J\amalg 5 \amalg 1\langle 3 \rangle\rangle$$^{\circ,*}$ &
   $\langle J\amalg 7 \amalg 1\langle 5 \rangle\rangle$$^{\circ,*}$  &  &


\\
 $\langle J\amalg 8 \amalg 1\langle 1 \rangle\rangle^\circ$  &  $\langle J\amalg 6 \amalg 1\langle 3 \rangle\rangle^\circ$  &
   $\langle J\amalg 8 \amalg 1\langle 5 \rangle\rangle^\circ$   &   &
 $\langle J\amalg 1\amalg 1\langle 1 \langle 1\rangle\rangle\rangle$$^{*}$

\\
 $\langle J\amalg 9 \amalg 1\langle 1 \rangle\rangle$$^{\circ,*}$  &   $\langle J\amalg 7 \amalg 1\langle 3 \rangle\rangle$$^{\circ,*}$ &
  $\langle J\amalg 9 \amalg 1\langle 5 \rangle\rangle$$^{*}$    &   %&
\end{tabular}
\end{table}
\end{prop}
\textit{Proof. }In order to apply the Viro method without change of coordinates, we consider here symmetry with respect to the line $\{y-z=0\}$. Consider the union of the line $\{x=0\}$ and three symmetric conics in $\mathbb RP^2$ tangent
to each other into the two symmetric points $[0:0:1]$ and $[0:1:0]$. Using the
 Viro method (see \cite{V1}, \cite{V2}) and the classification, up to isotopy, of the
curves of degree $7$ in $\mathbb RP^2$ with the only singular point $Z_{15}$ established by A. B. Korchagin in \cite{K1}, we perturb these reducible symmetric curves.
In order to obtain nonsingular symmetric curves, we have to perturb symmetrically the two singular points. That is to say, if we
perturb the singular point at $[0:0:1]$ gluing the chart of a polynomial $P(x,y)$, we have to perturb the singular point at
$[0:1:0]$ gluing the chart of the polynomial $y^7P(\frac{x}{y},\frac{1}{y})$.
~\findemo

\textbf{Remark. }Using this method, we constructed nonsingular symmetric algebraic curves of degree $7$ in $\mathbb RP^2$ realizing
the complex schemes $\langle J\amalg  4_+\amalg 5_-\amalg 1_+\langle  1_+\rangle\rangle_I$ and
 $\langle J\amalg 3_+\amalg 6_-\amalg 1_-\langle 1_+\rangle\rangle_I$. So unlike in the $M$-curves case, the real scheme of
a nonsingular symmetric curves of degree $7$ in $\mathbb RP^2$ does not determine its complex scheme.


\begin{prop}\label{restr thm 4 5}
The complex schemes $\langle J\amalg  1_-\langle  1_+\amalg 3_-\rangle\rangle_I$, $\langle J\amalg 2_+\amalg 3_-\amalg 1_-\langle 1_-\rangle\rangle_I$ and
 $\langle J\amalg 1_-\amalg 1_-\langle 2_+\amalg 3_-\rangle\rangle_I$ are realizable by nonsingular symmetric real algebraic dividing curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }In  \cite{It}, symmetric sextics realizing the complex schemes
 $\langle 1_-\langle  1_+\amalg 3_-\rangle\rangle_I$, $\langle 5\amalg 1_-\langle 1_-\rangle\rangle_I$ and
 $\langle 1 \amalg 1_-\langle 2_+\amalg 3_-\rangle\rangle_I$  are constructed. Consider the union of each of these
curves and a real line oriented and disposed in $\mathbb RP^2$ such that the (symmetric) pertubations according to the orientations satisfie the Rokhlin-Mischachev orientation formula. So, according
to Theorem $4.8$ in \cite{V3}, the obtained real algebraic symmetric curves of degree $7$ in $\mathbb RP^2$ are of type $I$ and
realize the announced complex schemes.
~\findemo

