text:
\section{Introduction and statement of main results} \label{intro}
In 2016, E. Brugall\'e and P. Georgieva in \cite{brugalle2016pencils} proposed a method to compute the  genus-$0$ Gromov-Witten and Welschinger invariants of $\mathbb{C}P^3$. The idea comes from the obsevation of J. Koll\'ar in \cite{kollar2015examples} about the properties of pencils of quadric surfaces in $\mathbb{C}P^3$.

In this paper, we generalize the results obtained in \cite{brugalle2016pencils} for high degree del Pezzo varieties $X$. More precisely, we compare the genus-$0$ Gromov-Witten (resp. Welschinger invariants) of three-dimensional (resp. real) del Pezzo varieties with that of (resp. real) non-singular surfaces which realize half of the first Chern class of
$X$, i.e. (resp. real) non-singular surfaces in the linear system $|-\frac{1}{2}K_X|$ where $K_X$ is the canonical divisor of $X$.
According to Proposition \ref{Prop-delPezzo.surfaces}, these surfaces are indeed del Pezzo surfaces.
The method used in this text restricts us to be concerned with three-dimensional del Pezzo varieties of degree among $\{6,7,8\}$. In 2018,  A. Zahariuc also adapted the same strategy to study genus-$0$ Gromov-Witten invariants in three-dimensional del Pezzo varieties of degree $2,3,4$ and $5$ (see \cite{zahariuc2018rational}). It is interesting if one can find out some relations between his formula and our formula in Theorem \ref{Theorem1-GW}. More general, we are looking forward to studying Gromov-Witten and Welschinger invariants' relations between (real) two and three dimensional del Pezzo varieties of any degree.

We first introduce some notations and main results of this paper.
\subsubsection*{Some notations}
In this text, the lower point $.$ stands for intersection between two homology classes and the upper point $\cdot$ stands for the cup product.
\subsection{Introduction and motivation}
A Fano variety of dimension $n$, introduced by Gino Fano in 1930s, is a non-singular complex projective variety $X_n$ of complex dimension $n$ whose anti-canonical divisor $-K_{X_n}$ is ample. The maximal positive integer $r>0$ such that $-K_{X_n}=rH$, where $H$ is a primitive divisor in $Pic(X_n)$, is called \textit{index of $X_n$} and denoted by $ind(X_n)$.

Non-singular $n$-dimensional Fano varieties are completely classified in 1980s by V. A. Iskovskikh (for $n=3$) and T. Fujita (for any $n$) (see
 \cite{iskovskikh2010algebraic}).

 A $n$-dimensional Fano variety of index divisible by $(n-1)$ is also called a \textbf{$n$-dimensional del Pezzo variety} for any $n\geq 2$.
 The degree of a $n$-dimensional del Pezzo variety $X_n$, denoted by $\deg(X_n)$, is the integer given by $(\frac{1}{n-1})^n(-K_{X_n})^n$.
   Note that the degree of a three-dimensional  del Pezzo variety is no more than $8$ (see Chapter 2 and Chapter 3 in \cite{iskovskikh2010algebraic}).  Note also that every Fano surface is a del Pezzo surface.

We consider a rational curve realizing homology class $\beta\in H_2(X_n;\mathbb{Z})$ as
 the image of a morphism $f: \mathbb{C}P^1\longrightarrow X_n$ such that $f_*([\mathbb{C}P^1])=\beta$.

Suppose that $(n-1)$ divides $\big(c_1(X_n)\cdot\beta+n-3\big)$. We denote the integer $\frac{1}{n-1} \big(c_1(X_n)\cdot\beta+n-3\big)$ by $k_\beta$.  One define the genus-$0$ Gromov-Witten invariant as follows.
\begin{definition} \label{definitionGW}
The \textbf{genus-$0$ Gromov-Witten invariant of a  $n$-dimensional del Pezzo variety $X_n$ ($n\in \{2,3\}$) of homology class $\beta\in H_2(X;\mathbb{Z})$},  denoted by $GW_{X_n}(\beta)$, is the number of irreducible rational curves representing homology class $\beta$ and passing through a generic configuration $\underline{x}_\beta$ of $k_\beta$ points in $X_n$.
\end{definition}

We denote by $(X_n,\tau)$ a real $n$-dimensional del Pezzo variety $X_n$ equipped with an anti-holomorphic involution $\tau$, so-called a real structure. The fixed locus of this involution is the real part of $X_n$, denoted by $\mathbb{R} X_n$. An algebraic curve in  $(X_n,\tau)$ is \textit{real} if it is preserved by the real structure $\tau$. A configuration of points in  $(X_n,\tau)$ is \textit{real} if it is preserved by the real structure $\tau$.

Let $\beta\in H_2^{-\tau}(X_n;\mathbb{Z})$ be a $\tau$-anti-invariant class, i.e. $\beta$ is such that $\tau_*(\beta)=-\beta$. Let $\underline{x}_\beta$ be a real  generic configuration of $k_\beta$ points in $X_n$ consisting of $r$ real points and $l$ pairs of complex conjugated points such that $k_\beta=r+2l$.

For each real rational curve in  $(X_n,\tau)$, we can associate to a sign $\pm 1$, called Welschinger's sign (see \cite{welschinger2005fourfolds}, \cite{welschinger2005spinor}) such that when counting such curves with this sign, we obtain an invariant independant of the choice of real configurations. Such invariant is called Welschinger invariant and is known as the real counter part of the above Gromov-Witten invariant. We will proceed the precise definitions of Welchinger's sign in Section \ref{Sect-Welschinger}. This sign in particular depends on the $Pin^\pm$- structure on the real part of the variety whose dimension is three. In the following, we describe the definition of the genus-$0$  Welschinger invariant.

\begin{definition} \label{definitionW}
Let $(X_n,\tau)$ be a real  $n$-dimensional ($n\in \{2,3\}$) del Pezzo variety whose real part is $\mathbb{R} X_n$.
The \textbf{genus-$0$ Welschinger invariant of $(X_n,\tau)$ of homology class $\beta\in H_2^{-\tau}(X_n;\mathbb{Z})$} is the number of irreducible rational curves counted with Welschinger's sign, representing homology class $\beta$ and passing through a real generic configuration $\underline{x}_\beta$ of $k_\beta$ points in $X_n$ consisting of $l$ pairs of complex conjugated points.

In particular, if $\mathbb{R} X_3$ is orientable with orientation $\mathfrak{o}_{X_3}$ and is equipped with  a $Spin_3$-structure $\mathfrak{s}_{X_3}$, we denote this invariant by $W_{\mathbb{R} X_3}^{\mathfrak{s}_{X_3}, \mathfrak{o}_{X_3}}(\beta,l)$.

When $n=2$, we denote this invariant simply by $W_{\mathbb{R} X_2}(\beta,l)$.
\end{definition}

\subsection{Main results}

Throughout the rest of this paper, unless otherwise stated, we denote by $X$ a three-dimensional del Pezzo variety.

Let $\Sigma\subset X$ be a non-singular surface  in the linear system $|-\frac{1}{2}K_X|$.

Let $\psi: H_2(\Sigma;\mathbb{Z})\longrightarrow H_2(X;\mathbb{Z})$ be the map induced by the inclusion from $\Sigma$ to $X$.  We have the following theorem.

\begin{theorem}\label{Theorem1-GW}
Let $X$ be a three-dimensional del Pezzo variety such that $\ker(\psi) \cong \mathbb{Z}$ where $\Sigma$ and $\psi$ are as described above. Let $S$ be a generator of $\ker (\psi)$.
For every homology class $d\in H_2(X;\mathbb{Z})$, one has
$$GW_X(d)=\frac{1}{2}\sum_{D\in \psi^{-1}(d)}(D.S)^2 GW_{\Sigma}(D).$$

\end{theorem}
Theorem \ref{Theorem1-GW} is a generalization of Theorem 2 in \cite{brugalle2016pencils}.

This theorem allows us to compute the genus-$0$ Gromov-Witten invariants for almost all three-dimensional  del Pezzo varieties of degree $6,7$ and $8$.

Suppose now that $(X,\tau)$ is a real three-dimensional del Pezzo variety equipped with a real structure $\tau$. Suppose also that $\mathbb{R} X$ is orientable with orientation $\mathfrak{o}_X$ and is equipped with a $Spin_3$-structure $\mathfrak{s}_X$.  Let $(\Sigma,\tau|_\Sigma) \in |-\frac{1}{2}K_X|$  be a non-singular real del Pezzo surface.

Let $\psi: H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z})$ be as in the Gromov-Witten case.

Let $g(D)=\frac{1}{2}(K_\Sigma . D +D^2+2) \in \mathbb{Z}$.

Let $\epsilon:H_2(\Sigma;\mathbb{Z})\to \mathbb{Z}_2$ be a map     given by     $$\epsilon(D)=\begin{cases}
  c_1(\Sigma)\cdot D \mod 2, \text{ if } \mathbb{R} f^* T \Sigma /T \mathbb{C} P^1=E^+,\\
  c_1(\Sigma)\cdot D-1 \mod 2,\text{ if } \mathbb{R} f^* T \Sigma /T \mathbb{C} P^1=E^-
  \end{cases}$$

 where $f:\mathbb{C} P^1\to \Sigma\subset X$ is a  real immersion such that $f_*[\mathbb{C} P^1]=D\in H_2(\Sigma;\mathbb{Z})$, and  $E^\pm$ are two isotopy classes of the real part of a holomorphic line subbundle of degree $(c_1(\Sigma)\cdot D-2)$ of the rank $2$ holomorphic normal  bundle $f^*T X/T \mathbb{C} P^1$ (see Section \ref{Sect-spinor.dim.3.with.normal.bundle} and Definition \ref{def-epsilon.function}).

Let  $s^{\mathfrak{s}_X|_\Sigma}:H_1(\mathbb{R} \Sigma;\mathbb{Z}_2)\to \mathbb{Z}_2$
be the quasi-quadratic enhancement in one-to-one correspondence to
    the $Pin^-_2$-structure $\mathfrak{s}_{X|\Sigma}$ over $\mathbb{R} \Sigma$, which is restricted from the $Spin_3$-structure $\mathfrak{s}_X$ over $\mathbb{R} X$ (see Proposition \ref{prop-quasi-quadratic.enhancement.def.prop} and Equation (\ref{eq-RelationW2W2})).

Let $\rho: H_2^{-\tau}(\Sigma;\mathbb{Z}) \to H_1(\mathbb{R} \Sigma;\mathbb{Z}_2)$ be the homomorphism given by the homology class realized by the real part of the $\tau$-anti-invariant representative of $D$.

We have the following theorem.

\begin{theorem}\label{Theorem2-W}
Let $((X,\tau),\mathfrak{o}_X,\mathfrak{s}_X)$ be a real three-dimensional del Pezzo variety such that $\ker(\psi) \cong \mathbb{Z}$, where $(\mathfrak{o}_X,\mathfrak{s}_X)$ are as described above. Let $S$ be a generator of $\ker(\psi)$.  Suppose $(\Sigma,\tau|_{\Sigma})\in |-\frac{1}{2}K_X|$ is  such that $S$ is $\tau|_{\Sigma}$-anti-invariant.  For every $d\in H_2^{-\tau}(X;\mathbb{Z})$ and $0\leq l< \frac{1}{2}k_d$,
 one has

$$W^{\mathfrak{s}_X,\mathfrak{o}_X}_{\mathbb{R} X}(d,l)=\frac{1}{2}\sum_{D\in \psi^{-1}(d)}(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_X|_\Sigma}(\rho D)} |D.S| W_{\mathbb{R} \Sigma} (D,l).$$

\end{theorem}

Theorem \ref{Theorem2-W} is a generalization of Theorem 1 in \cite{brugalle2016pencils}.

As a result, this theorem allows us to compute the genus-$0$ Welschinger invariants of almost all real three-dimensional del Pezzo varieties of degree $ 6,7$ and $8$.

 Since the Welschinger invariants count real curves with some sign $\pm 1$, in the case where these invariants vanish, it is not clear that there exists no such real curves. This gives rise to one of the optimal problems in real enumerative geometry: the sharpness of the Welschinger invariants, in which one shows that the lower bounds given by these invariants can be obtained. The first examples of vanishing genus-0 Welschinger invariants
in projective $3$-space was explored by G. Mikhalkin for rational curves of degree $4$ and then by J. Koll\'ar for any even degree, see \cite{kollar2015examples}. We extend Koll\'ar's results for some classes of del Pezzo varieties of dimension three, as shown in the following theorem.

\begin{theorem}\label{Theorem3-W}
Let $((X,\tau),\mathfrak{o}_X,\mathfrak{s}_X)$ be a real three-dimensional del Pezzo variety as in Theorem \ref{Theorem2-W}.

Let $d\in H_2^{-\tau}(X;\mathbb{Z})$ such that $D.S$ is even for any $D\in \psi^{-1}(d)$. Let $k_d$ be the associated integer of $d$.

Then, there exists a real configuration of $k_d$ points, consisting of at least one real point,  through which no real irreducible rational curve of homology class $d$ passes.
\end{theorem}

The structure of this paper is as follows. In Section \ref{Section-Pencils}, we recall some properties of complex and real pencils of surfaces in the linear system $|-\frac{1}{2}K_X|$ and we explain the monodromy action to such pencils. Then, we solve the \textit{$1$-dimensional problems} related to enumeration on pencils of surfaces, which contribute to linking terms of Theorems \ref{Theorem1-GW} and \ref{Theorem2-W}. In Section \ref{Section-ratcurve}, we investigate the properties of rational curves on three-dimensional del Pezzo variety $X$ and that on non-singular del Pezzo surfaces $\Sigma\in |-\frac{1}{2}K_X|$. As a consequence, we construct a comparison between genus-$0$ Gromov-Witten invariants of $X$ in one side and that of $\Sigma$ in the other side. In Section \ref{Sect-Welschinger}, we discuss about the problem of  Welschinger's signs $\{\pm 1\}$ arising when we count real curves in real surfaces and in  real three-dimensional varieties. Then, we propose a comparison between those signs via the quasi-quadratic enhancement corresponding to a particular $Pin_2^-$ structure on the real part of algebraic surfaces.
In Section \ref{Sect-proofs}, we give the proofs of Theorems \ref{Theorem1-GW},\ref{Theorem2-W} and \ref{Theorem3-W}.
In the last section, we first apply Theorem \ref{Theorem1-GW} in some particular cases (Theorems \ref{Theorem-GW.in.deg.8}, \ref{Theorem-GW.in.deg.7}, \ref{Theorem-GW.in.deg.6}) and we exhibit the explicit calculation tables of genus-$0$ Gromov-Witten invariants in $\mathbb{C} P^1\times\mathbb{C} P^1\times \mathbb{C} P^1$, using \textit{Maple program}. Then, we apply Theorem \ref{Theorem2-W} and Theorem \ref{Theorem3-W} in some particular cases (Theorems \ref{Theorem-W.in.deg.8}, \ref{Theorem-W.in.deg.7}, \ref{Theorem-W.in.deg.6.1}, \ref{Theorem-W.in.deg.6.2}, Propositions \ref{prop-application.deg8} \ref{prop-application.deg7},\ref{prop-application.deg6.1}, \ref{prop-application.deg6.2}). Finally, we exhibit the explicit calculation tables of genus-$0$ Welschinger invariants in $\mathbb{C} P^3\sharp \overline{\mathbb{C} P^3}$ with the standard real structure and in  $\mathbb{C} P^1\times\mathbb{C} P^1\times \mathbb{C} P^1$ with  two distinct real structures, also using \textit{Maple program}.

\subsubsection*{Acknowledgment.} The present article is contained in the author's  Ph.D. thesis which is written at Nantes University under supervision of Erwan Brugall\'e.  The author is sincerely grateful to him for his infinite support, fruitful discussions and  numerous indications from the implementation to the accomplishment of this article. She would like to thank Xujia Chen for her very helpful comments and calculations, and thank Duc Pham for his useful skill in programming which is used in the application parts. She also wants to thank the Henri Lebesgue Center for the financial support.

\section{Pencils of surfaces in the linear system $|-\frac{1}{2}K_X|$} \label{Section-Pencils}

Let $\mathcal{Q}$ be a pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$. Let $E$ be its base locus. Let  $\Sigma$ be a general element of the pencil $\mathcal{Q}$. Firstly, we prove that the base locus $E$ is an elliptic curve (Proposition \ref{Prop-E.is.elliptic}) and $\Sigma$ is a non-singular del Pezzo surface of the same degree as $\deg(X)$ (Proposition \ref{Prop-delPezzo.surfaces}).   Then, we explain how monodromy action relates to the enumerative problems. In the next subsection, we give the main ingredient to solve the \textit{$1$-dimensional problem} of Theorem \ref{Theorem1-GW} (Proposition \ref{Prop-degree.case.complex}). In the last subsection, by analysing the properties of real pencils of surfaces, we give the main ingredient to solve the \textit{$1$-dimensional real  problem} of Theorem \ref{Theorem2-W} (Proposition \ref{prop-corresponding.surface.picE}, Corollary \ref{coro-degree.case.real}).

\subsection{Properties of pencils of surfaces in the linear system $|-\frac{1}{2}K_X|$}

\begin{proposition}
\label{Prop-E.is.elliptic}
 If the pencil $\mathcal{Q}$ is generic enough, then its base locus is a non-singular elliptic curve. Moreover, this curve realizes the homology class $\frac{1}{4}K^2_X$ in $H_2(X;\mathbb{Z})$.
\end{proposition}

\begin{proof}

 By Bertini's theorem (see \cite{griffiths1978principles}) and Theorem 1.2 in \cite{vsokurov1980smoothness}, a general element $\Sigma \in |-\frac{1}{2}K_X|$ is a non-singular surface.

 The base locus $E$ of the pencil $\mathcal{Q}$ realizes the homology class $[\Sigma].[\Sigma]$, that is the self-intersection product of homology class $[\Sigma]\in H_4(X;\mathbb{Z})$.
 By the adjunction formula, we have:

$$K_\Sigma =(K_X + [\Sigma] ).[\Sigma] = -[\Sigma]^2.$$

Thus, the curve $E$ realizes the homology class $-K_\Sigma$, hence
$E$ realises the homology class $\frac{1}{4}K^2_X$ in $H_2(X;\mathbb{Z})$. Since two general elements in pencil have  traversal intersections, the curve  $E$ is in fact a non-singular representative of the class $-K_\Sigma$. By the adjunction formula, we deduce easily that the genus of the curve $E$ is $1$.

Hence, the proposition follows.
\end{proof}

Next proposition explains the interest of studying surfaces in the linear system $|-\frac{1}{2}K_X|$.
\begin{proposition}\label{Prop-delPezzo.surfaces}
Every non-singular surface $\Sigma\in |-\frac{1}{2}K_X|$ is a del Pezzo surface whose degree is equal to $\deg(X)$.
\end{proposition}
\begin{proof}

By Corollary 1.11 in \cite{iskovskikh1977fano}, every non-singular surface in $ |-\frac{1}{2}K_X|$  is a del Pezzo surface.

Moreover, by definition, the degree of del Pezzo surface $\Sigma$ is equal to $K_\Sigma^2$ and degree of del Pezzo space $X$ is equal to $\frac{1}{8}(-K_X)^3$.  The adjunction formula implies that $K_{\Sigma}^2=\frac{1}{4}K_X^2.[\Sigma]$.
Since $[\Sigma]=-\frac{1}{2}K_X$, we deduce that $\deg(\Sigma)=\deg (X)$.

\end{proof}

\begin{proposition} \label{prop-elliptic.implies.pencil}
Let $E$ be a non-singular elliptic curve in $X$  realising the homology class $\frac{1}{4}K^2_X$ in $H_2(X;\mathbb{Z})$ and contained in a non-singular surface $\Sigma\in |-\frac{1}{2}K_X|$.

Then, $E$ is the base locus of a pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$.
\end{proposition}
 \begin{proof}
By adjunction formula, the elliptic curve $E$ realizes the anti-canonical class $-K_\Sigma\in H_2(\Sigma;\mathbb{Z})$.
  It is sufficient to prove that $E$ is the complete intersection of $\Sigma$ with another non-singular surface $\Sigma'\in |-\frac{1}{2}K_X|$. In other words, we prove that

 $$H^0(X,\mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma)=H^0(\Sigma,\mathcal{O}_\Sigma(-K_\Sigma)).$$

It is evident that $H^0(\Sigma,\mathcal{O}_\Sigma(-K_\Sigma)) \subset H^0(X,\mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma)$ as subspace. We then prove that these two vector spaces have the same dimension.
 We consider the short exact sequence

 $$0\longrightarrow \mathcal{O}_X(0)\longrightarrow \mathcal{O}_X(-\frac{1}{2}K_X)\longrightarrow \mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma \longrightarrow 0$$
 which induces the long exact sequence of cohomology

\begin{tikzcd}
0 \arrow[r]& H^0(X,\mathcal{O}_X(0)) \arrow[r]
& H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X)) \arrow[r]
\arrow[d, phantom, ""{coordinate, name=Z}]
&  H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma) \arrow[dll,
"\delta", rounded corners, to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}] \\
&H^1(X,\mathcal{O}_X(0)) \arrow[r]
&  \cdots \quad \quad \quad \quad \quad \quad \quad
&
\end{tikzcd}

We have the following facts (see \cite{iskovskikh2010algebraic}, Section 3.2).

\begin{itemize}
\item Firstly, we have that $\dim (H^0(X,\mathcal{O}_X(0)))=1$ and $H^1(X,\mathcal{O}_X(0))=0$.
\item Secondly, by Riemann-Roch theorem, we have
 $$\dim H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X))=(-\frac{1}{2}K_X)^3+3-1=\deg (X) +2,$$
 and
 $$\dim H^0(\Sigma,\mathcal{O}_\Sigma(-K_\Sigma))=(-K_\Sigma)^2+2-1=\deg (\Sigma) +1.$$
\end{itemize}

 Therefore, we deduce that
 $$\dim H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma)=\dim H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X))-1=\deg (X)+1.$$

Remark that $\deg(X)=\deg(\Sigma)$. As a consequence, we obtain the identification
$$H^0(X, \mathcal{O}_X(-\frac{1}{2}K_X)|_\Sigma)= H^0(\Sigma,\mathcal{O}_\Sigma(-K_\Sigma)).$$

Thus, the statement follows.

  \end{proof}

\begin{proposition} \label{Prop-why.degree.is.678}
 For every $\Sigma\in |-\frac{1}{2}K_X|$ non-singular, one has a surjective map  $$\psi : H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z}),$$   which is induced by the inclusion of $\Sigma$ in $X$.

In particular, if  $X\in \{\mathbb{C} P^3, \mathbb{C} P^3\sharp \overline{ \mathbb{C} P^3},\mathbb{C} P^1\times \mathbb{C} P^1 \times \mathbb{C} P^1\}$ then $\ker(\psi)\cong \mathbb{Z}$.
\end{proposition}

\begin{proof}
One has the divisor $-\frac{1}{2} K_X$ is ample. Hence, the statement follows by Lefschetz hyperplane theorem (see for example Theorem C.8 in \cite{eisenbud20163264}).

In particular,
\begin{itemize}
\item if $X=\mathbb{C} P^3$ then $H_2(X;\mathbb{Z})\cong \mathbb{Z}$ and $H_2(\Sigma;\mathbb{Z})\cong \mathbb{Z}^2$ for any  $\Sigma \in  |-\frac{1}{2}K_X|$ non-singular;
\item if $X=\mathbb{C} P^3\sharp \overline{ \mathbb{C} P^3}$ then $H_2(X;\mathbb{Z})\cong \mathbb{Z}^2$ and $H_2(\Sigma;\mathbb{Z})\cong \mathbb{Z}^3$ for any  $\Sigma \in  |-\frac{1}{2}K_X|$ non-singular;
\item if $X=\mathbb{C} P^1\times \mathbb{C} P^1 \times \mathbb{C} P^1$ then $H_2(X;\mathbb{Z})\cong \mathbb{Z}^3$ and $H_2(\Sigma;\mathbb{Z})\cong \mathbb{Z}^4$ for any  $\Sigma \in  |-\frac{1}{2}K_X|$ non-singular.
\end{itemize}
By the first isomorphism theorem for groups, we have $H_2(X;\mathbb{Z})\cong H_2(\Sigma;\mathbb{Z})/\ker(\psi)$. Hence, if  $X$ is in the set  $\{\mathbb{C} P^3, \mathbb{C} P^3\sharp \overline{ \mathbb{C} P^3} ,\mathbb{C} P^1\times \mathbb{C} P^1 \times \mathbb{C} P^1\}$ and $\Sigma \in  |-\frac{1}{2}K_X|$ is a non-singular surface then  $\ker(\psi)\cong\mathbb{Z}$.
\end{proof}

\begin{lemma} \label{lem-4.singulars}
If $X$ is a three-dimensional del Pezzo variety of  $\deg(X)\in \{5,6,7,8\}$. There are exactly four singular surfaces in a generic pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$.
\end{lemma}
\begin{proof}
Let $\mathcal{Q}$ be a generic pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$ whose base locus is an elliptic curve $E$. Let $N$ be the number of singular elements in this pencil.

Let $\tilde{X}$ be the blown-up of $X$ along the elliptic curve $E$.  We will compute $N$ by comparing the Euler characteristic of $\tilde{X}$ in two equivalent ways.
\begin{enumerate}
    \item On the one hand, we have     $\chi (\tilde{X})=\chi (X)$     since the Euler characteristic of the blown-up locus is $\chi(E)=0$.

    \item On the other hand, we consider $\tilde{X}$ as blown-up of the pencil $\mathcal{Q}$ along its base locus $E$. The number $N$ is then equal to the intersection numbers of the general projective line defined by $\mathcal{Q}$ and the discriminant hypersurface $\Delta$ in $\mathbb{C}P^m$ (where $m=\dim (\mathbb{P}(\Gamma(\mathcal{O}_X(-\frac{1}{2}K_X))))$). Let $\Sigma_{t_1},\Sigma_{t_2},\ldots, \Sigma_{t_N}$ be $N$ singular surfaces of the pencil $\mathcal{Q}$. Then, $\tilde{X}-\{\bigcup\limits_{i=1}^{N}\Sigma_{t_i}\}$ is a fibration over $\mathbb{C} P^1-\{t_1,\ldots,t_N\}$, whose fibres are non-singular del Pezzo surfaces. Let $\Sigma$ be one of such fibres.
      We have
    $$\chi (\tilde{X}-\{\bigcup\limits_{i=1}^{N}\Sigma_{t_i}\})=\chi(\mathbb{C} P^1-\{t_1,\ldots,t_N\}) \times \chi (\Sigma).$$
    This implies that
    $$\chi (\tilde{X})=(2-N)\chi (\Sigma) + \sum_{i=1}^N\chi (\Sigma_{t_i}). \; (*)$$

    Let $d:=\deg(\Sigma)$ ($1\leq d\leq 9$). The Euler characteristics of $\Sigma$ and $\Sigma_{t_i}$, for $1\leq i\leq N$, are $12-d$ and $11-d$ respectively. Hence, the equality $(*)$ becomes
    $$\chi (\tilde{X})=(2-N)(12-d) + N (11-d).$$
\end{enumerate}

As a consequence from two ways of computing $\chi(\tilde{X})$ as above, we have:
$$\chi (X)=24-2d-N. \; (**)$$

 The values of $\chi(X)$ and $N$ inducing from the equality $(**)$ are shown in Table \ref{Tab:singular.surfaces}, for $d\in \{5,\ldots,8\}$.
\begin{table}[h!]
\centering
\begin{tabular}{ |c||c|c| }
 \hline
degree $d$ & $\chi(X)$ & N \\
 \hline
 \hline
    5 & 10 & 4 \\
 \hline
 6 & 8 & 4 \\
  \hline
 7 & 6 & 4 \\
  \hline
  8 & 4 & 4 \\
 \hline
\end{tabular}
\caption{Euler characteristic $\chi(X)$ of three-dimensional del Pezzo varieties of degree from $5$ to $8$ and the number $N$ of singular elements in a generic pencil of surfaces in $|-\frac{1}{2}K_X|$.} \label{Tab:singular.surfaces
\end{table}

The statement then follows.
\end{proof}

\subsection{Monodromy} \label{Sect-Monodromy}
We consider the Lefschetz pencil $\mathcal{Q}$. Let $f:\tilde{X} \to \mathbb{C}P^1$ be its associated Lefschetz fibration. Let $\Sigma$ be a non-singular fibre over $t\in \mathbb{C} P^1$ and $\Sigma_{t_i} (i\in \{1,\ldots,4\})$ be singular fibres over $t_i\in \mathbb{C} P^1$. The monodromy transformation  on $H_2(\Sigma;\mathbb{Z})$ corresponding to the vanishing cycle $S_i$ arises when we go along a path from $t$  to the critical value $t_i$ in the fibration $f$. This is given by  Picard-Lefschetz formula as follows:
 \begin{align*}
     T_{{S_i}_*}: H_2(\Sigma;\mathbb{Z})&\to H_2(\Sigma;\mathbb{Z})\\
   D&\mapsto D+(D.S_i)S_i.
 \end{align*}

Note that the vanishing cycle $S_i$ is mapped to its inverse  via monodromy transformation and it has self-intersection $-2$. The monodromy transformation does not depend on the choice of $S_i$ or $-S_i$, i.e. $T_{{(-S_i)}_*}(D)=T_{{S_i}_*}(D)$.

\begin{lemma} \label{Lemma-Vanishing.Cycle}
Let $X$ be a three-dimensional del Pezzo variety such that $$\ker(\psi : H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z}))\cong \mathbb{Z}$$
where $\Sigma\in |-\frac{1}{2}K_X|$ is a non-singular surface.
Then, the group $\ker (\psi)$ is generated by a vanishing cycle $S_i$, for every $i\in \{1,\ldots,4\}$.
\end{lemma}
\begin{proof}

By definition of a vanishing cycle $S_i\in H_2(\Sigma;\mathbb{Z})$, one can deduce that $S_i\in \ker (\psi)$ for every $i\in \{1,\ldots,4\}$.

Suppose that $S$ is a generator of $\ker (\psi)\cong \mathbb{Z}$, we can write $S_i=kS$ where $k\in \mathbb{Z}$. Since $S_i^2=-2$, we obtain $k^2=1$ and hence $k\in \{\pm 1\}$. This implies that $S_i=\pm S$ for any $i\in \{1,\ldots,4\}$.

\end{proof}

Suppose now that $X$ is as in Lemma \ref{Lemma-Vanishing.Cycle}. In this case, we can identify $S_i$ with a generator $S$ of  $\ker(\psi)$, for any $i\in \{1,\ldots,4\}$, when studying about monodromy transformations. In the sequel,
we simply denote by $T$ instead of $T_{{S_i}_*}$ for the monodromy transformation on $H_2(\Sigma;\mathbb{Z})$, i.e. $T(D):=D+(D.S)S$.

 We define the following equivalent relation $\sim$ on $H_2(\Sigma;\mathbb{Z})$ (where $\Sigma \in \mathcal{Q}$ non-singular):
 $$D_1\sim D_2 \text{ if } D_2=D_1+(D_1.S)S.$$

 We denote by $\overline{D}$  the equivalence class of $D$ in $H_2(\Sigma;\mathbb{Z})/_\sim$.

  For any  $\Sigma$ and $\Sigma'$ are non-singular elements in the pencil $\mathcal{Q}$, the corresponding Lefschetz fibration induces the following isomorphism
$$H_2(\Sigma;\mathbb{Z})/_\sim \;\cong H_2(\Sigma';\mathbb{Z})/_\sim .$$

\begin{example}
   In the case $X=\mathbb{C}P^3$, we consider the Lefschetz fibration on $\mathbb{C} P^1$ whose non-singular fibres are isomorphic to $\mathbb{C}P^1\times \mathbb{C}P^1$ (see Figure \ref{fig-monodromy}).
   \begin{figure}
 \centering
\includegraphics[scale=0.4]{figures/pencilquadric-article.png}
\caption{Example of monodromy
transformation where the general fibres $\Sigma$ are quadric surfaces and the singular ones are quadric cones in $\mathbb{C}P^3$.}
\label{fig-monodromy}
\end{figure}
   Let $(L_1,L_2)=([\mathbb{C} P^1\times\{pt\}],[\{pt\}\times \mathbb{C} P^1])$ be a basis of $H_2(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})$. Let  $S=L_1-L_2\in H_2(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})$ be the vanishing cycle. It is straightforward that two homology classes $aL_1+bL_2$ and $bL_1+aL_2$ differ up to monodromy transformation, for every $(a,b)\in\mathbb{Z}^2$. In other words, one has $bL_1+aL_2=T(aL_1+bL_2)$.

   Therefore, for any non-singular fibre $\Sigma$ in this fibration, we can identify the quotient $H_2(\Sigma;\mathbb{Z})/_{\sim}$ with $H_2(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})/_{aL_1+bL_2\sim bL_1+aL_2}$.

\end{example}

     In the sequel, we will use interchangeably the notations of homology class in $H_2(\Sigma;\mathbb{Z})$ and  divisor class in $Pic(\Sigma)$, where $\Sigma$ is a del Pezzo surface. We denote the restriction of a divisor $D$ in $Pic(\Sigma)$ to $Pic(E)$  by $D|_{E,\Sigma}\in Pic_{D.E}(E)$.

   Since $K_\Sigma.S=0$, for $K_\Sigma$ the anti-canonical divisor of any non-singular surface $\Sigma$ in the pencil $\mathcal{Q}$, and $[E]=-K_\Sigma$, we deduce that  the restriction of the vanishing cycle to $E$  is in $Pic_0(E)$.

\subsection{Enumeration on pencils of surfaces} \label{Sect-Pencils}
   Since $S.E=0$, for every divisor $D\in Pic(\Sigma)$, the restrictions $D|_{E,\Sigma}$ and $T(D)|_{E,\Sigma}$ are in $Pic_{D.E}(E)$. Given any divisor $D$ and its monodromy transformation $T(D)$, we can construct the following holomorphic map:
\begin{align*}
    \phi:  \mathcal{Q}& \longrightarrow Pic_{2D.E}(E)\\
    \Sigma_t &\longmapsto  (D+T(D))|_{E,\Sigma_t}
\end{align*}

    \begin{lemma} \label{lem-constant.map}
One has $\phi$ is a constant map.
\end{lemma}

\begin{proof}

This lemma follows directly from the fact that compact Riemann surfaces can be mapped holomorphically to surfaces of lower genus, but not to higher genus, except as constant maps.
\end{proof}

We denote by $$\phi(\overline{D}):=(D+T(D))|_{E,\Sigma_t}= 2D|_{E,\Sigma_t}+(D.S) S|_{E,\Sigma_t}.$$

With abuse of notation, we also use $\sim$ for an equivalent relation on $Pic_{D.E} (E)$, which is given by $x {\sim} \phi(\overline{D})-x$. We denote by  $\overline{x}$  the equivalence class of $x$ in $Pic_{D.E} (E)/_{x{\sim}\phi(\overline{D})-x}$.

Since  $D|_{E,\Sigma_t}=\phi(\overline{D})-T(D)|_{E,\Sigma_t}$, we have $\overline{D|_{E,\Sigma_t}}=\overline{T(D)|_{E,\Sigma_t}}$. Hence, we can define the following
holomorphic map between Riemann spheres:
    \begin{align*}
    \phi_{\overline{D}}: \mathcal{Q} &\longrightarrow Pic_{D.E} (E)/_{x {\sim}\phi(\overline{D})-x}\\
    \Sigma_t &\longmapsto \overline{D|_{E,\Sigma_t}}
\end{align*}
\begin{lemma} \label{lem-existence.of.L}
There exists $\{L,T(L)\}\subset H_2(\Sigma;\mathbb{Z})$ such that $(L.S)^2=1$.

In particular, classes $L$ and  $T(L)$ can be chosen to be both effective.
\end{lemma}
\begin{proof}
According to Claim (6) in page 26 of \cite{demazure2006seminaire}, we deduce that the vanishing cycle $S$ can be written as $L_1-L_2$ where $(L_1,L_2,L_3,\ldots, L_k)$ is a $k$-uplet of effective generators of $H_2(\Sigma;\mathbb{Z})\cong \mathbb{Z}^k$.

The intersection form on $H_2(\Sigma;\mathbb{Z})$ implies that we can choose $L=L_2$ and $T(L)=L_1$, or vice-verse.
\end{proof}

\begin{lemma}\label{lem-degree.phi(S)}
 One has $\deg (\phi_{\overline{S}})= 4$.
\end{lemma}
\begin{proof}
By Lemma \ref{lem-existence.of.L}, a couple of effective divisors $L|_E$ and $T(L)|_E$ determines a surface (non-singular if $L|_E \neq T(L)|_E$ and singular if $L|_E = T(L)|_E$) in a pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$ whose base locus is $E$. We have the following commutative diagram:

\begin{center}
 \begin{tikzcd}
Pic_{L.E} (E) \arrow[r, "\pi:\; x \mapsto \Sigma_x"] \arrow[d, "2:1"]
& \mathcal{Q} \arrow[dl, "\cong"] \\
Pic_{L.E} (E)/_{x\sim \phi(\overline{L}) -x} &
\end{tikzcd}
\end{center}
where $\pi$ is a  ramified double covering map. Note that $\Sigma_{\phi(\overline{L})-x}=\Sigma_x$.
By computing the ramification values of the map $\pi$, we get exactly $4$ singular surfaces in the pencil $\mathcal{Q}$. (This is compatible with Lemma \ref{lem-4.singulars}).

Then, we have the following commutative diagram

\begin{center}
 \begin{tikzcd}

&& \mathcal{Q} \arrow[d, "\phi_{\overline{S}}"] \arrow[dll] \\
Pic_{L.E} (E)/_{x\sim \phi(\overline{L}) -x}  & \arrow[r, "\overline{x}\mapsto \overline{\phi(\overline{L})-2 x}"]& \quad \quad Pic_{0} (E)/_{y\sim -y}
\end{tikzcd}
\end{center}

From these two diagrams, we deduce that each singular surface in the pencil $\mathcal{Q}$ corresponds to an element of the set $\{x\in Pic_{L.E}(E): x=\phi(\overline{L})-x\}$, which is sent to $\overline{0}\in Pic_0(E)/_{y\sim -y}$. Hence, the degree of the map $\phi_{\overline{S}}$  is exactly the number of singular surfaces in the pencil $\mathcal{Q}$, which is equal to $4$.

\end{proof}
\begin{corollary} \label{coro-isomorphism.pencil.picard}
Let $\{L,T(L)\}\subset H_2(\Sigma;\mathbb{Z})$ be such that $(L.S)^2=1$. Then, the corresponding map $\phi_{\overline{L}}$ is an isomorphism.
\end{corollary}

The following proposition is the main point for solving the \textit{$1$-dimensional problem} that we mentioned in the beginning of this section.

\begin{proposition} \label{Prop-degree.case.complex}
Let $X$ be a three-dimensional del Pezzo variety such that $$\ker(\psi: H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z}))\cong \mathbb{Z}$$ where $\Sigma\in \mathcal{Q}$ is a non-singular surface. Let  $S\in \ker(\psi)$ be a generator.

Then, the map $\phi_{\overline{D}}$ has degree $(D.S)^2$.
\end{proposition}

\begin{proof}

We consider the following diagram of holomorphic maps between Riemann spheres:

\begin{center}
    \begin{tikzcd}
\mathcal{Q} \arrow[r, "\phi_{\overline{D}}"] \arrow[d, "\phi_{\overline{S}}"]
& Pic_{D.E} (E)/_{x\sim \phi(\overline{D})-x} \arrow[d, "\theta_1"] \\
Pic_{0} (E)/_{y\sim -y} \arrow[r, "\theta_2"]
& Pic_{0} (E)/_{y\sim -y}
\end{tikzcd}
$(I)$
\end{center}

where

$$\theta_1(\overline{x})=\overline{\phi(\overline{D})-2x}\, ;$$
$$\theta_2(\overline{y})=(D.S)\overline{y}.$$

The diagram $(I)$ is commutative. Indeed, given $\Sigma_t\in \mathcal{Q}$, one has:
\begin{align*}
    \theta_1(\phi_{\overline{D}}(\Sigma_t))&=\theta_1(\overline{D|_{E,\Sigma_t}})=\overline{\phi(\overline{D})-2D|_{E,\Sigma_t}}=(D.S)\overline{S|_{E,\Sigma_t}}; \text{ and }\\
    \theta_2(\phi_{\overline{S}}(\Sigma_t))&=\theta_2(\overline{S|_{E,\Sigma_t}})=(D.S)\overline{S|_{E,\Sigma_t}}.
\end{align*}

Therefore, we have the following equality $$\deg(\phi_{\overline{D}})\times \deg(\theta_1)=\deg(\phi_{\overline{S}})\times \deg(\theta_2).$$

\textit{Claim 1.} One has $\deg (\theta_2)= (D.S)^2$.
\begin{proof}
Let $\overline{y_1} \in Pic_{0} (E)/_{y\sim -y}$ (the quotient which is in the right corner of Diagram $(I)$). One has
$$\deg(\theta_2)=|\{\overline{y}\in Pic_{0} (E)/_{y\sim -y} \; : \; (D.S)\overline{y}=\overline{ y_1}\}|.$$
The following commutative diagram
\begin{center}
    \begin{tikzcd}
Pic_{0} (E) \arrow[r, "y\mapsto (D.S)y"] \arrow[d, "2:1"]
& Pic_{0} (E) \arrow[d, "2:1"] \\
Pic_{0} (E)/_{y\sim -y} \arrow[r, "\theta_2"]
& Pic_{0} (E)/_{y\sim -y}
\end{tikzcd}
\end{center}
 implies that $\deg(\theta_2)$ is exactly the number of solutions $y\in Pic_0(E)$ of the equation $(D.S)y=y_1$ in $Pic_0(E)$ where divisor $y_1$ is given. Any two solutions $y$ of this equation differ by a torsion point of order $D.S$, then this equation has exactly $(D.S)^2$ solutions $y$. Hence, the degree of the map $\theta_2$ is equal to $(D.S)^2$.
\end{proof}

\textit{Claim 2.} One has  $\deg (\theta_1)= 4$.

\begin{proof} We apply the same argument as in the previous claim.

Indeed, let $\overline{y_1} \in Pic_{0} (E)/_{y\sim -y}$ (the quotient which is in the right corner of Diagram $(I)$). One has
$$\deg(\theta_1)=|\{\overline{x}\in Pic_{D.E} (E)/_{x\sim \phi(\overline{D}) -x} \; : \; \overline{ \phi(\overline{D})-2x}=\overline{ y_1}\}|.$$
The following commutative diagram
\begin{center}
    \begin{tikzcd}
Pic_{D.E} (E) \arrow[r, "x\mapsto \phi(\overline{D})-2x"] \arrow[d, "2:1"]
& Pic_{0} (E) \arrow[d, "2:1"] \\
Pic_{D.E} (E)/_{x\sim \phi(\overline{D}) -x} \arrow[r, "\theta_1"]
& Pic_{0} (E)/_{y\sim -y}
\end{tikzcd}
\end{center}
implies that $\deg(\theta_1)$ is exactly the number of solutions $x\in Pic_{D.E}(E)$ of the equation $\phi(\overline{D})-2x=y_1$ in $Pic_0(E)$  where divisors $y_1$ and $\phi(\overline{D})$ are given. Any two solutions $x$ of this equation differ by a torsion point of order $2$, then this equation has exactly $2^2$ solutions $x$.  Hence, the degree of the map $\theta_1$ is equal to $4$.
\end{proof}

\textit{Claim 3.}  One has $\deg (\phi_{\overline{S}})= 4$, see Lemma \ref{lem-degree.phi(S)}.\\

Conclusion: \textit{Claim 1}, \textit{Claim 2} and \textit{Claim 3} imply that the degree of the map $\phi_{\overline{D}}$ is exactly equal to $(D.S)^2$.
\end{proof}

\subsection{Real pencils of real del Pezzo surfaces}
Throughout this subsection,  we restrict attention to $(X,\tau)$ a real three-dimensional del Pezzo variety such that $\ker(\psi) \cong \mathbb{Z}$, for any $\Sigma \in |-\frac{1}{2}K_X|$ non-singular surface and for surjection $\psi: H_2(\Sigma;\mathbb{Z})	\twoheadrightarrow H_2(X;\mathbb{Z})$ induced by the inclusion $\Sigma\hookrightarrow X$. Henceforth, the notations of vanishing cycle and generator of $\ker (\psi)$ might be used interchangeably (see Lemma \ref{Lemma-Vanishing.Cycle}). Let $\mathcal{Q}$ be a pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$. Let $E$ be its base locus. Note that $E$ is an elliptic curve realizing the homology class $\frac{1}{4}K_X^2$ (see Proposition \ref{Prop-E.is.elliptic}). Suppose that the curve $E$ and the pencil $\mathcal{Q}$ are real with non-empty real parts.

We first study some  properties of the real pencil $\mathcal{Q}$, which make them distinct from the complex case.
\subsubsection{Properties of real pencils of real surfaces}

\begin{lemma} \label{lem-surfaces.in.real.pencil}
The surfaces in the real pencil $\mathcal{Q}$ containing a real rational curve in $(X,\tau)$ are in the real part of this pencil.
\end{lemma}
\begin{proof}
Let $C$ be a real rational curve in $(X,\tau)$. Suppose that $\Sigma\in \mathcal{Q}$ contains $C$. Since $\mathcal{Q}$ is a real pencil, we have $\tau(\Sigma)\in \mathcal{Q}$. Since $C$ is real, we have $\tau(C)=C$.  If $\Sigma\notin \mathbb{R} \mathcal{Q}$, i.e. $\tau(\Sigma)\neq \Sigma$, then $C\subset (\Sigma\cap \tau(\Sigma))=E$ which is not possible since $C$ is rational. Thus,  $\tau(\Sigma)=\Sigma$. In other words, $\Sigma$ is in the real part of the pencil $\mathcal{Q}$, i.e. $\Sigma\in  \mathbb{R} \mathcal{Q}$.
\end{proof}

Remark that if $(\Sigma_1,\tau|_{\Sigma_1})$ and $(\Sigma_2,\tau|_{\Sigma_2})$ are two real non-singular surfaces in $\mathbb{R} \mathcal{Q}$, such that $\Sigma_2$ is a surgery of $\Sigma_1$ along the vanishing cycle, then the Euler characteristics of the real parts of these two surfaces differ by two. Suppose that $\chi(\mathbb{R} \Sigma_2)=\chi(\mathbb{R} \Sigma_1)+ 2$.  One has the following property (see \cite{bru16surgery}).
 \begin{remark}
 The  vanishing cycle $S$ in $H_2(\Sigma_1;\mathbb{Z})$ is $\tau|_{\Sigma_1}$-anti-invariant if and only if it is $\tau|_{\Sigma_2}$-invariant.
 \end{remark}

Let $(\Sigma_t,\tau|_{\Sigma_t}) \in \mathbb{R} \mathcal{Q}$ be a real del Pezzo surface containing a real rational curve in $X$ ($t\in {\mathbb{Z}}$). We denote by $\psi_t: H_2(\Sigma_t;\mathbb{Z})\rightarrow H_2(X;\mathbb{Z})$ the surjective map  induced from the inclusion from $\Sigma_t$ to $X$.

\begin{proposition} \label{prop-tau.vs.mu}
Let $d\in H^{-\tau}(X;\mathbb{Z})$. Let $\alpha \in \{D,T(D)\}\subset \psi_t^{-1}(d)$ such that $D.S\neq 0$. In a real surface $(\Sigma_t,\tau|_{\Sigma_t})$, one has that the vanishing cycle is $\tau|_{\Sigma_t}$-anti-invariant if and only if $\alpha$ is $\tau|_{\Sigma_t}$-anti-invariant.
\end{proposition}
\begin{proof}
The sufficient condition is direct. Indeed, if
$$\begin{cases}
\tau|_{\Sigma_t*}(D)=-D\\
\tau|_{\Sigma_t*}(T(D))=-(D+(D.S)S)
\end{cases}$$
then $$(D.S)\tau|_{\Sigma_t*}(S)=\tau|_{\Sigma_t*}(T(D)-D)=-(D.S)\tau|_{\Sigma_t*}(S).$$
Hence, $S$ is $\tau|_{\Sigma_t}$-anti-invariant.

We prove the necessary condition by two steps.

In the first step, we prove that $\tau|_{\Sigma_t*}(\{D,T(D)\})=\{-D,-T(D)\}$.
Suppose that $\alpha=D$. It is sufficient to prove that if $D$ satisfying $\psi_t(D)=d\in H^{-\tau}(X;\mathbb{Z})$ then
\begin{itemize}
    \item either $\tau|_{\Sigma_t*}(D)=-D$, i.e. $D$ is $\tau|_{\Sigma_t}$-anti-invariant,
    \item or $\tau|_{\Sigma_t*}(D)=-(D+(D.S)S)$ where $S$ is a generator of $\ker(\psi_t)$.
\end{itemize}
Indeed, we have $\tau_* (d)=-d$ then $\tau_*(\psi_t(D))=-\psi_t (D)$. Since $\psi_t\circ\tau|_{\Sigma_t*}=\tau_*\circ \psi_t$, we obtain
$$\psi_t (\tau|_{\Sigma_t*}(D)+D)=0.$$
Moreover, $\ker (\psi_t) =\mathbb{Z} S$ then there exists an integer $k\in \mathbb{Z}$ such that $\tau|_{\Sigma_t*}(D)+D=kS$. Since the intersection form is preserved via $\tau|_{\Sigma_t*}$, i.e. $(\tau|_{\Sigma_t*}(D))^2=D^2$, we deduce that $k$ is  equal to either $0$ or $-D.S$. Then, the first step is done.