\subsection{Parametrisation of a rational curve}\label{param constr}
Here we apply the method used in \cite{O6} and \cite{O7}.  Namely, we construct a singular rational curve and perturb it using Shustin's results on the independant perturbations of generalized semi-quasihomogeneous singular points of a curve keeping the same Newton polygon
 (see \cite{Sh1} and \cite{Sh2}).
For the definition of the type of a singular point, one can refer to \cite{Ar} or \cite{V4}.
\begin{prop}\label{par}
There exists a rational algebraic curve of degree $4$ in $\mathbb RP^2$ situated with respect to the lines
$\{x=0\}$, $\{y=0\}$, $\{z=0\}$ and $\{y=-z\}$ as shown in Figure \ref{rational}a), with a singular point of type $A_4$ at $[0:0:1]$, a singular point of type $A_1$ at $p$
and a tangency point of order $2$ with the line $\{x=0\}$ at $[0:1:0]$.
\end{prop}
 \begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=4cm, angle=0]{Figures3/rationnal.eps}    &
    \includegraphics[height=2cm, angle=0]{Figures3/rationnal_sigma1.eps}&
\includegraphics[height=2cm, angle=0]{Figures3/rationnal_sigma2.eps}
\\ a)&b)&c)
 \end{tabular}
\caption{}
 \label{rational}
\end{figure}
\textit{Proof. }Consider the  map from $\mathbb C$ to
$\mathbb CP^2$ given by $t\mapsto [x(t):y(t):z(t)]$ where
\begin{center}
\begin{tabular}{lcl}
$\left\{ \begin{array}{l}
x(t)=t^2
\\y(t)=\alpha t^2(t-\gamma)(t+\gamma)
\\z(t)=-((t-1)(\epsilon t-1)(\delta t-1))
\end{array}\right .$
& &with $\alpha =\frac{1}{11}$, $\gamma =\delta =\frac{9}{10}$ and $\epsilon =\frac{99}{100}$.
\end{tabular}
\end{center}
The curve defined by this map has a singular point of type $A_4$ at $[0:0:1]$, as we can see using the following identity~:
$$y(t)z(t)+\alpha\gamma ^2x(t)z(t) - \alpha x(t)^2=- {\displaystyle \frac {81}{1000}} \,t^{7} + {\displaystyle
\frac {2781}{11000}} \,t^{6} - {\displaystyle \frac {289}{1100}}
\,t^{5}. $$
Moreover, it is clear from the equations that the curve has a tangency point of order $2$ with the line $\{x=0\}$ at $[0:1:0]$.
This map define an algebraic curve of degree $4$ in $\mathbb CP^2$
$C=Res_t(x(t)Y-y(t)X,x(t)Z-z(t)X)/X^3$.
Considering $C$ in the affine plane $\{Z=1\}$, we obtain
\begin{center}
\begin{tabular}{ll}
$C=$&${\displaystyle -\frac{1}{11}Y^2-\frac{13468021579}{1331000000000000}X^4+\frac{1666467523}{1210000000000}YX^3+\frac{275261}{3025000}X^2Y-\frac{65485017}{1100000000}X^2Y^2}$\\
\\&${\displaystyle +\frac{2781}{5500}XY^2-\frac{81}{6050}XY+\frac{793881}{1000000}XY^3+\frac{26346141}{6655000000}X^3-\frac{6561}{13310000}X^2}.$
\end{tabular}
\end{center}
Projecting this curve to the line $\{y=0\}$ and using the Budan Fourier theorem (see \cite{BPR}), one can check that $C$ verifies the hypothesis of the proposition.
~\findemo

\begin{cor}\label{restr thm 4 4}
The complex scheme $\langle J\amalg 5\amalg 1\langle 7\rangle \rangle_{II}$ is realizable by  nonsingular non-dividing symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{cor}
\textit{Proof. }The strict transform of the curve constructed in Proposition \ref{par} under the blow up
of $\mathbb CP^2$ at  the point $[0:1:0]$ is the rationnal algebraic curve of bidegree $(3,1)$ in $\Sigma_1$
depicted in Figure \ref{rational}b). Blowing up the point $q$ and blowing down the strict transform of the fiber, we obtain the rational curve of bidegree $(3,0)$
in $\Sigma_2$ with a singular point of type $A_6$ at the point $r$. Then according to Shustin's results
(see \cite{Sh1} and \cite{Sh2}), it is possible to smooth this curve
as depicted in Figure \ref{rational3}a). Perturbing the union of this curve and the fiber $G$, we obtain the nonsingular curve of
bidegree $(3,1)$ in $\Sigma_2$ arranged with the base $\{y=0\}$ as shown in Figure \ref{rational3}b). The corresponding symmetric curve realizes the
 real scheme $\langle J\amalg 5\amalg 1\langle 7\rangle \rangle$ and according to Proposition \ref{oval quotient} this is a non-dividing symmetric curve.
~\findemo