In the second step, we prove  that if $(\Sigma_t,\tau|_{\Sigma_t})$ is  such that the vanishing cycle $S$ is $\tau|_{\Sigma_t}$-anti-invariant in $H_2(\Sigma_t; \mathbb{Z})$, then $D$ and $T(D)$ have to be $\tau|_{\Sigma_t}$-anti-invariant.  Suppose that $\tau|_{\Sigma_t*}(D)=-T(D)$. One has $\tau|_{\Sigma_t*}(D)=-D-(D.S)S=\tau|_{\Sigma_t*}(T(D))-(D.S)S$. This implies that

$$\tau|_{\Sigma_t*}(T(D))-\tau|_{\Sigma_t*}(D)=(D.S)S.$$
Moreover, $$\tau|_{\Sigma_t*}(T(D))=\tau|_{\Sigma_t*}(D+(D.S)S)=\tau|_{\Sigma_t*}(D)+(D.S) \tau|_{\Sigma_t*}(S).$$
Hence, $S=\tau|_{\Sigma_t*}(S)$. This means that $S$ is not $\tau|_{\Sigma_t}$-anti-invariant. Thus, the second step is done, and the proposition follows.
\end{proof}

As a consequence, the enumeration of real curves representing  $\tau$-anti-invariant classes in $X$ can degenerate to the enumeration of real curves representing $\tau|_{\Sigma_t}$-anti-invariant  classes in such $\Sigma_t$ in Proposition \ref{prop-tau.vs.mu}.

It is known that the choice of a point $x_0\in E$ induces an isomorphism $E\cong Pic_k(E)$  for any $k\in \mathbb{N}$. Since $E$ is real and $\mathbb{R} E\neq \emptyset$, the real parts  $\mathbb{R} E$ and $\mathbb{R} Pic (E)$ have the same topological type, we then have a homeomorphism  $\mathbb{R} E \simeq \mathbb{R} Pic_k(E)$ for any $k\in \mathbb{N}$. Suppose that $x_0\in \mathbb{R} E$. The real part $\mathbb{R} E$ is either connected or disconnected with two connected components. In the former case, we write $\mathbb{R} E =\mathbb{S}^1$. In the later case, we write $\mathbb{R} E =\mathbb{S}^1_0\sqcup \mathbb{S}^1_1$ and suppose that $x_0\in\mathbb{S}^1_0$. We call $\mathbb{S}^1_0$ the \textbf{pointed component} and $\mathbb{S}^1_1$ the \textbf{non-pointed component} of $\mathbb{R} E$.

Let $\underline{x}_{d}$ be a real configuration of $D.E$ points in $E$. Suppose that $\alpha|_{E,\Sigma_t}=[\underline{x}_{d}]$ in $Pic_{D.E}(E)$ where $\alpha$ is as in Proposition \ref{prop-tau.vs.mu}.
We consider the following maps
    \begin{align*}
    \mu_t:Pic(\Sigma_t)&\to Pic(\Sigma_t)\\
    \sum_i n_i [C_i]&\mapsto \sum_i n_i [\tau|_{\Sigma_t}(C_i)]
\end{align*}
and
\begin{align*}
    \mu_E:Pic(E)&\to Pic(E)\\
    \sum_i n_i [P_i]&\mapsto \sum_i n_i [\tau|_{E}(P_i)].
\end{align*}

The restriction of a divisor $D=\sum_i n_i [C_i]\in Pic(\Sigma_t)$ to $Pic(E)$ is given by $$D|_{E,\Sigma_t}=\sum_i n_i [C_i\cap E].$$

The restriction of $Pic(\Sigma_t)$ to $Pic(E)$ induces the following commutative diagram:
 \begin{center}
    \begin{tikzcd}
Pic(\Sigma_t) \; \; \;  \arrow[rr, "\mu_t"] \arrow[d]
&& \; \; \;  Pic(\Sigma_t)\arrow[d] \\
Pic(E)  \; \; \;  \arrow[rr, "\mu_E"]
&& \; \; \;  Pic(E)
    \end{tikzcd}
        \end{center}

     \begin{lemma} \label{lem-anti-invariant.and.real.divisors}
         If a class $D$ is $\tau|_{\Sigma_t}$-anti-invariant then $D|_{E,\Sigma_t}\in \mathbb{R} Pic(E)$.
     \end{lemma}
\begin{proof}
   We show that if $D$ is
$\tau|_{\Sigma_t}$-anti-invariant then  $\mu_E(D|_{E,\Sigma_t})=D|_{E,\Sigma_t}$.  Indeed, since $\mu_t=-\tau|_{\Sigma_t*}$, one has $D|_{E,\Sigma_t}=\mu_t(D)|_{E,\Sigma_t}$. The above commutative diagram implies that $\mu_E(D|_{E,\Sigma_t})=\mu_t(D)|_{E,\Sigma_t}$. Hence, the lemma follows.
\end{proof}

We consider the following  ramified double covering map
$$\pi: Pic (E) \to Pic(E)/_{x \sim \phi(\overline{D})-x}\;,\; x\mapsto \overline{x}.$$
The critical points are the solutions of equation $\{x \in Pic (E) :\; 2x=\phi(\overline{D})\}$. One has a homeomorphism $\mathbb{R} (Pic(E)/_{x \sim \phi(\overline{D})-x})\simeq \mathbb{S}^1$, and moreover
$$\pi^{-1}(\mathbb{R} (Pic(E)/_{x \sim \phi(\overline{D})-x}))=\Big(\{x,\phi(\overline{D})-x\}\subset \mathbb{R} Pic(E)\Big) \cup \Big(\{y, \tau|_E(y)\}: y+\tau|_E(y)=\phi(\overline{D})\Big).$$
Two types of real divisors on  $\mathbb{R}(Pic(E)/_{x \sim \phi(\overline{D})-x})$ are as illustrated in  Figure \ref{fig-RealElliptic}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{figures/RealElliptic.png}
\caption{Real divisors in the Picard group $\mathbb{R}(Pic(E)/_{x \sim \phi(\overline{D})-x})$ of a real elliptic curve with non-empty real part.}
\label{fig-RealElliptic}
\end{figure}

\subsubsection{Enumerations on real pencils of real del Pezzo surfaces}
We keep the same notations as in  Section \ref{Sect-Pencils}: the vanishing cycle  $S$ of the Lefschetz fibration induced from the pencil $\mathcal{Q}$; the constant map $\phi$ whose image is the divisor  $\phi(\overline{D})=(2D+(D.S)S)|_{E,\Sigma_t}$ in  $Pic_{2D.E} (E)$ which is independent of $\Sigma_t \in \mathcal{Q}$; and the degree $(D.S)^2$ map $\phi_{\overline{D}}$ (see Proposition \ref{Prop-degree.case.complex}).

Suppose that $\{L,T(L)\}\subset H_2(\Sigma_t;\mathbb{Z})$ is as in  Lemma \ref{lem-existence.of.L}. Recall in particular that $\phi(\overline{L})$ is  in $Pic_{2L.E}(E)$ and $\phi_{\overline{L}}$ is an isomorphism (Corollary \ref{coro-isomorphism.pencil.picard}).

We have seen that the divisor $\phi(\overline{D})$ is real, for any $D \in H^{-\tau}(\Sigma_t;\mathbb{Z})$ (see Lemma \ref{lem-anti-invariant.and.real.divisors}). In particular, the divisor $\phi(\overline{L})$ is real, and we have $\phi_{\overline{D}}$ and $\phi_{\overline{L}}$ are real maps.
Let $\mathcal{R}(\underline{x}_d)$ be a set of real irreducible rational curves in $X$ representing a given homology class $d\in H^{-\tau}(X;\mathbb{Z})$ and passing through a real configuration $\underline{x}_d$ of $k_d$ points in $E$.

\begin{lemma} \label{lem-constant.divisor.L.D}
The divisor $\big(D-|D.S|L\big)|_{E,\Sigma_t}$ of degree $|(D.E)-|D.S|(L.E)|$ in $Pic(E)$, denoted by $\phi(\overline{L},\overline{D})$,  does not depend on the choice of surface $\Sigma_t$ in the pencil $\mathcal{Q}$.
\end{lemma}

\begin{proof}
Clearly that the divisor  $\big(D-|D.S|L\big)|_{E,\Sigma_t}$ has degree $|(D.E)-|D.S|(L.E)|$ in $Pic(E)$. Without loss of generality, suppose that $D.S> 0$. We have
  \begin{align*}
      2\big(D-(D.S)L\big)|_{E,\Sigma_t} &= \big(2D-2(D.S)L\big)|_{E,\Sigma_t}\\
      &= \big((2D+(D.S)S)-(D.S) (2L+S)\big) |_{E,\Sigma_t}.
  \end{align*}

Since $(2L+S)|_{E,\Sigma_t}=\phi(\overline{L})$ and $(2D+(D.S)S)|_{E,\Sigma_t}=\phi(\overline{D})$ are both constant with respect to $\Sigma_t$, one has $ 2\big(D-(D.S)L\big)|_{E,\Sigma_t}$ is constant with respect to $\Sigma_t$. Thus, by continuity, the divisor $ \big(D-(D.S)L\big)|_{E,\Sigma_t}$ does not depend on $\Sigma_t$ neither.
\end{proof}

The next proposition is the main ingredient to solve the \textit{1-dimensional real problem} of Theorem \ref{Theorem2-W}.

\begin{proposition} \label{prop-corresponding.surface.picE}
Fix $\Sigma\in \mathcal{Q}$ a real non-singular surface.
Let $d\in H^{-\tau}(X;\mathbb{Z})$ and $\alpha\in \{D,T (D)\}$ be in the preimage of $d$ via $\psi$. Then, the real surfaces $(\Sigma_t,\tau|_{\Sigma_t})\in \mathbb{R} \mathcal{Q}$, which contain  a curve in the set $\mathcal{R}(\underline{x}_d)$, satisfy that the vanishing cycle is $\tau|_{\Sigma_t}$-anti-invariant.

Furthermore, such real surfaces correspond to the real solutions $l\in \mathbb{R} Pic(E)$ of the equation
\begin{equation}\label{eq-solutions.in.PicE}
    |D.S|l=[\underline{x}_d]-\phi(\overline{L},\overline{D}).
\end{equation}
\end{proposition}

  \begin{proof}
  The first part is followed directly from the fact that if the vanishing cycle is $\tau|_{\Sigma_t}$-invariant then $D\notin  H_2^{-\tau}(\Sigma_t;\mathbb{Z})$, for any $D\in H_2(\Sigma_t;\mathbb{Z})$.

  We now prove the second part. Since the equality $D.S=-T(D).S$ holds, one can suppose that $D.S> 0$.
   Then, one has
  $$\phi(\overline{L},\overline{D})=\big(D-(D.S)L\big)|_{E,\Sigma_t}=\big(D+(T(D).S)L\big)|_{E,\Sigma_t}.$$

  For every $\Sigma_t \in \mathcal{Q}$ that contains a curve in $\mathcal{R}(\underline{x}_d)$, suppose that this curve represents homology class $D\in H_2^{-\tau}(\Sigma_t;\mathbb{Z})$. Then one has $[\underline{x}_d]=D|_{E,\Sigma_t}$ is a real divisor in $Pic_{D.E}(E)$.

  One can rewrite  $D=(D.S)L+D-(D.S)L$. Restricting to $E$, with a remark that $L$ is also $\tau|_{\Sigma_t}$-anti-invariant, one gets
  $$[\underline{x}_d]=(D.S)L|_{E,\Sigma_t}+\big(D-(D.S)L\big)|_{E,\Sigma_t}$$
 is an equation in $ Pic(E)$.

 By Lemma \ref{lem-constant.divisor.L.D}, the divisor $\big(D-(D.S)L\big)|_{E,\Sigma_t}=\phi(\overline{L},\overline{D})$ does not depend on the choice of surface $\Sigma_t$ in the pencil.
   Thus, every surface $\Sigma_t \in \mathcal{Q}$, which contains a curve in $\mathcal{R}(\underline{x}_d)$ of homology class $D\in H_2^{-\tau}(\Sigma_t;\mathbb{Z})$ where $D.S> 0$, satisfies  the following equation $$(D.S)L|_{E,\Sigma_t}=[\underline{x}_d]-\phi(\overline{L},\overline{D})$$
    in $ Pic(E)$.

 Therefore, a real surface $\Sigma_t \in  \mathbb{R} \mathcal{Q}$, on which a curve in $\mathcal{R}(\underline{x}_d)$ represents a $\tau|_{\Sigma_t}$-anti-invariant class $\alpha\in \{D,T (D)\}$,  satisfies the equation
 \begin{equation} \label{eq-proof-eq-solutions.in.PicE}
     |D.S|L|_{E,\Sigma_t}=[\underline{x}_d]-\phi(\overline{L},\overline{D})
 \end{equation}
  in $ Pic(E)$.

Moreover, the following isomorphism is real
   $$   \phi_{\overline{L}}: \mathcal{Q} \xrightarrow{\cong} Pic_{L.E} (E)/_{x\sim \phi(\overline{L})-x}, \quad    \Sigma_t \mapsto \overline{L|_{E,\Sigma_t}}.$$
Together with the remarks about real structures on the pencil $\mathcal{Q}$ and that on the quotient of the Picard group of its base locus, one deduces that such real surfaces  $\Sigma_t \in  \mathbb{R} \mathcal{Q}$ are exactly the  solutions in $\mathbb{R} Pic(E)$ of Equation (\ref{eq-proof-eq-solutions.in.PicE}). The proposition then follows.
  \end{proof}

As a consequence, the set of such real surfaces in the pencil $\mathcal{Q}$ is in one-to-one correspondence to the set of real solutions of Equation (\ref{eq-solutions.in.PicE}). Hence, the number of such real surfaces depends on the number of real $|D.S|$-torsion points on $\mathbb{R} Pic(E)$. This means that this number depends on the real structure on $E$ and the position of the real divisors  $\phi(\overline{L},\overline{D})$ and $[\underline{x}_d]$ on $\mathbb{R} Pic(E)$ if $\mathbb{R} E$ is not connected. More precisely, this number is described in the following corollary, see also Figure \ref{fig-real.solutions.in.PicE}.

\begin{corollary}\label{coro-degree.case.real}
  The set of real surfaces $(\Sigma_t,\tau|_{\Sigma_t})$ in $\mathbb{R} \mathcal{Q}$, in which each element   contains a real curve of class $\alpha\in \{D,T (D)\}\subset \psi^{-1}(d)$ in $\mathcal{R}(\underline{x}_d)$, has exactly
  \begin{itemize}
         \item[$a)$] $|D.S|$ elements if either $\mathbb{R} E$ is connected or $\mathbb{R} E$ is disconnected and $D.S$ is odd;
      \item[$b)$] $2|D.S|$ elements if $\mathbb{R} E$ is disconnected and $D.S$ is even and      both $[\underline{x}_d]$ and $\phi(\overline{L},\overline{D})$ are on the same component of $\mathbb{R} E$;
      \item[$c)$] $0$ element if $\mathbb{R} E$ is disconnected and $D.S$ is even and $[\underline{x}_d]$ and $\phi(\overline{L},\overline{D})$ are on different components of $\mathbb{R} E$.
  \end{itemize}

  \end{corollary}

 \begin{proof}
Firstly, we count the real $|D.S|$-torsion points, i.e. the real solutions of the equation $|D.S|l=0$, which depend on the real structure on $E$. More precisely,
  \begin{enumerate}
 \item if $\mathbb{R} E$ is connected then there are exactly $|D.S|$  real $|D.S|$-torsion points;
 \item if $\mathbb{R} E=\mathbb{S}^1_0\sqcup \mathbb{S}^1_1$ disconnected then
\begin{itemize}
\item if $|D.S|$ is odd, then there are exactly $|D.S|$ real $|D.S|$-torsion points which are in the pointed component $\mathbb{S}^1_0$,
\item if $|D.S|$ is even, then there are exactly $2|D.S|$  real $|D.S|$-torsion points such that half of which are in each connected component of $\mathbb{R} E$.
\end{itemize}

 \end{enumerate}
Then, we count the real solutions $l$ of Equation (\ref{eq-solutions.in.PicE}), which depend on the position of the real divisors $\phi(\overline{L},\overline{D})$ and $[\underline{x}_d]$ on $\mathbb{R} E$ when $\mathbb{R} E$ is disconnected. More precisely,
\begin{enumerate}
\item if $\mathbb{R} E=\mathbb{S}^1$ then there are exactly $|D.S|$ real solutions of Equation (\ref{eq-solutions.in.PicE});
\item if $\mathbb{R} E=\mathbb{S}^1_0\sqcup \mathbb{S}^1_1$ then
\begin{itemize}
\item[$\bullet$]  if $|D.S|$ is odd, then there are exactly $|D.S|$ real solutions of Equation (\ref{eq-solutions.in.PicE}) whether the subtraction $\phi(\overline{L},\overline{D})-[\underline{x}_d]$ is in the pointed component $\mathbb{S}^1_0$ or in $\mathbb{S}^1_1$;
\item[$\bullet$] if $|D.S|$ is even then
\begin{itemize}
\item there are exactly $2|D.S|$ real solutions of Equation (\ref{eq-solutions.in.PicE}) if $[\underline{x}_d]-\phi(\overline{L},\overline{D})$ is in the pointed component $\mathbb{S}^1_0$ (or equivalently, if $\phi(\overline{L},\overline{D})$ and $[\underline{x}_d]$ are in the same connected component of $\mathbb{R} E$), and
\item there is no real solution of Equation (\ref{eq-solutions.in.PicE}) if  $[\underline{x}_d]-\phi(\overline{L},\overline{D})$ is in $\mathbb{S}^1_1$ (or equivalently, if $\phi(\overline{L},\overline{D})$ and $[\underline{x}_d]$ are in the different connected components of $\mathbb{R} E$), see Figure \ref{fig-real.solutions.in.PicE}.
\end{itemize}

\end{itemize}

\end{enumerate}
  \begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{figures/real.solutions.in.PicE.png}
\caption{The real solutions of Equation (\ref{eq-solutions.in.PicE}) associate to three cases of Corollary \ref{coro-degree.case.real}.}
\label{fig-real.solutions.in.PicE}
\end{figure}
Thus, the corollary follows.
\end{proof}

\section{Rational curves on del Pezzo varieties of dimension thee and two: relationship } \label{Section-ratcurve}
Recall from the last section that a non-singular element $\Sigma\in |-\frac{1}{2}K_X|$ is a non-singular del Pezzo surface where $X$ is a del Pezzo variety of dimension three. Recall also that $\psi: H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z})$ is a surjective map induced by the inclusion $\Sigma\hookrightarrow X$.

 In this section, we also use the notation $\mathcal{Q}$ for a pencil of surfaces in the linear system $|-\frac{1}{2}K_X|$, whose base locus is an elliptic curve $E$ of class $\frac{1}{4}K_X^2$.  Let $d\in H_2(X;\mathbb{Z})$ and $k_d=\frac{1}{2}c_1(X)\cdot d \in \mathbb{Z}$. Let $D\in H_2(\Sigma;\mathbb{Z})$ and $k_D=c_1(\Sigma)\cdot D-1\in \mathbb{Z}$. We continue to study about the degenerate situation of configuration of points on the elliptic curve $E$. We first prove a generalization of a result of Koll\'ar (Proposition \ref{Prop-Kollar}). Then, we prove that this specialized configuration could gives us genus-$0$ Gromov-Witten invariants of $\sigma$ (Proposition \ref{Prop.Modulispace1}, Corollary \ref{Corollary.Modulispace1}) and of $X$ (Proposition \ref{Prop.Modulispace2}) and hence allows us to relate $GW_X(d)$ to $GW_\Sigma(D)$.

\subsection{Degeneration of configuration of points} \label{SectionDegenerationConfiguration}
 \begin{lemma} \label{lemkDandkd}
Given a three-dimensional del Pezzo variety $X$, if  $\Sigma\in |-\frac{1}{2}K_X|$ is a non-singular surface and $D\in \psi^{-1}(d)$ then one has $k_D=k_d-1$.
\end{lemma}
\begin{proof}

By the Poincar\'e duality $\frac{1}{2}c_1(X)=PD(-\frac{1}{2}K_X)$ and by hypothesis $\Sigma\in |-\frac{1}{2}K_X|$ is non-singular, one has $$k_d=\frac{1}{2}c_1(X)\cdot d=[\Sigma]. d .$$

By the adjunction formula, by hypothesis $D\in \psi^{-1}(d)$ and by the Poincar\'e duality $c_1(\Sigma) =PD(-K_\Sigma)$, one has
$$[\Sigma].d=-K_\Sigma.D=k_D+1.$$

Hence, the lemma follows.
\end{proof}

  Let $\underline{x}_d$ be a configuration of $k_d$ distinct points in the base locus $E$ of pencil $\mathcal{Q}$. Let $\mathcal{C}(\underline{x}_d)$ be the set of \textit{connected algebraic curves of arithmetic genus zero representing the homology class $d$ and passing through $\underline{x}_d$}.

The following proposition generalizes Proposition 3 by J. Koll\'ar  in \cite{kollar2015examples} for the case $\mathbb{C}P^3$.

\begin{proposition} \label{Prop-Kollar}
 Suppose that the $k_d$ points of $\underline{x}_d$ are in general position in $E$. Then every curve in the set $\mathcal{C}(\underline{x}_d)$ is irreducible and is contained in some non-singular surface of the pencil $\mathcal{Q}$.
 \end{proposition}

\begin{proof}
Let $C_d$ be a curve in the set $\mathcal{C}(\underline{x}_d)$. We then prove the following claims.

\textit{Claim 1:}
\textit{Suppose that $C_d$ is irreducible. Then,  the whole curve $C_d$ has to be contained in some non-singular surface of the pencil $\mathcal{Q}$. }
\begin{proof}
The first observation is that any point of $C_d\setminus\underline{x}_d$ is contained in some  surface, which is denoted by $\Sigma_t$, of the pencil $\mathcal{Q}$. Then, the number of intersection points between $C_d$ and  $\Sigma_t$ is $\sharp (C_d\cap \Sigma_t) > \frac{1}{2}c_1(X)\cdot d$. By B\'ezout's theorem, the whole curve $C_d$ has to be contained in this surface.

We show that such surface $\Sigma_t$ is non-singular if the chosen points of $\underline{x}_d$ are generic enough. Indeed, suppose that the curve $C_d$ (passing through the configuration $\underline{x}_d$ in $E$) is contained in a singular surface $\Sigma_{sing}$ of the pencil $\mathcal{Q}$. Let $rank(Pic(\Sigma_{sing}))=k-1$, $k\geq 2$. Note that  $rank(Pic(\Sigma_{sing}))=rank(Pic(\Sigma))-1$, where $\Sigma$ is a non-singular surface in $\mathcal{Q}$. Let $(L_1,L_2\ldots,L_k)$ be a basis of $ H_2(\Sigma;\mathbb{Z})$, such that the vanishing cycle $S=L_1-L_2\in H_2(\Sigma;\mathbb{Z})$. Let $(\tilde{L_{12}},\tilde{L_3}\ldots,\tilde{L_k})$ be a basis of $ H_2(\Sigma_{sing};\mathbb{Z})$. Consider a  homomorphism given by
\begin{align*}
    \gamma: H_2(\Sigma;\mathbb{Z}_2)&\to H_2(\Sigma_{sing};\mathbb{Z})\\
    L_1-L_2& \mapsto 0 \\
    L_1+L_2& \mapsto \tilde{L_{12}}\\
    L_i & \mapsto \tilde{L_i}, \forall i \in \{3,\ldots,k\}.
\end{align*}

Suppose that the curve $C_{d}$ realises  homology class $D=\alpha_{1}L_{1}+\alpha_1 L_2+\alpha_3 L_3\ldots + \alpha_k L_k$ in $H_2(\Sigma;\mathbb{Z})$. One has $\gamma(D)=\alpha_{1}\tilde{L_{12}}+\alpha_3 \tilde{L_3}+\ldots + \alpha_k \tilde{L_k}\in H_2(\Sigma_{sing};\mathbb{Z})$.
The configuration $\underline{x}_{d}=C_{d}\cap E$ realises the divisor class $\gamma(D)|_E$ in $Pic_{k_d}(E)$.
If we fix $k_{d}-1$ first points on $\underline{x}_{d}\subset E^{k_{d}}$ and move the last point (still in $E$), then we get a new configuration $\underline{x}'_{d}$, such that $[\underline{x}_{d}']\neq [\underline{x}_{d}]$. On the contrary, the divisor class $\gamma(D)|_E$ stays fixed. In other words, we can choose a new configuration $\underline{x}_{d}'\subset E^{k_{d}}$ in an open subset of $E^{k_{d}}$ such that it represents a divisor $[\underline{x}_{d}']\neq \gamma(D)|_E$  in $Pic_{k_{d}}(E)$. Thus, if $C_{d}\subset\Sigma_{sing}$ then the chosen configuration $\underline{x}_{d}$, through which $C_{d}$ passes, is not generic. Moreover, there are only finitely many singular surfaces in the pencil $\mathcal{Q}$ (see Lemma \ref{lem-4.singulars}) on which the curve $C_{d}$ can not be contained.

\end{proof}

We now prove that $C_d$ can not be reducible via following claims and remarks. Assume that $C_d$ can be decomposed into $n$ ($n\geq 0$) irreducible rational curves $C_{d_i}$, i.e. $C_d=\bigcup\limits_{i=1}^{n} C_{d_i}$, where a curve $C_{d_i}$ realizes a homology class $d_i\in H_2(X;\mathbb{Z})$  such that $\sum\limits_{i=1}^{n}d_i=d$ and passes through $\underline{x}_{d_i}$ such that $\bigcup\limits_{i=1}^{n}\underline{x}_{d_i}=\underline{x}_d$.

\textit{Claim 2A:}\label{claim2}
\textit{ Each component curve $C_{d_i}$ passes though exactly $k_{d_i}=\frac{1}{2}c_1(X)\cdot d_i$ points of configuration $\underline{x}_d$.}

\begin{proof}
Suppose that there exists a component of $C_d$, noted by $C_{d_1}$, passing through more than $\frac{1}{2}c_1(X)\cdot d_1$ points of $\underline{x}_d$. Then the curve $C_{d_1}$ intersects with every surface of the pencil $\mathcal{Q}$ at more than $\frac{1}{2}c_1(X)\cdot d_1$ points. Therefore,  the curve  $C_{d_1}$ must be contained in every surface of the pencil  $\mathcal{Q}$  (because of B\'ezout's theorem). This means that $C_{d_1}=E$, then $C_d$ contains $E$, this is impossible. Hence, every component curve $C_{d_i}$ passes through a configuration $\underline{x}_{d_i}$ of exactly $k_{d_i}$
points of $\underline{x}_d$. As a consequence, different $C_{d_i}$ passes though different points of $\underline{x}_d$ for all $i$, i.e. $\bigsqcup\limits_{i=1}^{n}\underline{x}_{d_i}=\underline{x}_d$ and $\sum\limits_{i=1}^{n}k_{d_i}=k_d$ where $k_d$, $k_{d_i}$ are the associated integers of classes $d$ and $d_i$.
\end{proof}

\textit{Claim 2B:}
\textit{The reducible curve $C_d=\bigcup\limits_{i=1}^{n} C_{d_i}$ is contained in one non-singular surface of the pencil $\mathcal{Q}$.}

\begin{proof}
Note that the curve $C_d$ is connected. If there exists the case where two component curves $C_{d_i}$ and $C_{d_j}$, whose intersection is not empty, are contained in two different surfaces $\Sigma_i\neq \Sigma_j$ of the pencil $\mathcal{Q}$, then $\emptyset \neq C_{d_i}\cap C_{d_j}\subset \Sigma_i\cap\Sigma_j=E$. This implies that two different curves $C_{d_i}$ and $C_{d_j}$ have some points of $\underline{x}_d$ in common, contradiction to \textit{Claim 2A}. Hence the reducible curve $C_d$ is contained in only one  surface of the pencil $\mathcal{Q}$. Moreover, such surface is non-singular by the genericity of the chosen configuration. One can suppose that $C_d\subset \Sigma_t\in \mathcal{Q}$ and one can write its homology class as $[C_d]=\sum\limits_{i=1}^{n} [C_{d_i}] \in H_2(\Sigma_t;\mathbb{Z})$.
\end{proof}

\begin{remark}
    Let $\Sigma\in \mathcal{Q}$. If $D\in H_2(\Sigma;\mathbb{Z})\cong Pic(\Sigma)$ is an effective divisor class then there exists only countably many decompositions of $D$ into a sum of  effective divisors.
\end{remark}

\textit{Claim 2C:}\label{lem44}
\textit{  If the configuration $\underline{x}_d$ is generic then the curve $C_d$ is irreducible. }
\begin{proof}
We prove this claim by contradiction. Fix an integer $k$ ($0<k<k_d$). Let $\underline{x}_{d_1}\subsetneq \underline{x}_d$ be a strict sub-configuration consisting of points in $E$ such that its cardinality is equal to $k$. The reducibility and genericity of the configuration $\underline{x}_d$ implies that we can vary $\underline{x}_d$  in an open subset of $E^{k_d}$ such that we get another configuration $\underline{x}_d'$ satisfying $[\underline{x}_d'] \sim [\underline{x}_d]$ and there exists $\underline{x}_{d_1}'\subsetneq \underline{x}_d'$ and $[\underline{x}_{d_1}]\sim [\underline{x}_{d_1}]$. By the  Riemann-Roch theorem, every effective divisor linearly equivalent to $[\underline{x}_d]$ is determined by $k_d-1$ points and that to $[\underline{x}_{d_1}]$ is determined by $k-1$ points. By  avoiding a set of configurations of $k_d-1$ points in $E$ which forms a subvariety in $E^{k_d-1}$ , we can vary $k_d-1$ points in $E$ until we get a new configuration $ \underline{x}_{d}'$ and its sub-configuration $ \underline{x}_{d_1}'$, such that  $[ \underline{x}_{d}']\sim [ \underline{x}_{d}]$  in $Pic_{k_d-1}(E)$ but $[ \underline{x}_{d_1}']\neq [ \underline{x}_{d_1}]$ in $Pic_{k-1}(E)$. By \textit{Claim 2B} and \textbf{Remark 2}, there exists a non-singular surface $\Sigma_t\in \mathcal{Q}$, which contains the reducible curve $C_d$, and the effective divisor $[C_d]\in H_2(\Sigma_t;\mathbb{Z})$ has countably many decompositions into effective divisors. Therefore, the configurations satisfying the above variation can be chosen in an open subset of $E^{k_d-1}$. As a result, if the curve $C_d$ is reducible, then we can only choose the configurations, through which $C_d$ passes, in a measure zero subset, hence such configurations are not generic.

\end{proof}

    The last claim finishes our proof of this proposition.

    \end{proof}

\subsection{Moduli spaces and enumeration of balanced rational  curves}
In this technical part, we show that we can recover the genus-$0$ Gromov-Witten invariants of the three-dimensional del Pezzo variety $X$, and of a non-singular del Pezzo surface in $|-\frac{1}{2}K_X|$ after specializing our configuration of points, see Section $5$ in \cite{brugalle2016pencils}.

\subsubsection{Moduli spaces of stable maps and balanced rational curves}

We consider the moduli space $\mathcal{M}_{0,k_d}(X,d)$  of irreducible genus-$0$ $k_d$-pointed stable maps $$f:(\mathbb{C}P^1;p_1,\ldots,p_{k_d})\to X$$ of homology class $d\in H_2(X;\mathbb{Z})$. We denote by $\mathcal{M}^*_{0,k_d}(X,d)$ the subspace of $\mathcal{M}_{0,k_d}(X,d)$ consisting of simple maps.  And, we denote by  $\overline{\mathcal{M}}_{0,k_d}(X,d)$ the compactification (due to M. Kontsevich) of $\mathcal{M}_{0,k_d}(X,d)$, and its corresponding total evaluation map $ev_{X}$.

By the genericity of the configuration of points $\underline{x}_d=f(\{p_1,\ldots,p_{k_d}\})$ in $X$, the invariant $GW_{X}(d)$ can be identified with the number of elements $[(\mathbb{C} P^1;p_1,\ldots,p_{k_d});f]$ in $\mathcal{M}^*_{0,k_d}(X,d)$, which are regular points of  $ev_{X}$.

\begin{definition}
 A rational curve parametrized by $f:\mathbb{C} P^1\to X$  is \textbf{balanced} if $f$ is balanced, i.e. $f$ is an immersion and the normal bundle  $f^*TX/T\mathbb{C} P^1$ is isomorphic to the direct sum of two holomorphic line bundles of the same degree.
\end{definition}

\begin{lemma}(Lemma 1.2 \cite{welschinger2005spinor}) \label{Lemma-Regular.Balanced}
An element $[(\mathbb{C} P^1;p_1,\ldots,p_{k_d});f]$ in $\overline{\mathcal{M}}_{0,k_d}(X,d)$ is a regular point of the map $ev_{X}$ if and only if $f$ is balanced.

\end{lemma}

 \begin{lemma} \label{Lemma-not.intersection.balanced}
  Let $f:\mathbb{C}P^1\to X$ be an
  immersion such that $f(\mathbb{C} P^1)\subset \Sigma$. If $f(\mathbb{C} P^1)$ is not a complete intersection in $X$, then $f$ is balanced.
   \end{lemma}

\begin{proof}

 Let $\mathcal{N}_f=f^*TX/T\mathbb{C} P^1$ and   $\mathcal{N}'_f=f^*T\Sigma/T\mathbb{C} P^1$ be normal bundles.
We consider the following short exact sequence of holomorphic vector bundles over $\mathbb{C} P^1$:
$$0\longrightarrow \mathcal{N}'_f \longrightarrow \mathcal{N}_f  \longrightarrow f^*TX/T\Sigma \longrightarrow 0. \; (*) $$

\textit{Claim 1.} The map $f$ is balanced if and only if the short exact sequence $(*)$ does not split.

Suppose that $f_*[\mathbb{C} P^1]=D\in H_2(\Sigma;\mathbb{Z})$.
We have seen that the rank $2$ normal bundle $\mathcal{N}_f$ has degree $c_1(X)\cdot d-2=2k_d-2$ where $d=\psi(D)\in H_2(X;\mathbb{Z})$. By definition, we have $f$ is balanced if the normal bundle $\mathcal{N}_f$ is isomorphic to $\mathcal{O}_{\mathbb{C} P^1}(k_d-1)\oplus \mathcal{O}_{\mathbb{C} P^1}(k_d-1)$. We also have an isomorphism $$\mathcal{N}'_f\cong \mathcal{O}_{\mathbb{C} P^1}(c_1(\Sigma)\cdot D -2).$$ This implies that the normal line bundle $\mathcal{N}'_f$ has degree $c_1(\Sigma)\cdot D -2=k_d-2$. Hence, the normal line bundle $f^*TX/T\Sigma$ has degree  $(2k_d-2)-(k_d-2)=k_d$. Therefore, the map $f$ is balanced implies that the short exact sequence $(*)$ does not split. Conversely, if the short exact sequence $(*)$ does not split then $f$ is balanced. Indeed, we suppose that $\mathcal{N}_f\cong\mathcal{O}_{\mathbb{C} P^1}(k_1)\oplus \mathcal{O}_{\mathbb{C} P^1}(k_2)$ such that $k_1>k_2$, i.e. $k_1>c_1(\mathcal{N}'_f)$. This implies the null-morphism $\mathcal{O}_{\mathbb{C} P^1}(k_1)\to \mathcal{N}'_f$. Then, there is an isomorphism $\mathcal{O}_{\mathbb{C} P^1}(k_1)\to \mathcal{N}_f/\mathcal{N}'_f$, where $\mathcal{N}_f/\mathcal{N}'_f\cong \mathcal{O}_{\mathbb{C} P^1}(k_d)$. Hence, the short exact sequence $(*)$ splits (see page 252 \cite{griffiths1983infinitesimal}).

\textit{Claim 2.} The exact sequence $(*)$  splits if and only if $f(\mathbb{C} P^1)$ is a complete intersection in $X$.

An idea of a proof is using the first-order deformation of the surface $\Sigma$ in a pencil $\mathcal{Q}$ generated by an elliptic curve $E$ which realizes the homology class $\frac{1}{4}K^2_X$; and considering the divisor classes restricted on $E$ by the first-order deformation of  $f(\mathbb{C}P^1)$ in $\Sigma$.
By extending a result
in the proof of
Theorem $(4.f.3)$ in  \cite{griffiths1983infinitesimal}, the following two statements hold.
\begin{enumerate}
\item If the exact sequence $(*)$ splits then the section of $\mathcal{N}_f$ which vanishes at $D.E$ points in the intersection $f(\mathbb{C} P^1)\cap E$ implies that restricted divisor class  $[f(\mathbb{C}P^1)]|_E$ is constant.
\item If $f(\mathbb{C} P^1)$ is not a complete intersection with $\Sigma$ in $X$, then the restricted divisor class $[f(\mathbb{C}P^1)]|_E$ in $Pic(E)$ must vary as we deform $f(\mathbb{C} P^1)$. Indeed, let $f_\epsilon(\mathbb{C} P^1)$ be a first-deformation of $f(\mathbb{C} P^1)$ in the pencil, i.e. $f_\epsilon(\mathbb{C} P^1)\subset \Sigma_\epsilon\in \mathcal{Q}$. One has $$2[f_\epsilon(\mathbb{C} P^1)]|_{E,\Sigma_\epsilon}\sim \phi(\overline{D})-(D.S)S|_{E,\Sigma_\epsilon}=\phi(\overline{D})-(D.S)\phi(\overline{L})+ 2(D.S)L|_{E,\Sigma_\epsilon}.$$
  By Lemma \ref{lem-constant.map}, Lemma \ref{lem-existence.of.L} and the fact that $L|_{E,\Sigma_\epsilon}$ is not a constant as we deform $\Sigma_\epsilon$, the statement follows.
\end{enumerate}

The proposition follows by the sufficient condition of \textit{Claim 1} and necessary condition of \textit{Claim 2}.
In other words, if $f(\mathbb{C} P^1)$ is not a complete intersection, then the exact sequence  $(*)$ does not split, and hence $f$ is balanced.

 \end{proof}
    \begin{remark}

    If an irreducible curve $C$ is a complete intersection of a non-singular surface of the  pencil $\mathcal{Q}$ in $X$ then $[C].S=0$.
    \end{remark}

\subsubsection{Enumeration of balanced rational curves}

 In this part, we show that we can choose a configuration of points on $E$, so that every irreducible balanced rational curve passing though such configuration is a regular point of the corresponding total evaluation map.

     We denote by $V_{k_d}$ the set of configurations of $k_d$ distinct points on $E$.

     Recall that for each homology class $D\in H_2(\Sigma;\mathbb{Z})$ satisfying $D\in \psi^{-1}(d)$, we have $k_D=k_d-1$ (Lemma \ref{lemkDandkd}).
     Let $\underline{y}_D\subset E^{k_D}$ be a configuration of $k_D$ distinct points on $E$.

    For every non-singular surface $\Sigma_t$ in the pencil $\mathcal{Q}$, let $\mathcal{C}_{\Sigma_t,D}(\underline{y}_D)$ be the set of \textit{$k_D$-pointed stable maps $f:(C_0;p_1,\ldots,p_{k_D})\to \Sigma_t$, where $C_0$ is a connected nodal curve of arithmetic genus zero,
   such that $f(C_0)$ represents the homology class $D\in H_2(\Sigma_t;\mathbb{Z})$     and $f(\{p_1,\ldots,p_{k_D}\})=\underline{y}_D$.}

\begin{proposition} \label{Prop.Modulispace1}
There exists a dense open subset $U_{k_D}$ in the set $V_{k_D}$, such that for every configuration $\underline{y}_D$ in $U_{k_D}$ and for every element $f$ in $\mathcal{C}_{\Sigma_t,D}(\underline{y}_D)$, one has $C_0= \mathbb{C} P^1$ and $f$ is an immersion.
\end{proposition}
 A proof of this proposition is similar to the one of Proposition $5.2$  in \cite{brugalle2016pencils}, see also Lemma 2.1 in \cite{zahariuc2018rational}.
 \begin{corollary}\label{Corollary.Modulispace1}
 Let $\underline{y}_D\in U_{k_D}$ be a configuration of points as in Proposition \ref{Prop.Modulispace1}.
 Then, every non-singular surface $\Sigma_t$ in the pencil $\mathcal{Q}$, which contains a curve of class $D$ of the set $\mathcal{C}(\underline{x}_d)$,  contains  exactly $GW_{\Sigma_t}(D)$ elements of $\mathcal{C}(\underline{x}_d)$.

    \end{corollary}
    \begin{proof}
We consider the moduli space of stable maps $\mathcal{M}^*_{0,k_D}(\Sigma_t,D)$ and its corresponding total evaluation map $ev_{\Sigma_t}$. Proposition \ref{Prop.Modulispace1} and Lemma \ref{Lemma-Regular.Balanced} imply that for every map $f\in\mathcal{C}_{\Sigma_t,D}(\underline{y}_D)$ such that $\underline{y}_D\in U_{k_D}$, one has $[(\mathbb{C} P^1;p_1,\ldots,p_{k_D});f]$ is a regular point of  $ev_{\Sigma_t}$ if $f$ is an immersion. Hence, we get $$|\mathcal{C}_{\Sigma_t,D}(\underline{y}_D)|=GW_{\Sigma_t}(D).$$

    \end{proof}

     \begin{proposition}\label{Prop.Modulispace2}
Regular values of the total evaluation map $ev_{X}$ contained in $V_{k_d}$ form a dense open subset $U_{k_d}\subset V_{k_d}$.
In particular, if $\underline{x}_d\in U_{k_d}$ then the cardinality of the set $\mathcal{C}_{d}(\underline{x}_d)$ is equal to $GW_X(d)$.
    \end{proposition}

    \begin{proof}

    Let $\underline{x}_d$ be a configuration of $k_d$ distinct points on $E$ for which the conclusion of Proposition \ref{Prop-Kollar} holds, i.e. every curve in $\mathcal{C}(\underline{x}_d)$ is irreducible of arithmetic genus $0$ and is contained in some non-singular surface $\Sigma$ of the pencil $\mathcal{Q}$.

    By Lemma \ref{Lemma-Regular.Balanced}, an element $[(\mathbb{C} P^1;p_1,\ldots,p_{k_d});f]$ is a regular point of $ev_X$ if and only if $f$ is a balanced immersion.

    Note that if we replace $\underline{x}_d$ by another linearly equivalent configuration of points $\tilde{\underline{x}_d}$ on $E$, then the set of surfaces of the pencil $\mathcal{Q}$ containing a curve in $\mathcal{C}(\tilde{\underline{x}_d})$ is the same as that containing a curve in $\mathcal{C}({\underline{x}_d})$.

By the  Riemann-Roch theorem and by Proposition \ref{Prop.Modulispace1}, we can choose ${\underline{x}_d}$ generically such that every curve in $\mathcal{C}({\underline{x}_d})$  is parametrized by an algebraic immersion (see also the proof of Proposition 5.3 in \cite{brugalle2016pencils}).

 It remains to show that $f$ is balanced. Indeed, since the configuration of points ${\underline{x}_d}$ is generic, we can chose such configuration that is outside an algebraic subset with related to complete intersections of $\Sigma$ restricted to $E$.
 Hence, the balancity of $f$ follows from Lemma \ref{Lemma-not.intersection.balanced}.
    \end{proof}

  \section{The sign problems}\label{Sect-Welschinger}

  \subsection{Welschinger's signs of  real curves on real del Pezzo surfaces} \label{subsect-Welschinger.surfaces}

Let $(\Sigma,\tau)$ be a real del Pezzo surface.
Let $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$ and let $k_D=c_1(X)\cdot D-1$ be its associated integer.
Let $C$ be an irreducible real rational curve in $X$ representing the homology class $D$ and nodal, i.e. having only ordinary real nodes as singularities.
 By adjunction formula, the total number of complex nodes of $C$, which consists of real nodes and pairs of complex conjugated nodes, is equal to $$g(D)=\frac{1}{2}(-c_1(\Sigma)\cdot D+D^2+2).$$
 A real node of $C$ is either hyperbolic or elliptic in the following sense.
\begin{definition}
A \textbf{hyperbolic real node} of a curve is  a real point whose local equation of the curve is given by $x^2-y^2=0$ over $\mathbb{R}$ (i.e. it is the intersection of two real branches).
An \textbf{elliptic real node} (or an isolated real node) of a curve is a real point whose local equation of the curve is given by $x^2+y^2=0$ over  $\mathbb{R}$ (i.e. it is the intersection of two complex conjugated branches).
\end{definition}

Let  $\delta_E(C)$ be the numbers of elliptic real nodes of $C$. Let $\delta_H(C)$ be the numbers of hyperbolic real nodes of $C$. And let $\delta_{\mathbb{C}}(C)$ be the numbers of pairs of complex conjugated nodes of $C$.  One has
$$\delta_E(C)+\delta_H(C)+2\delta_{\mathbb{C}}(C)=g(D).$$
\begin{definition} \label{def-sign.curve.w.node.dim2}
The \textbf{Welschinger's sign} of an irreducible real rational curve $C$ in $(\Sigma,\tau)$, denoted by $s_{\mathbb{R} \Sigma}(C)$, is defined as
$$s_{\mathbb{R} \Sigma}(C):=(-1)^{\delta_E(C)}.$$
\end{definition}

Let $\underline{x}_D$ be a real generic  configuration of $k_D$ points consisting of $r$ real points and $l$ pairs of complex conjugated points in $X$, such that $r+2l=k_D$.
Let $\mathcal{R}(\underline{x}_D)$ be a set of real  rational curves in $\Sigma$ representing the homology class $D$ and passing through $\underline{x}_D$.
One has moreover, the genericity of $\underline{x}_D$  implies that every real curve in $\mathcal{R}(\underline{x}_D)$ is nodal and irreducible.
\begin{definition}\label{def-Welschinger.Inv.Nodes}
Let $(\Sigma,\tau)$ be a real del Pezzo surface and $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$. We associate each real curve $C\in \mathcal{R}(\underline{x}_D)$ to its Welschinger's sign $s_{\mathbb{R} \Sigma}(C)$.

The \textbf{Welschinger invariant}  of $(\Sigma,\tau)$ of homology class $D$ is defined as
 $$W_{\mathbb{R} \Sigma}(D,l):=\sum_{C\in \mathcal{R}(\underline{x}_D)}s_{\mathbb{R} \Sigma}(C).$$
\end{definition}

\begin{remark}
If either $D\in H_2(\Sigma;\mathbb{Z}) \setminus H_2^{-\tau}(\Sigma;\mathbb{Z})$ or $k_D\leq 0$, then the corresponding Welschinger invariants $W_{\mathbb{R} \Sigma} (D,l)$ vanish.
\end{remark}
\begin{theorem}(Welschinger \cite{welschinger2005fourfolds}, Brugall\'e \cite{brugalle2021invariance})
The Welschinger invariants in Definition \ref{def-Welschinger.Inv.Nodes} do not depend on the choice of a real generic  configuration of points, provided its number of real points (or equivalently, its number of complex conjugated points).
\end{theorem}

\subsection{Welschinger's signs of  real curves on real del Pezzo varieties of dimension three}\label{Sect-spinor.dim.3.with.normal.bundle}
According to J-Y. Welschinger (\cite{welschinger2005spinor}, \cite{welschinger2005stongly}), given a $Spin_n$- or $Pin^\pm_n$-structure on the real part of real projective Fano varieties\footnote{They are examples of strongly semipositive real symplectic manifolds as described in \cite{welschinger2005stongly}.}, we can define a sign (also known as spinor state) $\pm 1$ for each immersed balanced real rational curve on such varieties.

\subsubsection{Spinor states of balanced real rational curves on a real three-dimensional del Pezzo variety} \label{subsect-Spinor.states}
Let $(X,\tau)$ be a real three-dimensional del Pezzo variety with $\mathbb{R} X\neq \emptyset$. We equip the real part $\mathbb{R} X$ with a Riemannian metric $g_X$ and consider the tangent vector bundle $T \mathbb{R} X$ over $\mathbb{R} X$. We restrict attention to the cases $\mathbb{R} X$ is orientable and we equip $\mathbb{R} X$ with a $Spin_3$-structure, denoted by $\mathfrak{s}_X:=(Spin(T \mathbb{R} X,\mathfrak{o}_X),q_X)$. With abuse of notation, here we use the notation $\mathfrak{o}_X$ for an orientation on $T \mathbb{R} X$.

 Let $d \in H_2^{-\tau}(X;\mathbb{Z})$ and let $k_d=\frac{1}{2}c_1(X)\cdot d$ be its associated integer.