 \begin{figure}[h]
      \centering
 \begin{tabular}{cc}
\includegraphics[height=2cm, angle=0]{Figures3/J10_2.eps}    &
 \includegraphics[height=2cm, angle=0]{Figures3/5_7_II.eps}
\\ a)&b)
 \end{tabular}
\caption{}
 \label{rational3}
\end{figure}

\begin{cor}\label{constr thm1 1}
The real schemes $\langle J\amalg 5\amalg 1\langle 8\rangle \rangle$ and
$\langle J\amalg 4\amalg 1\langle 7\rangle \rangle$ are realizable by nonsingular symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{cor}
\textit{Proof. }One obtains these two curves modifying slightly the previous construction. To obtain the real scheme $\langle J\amalg 5\amalg 1\langle 8\rangle \rangle,$ one can keep all the ovals above the base depicted in Figure \ref{rational3}b). To obtain the real scheme $\langle J\amalg 4\amalg 1\langle 7\rangle \rangle$, one can consider the line $L$ instead of the line $\{y+z=0\}$ in Figure \ref{rational}a).
~\findemo



\subsection{Change of coordinates in $\Sigma_2$}

\begin{prop}\label{change1}
The real schemes $\langle J\amalg 7\amalg 1\langle 4\rangle \rangle$ and $\langle J\amalg 5\amalg 1\langle 6\rangle \rangle $ are realizable by nonsingular symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }
In section \ref{viro}, we constructed symmetric curves in $\mathbb RP^2$  shown in Figure \ref{constr1}a).
According to Lemma \ref{lem bezout1}, their quotient curve $X$ is depicted in Figure \ref{constr1}b).
\begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=2cm, angle=0]{Figures3/Constr1_a.eps}    &
\includegraphics[height=2cm, angle=0]{Figures3/Constr1_b.eps}    &   \includegraphics[height=2cm, angle=0]{Figures3/Constr1_c.eps}
\\ a)&b)&c)
\\&$(\alpha,\beta)=(2,4)$ or $(3,3)$&
 \end{tabular}
\caption{}
 \label{constr1}
\end{figure}
\begin{figure}[h]
      \centering
 \begin{tabular}{ccc}
\includegraphics[height=2cm, angle=0]{Figures3/Constr1_d.eps}    &
\includegraphics[height=2cm, angle=0]{Figures3/Constr1_e.eps}    &   \includegraphics[height=2cm, angle=0]{Figures3/Constr1_f.eps}
\\ a)&b)&c)
\\&$(\alpha,\beta)=(2,4)$ or $(3,3)$&
 \end{tabular}
\caption{}
 \label{constr2}
\end{figure}

Let $H$ be the base which passes through the points $a$, $b$ and $c$.
All possible mutual arrangements for $H$ and the quotient curves which do not contradict the Bezout theorem and Lemma \ref{convexe} are depicted in Figures \ref{constr1}c) and \ref{constr2}.

First, we prohibit pseudo-holomorphically the $\mathcal L$-schemes realized by the union of $X$ and $H$ in Figure \ref{constr2}a) and by $X$ in Figure \ref{constr2}b). Choose $L$ as the starting line for the pencil $\mathcal L$, and the braid corresponding to these $\mathcal L$-schemes are~:
\begin{quote}
$ b^{23}_{\alpha,\beta}= \sigma_3^{-(\beta-1)}\sigma_1^{-1}\sigma_2^{-1}\sigma_3\sigma_2^{-\alpha}\sigma_3^{-1}\sigma_2\sigma_1^{-4}
\sigma_2^{-1}\sigma_3^{2}\sigma_2\sigma_1\sigma_2^{-2}\sigma_3^{-1}\Delta_4^2
  \textrm{ with  }
(\alpha,\beta)=(3,3),(2,4)$,
\\$b^{24}=  \sigma_1^{-7}\sigma_2\sigma_1\Delta_3^2$.
\end{quote}
The braid $b^{23}_{3,3}$ was already shown to be not quasipositive in section \ref{prohib1}.
The computation of the Alexander polynomials of the remaining braids gives
\begin{center}
\begin{tabular}{ll}
$p^{24}=(-1+t)(t^4-t^3+t^2-t+1)$,&
$p^{23}_{4,2}=(t^2-t+1)(-1+t)^3 $.
\end{tabular}
\end{center}
Since $e(b^{24})=1$ and $e(b^{23}_{\alpha,\beta})=2$,  according to Proposition \ref{alex},  none of these braids is quasipositive.