 We consider a real balanced immersion $f:(\mathbb{C} P^1,\tau_1)\to (X,\tau)$ and its associated short exact sequence of holomorphic vector bundles over $\mathbb{C} P^1$:
 $$0\to T \mathbb{C} P^1 \to f^*TX\to \mathcal{N}_f\to 0 $$
 where $\mathcal{N}_f=f^*TX/T\mathbb{C} P^1$ is a rank $2$ holomorphic normal vector bundle  over $\mathbb{C} P^1$.
 Since $X$ and $f$ are real, we also have the corresponding real bundles  fitting in a  short exact sequence of vector bundles over $\mathbb{R} P^1$
$$0\to T \mathbb{R} P^1 \to f|_{\mathbb{R} P^1}^*T\mathbb{R} X \to \mathbb{R} \mathcal{N}_f\to 0. $$
 Up to homotopy,  one has the decomposition  $f|_{\mathbb{R} P^1}^*T\mathbb{R} X = T \mathbb{R} P^1 \oplus \mathbb{R} \mathcal{N}_f$. Suppose that $f_*[\mathbb{C} P^1]=d$.
 Let $C= Im (f)$ with $\mathbb{R} C \neq \emptyset$. One has  $f(\mathbb{R} P^1) \subset \mathbb{R} X$ is an immersed knot. In order to define a spinor state for the curve $C$, we first show a consistent way to decompose the real vector bundle $f|_{\mathbb{R} P^1}^*T\mathbb{R} X$ into sum of real line subbundles, and then define a loop of direct orthogonal frames of this bundle as the following.

One should notice that the balancing property  of $\mathcal{N}_f$ implies its projectivization  $\mathbb{P}(\mathcal{N}_f)\cong \mathbb{C} P^1 \times \mathbb{C} P^1$.  Up to real isotopy, there are two holomorphic real line subbundles of $ \mathcal{N}_f$ of degree $(k_d-2)$ such that its projectivization in $ \mathbb{P}(\mathcal{N}_f)$ is a section of bidegree $(1,1)$ in the basis $(L_1,L_2)=([\mathbb{C} P^1\times\{p\}],[\{q\}\times \mathbb{C} P^1])$, where $p,q\in \mathbb{C} P^1$. Note that the orientation on $\mathbb{R} X$ and the orientation on $T \mathbb{R} P^1$ induces an orientation on  $\mathbb{R} \mathcal{N}_f$. Then, these two real isotopy classes are characterised by the direction in which the real part of a fiber rotates in $\mathbb{R} \mathcal{N}_f$. We denote by $E^+$ the isotopy class of the real part of the real line subbundle of  $\mathcal{N}_f$ whose real part of a fiber rotates positively  in $\mathbb{R} \mathcal{N}_f$ and by $E^-$ otherwise. We choose a real line subbundle  $L\subset \mathbb{R} \mathcal{N}_f$ as follows.
\begin{itemize}
    \item In the case $k_d$ is even,  we choose $L$ in the isotopy class of $E^+$.
    \item In the case {$k_d$ is odd},    we choose $L$ in the isotopy class of $E^-$.
\end{itemize}

Then, there exists a unique choice of a real line subbundle $\tilde{L}$ in $\mathbb{R} \mathcal{N}_f$ such that $$f|_{\mathbb{R} P^1}^*T \mathbb{R} X=T \mathbb{R} P^1 \oplus L \oplus \tilde{L}.$$

Now we choose some trivialization of each factor of the above decomposition of $f|_{\mathbb{R} P^1}^*T \mathbb{R} X$ given by non-vanishing sections $v_i\;(i \in \{1,2,3\})$ and then define a  loop of direct orthogonal frames of the vector bundle $ f|_{\mathbb{R} P^1}^*T \mathbb{R} X$. Assuming that  $v_1(\xi)\in T\mathbb{R} P^1$ ($\xi\in \mathbb{R} P^1$)  and $(v_1,v_2,v_3)$ forms a direct basis. In the case $k_d$ is even, we can choose\footnote{ This choice is not unique since we can also choose $-v_2(\xi)\in \mathbb{R} L$.}
a non-vanishing section $v_2(\xi)\in L$ such that $(v_1(\xi),v_2(\xi))_{\xi\in \mathbb{R} P^1}$  is an orthogonal section of $T \mathbb{R} P^1 \oplus L$. Then, the unique non-vanishing section $v_3(\xi)\in \tilde{L}$ forms a loop of direct orthogonal frames $(v_1(\xi),v_2(\xi),v_3(p))_{\xi\in \mathbb{R} P^1}$ of the vector bundle $ f|_{\mathbb{R} P^1}^*T \mathbb{R} X$, see Figure \ref{fig-spinor}. In the case $k_d$ is odd, we first choose the non-vanishing sections $(v_1(\xi),v_2(\xi),v_3(\xi))_{\xi\in [0,\pi]\subset \mathbb{R} P^1}$, then we extend this trivialization over $\mathbb{R} P^1$ by applying the action of $\begin{pmatrix} 1& 0&0\\ 0&\cos(\xi) & \sin(\xi) \\ 0 & -\sin(\xi) & \cos(\xi) \end{pmatrix} \in SO_3(\mathbb{R})$.

\begin{figure}[h!]
 \centering
\includegraphics[width=0.3\textwidth]{figures/SpinorState.png}
\caption{A direct orthogonal frame $(v_1(x_0),v_2(x_0)),v_3(x_0)$ at $x_0=f(\xi)\in \mathbb{R} C$ of the vector bundle $T \mathbb{R} X|_{\mathbb{R} C}$.}
\label{fig-spinor}
\end{figure}

  Note that the pull-back via the immersion $f$ of a principal $SO_3$-bundle $SO(T\mathbb{R} X, \mathfrak{o}_{X})$ over $\mathbb{R} X$ is a principal $SO_3$-bundle $f^*SO(T\mathbb{R} X,\mathfrak{o}_{X})$ over $\mathbb{R} P^1$, and a $Spin_3$-structure on $SO(T\mathbb{R} X,\mathfrak{o}_X)$ induces a $Spin_3$-structure on this bundle. Hence, we call a loop in the above construction a \textit{loop of the principal $SO_3$-bundle} $f^*SO(T\mathbb{R} X,\mathfrak{o}_X)$.

  Then, we are ready to define spinor states of the curve  $C$.
 \begin{definition} \label{def-spinor3dim}
 Let  $((X,\tau),\mathfrak{s}_X,\mathfrak{o}_X))$ be a real three-dimensional del Pezzo variety whose real part is orientable and equipped  with a $Spin_3$-structure $\mathfrak{s}_X=(Spin(T \mathbb{R} X,\mathfrak{o}_X),q_X)$. Let $C=Im(f)$ be a real balanced irreducible rational curve immersed in $X$. Let $(v_1(\xi),v_2(\xi),v_3(\xi))_{\xi\in \mathbb{R} P^1}$ be a loop of the principal $SO_3$-bundle $f^*SO(T\mathbb{R} X,\mathfrak{o}_X)$ defined as above. The \textbf{spinor state} of $C$, denoted by $sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)$, is given by
  $$ sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)=\begin{cases}
		+1, & \text{ if this loop lifts to a loop of } f^*Spin(T\mathbb{R} X,\mathfrak{o}_X)\\
           -1, & \text{ otherwise}.
		 \end{cases}$$
   \end{definition}

\begin{remark}
The spinor state of a real rational curve in Definition \ref{def-spinor3dim} does not depend  on the choice of neither the sections $v_i \;(i\in \{1,2,3\})$ nor the metric $g_X$ or the parametrization $f$.
This spinor state  only depends on the choice of a $Spin_3$-structure on $\mathbb{R} X$ and of an orientation on $\mathbb{R} X$.
\end{remark}
Geometrically, if we change the orientation of $\mathbb{R} X$ then we interchange the two isotopy classes $E^+$ and $E^-$. Since $E^+$ and $E^-$ differ by exactly one full rotation, i.e. a generator of $\pi_1(SO_2(\mathbb{R}))$, the associating loops in $\pi_1(SO_3(\mathbb{R}))$ differ by a non-trivial element. Moreover, recall that
the set of $Spin_3$-structures on the bundle $SO(T\mathbb{R} X,\mathfrak{o}_X)$ are in one-to-one correspondence with the group $H^1(\mathbb{R} X;\mathbb{Z}_2)$, which is determined by mapping in circles.
We resume these properties in the following proposition.

\begin{proposition} (Lemma 1.7 \cite{welschinger2005stongly})\label{Prop-SpinorStates1}
Let $\eta\in H^1(\mathbb{R} X;\mathbb{Z}_2)$. One has

 $$sp_{\mathfrak{s}_X,\mathfrak{-o}_X}(C)=-sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C),$$
 and
 $$sp_{\eta\mathfrak{s}_X,\mathfrak{o}_X}(C)=(-1)^{\eta\cdot (f_*[\mathbb{R} P^1])}sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C).$$
\end{proposition}

  \begin{example}
   Let $(X,\tau)$ be $(\mathbb{C}P^3,\tau_3)$ with its real part $\mathbb{R} X=\mathbb{R} P^3$.    Since $H_1(\mathbb{R} P^3;\mathbb{Z}_2)= \mathbb{Z}_2$, there are two $Spin_3$-structures on $SO(T \mathbb{R} P^3,\mathfrak{o}_{\mathbb{R} P^3})$.

   We can choose a $Spin_3$-structure $\mathfrak{s}_{\mathbb{R} P^3}$ on $SO(T \mathbb{R} P^3,\mathfrak{o}_{\mathbb{R} P^3})$ such that the spinor state of a line $L$ in $\mathbb{C}P^3$ is $sp_{\mathfrak{s}_{\mathbb{R} P^3},\mathfrak{o}_{\mathbb{R} P^3}}(L)=0$. This choice is the same as the choice of a trivialisation on $T \mathbb{R} P^3$ by Brugall\'e-Georgieva in \cite{brugalle2016pencils}.

\end{example}

 Let $\underline{x}_d$ be a real generic configuration of $k_d$ points consisting of $r$ real points and $l$ pairs of complex conjugated points in $X$ such that $r+2l=k_d$

 Denote by $\mathcal{R}(\underline{x}_d)$ the set of real balanced irreducible  rational curves in $X$ representing the homology class $d$ and passing through $\underline{x}_d$.

   \begin{definition} \label{def-W.inv.dim3}
   Let $((X,\tau),\mathfrak{s}_X,\mathfrak{o}_X))$ be a real three-dimensional del Pezzo variety and $d\in H_2^{-\tau}(X;\mathbb{Z})$. We associate each real curve $C\in \mathcal{R}(\underline{x}_d)$ to its spinor state $sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)$ as in Definition \ref{def-spinor3dim}.

   The \textbf{Welschinger invariant} of  $((X,\tau),\mathfrak{s}_X,\mathfrak{o}_X))$ of homology class $d$ is defined as
   $$W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l):=\sum_{C\in \mathcal{R}(\underline{x}_d)}sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C).$$
   \end{definition}

   \begin{remark}
If either $d\in H_2(X;\mathbb{Z})\setminus H_2^{-\tau}(X;\mathbb{Z})$ or $k_d\leq 0$, then the corresponding Welschinger invariants $W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)$ vanish.
\end{remark}
\begin{theorem}(Welschinger \cite{welschinger2005spinor})
The Welschinger invariant in Definition \ref{def-W.inv.dim3} does not depend on the choice of a real generic  configuration $\underline{x}_d$  up to isotopy, provided its number of real points.

This invariant only depends on the $Spin_3$-structure $\mathfrak{s}_X$ on $SO(T\mathbb{R} X,\mathfrak{o}_X)$,
on the homology class $d$ and on the number $l$.
\end{theorem}

\subsubsection{Spinor states of rational curves: revisited} \label{subsect-Spinorstates.revisited}
In this subsection, we suppose that $C$ is a real balanced rational curve parametrized by a real balanced immersion $f:(\mathbb{C} P^1,\tau_1)\to (X,\tau)$ such that $$f_*[\mathbb{C} P^1]=d\in  H_2^{-\tau}(X;\mathbb{Z})$$ and $f (\mathbb{C} P^1)\subset \Sigma$, where $\Sigma\in |-\frac{1}{2}K_X|$ is a real non-singular surface. The curve $C$ also represents a homology class $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$, where $D\in \psi^{-1}(d)$ and $\psi: H_2(\Sigma;\mathbb{Z})\twoheadrightarrow H_2(X;\mathbb{Z})$.
We will study some properties of the holomorphic line subbundle $\mathcal{N}'_f=f^*T\Sigma/T\mathbb{C} P^1$, which is a holomorphic line subbundle of the normal bundle $\mathcal{N}_f$,  and then we relate such subbundle to the definition of spinor states of curves.
\begin{remark}
  The isotopy class of the normal bundle of $C$ in $\Sigma$ does not depend on the curve $C$ but on its homology class in $H_2(\Sigma;\mathbb{Z})$, then we can write it as $\mathcal{N}_{D/\Sigma}$ instead of $\mathcal{N}_{C/X}$. With this notation, we have $\mathcal{N}'_f=f^*\mathcal{N}_{D/\Sigma}$.
\end{remark}

By the adjunction formula, one has $$\deg(\mathcal{N}_{D/\Sigma})=c_1(\Sigma)\cdot D-2=k_D-1.$$

This implies that $\deg(\mathcal{N}'_f)=k_d-2$. Hence, the real line bundle $\mathbb{R} \mathcal{N}'_f$  is orientable if  $k_D$ is odd and is non-orientable otherwise.

Recall in Section \ref{subsect-Spinor.states} that there are two isotopy classes, denoted by  $E^+$ and $E^-$, of the real part of a holomorphic real line subbundle  of degree $(k_d-2)$ of $\mathcal{N}_{f}$ which are characterised by the direction in which the real part of a fiber rotates in $\mathbb{R} \mathcal{N}_{f}$.

Note that $k_D=k_{T(D)}$ for any $\{D, T(D)\}\subset H_2(\Sigma;\mathbb{Z})$, where $T$ is the monodromy transformation $T(D)=D+(D.S)S$.
\begin{definition} \label{def-epsilon.function}
Let $f: \mathbb{C} P^1 \to \Sigma\subset X$ be a real immersion such that $f_*[\mathbb{C} P^1]=D$ in $H_2^{-\tau}(\Sigma;\mathbb{Z})$ and let $\mathbb{R} \mathcal{N}'_f=\mathbb{R} f^*\mathcal{N}_{D/\Sigma}$ be as described above. We define the following  homomorphism:
\begin{align*}
    \epsilon:H_2(\Sigma;\mathbb{Z})&\to \mathbb{Z}_2\\
    D&\mapsto \begin{cases}
  k_D+1 \mod 2, \text{ if } \mathbb{R} \mathcal{N}'_f=E^+\\
  k_D \mod 2, \text{ if } \mathbb{R} \mathcal{N}'_f=E^-.
  \end{cases}
\end{align*}
\end{definition}
\begin{lemma} \label{lem-epsilon.L.and.TL}
Let $\{L,T (L)\}\subset H_2(\Sigma;\mathbb{Z})$ be such that $|L.S|=1$.
One has
$$\epsilon(T(L))\neq \epsilon(L).$$
\end{lemma}
\begin{proof}

  We have seen that a couple of divisors $L_t|_E\neq T(L_t)|_E$ such that $|L_t.S|=1$ determines a unique non-singular surface $\Sigma_t$ in the pencil $\mathcal{Q}$ (Corollary \ref{coro-isomorphism.pencil.picard}).

  Let $\{L_0,T(L_0)\}\subset H_2^{-\tau}(\Sigma_0;\mathbb{Z})$.  Let $\mathbb{R} \mathcal{N}_a'=\mathbb{R} f^*\mathcal{N}_{L_0/\Sigma}$ and  $\mathbb{R} \mathcal{N}_b'=\mathbb{R} f^*\mathcal{N}_{T(L_0)/\Sigma}$ be the real part of the normal bundles of $L_0$ and $T(L_0)$ respectively.
  By monodromy phenomenon, the role of $L_0$ and $T(L_0)$ in $H_2^{-\tau}(\Sigma_0;\mathbb{Z})$ interchanges when we make a loop around a singular fiber. In particular, $L_0$ and $T(L_0)$ rotate in the different direction in $\Sigma_0$ when we go along the pencil from $\Sigma_0$ to itself around a singular surface. Hence, we deduce that $\mathbb{R} \mathcal{N}_a'$ and $\mathbb{R} \mathcal{N}_b'$ rotate in different direction in $\mathbb{R} X$, see Figure \ref{fig-opposite.rotations}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{figures/opposite.rotations}
\caption{$L_0$ and $T(L_0)$ rotate in different direction in $\mathbb{R} X$.}
\label{fig-opposite.rotations}
\end{figure}
\end{proof}
Now, we can suppose $L$ and $T(L)$ are such that $L.S=-1$ and $\mathbb{R} \mathcal{N}'_{L/\Sigma}=E^+$. As a consequence,  $T(L).S=1$ and $\mathbb{R} \mathcal{N}'_{T(L)/\Sigma}=E^-$.
 \begin{proposition}\label{prop-function.epsilon1}
For any $\{D,T(D)\}\subset H_2^{-\tau}(\Sigma;\mathbb{Z})$ such that $D.S\neq 0$, one has
$$\epsilon(T(D))\equiv \epsilon(D)+1 \mod 2.$$
\end{proposition}
\begin{proof}
 By Lemma \ref{lem-epsilon.L.and.TL}, this proposition is true for $D=L$.

Let $\Sigma_0$ be a real non-singular surface in the pencil $\mathcal{Q}$. Let  $\Sigma_t$ be the first order real deformation of $\Sigma_0$ in $\mathcal{Q}$.

\textbf{Claim 1:}
The isotopy class of the real line subbundle $\mathbb{R} f^*\mathcal{N}_{D/\Sigma_0}$ in $\mathbb{R} \mathcal{N}_f$  is determined by the direction of the vector $\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}$.

A proof of Claim 1 is similar to the proof of Proposition 3.4 in \cite{brugalle2016pencils}, we stress here on some following main points.

We fix a point $x_0$ in the transverse intersection of $f(\mathbb{C} P^1)$ and the base locus $E$ of the pencil $\mathcal{Q}$ such that $p_0=f^{-1}(x_0) \in \mathbb{R} P^1$.
 By considering the first order real deformation $f_t$ of $f$ in the pencil $\mathcal{Q}$ such that $f_t(\mathbb{C} P^1)$ passes through all points of intersection $f(\mathbb{C} P^1)\cap E$ except $x_0$ for all $t$, we define a degree $(k_d-1)$ holomorphic line subbundle $M$ in $\mathcal{N}_{f}$ such that the fibre $M|_{p_0}=\sigma(p_0)$, where $\sigma:\mathbb{C} P^1\to \mathcal{N}_f$ is a non-null holomorphic section vanishing on $f^{-1}(E\setminus \{x_0\})$.  Since this deformation stays in the base locus $E$, one has  $\sigma(p_0)\in f^*\mathcal{N}_{D/\Sigma_0}$. Then, we can compare the mutual position between fibres $\mathbb{R} M|_{p_t}$ and $\mathbb{R} f^*\mathcal{N}_{D/\Sigma_0}|_{p_t}$, where $p_t\in \mathbb{R} P^1$ in the neighborhood of $p_0$, by proving that the isotopy class of $\mathbb{R} f^*\mathcal{N}_{D/\Sigma_0}$ is determined by the direction of the vector $\sigma(p_0)$.
 Moreover, the deformation $x_t=f(p_t)$ of $x_0$ as an intersection point of $f_t(\mathbb{C} P^1)\cap E$ is determined by the deformation of the divisor class $D|_{E,\Sigma_t}$ when $t$ approaches to $0$. Hence, Claim 1 follows.

 We now consider the deformation of divisors $D|_{E,\Sigma_t}$ and $T(D)|_{E,\Sigma_t}$ on $Pic_{D.E}(E)$ when $t$ approaches to $0$.

\textbf{Claim 2:} The direction of vectors $\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}$ and $\frac{dT(D)|_{E,\Sigma_t}}{dt}\Big|_{t=0}$  are opposite.

Indeed, this claim follows directly from the fact that $D|_{E,\Sigma_t}+T(D)|_{E,\Sigma_t}=\phi(\overline{D})$ is a constant divisor class for all $t$ (see Figure \ref{fig-TwoIsotopyClasses}).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{figures/TwoIsotopyClasses.png}
\caption{Two isotopy classes of a real line subbundle of $\mathbb{R} \mathcal{N}_{f}$ are represented in  $Pic(E)$ in the case $\mathbb{R} E$ is connected.}
\label{fig-TwoIsotopyClasses}
\end{figure}

\textbf{Claim 3:} The class $D$ is in the same isotopy class as class $L$ if and only if $T(D)$ is not.

This claim follows by Lemma \ref{lem-constant.divisor.L.D}. Indeed, one has

$$\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}=|D.S|\frac{dL|_{E,\Sigma_t}}{dt}\Big|_{t=0}.$$
Since $L$ and $T(L)$ are not in the same isotopy class, then so are $D$ and $T(D)$.

Hence, the statement of the proposition holds.

\end{proof}

\begin{proposition}\label{prop-function.epsilon2} For any $D,D'\in H_2(\Sigma;\mathbb{Z})$ such that $D.S\neq 0$ and $D'.S\neq 0$, one has
$$\epsilon(D)=\epsilon(D')\Leftrightarrow (D.S)(D'.S)>0.$$
\end{proposition}
\begin{proof}
On the pencil $\mathcal{Q}$, we suppose that $\Sigma_0$ is the fiber at $t_0\in \mathbb{C} P^1$ and $\Sigma_t$ is the fiber at $t\in \mathbb{C} P^1$ in the neighborhood of $t_0$. We have two following claims.

\textit{Claim 1:}
$\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}=\frac{-D.S}{2}\frac{dS|_{E,\Sigma_t}}{dt}\Big|_{t=0}.$

Indeed, this claim is directly implied from the fact that
$$2D|_{E,\Sigma_0}+(D.S)S|_{E,\Sigma_0}=2D|_{E,\Sigma_t}+(D.S)S|_{E,\Sigma_t}(=\phi(\overline{D})).$$

\textit{Claim 2:} $\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}\times \frac{dD'|_{E,\Sigma_t}}{dt}\Big|_{t=0}>0 \Leftrightarrow (D.S)(D'.S)>0$.

Indeed, \textit{Claim 1} implies that
$$\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}\times \frac{dD'|_{E,\Sigma_t}}{dt}\Big|_{t=0}= \Big( \frac{-D.S}{2}\frac{dS|_{E,\Sigma_t}}{dt}\Big|_{t=0}\Big)\times \Big( \frac{-D'.S}{2}\frac{dS|_{E,\Sigma_t}}{dt}\Big|_{t=0}\Big).$$

Hence,
$$\frac{dD|_{E,\Sigma_t}}{dt}\Big|_{t=0}\times \frac{dD'|_{E,\Sigma_t}}{dt}\Big|_{t=0}= \frac{(D.S)(D'.S)}{4}\Big(\frac{dS|_{E,\Sigma_t}}{dt}\Big|_{t=0}\Big)^2.$$

Then, the statement follows.
\end{proof}

\subsection{Welschinger's signs of real rational curves on real del Pezzo varieties of dimension two and three: comparision} \label{Sect-signs.in.comparison}
We use the following convention of notations in this part: if we consider a rank $n$ vector bundle $V$  equipped with a Riemannian metric over a topological space $B$, we denote by  $O(V)$ a principal $O_n$-bundle over $B$. If $V$ is orientable with orientation $\mathfrak{o}_V$, we denote by  $SO(V,\mathfrak{o}_{V})$ a principal $SO_n$-bundle over $B$.
We also denote by $\mathcal{P}_n^\pm(V)$ the set of $Pin^\pm_n$-structures on $O(V)$, and by $\mathcal{SP}_n(V,\mathfrak{o}_V)$  the set of $Spin_n$-structures on $SO(V,\mathfrak{o}_V)$. If $V=TB$ is the tangent bundle, we sometimes say $Pin^\pm$ (or $Spin$) structures over $B$ instead of on  $O(TB)$ (or on $SO(TB,\mathfrak{o}_{TB})$).

In this section, we study the properties of the $Pin_2^-$-structure over $\mathbb{R} \Sigma$ (where $\Sigma\in |-\frac{1}{2}K_X|$ is a real non-singular surface), which is restricted from a $Spin_3$-structure over $\mathbb{R} X$  (Proposition \ref{Prop-Spin3.Pin2}). The corresponding quasi-quadratic enhancement allows us to relate spinor states of rational curves in $(X,\tau)$ and Welschinger's signs of these curves in  $(\Sigma,\tau|_\Sigma)$ (Proposition \ref{Prop-RelationW2W3}).

\subsubsection{Construction of quasi-quadratic enhancement corresponding to a $Pin^-_2$-structure on a  topological surface}
\begin{definition} \label{def-quasiquadratic.enhancement}
A function $s: H_1(\mathbb{R} \Sigma;\mathbb{Z}_2) \to \mathbb{Z}_2$ is called \textbf{a quasi-quadratic enhancement} if it satisfies $$s(x+y)=s(x)+s(y)+x.y+(w_1(\mathbb{R} \Sigma)\cdot x)(w_1(\mathbb{R} \Sigma)\cdot y).$$
\end{definition}
It is known that every non-singular topological surface has a $Pin_2^-$-structure on its tangent bundle. We first equip $T\mathbb{R} \Sigma$ with a $Pin_2^-$-structure $\mathfrak{p}_{T\mathbb{R} \Sigma}:=(Pin^-(T\mathbb{R} \Sigma),q_{T\mathbb{R} \Sigma})$. Then, we proceed to construct a quasi-quadratic enhancement corresponding to such $Pin^-_2$-structure.

Let  $\alpha:\mathbb{S}^1\to \mathbb{R} \Sigma$ be an immersion and let $\mathcal{N}_\alpha:=
\alpha^*T \mathbb{R} \Sigma / T \mathbb{S}^1$ be the normal bundle of this map.
We consider the short exact sequence of real vector bundles over $\mathbb{S}^1$ determined by $\alpha$

$$0\to T \mathbb{S}^1 \to \alpha^* T \mathbb{R} \Sigma \to \mathcal{N}_\alpha \to 0, \quad (ses)_\alpha.$$

Up to homotopy, one has the splitting $\alpha^*T \mathbb{R} \Sigma=T \mathbb{S}^1\oplus \mathcal{N}_\alpha$.  We first choose the bounding $Spin_1$-structure on $T \mathbb{S}^1$, denoted by $\mathfrak{s}_0$, which corresponds to the disconnected double cover of $\mathbb{S}^1$. Then, on the normal line bundle $\mathcal{N}_\alpha$, there is a canonical $Pin^-_1$-structure, denoted by $\mathfrak{p}_0$, which maps to $0\in \mathbb{Z}_2$ by the natural bijection $\mathcal{P}_1^-(\mathcal{N}_\alpha)\to \mathbb{Z}_2$ (see Lemma 8.2 \cite{solomon2006intersection} or Section 11 \cite{chenzinger}). We say that a given $Pin^-_2$-structure on $\mathbb{R} \Sigma$ is \textit{compatible} with the short exact sequence $(ses)_\alpha$ if $\alpha^*\mathfrak{p}_{\mathbb{R} \Sigma}$ is mapped from $<\mathfrak{s}_0,\mathfrak{p}_0>_\alpha$ via the natural $H^1(\mathbb{S}^1;\mathbb{Z}_2)$-biequivariant map

$$<.,.>_\alpha: \mathcal{SP}_1(T \mathbb{S}^1)\times \mathcal{P}^-_1(\mathcal{N}_\alpha)\to \mathcal{P}_2^-(\alpha^*\mathbb{R} \Sigma).$$

 We define the following function:
\begin{align*}
  s_{\mathfrak{p}_{\mathbb{R} \Sigma}}:\{ \text{immersed circles in }\mathbb{R} \Sigma \} &\to \mathbb{Z}_2\\
     \alpha&\mapsto \begin{cases}
  1, \text{ if } \alpha^*\mathfrak{p}_{\mathbb{R} \Sigma}=<\mathfrak{s}_0,\mathfrak{p}_0>_\alpha; \\
  0, \text{ otherwise. }
  \end{cases}
\end{align*}

\begin{remark}
Since  a loop $\tilde{\alpha}$, which lifts $\alpha$ in the principal $O_2$-bundle $\alpha^*(O(T\mathbb{R} \Sigma))$, can be written as  $r$ ($r\in \mathbb{Z}$) times a generator of $\pi_1(O_2(\mathbb{R}))\cong \mathbb{Z}$, we can also interpreter $s_{\mathfrak{p}_{\mathbb{R} \Sigma}}(\alpha)\in \{0,1\}$ as
\begin{align*}
     s_{\mathfrak{p}_{\mathbb{R} \Sigma}}(\alpha)= 1+r \mod 2.
\end{align*}

It can be shown that the liftings of $\alpha$ in the principal $Pin^-_2$-bundle $\alpha^*(Pin^-(T\mathbb{R} \Sigma))$ are either all loops or all non-closed paths.
\end{remark}
By definition, the function $s_{\mathfrak{p}_{\mathbb{R} \Sigma}}$ is additive on disjoint unions of embedded circles and does not depend on the orientation of $\alpha$.
We can deform an union of immersed circles in $\mathbb{R} \Sigma$ such that all their intersections and self-intersections are transverse.
The rule of smoothing a transverse intersection is shown in Figure \ref{fig-smoothing}.

\begin{figure}[h!]
 \centering
\includegraphics[width=0.4\textwidth]{figures/smoothing.JPG}
\caption{Smoothing a transverse intersection.}
\label{fig-smoothing}
\end{figure}

 After smoothing an union $\bigcup\limits_{i=1}^{k} C_{i}$ of $k$ immersed circles in $\mathbb{R} \Sigma$, we get a disjoint union $\bigsqcup\limits_{i=1}^{k'} C'_{i}$ of $k'$ embedded circles in $\mathbb{R} \Sigma$.
 One has $$\sum\limits_{i=1}^k[C_i]=\sum\limits_{i=1}^{k'}[C_i']\in H_1(\mathbb{R} \Sigma;\mathbb{Z}_2).$$
 Let $\delta (\alpha)$ be the number of self-intersection points of $C=Im(\alpha)$,
 one has
$$k' \equiv k+\sum_{i=1}^k \delta(\alpha_i)+\sum_{i<j}[C_i].[C_j] \mod 2.$$

One has the following proposition.
\begin{proposition} \label{prop-quasi-quadratic.enhancement.def.prop} (Theorem 11.1 \cite{chenzinger})
The function $s_{\mathfrak{p}_{\mathbb{R} \Sigma}}$ induces a quasi-quadratic enhancement  $$s^{\mathfrak{p}_{\mathbb{R} \Sigma}}:H_1(\mathbb{R} \Sigma;\mathbb{Z}_2)\to \mathbb{Z}_2$$ given by
$$s^{\mathfrak{p}_{\mathbb{R} \Sigma}}([\alpha])\equiv s_{\mathfrak{p}_{\mathbb{R} \Sigma}}(\alpha)+\delta(\alpha) \mod 2.$$
\end{proposition}

\begin{remark}
Given any basis of $H_1(\mathbb{R} \Sigma;\mathbb{Z}_2)$, we can make $s^{\mathfrak{p}_{\mathbb{R} \Sigma}}$ take any specified values on the elements of the basis by choosing an
appropriate $\mathfrak{p}_{\mathbb{R} \Sigma}$.
\end{remark}

\subsubsection{Quasi-quadratic enhancements and restricted $Pin^-_2$-structures} \label{subsect-quasi.quad-pin2}

The restriction from $Spin_3$ to $Pin_2^-$ structures is described in the  following proposition.

\begin{proposition}\label{Prop-Spin3.Pin2}

  A $Spin_3$-structure on $SO(T \mathbb{R} X,\mathfrak{o}_{T\mathbb{R} X})$ can be restricted naturally to a $Pin^-_2$-structure on $O(T \mathbb{R} \Sigma)$.

 In particular, if $\mathbb{R} X$ and $\mathbb{R} \Sigma$ are orientable then $SO(T\mathbb{R} \Sigma,\mathfrak{o}_{T \mathbb{R} \Sigma})$ inherits a $Spin_2$-structure from a $Spin_3$-structure on $SO(T\mathbb{R} X,\mathfrak{o}_{T \mathbb{R} X})$.
\end{proposition}
\begin{proof}

It is sufficient to prove that $O(T\mathbb{R} \Sigma)$ can be embedded as a subspace of $SO(T \mathbb{R} X,\mathfrak{o}_{T \mathbb{R} X})|_{\mathbb{R} \Sigma}$. A proposition of Kirby-Taylor (\cite{kirby1990p}) implies a natural bijection
$$\mathcal{P}_2^-(T \mathbb{R} \Sigma)\cong \mathcal{SP}_3(T \mathbb{R} \Sigma \oplus \det T \mathbb{R} \Sigma,\mathfrak{o}_{T \mathbb{R} \Sigma \oplus \det T \mathbb{R} \Sigma}).$$
Let $ \mathcal{N}_{\Sigma/X}$ be the holomorphic normal line bundle of $\Sigma$ in  $X$.
The Riemannian metric on $\mathbb{R} X$ allows us to identify the real bundles $\mathbb{R} \mathcal{N}_{\Sigma/X}$ and $ \mathcal{N}_{\mathbb{R} \Sigma/\mathbb{R} X}$.
By adjunction formula, we have the bundle isomorphism  $\mathcal{N}_{\mathbb{R} \Sigma / \mathbb{R} X} \cong \det T\mathbb{R} \Sigma$. Up to homotopy, this implies the decomposition
$$T\mathbb{R} X= T \mathbb{R} \Sigma \oplus \det T \mathbb{R} \Sigma.$$
Let $(x,(v_1,v_2))\in O(T\mathbb{R} \Sigma)$.
Then one can construct a unique vector $v_3 := v_1 \wedge v_2$ in $\det T \mathbb{R} \Sigma$  such that $(x,(v_1,v_2,v_3))$ forms an element in $SO(T \mathbb{R} X,\mathfrak{o}_{T \mathbb{R} X})|_{\mathbb{R} \Sigma}$, whose orientation is agree with that coming from the $Spin_3$-structure on $SO(T \mathbb{R} X,\mathfrak{o}_{T \mathbb{R} X})$.

\end{proof}

  Let $\mathfrak{s}_{X}:=(Spin (T\mathbb{R} X,\mathfrak{o}_{T\mathbb{R} X}), q_{T \mathbb{R} X})$ be a $Spin_3$-structure on $SO(T \mathbb{R} X,\mathfrak{o}_{T \mathbb{R} X})$.
 We denote by $$\mathfrak{s}_{X|\Sigma}=(Pin^-(X|\Sigma),q_{X|\Sigma})$$ the restricted  $Pin^-_2$-structure to $O(T \mathbb{R} \Sigma)$ from the $Spin_3$-structure $\mathfrak{s}_{X}$ by Proposition \ref{Prop-Spin3.Pin2}.

Recall that we have a natural homomorphism
$$\rho: H_2^{-\tau}(\Sigma;\mathbb{Z}) \to H_1(\mathbb{R} \Sigma;\mathbb{Z}_2), \quad D\mapsto \rho D.$$

Hence, for every $D \in H_2^{-\tau}(\Sigma;\mathbb{Z})$ and every $Pin^-_2$-structure ${\mathfrak{p}_{T\mathbb{R} \Sigma}}$ on $O(T \mathbb{R} \Sigma)$, we can associate a quasi-quadratic enhancement $s^{\mathfrak{p}_{T\mathbb{R} \Sigma}}$ valuating $\rho D$, as described in the previous subsection. In particular, the $Pin^-_2$-structure $\mathfrak{s}_{X|\Sigma}$ on $O(T \mathbb{R} \Sigma)$ implies the following properties on $s^{\mathfrak{s}_{X|\Sigma}}$.
\begin{proposition} \label{prop-quasi-quad.and.monodromy}
Suppose that $\ker (\psi: H_2(\Sigma;\mathbb{Z})\to H_2(X;\mathbb{Z}))\cong \mathbb{Z}$.

Let $S\in H_2^{-\tau}(\Sigma;\mathbb{Z})$ be a generator of $\ker (\psi)$.
One has
\begin{enumerate}
    \item $s^{\mathfrak{s}_{X|\Sigma}}(\rho S)=0.$
    \item Furthermore,  for all $D\in H_2^{-\tau }(\Sigma;\mathbb{Z})$, one has  $$s^{\mathfrak{s}_{X|\Sigma}}(\rho ({D+(D.S)S})) \equiv s^{\mathfrak{s}_{X|\Sigma}}(\rho D) +(D.S) \mod 2.$$
\end{enumerate}

\end{proposition}
\begin{proof}
\begin{enumerate}
    \item     One has $\psi  (S)=0\in H_2^{-\tau}(X;\mathbb{Z})$ and a natural homomorphism
$$\rho: H_2^{-\tau}(X;\mathbb{Z}) \to H_1(\mathbb{R} X;\mathbb{Z}_2), \quad d\mapsto \rho d.$$

Hence,     $\rho \psi (S)=0\in H_1(\mathbb{R} X;\mathbb{Z}_2)$.

    Since $S$ is also the vanishing cycle of a Lefschetz fibration, $\rho \psi (S)=\alpha_*[\mathbb{S}^1]$, where $\alpha: \mathbb{S}^1\to \mathbb{R} X$ is an immersion and $Im(\alpha)$ bounds a $2$-dimensional disk $\mathbb{D}^2$ in $\mathbb{R} X$. Let $$\mathfrak{s}_{\mathbb{D}^2}=(Spin(T \mathbb{D}^2,\mathfrak{o}_{T \mathbb{D}^2}),q_{T \mathbb{D}^2})$$ be the  unique $Spin_2$-structure on $\mathbb{D}^2$. From example 11.3 in \cite{chenzinger}, we deduce that  $\mathfrak{s}_{\mathbb{D}^2}$ is not compatible with the short exact sequence of real vector bundles over $\mathbb{S}^1$:
$$0\to T \mathbb{S}^1 \to \alpha^* T \mathbb{D}^2 \to \mathcal{N}_\alpha \to 0.$$

Hence, we get $s_{\mathfrak{s}_{\mathbb{D}^2}}(\alpha)=0$, and then $s^{\mathfrak{s}_{\mathbb{D}^2}}([\alpha])=0$. The statement follows with a remark that the $Spin_3$-structure over $\mathbb{D}^2$ is induced canonically from the $Spin_2$-one.

    \item
    \textit{Claim (1)} and the properties of a quasi-quadratic enhancement  imply that (with abuse of notation, in this proof, we write $\omega_1(\rho D)$ for the dual pairing $\omega_1(\mathbb{R} \Sigma)\cdot (\rho D)$)
\begin{align*}
    \;&s^{\mathfrak{s}_{X|\Sigma}}(\rho (D+(D.S)S)\\
    = & s^{\mathfrak{s}_{X|\Sigma}}(\rho D+(D.S)\rho S)\\
    = & s^{\mathfrak{s}_{X|\Sigma}}(\rho D)+(D.S)s^{\mathfrak{s}_{X|\Sigma}}(\rho S)+(D.S)(\rho D. \rho S)+(D.S)\omega_1(\rho D)\omega_1(\rho S)\\
    \equiv & s^{\mathfrak{s}_{X|\Sigma}}(\rho D)+0+(D.S)^2+(D.S)(D)^2(-2) \mod 2\\
    \equiv & s^{\mathfrak{s}_{X|\Sigma}}(\rho D)+(D.S)^2 \mod 2\\
    \equiv & s^{\mathfrak{s}_{X|\Sigma}}(\rho D)+(D.S) \mod 2.
\end{align*}
\end{enumerate}
Thus, the proposition follows.
\end{proof}

\subsubsection{Quasi-quadratic enhancements and spinor states}
As in Subsection \ref{subsect-Spinorstates.revisited}, we suppose that $C$ is a real balanced rational curve parametrized by a real balanced immersion $f:(\mathbb{C} P^1,\tau_1)\to (X,\tau)$  such that $f_*[\mathbb{C} P^1]=d\in  H_2^{-\tau}(X;\mathbb{Z})$. We also suppose that $f (\mathbb{C} P^1)\subset \Sigma$, where $\Sigma\in |-\frac{1}{2}K_X|$ is a real non-singular surface and that $C$ represents homology class $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$. Since $f$ is real, we have an immersion $f|_{\mathbb{R} P^1}: \mathbb{R} P^1 \to \mathbb{R} X$  such that $f|_{\mathbb{R} P^1}(\mathbb{R} P^1)\subset \mathbb{R} \Sigma$.

Let $k_d$ and $k_D$ be the associated integers of classes $d$ and $D$ respectively. Recall that $k_D=k_d-1$.

For every $Pin_2^-$-structure $\mathfrak{p}_{T\mathbb{R} \Sigma}$ on $O(T \mathbb{R} \Sigma)$, we define
$$s_{\mathfrak{p}_{T\mathbb{R} \Sigma}}(C):=s_{\mathfrak{p}_{T\mathbb{R} \Sigma}}(f|_{\mathbb{R} P^1}) \in \{0,1\}.$$
One has the number of hyperbolic real nodes $\delta_H(C)$ of $C$ is exactly equal to $\delta(f|_{\mathbb{R} P^1})$. From Section \ref{subsect-Welschinger.surfaces}, we deduce that
$$\delta_H(C)\equiv g(D)+\delta_E(C) \mod 2,$$
 where $\delta_E(C)$ is the number of elliptic real nodes of $C$ and $g(D)=\frac{1}{2}(K_\Sigma.D+D^2)+1$.
 Then, from Proposition \ref{prop-quasi-quadratic.enhancement.def.prop}, we  deduce a quasi-quadratic enhancement
 $$s^{\mathfrak{p}_{T\mathbb{R} \Sigma}}:H_1(\mathbb{R} \Sigma;\mathbb{Z}_2)\to \mathbb{Z}_2$$
 satisfying
 \begin{equation} \label{eq-RelationW2W2}
     s^{\mathfrak{p}_{T\mathbb{R} \Sigma}}(\rho D)= g(D)+\delta_E(C)+s_{\mathfrak{p}_{T\mathbb{R} \Sigma}}(C) \mod 2.
 \end{equation}

The quasi-quadratic enhancement $s^{\mathfrak{p}_{T\mathbb{R} \Sigma}}(\rho D)$ allows us to relate spinor state $sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)$ in  Definition \ref{def-spinor3dim} to Welschinger's sign $s_{\mathbb{R} \Sigma}(C)$ in Definition \ref{def-Welschinger.Inv.Nodes}, as shown in the following proposition.
 \begin{proposition}\label{Prop-RelationW2W3}
Given $\mathfrak{s}_X$  a $Spin_3$-structure on $SO(T\mathbb{R} X,\mathfrak{o}_X)$ and $\mathfrak{s}_{X|\Sigma}$ the restricted $Pin_2^-$-structure over $\mathbb{R} \Sigma$, one has

 \begin{equation} \label{eq-Prop-RelationW2W3}
     sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)=(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}s_{\mathbb{R} \Sigma}(C)
 \end{equation}
 where  $\epsilon$ is the homomorphism defined in Definition \ref{def-epsilon.function}.

 \end{proposition}
 \begin{proof}

At the left hand side of (\ref{eq-Prop-RelationW2W3}), the spinor state $sp_{\mathfrak{s}_X,\mathfrak{o}_X}(C)$ is determined by two parameters: one is the $Spin_3$-structure $\mathfrak{s}_X$ and the other is the isotopy class of the real line subbundle  $L \subset \mathbb{R} \mathcal{N}_f$.

On the one hand, the $Spin_3$-structure $\mathfrak{s}_X$ induces a canonical $Pin^-_2$-structure $\mathfrak{s}_{X|\Sigma}$ on $O(T \mathbb{R} \Sigma)$ and hence its corresponding quasi-quadratic enhancement $s^{\mathfrak{s}_{X|\Sigma}}$. As a consequence, this enhancement contributes the difference of $(-1)^{g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}$ in Equation (\ref{eq-Prop-RelationW2W3}) (see also Equation (\ref{eq-RelationW2W2})).

On the other hand, recall that the holomorphic normal line bundle $\mathcal{N}'_f$ is of the same degree as the one, whose real part is $L$, and also is a subbundle of the holomorphic normal plane bundle $\mathcal{N}_f$.
The map $\epsilon$ (Definition \ref{def-epsilon.function}) allows us to compare the isotopy class of two real line subbundles $\mathbb{R} \mathcal{N}_f'$ and $L$ in $\mathbb{R} \mathcal{N}_f$. Hence, the difference of sign by $(-1)^{\epsilon(D)}$ in Equation (\ref{eq-Prop-RelationW2W3}) is deduced.

Therefore, Equation (\ref{eq-Prop-RelationW2W3}) follows.

 \end{proof}

\section{Proofs of main results}
\label{Sect-proofs}

\subsection{Proof of Theorem \ref{Theorem1-GW}}\label{Sect-Proof.GW}
\begin{remark}
By Proposition \ref{Prop-why.degree.is.678} and  Proposition \ref{Prop-Kollar}, we can degenerate the problem of computing genus-$0$ Gromov-Witten invariants in del Pezzo varieties of dimension three into that of dimension two.
\end{remark}
Remind about our hypotheses and notations:
\begin{itemize}
    \item $X$ is a del Pezzo variety of dimension three  such that $\ker(\psi) \cong \mathbb{Z}$, where $\Sigma\in |-\frac{1}{2}K_X|$ is any non-singular surface and $\psi:H_2(\Sigma;\mathbb{Z})\twoheadrightarrow H_2(X;\mathbb{Z})$ induced by the inclusion $\Sigma\hookrightarrow X$;
    \item $\mathcal{Q}$ is a pencil of surfaces  in the linear system $|-\frac{1}{2}K_X|$ whose base locus is $E$;
    \item $E\subset X$ is an elliptic curve realizing the homology class $\frac{1}{4}K^2_X$.
\end{itemize}

Let $d\in H_2(X;\mathbb{Z})$ and let $k_d$ be its associated integer.  Recall that $\mathcal{C}(\underline{x}_d)$ denotes the set of connected algebraic curves of arithmetic genus $0$ representing the given homology class $d$ and passing through  a generic configuration $\underline{x}_d$ of $k_d$ points in $E$ (see Proposition \ref{Prop-Kollar}).

  Let ${U}_{k_d}\subset E^{k_d}$ be as in Proposition \ref{Prop.Modulispace2}. Choose  $\underline{x}_d\in {U}_{k_d}$ and a subconfiguration $\underline{y}_D\subset \underline{x}_d$ of $k_D=k_d-1$ points in $\underline{x}_d$.
  By Proposition \ref{Prop-Kollar} and by the  Riemann-Roch theorem, every irreducible rational curve, which is contained in a non-singular surface of the pencil $\mathcal{Q}$ and  passes through  $\underline{y}_D$,
  intersects the elliptic curve $E$ at the $k_d^{th}$-point $x_{k_d}$, which is exactly $\underline{x}_d\setminus \underline{y}_D$.  Let $\Sigma$ be a surface of $\mathcal{Q}$ which contains a curve in $\mathcal{C}(\underline{x}_d)$.    As mentioned in Section \ref{Sect-Monodromy}, up to monodromy, we can identify canonically the second homology group $H_2(\Sigma_t;\mathbb{Z})$ with $H_2(\Sigma;\mathbb{Z})$  for every non-singular surface $\Sigma_t$ in the pencil $\mathcal{Q}$.
    Let $S$ be a generator of $\ker(\psi)$. Note that $S$ can be identified with the vanishing cycle in the monodromy transformation $T(D)=D+(D.S)S$.  Let $\overline{D}\in H_2(\Sigma;\mathbb{Z})/_ {D\sim T(D)}$.     The surjectivity of $\psi$ implies a surjective map $\overline{\psi}:  H_2(\Sigma;\mathbb{Z})/_ {D\sim T(D)}\twoheadrightarrow H_2(X;\mathbb{Z})$.

    By Proposition \ref{Prop-Kollar}, we deduce the following equality
   $$|\mathcal{C}(\underline{x}_d)|=\sum_{\overline{D}\in \overline{\psi}^{-1}(d)}\sum_{\substack{\Sigma_t\in\mathcal{Q}:\\
                 \overline{D}|_{E,\Sigma_t}\sim [\underline{y}_D\sqcup \{x_{k_d}\}]
                 }}|\{C\in \mathcal{C}(\underline{x}_d): C \subset \Sigma_t \text{ and } \overline{[C]}=\overline{D} \}|\;\;\; (\star)$$

By Proposition \ref{Prop.Modulispace2}, we have $$|\mathcal{C}(\underline{x}_d)|=GW_X(d).$$

 Moreover, the total evaluation map  $ev_{X}$ is regular at every $[(\mathbb{C} P^1;p_1,\ldots,p_{k_d});f]$ such that $$f(\{p_1,\ldots,p_{k_d}\})=\underline{x}_d,$$ then every curve in $\mathcal{C}({\underline{x}_d})$  is parametrized by an algebraic immersion $f:\mathbb{C}P^1\to X$.