Thus, the two remaining possibilities for the mutual arrangement of $X$ and $H$ are depicted in Figures \ref{constr1}c) and \ref{constr2}c).

In the first case, let $H'=H$.

In the second case,
consider the base $G$
passing through the points $d$, $e$ and $f$ where $f$ is some point on the fiber $L$. For some $f$,
the base $G$ passes through an oval of $X$. Since $G$ cannot have more than $7$ common points with $X$, there exist $f$ for
which the mutual arrangement of $G$ and $X$ is as shown in Figure \ref{constr1}c). Let $H'$ be the base corresponding to such an $f$.

The symmetric curves of degree $7$ in $\mathbb RP^2$ corresponding to the mutual arrangement of $H'$ and $X$ realize the real schemes
$\langle J\amalg 7\amalg 1\langle 4\rangle \rangle$ and $\langle J\amalg 5\amalg 1\langle 6\rangle \rangle$.
~\findemo

\begin{prop}\label{constr thm1 3}
The real schemes $\langle J\amalg 1\amalg 1\langle 12\rangle \rangle$ and $\langle J\amalg 9\amalg 1\langle 4\rangle \rangle $ are realizable by nonsingular symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }In the section \ref{viro},
 we constructed symmetric curves of degree $7$ in $\mathbb RP^2$
depicted in Figure \ref{constr5}a).
These are $M$-curves, so according to Lemma \ref{max sym}, the  $\mathcal L$-scheme realized
by their quotient curve $X$ is  one of these  depicted in Figures
 \ref{constr5}b) and c).
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2cm, angle=0]{Figures3/Constr5_a.eps}    &
\includegraphics[height=2cm, angle=0]{Figures3/Constr5_b.eps}    &   \includegraphics[height=2cm, angle=0]{Figures3/Constr5_c.eps}  &
\includegraphics[height=2cm, angle=0]{Figures3/Constr5_d.eps}
\\ a)&b)&c)&d)
\end{tabular}

$(\alpha,\beta)=(6,0)$ or $(2,4)$
\caption{}
 \label{constr5}
\end{figure}
The braids associated to the $\mathcal L$-schemes shown in Figure \ref{constr5}c) are
\begin{quote}
$ b^{25}_{\alpha,\beta}= \sigma_2\sigma_1^{-\alpha}\sigma_2^{-1}\sigma_1\sigma_2^{-\beta}\Delta_3^2$ with $(\alpha,\beta)=(6,0)$ or $(2,4)$.
\end{quote}
The computation of the corresponding Alexander polynomials gives
\begin{center}
\begin{tabular}{ll}
$p^{25}_{6,0}=(-1+t)(t^4-t^3+t^2-t+1)$,
&$p^{25}_{2,4}=(-1+t)(t^2-t+1) $.
\end{tabular}
\end{center}
Since $e(b^{25}_{\alpha,\beta})=1$, according to Proposition \ref{alex}, these two braids are not quasipositive.

Hence, $X$ realizes the $\mathcal L$-scheme shown $  $in Figure \ref{constr5}b).
Let $H$ be the base which passes through the points $a$, $b$ and $c$ in Figure \ref{constr5}b).
The only possible mutual arrangement for $H$ and $X$ which does not contradict the Bezout theorem, Proposition \ref{max sym} and Lemma \ref{convexe} is depicted in Figure \ref{constr5}d). The corresponding symmetric curves realize the real schemes
$\langle J\amalg 1\amalg 1\langle 12\rangle \rangle$ and
$\langle J\amalg 9\amalg 1\langle 4\rangle \rangle$.~\findemo