Remark that there is a symmetry property of genus-$0$ Gromov-Witten invariants via monodromy transformation. That is for every $D\in H_2(\Sigma;\mathbb{Z})$, one has
  $$GW_{\Sigma}(D)=GW_{\Sigma}(D+(D.S)S).$$

We denote by $$GW_{\Sigma_t}(\overline{D}):= GW_{\Sigma_t}(D)=GW_{\Sigma_t}(T(D)).$$ Hence, the cardinality of the set $\{C\in \mathcal{C}(\underline{x}_d): C \subset \Sigma_t \text{ and } \overline{[C]}=\overline{D} \}$ is exactly equal to $GW_{\Sigma_t}({D})$, and hence is equal to $GW_{\Sigma_t}(\overline{D})$  (see Proposition \ref{Prop.Modulispace1} and Corollary \ref{Corollary.Modulispace1}).

    Moreover, $\overline{D}|_{E,\Sigma_t}=\overline{D|_{E,\Sigma_t}}\in Pic_{D.E} (E)/_{x\sim \phi(\overline{D})-x}$. Then, the cardinality of the set $\{\Sigma_t\in\mathcal{Q}: \overline{D}|_{E,\Sigma_t}\sim [\underline{y}_D\sqcup \{x_{k_d}\}]\}$ is exactly the degree of  the map  $\phi_{\overline{D}}$ (see  Proposition \ref{Prop-degree.case.complex}). Since we are considering the cases where a generator of $\ker (\psi)$ can be identified with the vanishing cycle (Lemma \ref{Lemma-Vanishing.Cycle}), the degree of the corresponding map  $\phi_{\overline{D}}$ is exactly $(D.S)^2$. Therefore, one has
    \begin{align*}
        GW_X(d) &=\sum_{\overline{D}\in \overline{\psi}^{-1}(d)} GW_{\Sigma_t}(\overline{D}) \times  |\{\Sigma_t\in\mathcal{Q}: \overline{D}|_{E,\Sigma_t}\sim [\underline{y}_D\sqcup \{x_{k_d}\}]\}| \\
       &=\sum_{\overline{D}\in \overline{\psi}^{-1}(d)}  (D.S)^2 GW_{\Sigma_t}(\overline{D}) \;\;\; (\star\star).
    \end{align*}

By an easy calculation, one has $(D.S)^2=(T(D).S)^2$. Then, the equality $(\star\star)$ becomes

$$
    GW_X(d)=\sum_{\substack{\{D,T(D)\}\subset H_2(\Sigma;\mathbb{Z}):\\D\in \psi^{-1}(d)}
      }(D.S)^2 GW_{\Sigma}(D).$$

Thus,
          $$GW_X(d)=\frac{1}{2}\sum_{\substack{D\in H_2(\Sigma;\mathbb{Z}):\\
          D\in \psi^{-1}(d)}}(D.S)^2 GW_{\Sigma}(D).$$

\subsection{Proof of Theorem \ref{Theorem2-W}}
 \begin{remark}
By Proposition \ref{Prop-Kollar} and Proposition \ref{prop-tau.vs.mu}, we can degenerate the problem of computing genus-$0$ Welschinger invariants in real del Pezzo varieties of dimension three into that of dimension two. In addition to the Gromov-Witten case, we have to solve a \textit{sign problem} which comes from the difference between two definitions of Welschinger's sign for curves in those varieties.
\end{remark}
Remind about our hypotheses and notations:
\begin{itemize}
    \item $(X,\tau)$ is a real del Pezzo variety of dimension three such that $\ker(\psi) \cong \mathbb{Z}$, where $\Sigma \in |-\frac{1}{2}K_X|$ is any non-singular surface and $\psi: H_2(\Sigma;\mathbb{Z})\twoheadrightarrow H_2(X;\mathbb{Z})$ is induced by the inclusion $\Sigma\hookrightarrow X$;
    \item  $\mathcal{Q}$ is a real pencil of surfaces, whose real part is $\mathbb{R} \mathcal{Q}\cong \mathbb{R} P^1$,  in the linear system $|-\frac{1}{2}K_X|$, whose base locus is $E$;
    \item  $E\subset X$ is a real elliptic curve, whose real part is $\mathbb{R} E\neq \emptyset$,  realizing the homology class $\frac{1}{4}K^2_X$.
\end{itemize}
Let $d\in H_2^{-\tau}(X;\mathbb{Z})$ and let $k_d$ be its associated integer.
Let $\mathcal{R}(\underline{x}_d)$ be the set of real irreducible rational curves representing the homology class $d$ and passing through a real generic configuration $\underline{x}_d$ of $k_d$ points in $E$ consisting of $l$ pairs of complex conjugated points. Suppose that $\underline{x}_d$ has at least one real point.

Let $U_{k_d}\subset E^{k_d}$ be as in Proposition \ref{Prop.Modulispace2}. Choose $\underline{x}_d\in U_{k_d}$ and a subconfiguration $\underline{y}_D\subset \underline{x}_d$ of $k_D=k_d-1$ points in $\underline{x}_d$. By Proposition \ref{Prop.Modulispace2} and Lemma \ref{lem-surfaces.in.real.pencil}, there exists $(\Sigma,\tau|_{\Sigma}) \in \mathbb{R} \mathcal{Q}$ a real non-singular del Pezzo surface containing a real curve in $\mathcal{R}(\underline{x}_d)$.

Let $S$ be a generator of $\ker(\psi)$. By Proposition \ref{prop-tau.vs.mu}, in such real surface the vanishing cycle realizes a $\tau|_{\Sigma}$-anti-invariant class.
 An analogous remark about monodromy transformation as in the proof of Theorem \ref{Theorem1-GW} is also applied. Let $\overline{D}\in H_2^{-\tau}(\Sigma;\mathbb{Z})/_{D \sim T(D)}$. One consider the surjective map $\overline{\psi}: H_2^{-\tau}(\Sigma;\mathbb{Z})/_{D \sim T(D)}\twoheadrightarrow H_2^{-\tau}(X;\mathbb{Z})$.

Remark also that there is a symmetry property of genus-$0$ Welschinger invariants via monodromy transformation. That is for any $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$, one has
  $$W_{\mathbb{R} \Sigma}(D,l)=W_{\mathbb{R} \Sigma}(D+(D.S)S,l).$$
  We denoted by $$W_{\mathbb{R} \Sigma}(\overline{D},l):=W_{\mathbb{R} \Sigma}(D,l)=W_{\mathbb{R} \Sigma}(T(D),l).$$

From Proposition \ref{Prop-RelationW2W3}, we deduce that: for every $D\in H_2^{-\tau}(\Sigma;\mathbb{Z})$ such that $\psi(D)=d$, the contribution to $W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)$ of elements of $\mathcal{R}(\underline{x}_d)$ which are contained in $\Sigma$ is

$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}W_{\mathbb{R} \Sigma}(\overline{D},l).$$

Suppose that $D.S$ is odd. From Corollary \ref{coro-degree.case.real}, we deduce that there are exactly $|D.S|$ such real surfaces $(\Sigma,\tau|_{\Sigma})$ in $\mathbb{R} \mathcal{Q}$.

 Therefore, we have
$$W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)=\sum\limits_{\overline{D}\in \overline{\psi}^{-1}(d)}(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)} |D.S| W_{\mathbb{R} \Sigma}(\overline{D},l).$$

Moreover, we also have
\begin{itemize}
         \item[$\bullet$] $\epsilon(T(D))=\epsilon(D)+1$
 (see Proposition  \ref{prop-function.epsilon1});
 \item[$\bullet$] $s^{\mathfrak{s}_{X|\Sigma}}(\rho {T(D)}) \equiv s^{\mathfrak{s}_{X|\Sigma}}(\rho D) +(D.S) \mod 2$. Thus, $s^{\mathfrak{s}_{X|\Sigma}}(\rho {T(D)}) \equiv s^{\mathfrak{s}_{X|\Sigma}}(\rho D) +1 \mod 2$  since $D.S$ is odd
 (see Proposition \ref{prop-quasi-quad.and.monodromy}.2);
     \item[$\bullet$] $g(T(D))=g(D)$ by a straightforward calculation;
     \item[$\bullet$] $T(D).S=-D.S$ by another straightforward calculation.
\end{itemize}

Thus,
\begin{equation*}
\begin{aligned}
      &(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}|D.S|W_{\mathbb{R} \Sigma}(D,l)\\
      &+(-1)^{\epsilon(T(D))+g(T(D))+s^{\mathfrak{s}_{X|\Sigma}}(\rho T(D))} |T(D).S|W_{\mathbb{R} \Sigma}(T(D),l)\\
       = &\Big((-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}    +(-1)^{\epsilon(D)+1+g(D)+s_{\mathfrak{s}_{X|\Sigma}}(\rho D)+1} \Big) |D.S|W_{\mathbb{R} \Sigma}(D,l)\\
    =&2 \times (-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)}|D.S|W_{\mathbb{R} \Sigma}(D,l).
\end{aligned}
\end{equation*}

Then, the equality

$$
  W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)=\sum_{\substack{\{D,T(D)\}\subset H_2^{-\tau}(\Sigma;\mathbb{Z}):\\D\in \psi^{-1}(d)}
      }(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)} |D.S| W_{\mathbb{R} \Sigma}(D,l).$$

     can be written as
$$W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)=\frac{1}{2}\sum_{\substack{D\in H_2^{-\tau}(\Sigma;\mathbb{Z}):\\
          D\in \psi^{-1}(d)}}(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{X|\Sigma}}(\rho D)} |D.S| W_{\mathbb{R} \Sigma}(D,l).$$

If $D.S$ is even then, by symmetry property we deduce that $W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)=0$.
\subsection{Proof of Theorem \ref{Theorem3-W}}
Firstly, from the proof of Theorem \ref{Theorem2-W}, we obtain directly the following consequence.
\begin{corollary} \label{coro-Theorem2-W}
If $d\in H_2^{-\tau}(X;\mathbb{Z})$ such that $D.S$ is even, where $D\in \psi^{-1}(d)$, then  the genus-$0$ Welschinger invariants  $W_{\mathbb{R} X}^{\mathfrak{s}_X,\mathfrak{o}_X}(d,l)$ vanish for any $Spin_3$-structure $\mathfrak{s}_X$ on $SO(T \mathbb{R} X,\mathfrak{o}_X)$.
\end{corollary}

Let $\phi(\overline{L},\overline{D})\in Pic_{|D.E-|D.S|L.E|}(E)$ be a divisor as in Lemma \ref{lem-constant.divisor.L.D}.

Proposition \ref{prop-corresponding.surface.picE} and Corollary \ref{coro-degree.case.real}.c) imply that if $\mathbb{R} E$ is disconnected and if  moreover the divisor of real configurations of points $\underline{x}_d$ and the divisor $\phi(\overline{L},\overline{D})$ do not belong to the same component of $\mathbb{R} E$, then there is no real irreducible rational curve of class $d$ passing through $\underline{x}_d$.

A choice of such a configuration $\underline{x}_d$ is described as follows.
Suppose that the divisor $\phi(\overline{L},\overline{D})\in \mathbb{S}^1_0$ is in the pointed component of $\mathbb{R} E$, then we choose the configuration $\underline{x}_d$ such that it has an odd number of real points on the non-pointed component of $\mathbb{R} E$.
If the divisor $\phi(\overline{L},\overline{D})\in \mathbb{S}^1_1$ is in the non-pointed component of $\mathbb{R} E$, then we choose the configuration $\underline{x}_d$ such that it has at least one real point and that an even  number (possible be zero) of its real points is on the non-pointed component of $\mathbb{R} E$. Moreover, in both cases, such real configurations of points are contained in a semi-algebraic open set in $\mathbb{R} (Sym^{k_d}(X))$ the space of real points of the symmetric power of $X$. Indeed, this follows by a natural extension for the case $X=\mathbb{C} P^3$ (Theorem 10 in \cite{kollar2015examples}).

Hence, Theorem \ref{Theorem3-W} follows.
\section{Applications}
\label{applications}
\subsection{Applications of Theorem \ref{Theorem1-GW}} \label{sect-applications.GW}

\subsubsection{Three-dimensional projective space}
One has a non-singular element $\Sigma\in |-\frac{1}{2}K_{\mathbb{C}P^3}|$ is a non-singular quadric surface in $\mathbb{C}P^3$ and $\Sigma$ is isomorphic to $\mathbb{C}P^1\times \mathbb{C}P^1$.

    We identify $H_2(\Sigma;\mathbb{Z})$ with  $\mathbb{Z}^2$ by considering the standard basis $(L_1,L_2)$ of $H_2(\Sigma;\mathbb{Z})$ given by
\begin{itemize}
    \item[$\bullet$] $L_1=[\mathbb{C} P^1\times\{p\}]$,
    \item[$\bullet$] $L_2=[\{q\}\times \mathbb{C} P^1]$,
\end{itemize}
 where $p,q$ are points in $\mathbb{C} P^1$. Then, the class $a L_1+b L_2\in H_2(\Sigma;\mathbb{Z})$ can be identified with the pair $(a,b)\in \mathbb{Z}^2$.

We identify $H_2(\mathbb{C} P^3;\mathbb{Z})$ with  $\mathbb{Z}$ by considering the basis  $\{H\}$ of $H_2(\mathbb{C} P^3;\mathbb{Z})$  which is the class of a line in $\mathbb{C} P^3$.

\begin{theorem} \label{Theorem-GW.in.deg.8}
For every positive integer $d$, one has
\begin{align*}
        GW_{\mathbb{C}P^3}(d) &=\frac{1}{2}\sum_{a\in \mathbb{Z}}(d-2a)^2 GW_{\mathbb{C}P^1\times \mathbb{C}P^1}(a,d-a).
        \end{align*}
\end{theorem}
This formula is in agreement with Theorem 2 in \cite{brugalle2016pencils}.
\begin{proof}
With the above identifications and the natural inclusion of a quadric surface into three-dimensional projective space, the corresponding surjective map $\psi$ is given by:
    \begin{align*}
        \psi:H_2(\Sigma;\mathbb{Z})&\longrightarrow H_2(\mathbb{C} P^3;\mathbb{Z})\\
        (a,b)&\longmapsto a+b.
    \end{align*}

This implies that $\ker(\psi)=\mathbb{Z}(1,-1)$, and that $(D.S)^2=(d-2a)^2$ for any $D=(a,d-a)\in \psi^{-1}(d).$

Hence, the statement follows by Theorem \ref{Theorem1-GW}.
\end{proof}

\subsubsection{Three-dimensional projective space blown-up at a point}
   Let $X=\mathbb{C}P^3\sharp \overline{\mathbb{C}P^3}$. Let us denote the blowing-up point by $x$. We have seen that a non-singular surface $\Sigma \in |-\frac{1}{2}K_X|$ is a non-singular del Pezzo surface of degree $7$, and moreover $\Sigma$ is isomorphic to the blow-up of a quadric surface at $x$. One can write $\Sigma\cong \mathbb{C}P^1\times \mathbb{C}P^1 \sharp \overline{\mathbb{C}P^2}$.

We identify $H_2(\Sigma;\mathbb{Z})$ with  $\mathbb{Z}^3$ by considering a basis $(L_1,L_2;E)$ of $H_2(\Sigma;\mathbb{Z})$ given by
\begin{itemize}
    \item[$\bullet$] $L_1=[\mathbb{C} P^1\times\{p\}]$,
    \item[$\bullet$] $L_2=[\{q\}\times \mathbb{C} P^1]$,
    \item[$\bullet$]  $E$ is the exceptional divisor class
\end{itemize}
 where $p,q$ are points in $\mathbb{C} P^1$. Then, the class $a L_1+b L_2-kE\in H_2(\Sigma;\mathbb{Z})$ can be identified with the triplet $(a,b;k)\in \mathbb{Z}^3$.

     We identify $H_2(X;\mathbb{Z})$ with  $\mathbb{Z}^2$ by considering a basis $(H;E)$ of $H_2(X;\mathbb{Z})$ given by
\begin{itemize}
    \item[$\bullet$] $H$ is the class of a line in $\mathbb{C}P^3$,
    \item[$\bullet$]  $E$ is the class of a line in the exceptional divisor.
\end{itemize}

Then, the class $d H -kE \in H_2(X;\mathbb{Z})$ can be identified with the point $(d;k)\in \mathbb{Z}^2$.
    \begin{theorem} \label{Theorem-GW.in.deg.7}
    For every positive integer $d$, every non-negative integer $k$ such that $k\leq d$, one has

$$GW_{\mathbb{C}P^3\sharp \overline{\mathbb{C}P^3}}(d;k)
        = \frac{1}{2}\sum_{a\in \mathbb{Z}}(d-2a)^2 GW_{\mathbb{C}P^1\times \mathbb{C}P^1\sharp \overline{\mathbb{C}P^2}}(a,d-a;k).$$
    \end{theorem}
 This formula is in agreement with Theorem 1.2 in \cite{ding2019remark}.

 \begin{proof}

With the above identifications and the natural inclusion of a blown-up quadric surface into blown-up projective space,  the corresponding surjective map $\psi$ is given by:
    \begin{align*}
        \psi:H_2(\Sigma;\mathbb{Z})&\longrightarrow H_2(X;\mathbb{Z})\\
        (a,b;k)&\longmapsto (a+b;k).
    \end{align*}

    This implies that     $\ker(\psi)=\mathbb{Z}(1,-1;0)$, and that $(D.S)^2=(d-2a)^2$ for any $$D=(a,d-a;k)\in \psi^{-1}(d;k).$$

    Hence, the statement follows by Theorem \ref{Theorem1-GW}.
 \end{proof}
  \subsubsection{Threefold product of the projective line} \label{Subsect-exam.deg.6.GW}
 Let $X=\mathbb{C} P^1\times \mathbb{C} P^1 \times \mathbb{C} P^1$.  We have seen that a non-singular surface $\Sigma\in |-\frac{1}{2}K_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}|$, i.e. a non-singular surface of tridegree $(1,1,1)$ in the variety $\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1$, is a non-singular del Pezzo surface of degree $6$.

It can be shown that a non-singular $(1,1,1)$-surface $\Sigma$  in $\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1$ is isomorphic to the blow-up of a quadric surface at two distinct points, denoted by $x_1$ and $x_2$. One can write $\Sigma\cong \mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2$.

We identify $H_2(\Sigma;\mathbb{Z})$ with  $\mathbb{Z}^4$ by considering a basis $(L_1,L_2;E_1,E_2)$ of $H_2(\Sigma;\mathbb{Z})$ given by
\begin{itemize}
    \item[$\bullet$] $L_1=[\mathbb{C} P^1\times\{p\}]$,
    \item[$\bullet$] $L_2=[\{q\}\times \mathbb{C} P^1]$,
    \item[$\bullet$]  $E_1$ is the exceptional divisor class over $x_1$,
     \item[$\bullet$]  $E_2$ is the exceptional divisor class over $x_2$
\end{itemize}
 where $p,q$ are points in $\mathbb{C} P^1$.
 Then, the class $a L_1+b L_2-\alpha E_1 - \beta E_2\in H_2(\Sigma;\mathbb{Z})$ can be identified with the quadruple $(a,b;\alpha,\beta)\in \mathbb{Z}^4$.

 We identify $H_2(X;\mathbb{Z})$ with  $\mathbb{Z}^3$ by considering a basis $(M_1,M_2,M_3)$ of $H_2(X;\mathbb{Z})$ given by
\begin{itemize}
    \item[$\bullet$] $M_1=[\mathbb{C} P^1\times\{p_1\}\times\{p_2\}]$,
    \item[$\bullet$] $M_2=[\{q_1\}\times \mathbb{C} P^1\times \{q_2\}]$,
    \item[$\bullet$] $M_3=[\{r_1\}\times \{r_2\}\times \mathbb{C} P^1]$
\end{itemize}
 where $p_i,q_i,r_i\in \mathbb{C} P^1$ ($i \in \{1,2\}$). Then,  the class $a_1 M_1 +a_2 M_2+a_3 M_3\in H_2(X;\mathbb{Z})$ can be identified with the point $(a_1,a_2,a_3)\in \mathbb{Z}^3$.

 \begin{theorem}\label{Theorem-GW.in.deg.6}
 For every triplet of non-negative integers $(a, b, c)$, one has
 \begin{equation}
      \begin{aligned}
      & GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(a,b,c)\\
     &=\frac{1}{2}\sum_{\alpha \in \mathbb{Z}}(a+b-c-2\alpha)^2 GW_{\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2}(a,b;\alpha, a+b-c-\alpha)
 \end{aligned}
 \label{Eq:GW.in.deg.6}
 \end{equation}
 \end{theorem}
\begin{proof}

With the above identifications, one can show that the surjective map $\psi$ induced from the inclusion of $\Sigma$ in $X$ is given by:
    \begin{align*}
        \psi:H_2(\Sigma;\mathbb{Z})&\longrightarrow H_2(X;\mathbb{Z})\\
       (a,b;\alpha,\beta)&\longmapsto (a,b,a+b-\alpha-\beta)
    \end{align*}

    This implies that    $\ker(\psi)=\mathbb{Z}(0,0;1,-1)$, and that $(D.S)^2=(a+b-c-2 \alpha)^2$ for any $$D=(a,b;\alpha,a+b-c-\alpha)\in \psi^{-1}(a,b,c).$$

 Hence, the statement follows by Theorem \ref{Theorem1-GW}.
\end{proof}

\begin{proposition} \label{Prop-vanishing.GW}
One has $ GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(a,b,c)=0$ if $a+b+c>1$ and either one of the following cases holds:
    \begin{itemize}
        \item[$\bullet$] $a\geq b+c$,
        \item[$\bullet$] $b\geq c+a$,
        \item[$\bullet$] $c\geq a+b$.
    \end{itemize}
\end{proposition}
\begin{proof}
We first compare genus-$0$ Gromov-Witten invariants of  $\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2$ and that of $\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2$.
\begin{remark}
By L. Gottsche and R. Pandharipande  \cite{gottsche1996quantum}, we can compute the genus-$0$ Gromov-Witten invariants of $\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2$ by recursive formulas.
\end{remark}
We identify  $H_2(\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2;\mathbb{Z})$ with $\mathbb{Z}^4$ by considering a basis $(L;F_1,F_2,F_3)$ of  $H_2(\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2;\mathbb{Z})$  given by
\begin{itemize}
    \item[$\bullet$] $L$ is the class of a line in $\mathbb{C} P^2$,
    \item[$\bullet$]  $F_1$ is the exceptional divisor class over $x_1$,
     \item[$\bullet$]  $F_2$ is the exceptional divisor class over $x_2$,
     \item[$\bullet$] $F_3$ is the exceptional divisor class over $x_3$,
\end{itemize}
 where $x_1,x_2,x_3$ are points in $\mathbb{C} P^2$.

We consider the map $ \kappa_*: H_2(\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2;\mathbb{Z}) \to H_2(\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2;\mathbb{Z}) $ sending
\begin{itemize}
    \item[$\bullet$] $L_1$ to $L-F_1$,
    \item[$\bullet$]  $L_2$ to $L-F_2$,
     \item[$\bullet$]  $E_1$ to $L-F_1-F_2$,
     \item[$\bullet$] $E_2$ to $F_3$.
\end{itemize}

Hence, one can deduce that $\kappa_*$ sends the class $(a,b;\alpha,\beta)$ to $(a+b-\alpha;a-\alpha,b-\alpha,\beta)$. One has the following equality
\begin{equation}\label{Eq:GW.P1P1.and.P2}
    GW_{\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2}(a,b;\alpha, \beta)=GW_{\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2}(a+b-\alpha;a-\alpha, b-\alpha,\beta).
\end{equation}

\begin{remark}
Theorem \ref{Theorem1-GW} implies that if either $|D.S|=0$ or $GW_\Sigma(D)=0$ for all $D\in \psi^{-1}(d)$ then $GW_X(d)$ vanishes.
\end{remark}

In particular, we have
\begin{itemize}
    \item if $|D.S|=0$, i.e. $|(a,b;\alpha, a+b-c-\alpha)(0,0;1,-1)|=0$ or equivalently $a+b-c-\alpha=\alpha$, then $2\alpha={a+b-c}$ and  $D= (a,b;\alpha,\alpha) \in H_2(\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2;\mathbb{Z})$;
    \item if $GW_\Sigma(D)=GW_\Sigma(T(D))=0$, then Equation (\ref{Eq:GW.P1P1.and.P2}) implies that
 \begin{equation*}
     \begin{cases}
    GW_{\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2}(a,b;\alpha, a+b-c-\alpha)=0\\
    GW_{\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2}(a,b; a+b-c-\alpha,\alpha)=0
    \end{cases}
 \end{equation*}

    and hence
       \begin{equation} \label{Eq:GW.P2.blowup}
        \begin{cases}
    GW_{\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2}(a+b-\alpha;a-\alpha, b-\alpha,a+b-c-\alpha)=0\\
    GW_{\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2}(c+\alpha;c-b+\alpha, c-a+\alpha,\alpha)=0.
    \end{cases}
    \end{equation}

    Moreover, it is known that $GW_{\mathbb{C} P^2 \sharp 3 \overline\mathbb{C}P^2}(d;a_1,a_2,a_3)=0$ if there exists a couple $a_i,a_j$ such that $a_i+a_j>d$, unless $(d;a_1,a_2,a_3)\in \{(1;1,1,0),(1;1,0,1),(1;0,1,1)\}$ (see  \cite{gottsche1996quantum}).