\subsection{Construction of auxiliary curves}
The reader can refer to \cite{V1} or \cite{V2} for the definition of the chart and a truncation of a polynomial.
\begin{lemma}\label{curve 3 1}
For any real positive numbers $\alpha,\beta,\gamma$, there exist real curves of degree $3$  in $\mathbb RP^2$
having the charts and the arrangement
with respect to the axis $\{y=0\}$ shown in Figures \ref{degree 3 1}a) and b) with truncation on the segment $[(0,3),(3,0)]$ equal to
$(x-\alpha y)(x-\beta y)(x-\gamma y)$.
\end{lemma}
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[width=2.5cm, angle=0]{Figures3/Cubic1a.eps}
 &\includegraphics[width=2.5cm,angle=0]{Figures3/Cubic2a.eps}&
 \includegraphics[width=4cm, angle=0]{Figures3/Cubic1b.eps}
 &\includegraphics[width=4cm,angle=0]{Figures3/Cubic2b.eps}  \\
a)&b)&c)&d)\\
 \end{tabular}
\caption{}
 \label{degree 3 1}
\end{figure}
\textit{Proof. }Consider the points $A=[\alpha:1:0],B=[\beta:1:0],C=[\gamma:1:0]$ and
four lines $L_1,L_2,L_3,L_4$  as shown in Figure  \ref{degree 3 1}c). For $t$ small enough and of suitable sign,
 the curve $yzL_1+tL_2L_3L_4$
is arranged with respect to the coordinate axis and the lines $L_1,L_2,L_3,L_4$ as shown in Figure  \ref{degree 3 1}c).

To construct the curve with the chart depicted in Figure \ref{degree 3 1}b), we perturb the third degree curve $yzL_1$ as shown in Figure \ref{degree 3 1}d).
~\findemo

\begin{cor}
For any real positive numbers $\alpha,\beta,\gamma$, there exists  real symmetric dividing curves of degree $6$ in $\mathbb RP^2$
with a singular point of type $J_{10}$ at $[1:0:0]$ having  the charts, the arrangement
with respect to the axis $\{y=0\}$ and the complex orientations
shown in Figures \ref{degree 6}a), b) and c) with truncation on the segment $[(0,3),(6,0)]$ equal to
$(x-\alpha y^2)(x-\beta y^2)(x-\gamma y^2)$.
\end{cor}
\begin{figure}[h]
      \centering
 \begin{tabular}{ccccc}
\includegraphics[width=2cm,  angle=0]{Figures3/Sext_sym1.eps}&
\includegraphics[width=2cm,  angle=0]{Figures3/Sext_sym3.eps}&
\includegraphics[width=2cm,  angle=0]{Figures3/Sext_sym2.eps}&
\includegraphics[width=2cm,  angle=0]{Figures3/Sext_sym4.eps}&
\includegraphics[height=4cm,angle=0]{Figures3/Sym_quarticb.eps}
\\a)&b)&c)&d)
\end{tabular}
\caption{}
\label{degree 6}
\end{figure}
\textit{Proof. } The Newton polygon of the third degree curves constructed in Lemma \ref{curve 3 1} lies inside the triangle with vertices
$(0,3),(0,0)$ and $(6,0)$, so these curves can be seen as a (singular) curve of bidegree $(3,0)$ in $\Sigma_2$. The corresponding symmetric
 curves are of degree $6$ and has the chart and the arrangement with respect to the axis $\{y=0\}$ shown in Figures \ref{degree 6}a) and c). Moreover,
it is well known that such curves are of type I, and we deduce their complex orientations from their quotient curve.
\\ If we perform the coordinate changes $(x,y)\mapsto (-x+\delta y^2,y)$  with $\delta\in \mathbb R$ to the curves with chart depicted in Figure \ref{degree 6}a), we obtain curves
with the chart depicted in Figure \ref{degree 6}b).
~\findemo

The following lemma can be proved using the same technique.
\begin{lemma}
For any real positive numbers $\alpha$ and $\beta$, there exists real symmetric dividing curves of degree $4$ in $\mathbb RP^2$
with a singular point of type $A_3$ at $[1:0:0]$ having the charts, the arrangement
with respect to the axis $\{y=0\}$ and the complex orientations
shown in Figure \ref{degree 6}d) with truncation on the segment $[(0,2),(4,0)]$ equal to
$(x-\alpha y^2)(x-\beta y^2)$.\findemo
\end{lemma}

\subsection{Perturbation of irreducible singular symmetric curves}

\begin{prop}\label{restr thm 4 6}
The complexe schemes $\langle J\amalg 2_-\amalg 1_+\langle 4_+\amalg 2_-\rangle\rangle_I$ and $\langle J\amalg 2_+\amalg 4_-\amalg 1_+\langle 2_+\rangle\rangle_I $ are realizable by nonsingular symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }First, we construct the symmetric singular dividing curve of degree $7$ with two singular points $J_{10}$ depicted in Figure \ref{2_6_I}d).
In order to do this, we use the Hilbert method (as in \cite{V1})~: we symmetrically perturb the union of a symmetric conic and a disjoint symmetric line
(Figure \ref{2_6_I}a)) keeping the tangency points with the conic.
 Next, we symmetrically perturb the union of the third degree curve obtained and the same conic (Figure \ref{2_6_I}b))
keeping the tangency points of order $4$ with the conic. Finally, we symmetrically perturb the union of the curve of degree $5$
obtained and the same conic (Figure \ref{2_6_I}c)) keeping the tangency points of order $6$ with the conic and we obtained
the desired curve.
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[width=3.7cm, angle=0]{Figures3/2_6_Ib.eps}    &
 \includegraphics[width=3.7cm, angle=0]{Figures3/2_6_Ic.eps} &
\includegraphics[width=3.7cm, angle=0]{Figures3/2_6_Id.eps}
&\includegraphics[width=3.7cm,angle=0]{Figures3/2_6_Ia.eps}   \\
a)&b)&c)&d)\\
 \end{tabular}
\caption{}
 \label{2_6_I}
\end{figure}

Now we symmetrically perturb the singular points using the chart shown in Figure \ref{degree 6}b) (resp. d)) in $P_1$ and  \ref{degree 6}c) (resp. a)) in $P_2$
and obtain the desired curves.
~\findemo

\begin{prop}\label{restr thm 4 7}
The complex scheme $\langle J\amalg  1_+\langle 6_+\amalg 6_-\rangle\rangle_I $ is realizable by nonsingular  symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Consider the curve of degree $4$ with a C-shapped oval constructed in \cite{K1} and the coordinate system shown in Figure \ref{12_I}a). In this
coordinate system, the Newton polygon of the curve is the trapeze with vertices $(0,0),(0,3),(1,3)$ and $(4,0)$, and its chart is depicted in Figure  \ref{12_I}b).
Thus, we can see this curve as a singular curve of bidegree $(3,1)$ in the surface $\Sigma_2$. The corresponding symmetric curve of degree $7$
 has a singular point $J_{10}$ at $[1:0:0]$ and is depicted in Figure \ref{12_I}c). According to Proposition \ref{lattice}, this curve is of type I.
 Looking at its quotient curve,
we see that the complex orientations of the symmetric curve is as shown in Figure \ref{12_I}c). Finally, we symmetrically smooth the singular point
with the chart depicted in Figure \ref{degree 6}a) and obtain the desired curve.
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=3cm, angle=0]{Figures3/12_Ia.eps}    &
 \includegraphics[height=3cm, angle=0]{Figures3/12_Ib.eps} &
\includegraphics[height=3cm, angle=0]{Figures3/12_Ic.eps}
\\ a)&b)&c)%&d)\\
 \end{tabular}
\caption{}
 \label{12_I}
\end{figure}
~\findemo

Denote by $f_\alpha$ the real birational transform of $\mathbb CP^2$ given by
$(x,y)\mapsto (x,y-\alpha x^2)$ in the affine coordinate $\{z=1\}$, with $\alpha$ a real number.

\begin{prop}\label{restr thm 4 8}
The complex scheme $\langle J\amalg 6_+\amalg 4_-\amalg  1_+\langle 1_+\amalg 1_-\rangle\rangle_I$ is realizable by nonsingular  symmetric real algebraic curves of degree $7$ in $\mathbb RP^2$.
\end{prop}
\textit{Proof. }Consider the nodal curve of degree $3$ depicted in Figure \ref{10_2_IA}a) with a contact of order $3$ at
the point $[0:1:0]$ with the line $\{z=0\}$.
 Then there exists a unique real $\alpha$ such that the image of the cubic under $f_\alpha$ is the curve of
degree $4$ depicted in Figure \ref{10_2_IA}b), with a singular point of type $A_4$ at $[0:1:0]$ and a contact of order $4$
 at this point with the line $\{z=0\}$. Moreover, the line $\{y=0\}$ intersects the quartic in two points, one
of them is the node, and this line is tangent at one of the local branches at the node
and at the second intersection point, a line $\{y=az\}$ is tangent at the curve of degree $4$. Perform the change of coordinates
of $\mathbb CP^2$ $[x:y:z]\mapsto [y:x:y-az]$.
For this new coordonate system, there exists a real $\beta$ such that the image of the quartic under $f_\beta$ is the curve of
degree $5$ depicted in Figure \ref{10_2_IA}c), with a singular point of type $A_{10}$ at $[0:1:0]$ and a contact of order $4$
 at this point with the line $\{z=0\}$. Applying the change of coordinates of $\mathbb CP^2$ $[x:y:z]\mapsto [y:z:x]$ and using Shustin's results on the independant perturbations of generalized semi-quasihomogeneous singular points of a curve keeping the same Newton polygon(see \cite{Sh1} and \cite{Sh2}), we can
smooth the singular point $A_{10}$ in order to obtain a curve
with the chart shown in Figure \ref{10_2_IB}a).

Hence, we can see this curve as a  singular curve of bidegree $(3,1)$ in the surface $\Sigma_2$. The corresponding symmetric
 curve of degree $7$ has a singular point $A_{3}$ at $[1:0:0]$ and is depicted in Figure \ref{10_2_IB}b). This curve is maximal according
to Propositoin \ref{lattice},
so of type I, and we can deduce its complex orientations from its quotient curve.
Finally, we symmetrically smooth the singular point
with the chart depicted in Figure \ref{degree 6}e) and obtain the desired curve.
\begin{figure}[h]
      \centering
 \begin{tabular}{cccc}
\includegraphics[height=2.5cm, angle=0]{Figures3/10_2_Ia.eps}    &
 \includegraphics[height=2.5cm, angle=0]{Figures3/10_2_Ib.eps} &
\includegraphics[height=2.5cm, angle=0]{Figures3/10_2_Id.eps}
\\ a)&b)&c)\\
 \end{tabular}
\caption{}
 \label{10_2_IA}
\end{figure}
\begin{figure}[h]
      \centering
 \begin{tabular}{cc}
\includegraphics[height=3cm, angle=0]{Figures3/10_2_If.eps}    &
 \includegraphics[height=3cm, angle=0]{Figures3/10_2_Ig.eps}
\\ a)&b)
 \end{tabular}
\caption{}
 \label{10_2_IB}
\end{figure}
~\findemo



\section{Recapitulation}\label{recap}
Here we recapitulate the proof of the four theorems announced in section \ref{statement}.

\textit{Proof of Theorem \ref{Main thm1}. }
In Propositions \ref{first M} and \ref{prohib thm 1}, the real schemes announced in Theorem \ref{Main thm1}
are prohibited pseudo-holomorphically. All the other curves are constructed
algebraically in Propositions \ref{main constr},
\ref{change1}, \ref{constr thm1 3}, \ref{constr thm1 3} and in Corollary \ref{constr thm1 1}.
~\findemo

\textit{Proof of Theorem \ref{Main thm2}. }
If $C$ is a maximal nonsingular symmetric curve in $\mathbb RP^2$ of degree $7$,
all the  possibilities for  the  $\mathcal L$-scheme of the arrangement of its quotient curve and
the section $\{y=0\}$ of $\Sigma_2$ are described in  Propositions \ref{M scheme 1} and \ref{M scheme 2}.
 Such an arrangement determines uniquely the complex orientations
of the maximal symmetric curves of degree $7$.
Now, it is not difficult to check that the complex orientations of the maximal nonsingular symmetric curves
of degree $7$ in $\mathbb RP^2$ are exactly those of Theorem \ref{Main thm2}.
~\findemo

\textit{Proof of Theorem \ref{Main thm3}. }
All the pseudo-holomorphic prohibitions are proved in Propositions \ref{restr thm 4 3} and \ref{non 4 4 I}. All the constructions are done in Propositions \ref{restr thm 4 5},
\ref{restr thm 4 6}, \ref{restr thm 4 7}, \ref{restr thm 4 8} and in Corollary \ref{restr thm 4 4}, and \ref{pseudo non alg 2}.
~\findemo

\textit{Proof of Theorem \ref{Main thm4}. }
The two algebraic prohibitions are proved in Propositions \ref{only possibility}, \ref{prop 84} and \ref{prop 48}. All the constructions are done in Propositions \ref{restr thm 4 5},
\ref{restr thm 4 6}, \ref{restr thm 4 7}, \ref{restr thm 4 8}, and in Corollary \ref{restr thm 4 4}.
~\findemo


\small
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\hspace*{2 ex}
\textbf{\\Erwan Brugall\'e}\\
IRMAR-Universite de Rennes 1\\
Campus de Beaulieu\\
35042 Rennes cedex\\
FRANCE\\
\\
E-mail : erwan.brugalle@univ-rennes1.fr
\end{document}