      Therefore, Equations (\ref{Eq:GW.P2.blowup}) holds if $\alpha$ satisfies
        \begin{equation*}
     \begin{cases}
   \alpha < max \{0,b-c,a-c\}, and\\
    \alpha > min \{a,b,a+b-c\}.
    \end{cases}
 \end{equation*}
\end{itemize}
Together with the symmetrical property
$$ GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(a,b,c)=GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(\sigma(a),\sigma(b),\sigma(c))$$
where $\sigma$ is any permutation of the set $\{a,b,c\}$, the proposition follows.
\end{proof}

    In Table \ref{Tab:GW.in.deg.6}, we show the first non-vanishing genus-$0$ Gromov-Witten invariants $GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(a,b,c)$, i.e. either $a+b+c=1$ or  $a<b+c$,
    where  $4\geq a\geq b\geq c$ and $a+b+c$ increases up to $12$.
\begin{table}[h!]
\centering
\begin{tabular}{ |c|c||c|c|c|}
 \hline
$d$ & $GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(d)$ & $D$ & $|D.S|$ & $ GW_{\mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2}(D)$\\
$(a,b,c)$ &  & $(a,b;\alpha, a+b-c-\alpha)$ &  & \\
 \hline
  \hline
     $(1,0,0)^*$ & 1 & (1,0;0,1)&1&1 \\
 \hline
    (1,1,1) & 1 & (1,1;0,1)&1&1 \\

 \hline
     (2,2,1) & 1 & (2,2;0,3)&3&0 \\
      &  & (2,2;1,2)&1&1 \\

 \hline
     (2,2,2) & 4 & (2,2;0,2)&2&1 \\

 \hline
     (3,2,2) & 12 & (3,2;0,3)&3&0 \\
      &  & (3,2;1,2)&1&12 \\
 \hline
     (3,3,1) & 1 & (3,3;0,5)&5& 0\\
       &  & (3,3;1,4)&3& 0\\
       &  & (3,3;2,3)&1& 1\\
 \hline
     (3,3,2) & 48 & (3,3;0,4)& 4&0\\
       &  & (3,3;1,3)&2& 12\\
 \hline
     (3,3,3) & 728 & (3,3;0,3)&3& 12\\
       &  & (3,3;1,2)&1& 620\\

  \hline
     (4,3,2) & 96 & (4,3;0,5)&5& 0\\
       &  & (4,3;1,4)&3& 0\\
        &  & (4,3;2,3)&1& 96\\
 \hline
      (4,3,3) & 2480 & (4,3;0,4)&4& 0\\
       &  & (4,3;1,3)&2& 620\\
 \hline
      (4,4,1) & 1 & (4,4;0,7)&7& 0\\
       &  & (4,4;1,6)&5& 0\\
        &  & (4,4;2,5)&3& 0\\
        &  & (4,4;3,4)&1& 1\\
 \hline
    (4,4,2) & 384 & (4,4;0,6)&6& 0\\
       &  & (4,4;1,5)&4& 0\\
        &  & (4,4;2,4)&2& 96\\
 \hline
     (4,4,3) & 23712 & (4,4;0,5)&5& 0\\
       &  & (4,4;1,4)&3& 620\\
        &  & (4,4;2,3)&1& 18132\\
 \hline
     (4,4,4) & 359136 & (4,4;0,4)&4& 620\\
       &  & (4,4;1,3)&2& 87304\\
 \hline
\end{tabular}
\caption{Genus-$0$ Gromov-Witten invariants of $\mathbb{C} P^1 \times \mathbb{C} P^1 \times \mathbb{C} P^1$ via Formula (\ref{Eq:GW.in.deg.6}) with $a+b+c \leq 12$.}
\label{Tab:GW.in.deg.6}
\end{table}
\begin{remark}
The values of $ GW_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}(a,b,c)$ in this computation are compatible with that given by X. Chen and A. Zinger using Solomon's WDVV-type
relations in \cite{chen2021wdvv}.
\end{remark}

\subsection{Applications of Theorem \ref{Theorem2-W} and Theorem \ref{Theorem3-W}} \label{sect-applications.W}
In this section, we use the notation
$$\tau_n : \mathbb{C} P^n \to \mathbb{C} P^n , \quad \tau_n ([z_0:z_1:\ldots :z_n])=[\overline{z_0}:\overline{z_1}:\ldots :\overline{z_n}]$$
for the standard conjugation on the projective space $\mathbb{C} P^n$ ($n\geq 1$).

\subsubsection{Real three-dimensional projective space $X=(\mathbb{C} P^3,\tau_3$).}

 We consider the non-singular real quadric surface $\Sigma$ in $X$ with non-empty real part and real Lagrangian sphere realizing a $\tau_3|_{\Sigma}$-anti-invariant class. In other words,  $\Sigma$  is isomorphic to $(\mathbb{C}P^1\times \mathbb{C}P^1, \tau_3|_{\Sigma})$, where $$\tau_3|_{\Sigma}(p,q)= (\tau_1(p),\tau_1(q))$$ for all $(p,q)\in \mathbb{C}P^1\times \mathbb{C}P^1$.

Hence, the fixed loci of $\tau_3$ and $\tau_3|_{\Sigma}$ are real projective space $\mathbb{R} P^3$ and $2$-torus $\mathbb{R} P^1\times \mathbb{R} P^1$ respectively.

The group $H_2^{-\tau_3}(\mathbb{C} P^3;\mathbb{Z})$ is freely generated by $L_{\mathbb{C} P^3}$ the homology class of a line embedded in $\mathbb{C} P^3$. The group $H_1(\mathbb{R} P^3;\mathbb{Z}_2)$ is generated by $\rho L_{\mathbb{C} P^3}$.

We choose a basis for the group $H_2^{-\tau_3 }(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})$ as
$$(L_1,L_2)=([\mathbb{C} P^1\times \{p\}],[\{q\}\times \mathbb{C} P^1]), \quad (p,q \in \mathbb{C} P^1)$$
and a basis for the group  $H_1(\mathbb{R} P^1 \times \mathbb{R} P^1;\mathbb{Z}_2)$ as
$$(\rho L_1, \rho L_2)=([\mathbb{R} P^1\times \{p_\mathbb{R}\}],[\{ q_\mathbb{R} \}\times \mathbb{R} P^1]), \quad (p_\mathbb{R} , q_\mathbb{R} \in \mathbb{R} P^1).$$

 We choose the $Spin_3$-structure $\mathfrak{s}_{\mathbb{R} P^3}$ over $\mathbb{R} P^3$ such that the spinor state of a real line in $\mathbb{C} P^3$ is $+1$. In other words, $\mathfrak{s}_{\mathbb{R} P^3}$ is chosen such that $sp_{\mathfrak{s}_{\mathbb{R} P^3},\mathfrak{o}_{\mathbb{R} P^3}}(f(\mathbb{C} P^1))=+1$ where $f: (\mathbb{C} P^1,\tau_1) \to (\mathbb{C} P^3,\tau_3)$ is a real immersion and $f_*[\mathbb{C} P^1]=L_{\mathbb{C} P^3}$.

 One has the following theorem.
\begin{theorem} \label{theo-application.degree.8} \label{Theorem-W.in.deg.8}
For every odd positive integer $d$ and $0 \leq l \leq d-1$ one has
\begin{equation} \label{eq-Welschinger.formula.dim.8}
    W_{\mathbb{R} P^3}^{\mathfrak{s}_{\mathbb{R} P^3},\mathfrak{o}_{\mathbb{R} P^3}}(d,l)=\frac{1}{2}\sum_{a\in \mathbb{Z}}(-1)^{a} (d-2a) W_{\mathbb{R} P^1\times \mathbb{R} P^1} ((a,d-a),l).
\end{equation}

\end{theorem}
The Formula (\ref{eq-Welschinger.formula.dim.8}) can be reduced to Theorem 1 in \cite{brugalle2016pencils}.
\begin{proof}

For any $D=(a,b)\in H_2^{-\tau_3}(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z}_2)$, one has
$$k_D=(2,2).(a,b)-1\equiv 1 \mod 2,$$
and
$$g(D)=\frac{1}{2}((2,2).(a,b)+(a,b)^2+2)\equiv (a+1)(b+1) \mod 2.$$

We consider a real balanced immersion $f:(\mathbb{C} P^1,\tau_1) \to (\mathbb{C} P^3,\tau_3)$ such that
$$Im(f)\subset \mathbb{C}P^1\times \mathbb{C}P^1, \; \text{ and } f_*[\mathbb{C} P^1]\in H_2^{-\tau_3}(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z}).$$

Remark that the homology class $L_{\mathbb{C} P^3} \in H_2^{-\tau_3}(\mathbb{C} P^3;\mathbb{Z})$ has two effective classes\footnote{They are homology classes can be realized by algebraic curves.} in its preimages via the surjective homomorphism $\psi:H_2(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})\to H_2(\mathbb{C} P^3;\mathbb{Z})$, they are classes $(1,0)$ and $(0,1)$ in $H_2^{-\tau_3}(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})$.
\begin{lemma} \label{lemma1}
Let $L= (1,0) \in H_2^{-\tau_3}(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})$.

Suppose that $\mathbb{R} f^*\mathcal{N}_{L/\mathbb{C}P^1\times \mathbb{C}P^1}=E^+$.
Then, the function $\epsilon: H_2(\mathbb{C}P^1\times \mathbb{C}P^1;\mathbb{Z})\to \mathbb{Z}_2$ (see Definition \ref{def-epsilon.function}) satisfies $\epsilon(a,b)=0$ if and only if $a>b$.
\end{lemma}
\begin{proof}
Recall that $S\in \mathbb{Z}(1,-1)=\ker (\psi) $.

By an easy calculation one has $(D.S)(L.S)=a-b$. By definition, one has $\epsilon(L)=k_L
+1 \mod 2$ then $\epsilon(L)=0 \mod 2$. Thus, the lemma follows by Proposition \ref{prop-function.epsilon2}.
\end{proof}
Moreover, the restricted $Pin^-_2$-structure  on $O(T (\mathbb{R} P^1\times \mathbb{R} P^1))$ makes  $s^{\mathfrak{s}_{\mathbb{R} P^3|\mathbb{R} P^1\times \mathbb{R} P^1}}(1,0)=0$, and so $s^{\mathfrak{s}_{\mathbb{R} P^3|\mathbb{R} P^1\times \mathbb{R} P^1}}(0,1)=1$. Hence,
$$s^{\mathfrak{s}_{\mathbb{R} P^3|\mathbb{R} P^1\times \mathbb{R} P^1}}(a,b) \equiv b+ab \mod 2.$$

Thus,
\begin{itemize}
    \item if $D=(a,b)$ such that $a>b$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{\mathbb{R} P^3|\mathbb{R} P^1\times \mathbb{R} P^1}}(\rho D)}=(-1)^{0+(a+1)(b+1)+b+ab};$$
by simplifying, this sign is equal to $(-1)^{a+1};$
    \item if $D=(a,b)$ such that $a<b$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{\mathbb{R} P^3|\mathbb{R} P^1\times \mathbb{R} P^1}}(\rho D)}=(-1)^{1+(a+1)(b+1)+b+ab};$$
by simplifying, this sign is equal to $(-1)^{a};$
\end{itemize}

In conclusion, if $a+b \equiv 1 \mod 2$ then
\begin{align*}
     W_{\mathbb{R} P^3}^{\mathfrak{s}_{\mathbb{R} P^3},\mathfrak{o}_{\mathbb{R} P^3}}(d,l)
    = & \frac{1}{2}\Big(\sum_{0\leq a<\frac{d}{2}}(-1)^{a} (d-2a) W_{\mathbb{R} P^1\times \mathbb{R} P^1} ((a,d-a),l)\\
    &+ \sum_{d\geq a>\frac{d}{2}}(-1)^{a+1} (-(d-2a)) W_{\mathbb{R} P^1\times \mathbb{R} P^1} ((a,d-a),l) \Big)\\
    = & \sum_{0\leq a<\frac{d}{2}}(-1)^{a} (d-2a) W_{\mathbb{R} P^1\times \mathbb{R} P^1} ((a,d-a),l).
\end{align*}

Since the genus-$0$ Welschinger invariants $ W_{\mathbb{R} P^1\times \mathbb{R} P^1} ((a,d-a),l)$ vanish when $a \notin [0,d]$, the Formula (\ref{eq-Welschinger.formula.dim.8}) follows.
\end{proof}

\begin{remark}
Directly from Corollary \ref{coro-Theorem2-W} and the fact that $$|D.S|=|(a,d-a).(1,-1)|=| d-2a|\equiv d\mod 2,$$
one deduces that the invariants $ W_{\mathbb{R} P^3}^{\mathfrak{s}_{\mathbb{R} P^3},\mathfrak{o}_{\mathbb{R} P^3}}(d,l)$ vanish if $d$ is even for any $Spin_3$-structure $\mathfrak{s}_{\mathbb{R} P^3}$ over $\mathbb{R} P^3$. This result was obtained by G. Mikhalkin \cite{brugalle2007enumeration} by symmetry reasons.
\end{remark}

\begin{proposition} \label{prop-application.deg8}
Suppose that $\mathbb{R} E$ is disconnected and the hyperplane class is on its pointed-component. If $d$ is even and if the configuration $\underline{x}_d$ has an odd number of real points on each component of $\mathbb{R} E$, then there is no real irreducible rational degree $d$ curve in  $\mathbb{C} P^3$ passing through $\underline{x}_d$.
\end{proposition}

\begin{proof}
This is a consequence of Theorem \ref{Theorem3-W}     and the following remarks.
    \begin{itemize}
      \item The base locus of the pencil of surfaces in the linear system $|-\frac{1}{2}K_{\mathbb{C} P^3}|$ is an elliptic curve $E$ of degree $4$ in $\mathbb{C} P^3$. Moreover, $E$ is of homology class
        $$(2,2)\in  H_2(\mathbb{C} P^1\times \mathbb{C} P^1;\mathbb{Z}).$$
        \item By the intersection form on $\mathbb{C} P^3$ and the surjective morphism $\psi$, one has the hyperplane section is of class $$H=(1,1) \in H_2(\mathbb{C} P^1\times \mathbb{C} P^1;\mathbb{Z}) \cong Pic(\mathbb{C} P^1\times \mathbb{C} P^1),$$ Moreover, one has $H=L+T(L)=L_1+L_2$.
        \item  Let $h:=H|_E \in Pic_4(E)$ be the hyperplane class on $E$. One can precise the  divisor $ \phi(\overline{L},\overline{D})$ whose degree is $|D.E-|D.S|(L.E)|=|4(d-a)|$ as
        \begin{align*}
            \phi(\overline{L},\overline{D})&=(a,d-a)|_E-(d-2a) (1,0)|_E\\
            &=(d-a)h.
        \end{align*}

         Hence, the hyperplane class on $E$ and the divisor         $\phi(\overline{L},\overline{D})$ are in the same connected component of $\mathbb{R} E$, i.e. in the pointed component $\mathbb{S}^1_0$ of  $\mathbb{R} E$ by hypothesis.

         If the configuration $\underline{x}_d$ has an odd number of real points on each component of $\mathbb{R} E$ then its divisor class $[\underline{x}_d]$ is in the non-pointed component $\mathbb{S}^1_1$ of $\mathbb{R} E$. Thus, by  Corollary \ref{coro-degree.case.real}, the statement follows.
               \end{itemize}

\end{proof}
This vanishing result was obtained by J. Koll\'ar in \cite{kollar2015examples}.
\subsubsection{Real three-dimensional projective space blown-up at a real point.}

We consider $(\mathbb{C}P^3\sharp\overline{\mathbb{C}P^3},\tau)$  the real three-dimensional projective space blown-up at a real point $x$, with the real structure $\tau$ is chosen such that its fixed locus is the blow-up at a real point of the real projective space $\mathbb{R} P^3\sharp{\overline{\mathbb{R} P^3}}$.

We consider the immersed non-singular real del Pezzo surfaces $(\Sigma, \tau|_\Sigma)$ of degree $7$ in $\mathbb{C}P^3\sharp\overline{\mathbb{C}P^3}$,  which are isomorphic to the blow-ups at a real point of $\mathbb{C}P^1\times \mathbb{C}P^1$. Moreover, the fixed locus of $\tau|_\Sigma$ is the blow-up at a real point of a $2$-torus $\mathbb{R} P^1\times \mathbb{R} P^1 \sharp {\mathbb{R} P^2}$.

We can choose a basis for the group $H_2^{-\tau }(\mathbb{C} P^3\sharp\overline{\mathbb{C} P^3};\mathbb{Z})$ as
\begin{itemize}
    \item[$\bullet$]  $L_{\mathbb{C} P^3}:= \text{ the homology class of a line embedded in } \mathbb{C} P^3$,
      \item[$\bullet$] $F$ is the class of a line in the exceptional divisor,
\end{itemize}
and a basis for the group  $H_1(\mathbb{R} P^3  \sharp \overline{\mathbb{R} P^3};\mathbb{Z}_2)$ as
\begin{itemize}
       \item[$\bullet$]  $\rho L_{\mathbb{C} P^3}:= \text{ the real part of homology class of a line embedded in } \mathbb{C} P^3$,
      \item[$\bullet$] $\rho F$ is the real part of homology class of a line in the exceptional divisor.
\end{itemize}

We can choose a basis for the group $H_2^{-\tau }(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})$ as
\begin{itemize}
    \item[$\bullet$]  $L_1=[\mathbb{C} P^1\times \{p\}]$,
     \item[$\bullet$] $L_2=[\{q\}\times \mathbb{C} P^1]$,
      \item[$\bullet$] $\hat{E}$ is the exceptional divisor class,
\end{itemize}
where $p,q \in \mathbb{C} P^1$,
and a basis for the group  $H_1(\mathbb{R} P^1 \times \mathbb{R} P^1\sharp {\mathbb{R} P^2};\mathbb{Z}_2)$ as
\begin{itemize}
    \item[$\bullet$]  $\rho L_1=[\mathbb{R} P^1\times \{p_\mathbb{R}\}]$,
     \item[$\bullet$] $\rho L_2=[\{q_\mathbb{R}\}\times \mathbb{R} P^1]$,
      \item[$\bullet$] $\rho \hat{E}$ is the real part of the exceptional divisor class,
\end{itemize}
where $p_\mathbb{R} , q_\mathbb{R} \in \mathbb{R} P^1.$

We choose the $Spin_3$-structure $\mathfrak{s}_{\mathbb{R} P^3\sharp \overline{\mathbb{R} P^3}}$ over $\mathbb{R} P^3\sharp \overline{\mathbb{R} P^3}$ such that $sp_{\mathfrak{s}_{\mathbb{R} P^3\sharp \overline{\mathbb{R} P^3}},\mathfrak{o}_{\mathbb{R} P^3\sharp \overline{\mathbb{R} P^3}}}(f(\mathbb{C} P^1))=+1$ where $f: (\mathbb{C} P^1,\tau_1) \to (\mathbb{C}P^3\sharp\overline{\mathbb{C}P^3},\tau)$ is a real immersion and $f_*[\mathbb{C} P^1]=L_{\mathbb{C} P^3}$.

To simplify the notations, in the following we denote by
$$(\mathbb{R} P^3)_1:=\mathbb{R} P^3\sharp \overline{\mathbb{R} P^3}$$

and by
$$(\mathbb{R} P^1)^2_1:=\mathbb{R} P^1\times \mathbb{R} P^1 \sharp {\mathbb{R} P^2}.$$

One has the following theorem.
\begin{theorem} \label{Theorem-W.in.deg.7}
 For every odd positive integer $d$, every non-negative integer $k \leq d$ and every integer $l$ such that $0\leq l \leq \frac{1}{2}(2d-1-k)$, one has
\begin{equation}
    \label{eq-Welschinger.formula.dim.7}
    W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)=\frac{1}{2}\sum_{a\in \mathbb{Z}}(-1)^{a+\frac{1}{2}(k+k^2)} (d-2a) W_{(\mathbb{R} P^1)^2_1} ((a,d-a;k),l).
\end{equation}
\end{theorem}
If $k$ is \textit{even} then the Formula (\ref{eq-Welschinger.formula.dim.7}) can be reduced to Theorem 1.1 in \cite{ding2019remark}.

\begin{proof}

A similar proof as the one of Theorem \ref{theo-application.degree.8} can be applied. We emphasize some main points as follows.

For every $D=(a,b;k)\in H_2(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})$, one has the following topological invariants

    $$k_D=(2,2;1).(a,b;k)-1\equiv k+1 \mod 2;$$
and
    \begin{align*}
        g(D)&=\frac{1}{2}\Big((2,2;1).(a,b;k)+(a,b;k)^2+2\Big)\\
        &\equiv a+b+ab+ \frac{1}{2}(k + k^2 )+1 \mod 2.
    \end{align*}

Recall that $S\in \mathbb{Z}(1,-1;0)$.

We consider a real balanced immersion $f:(\mathbb{C} P^1,\tau_1) \to (\mathbb{C} P^3 \sharp \overline{\mathbb{C} P^3},\tau)$ such that $Im(f)$ is contained in $\mathbb{C}P^1\times \mathbb{C}P^1 \sharp \overline{\mathbb{C} P^2}$ and $f_*[\mathbb{C} P^1]\in H_2^{-\tau}(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})$.

Remark that the homology class $L_{\mathbb{C} P^3}=(1;0)\in H_2^{-\tau}(\mathbb{C} P^3 \sharp \overline{\mathbb{C} P^3};\mathbb{Z})$ has two effective classes in its preimages via the surjective homomorphism $$\psi:H_2(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})\to H_2(\mathbb{C} P^3 \sharp \overline{\mathbb{C} P^3};\mathbb{Z}).$$
These are classes $(1,0;0)$ and $(0,1;0)$ in $H_2^{-\tau}(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})$.

Lemma \ref{lemma1} implies directly the following lemma.
\begin{lemma}
Let  $L= (1,0;0)\in H_2^{-\tau}(\mathbb{C}P^1\times \mathbb{C}P^1\sharp\overline{\mathbb{C} P^2};\mathbb{Z})$.

Suppose that $\mathbb{R} f^*\mathcal{N}_{L/\mathbb{C}P^1\times \mathbb{C}P^1\sharp \overline{\mathbb{C} P^2}}=E^+$.
Then, the function $$\epsilon: H_2(\mathbb{C}P^1\times \mathbb{C}P^1\sharp \overline{\mathbb{C} P^2};\mathbb{Z})\to \mathbb{Z}_2$$ (see Definition \ref{def-epsilon.function}) satisfies
$\epsilon(a,b;k)=0$ if and only if $a>b$.
\end{lemma}

Note that $\epsilon(L)=k_L+1 \equiv 0\mod 2$.

Moreover, the restricted $Pin^-_2$-structure  on $O(T ((\mathbb{R} P^1)^2_1))$ makes  $s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(1,0;0)=0$, and so $s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(0,1;0)=1$. By properties of quasi-quadratic enhancements, we also have $$s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(0,0;1)=0.$$
Hence, for all $a,b,k \in \mathbb{Z}$ one has
$$s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(a,b;k)
\equiv b+ab\mod 2.$$

Thus,
\begin{itemize}
    \item if $D=(a,b;k)$ such that $a>b$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(\rho D)}=(-1)^{0+(a+b+ab+\frac{1}{2}(k+k^2)+1)+(b+ab)};$$
by simplifying, this sign is equal to $(-1)^{a+1+\frac{1}{2}(k+k^2)};$
    \item if $D=(a,b;k)$ such that $a<b$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{(\mathbb{R} P^3)_1|(\mathbb{R} P^1)^2_1}}(\rho D)}=(-1)^{1+(a+b+ab+\frac{1}{2}(k+k^2)+1)+(b+ab)};$$
by simplifying, this sign is equal to $(-1)^{a+\frac{1}{2}(k+k^2)};$
\end{itemize}

In conclusion, if $a+b = 1 \mod 2$ then

\begin{align*}
    W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)
    = & \frac{1}{2}\Big(\sum_{0\leq a<\frac{d}{2}}(-1)^{a+\frac{1}{2}(k+k^2)} (d-2a) W_{(\mathbb{R} P^1)^2_1} ((a,d-a;k),l)\\
    & + \sum_{d\geq a>\frac{d}{2}}(-1)^{a+1+\frac{1}{2}(k+k^2)} (-(d-2a)) W_{(\mathbb{R} P^1)^2_1} ((a,d-a;k),l) \Big)\\
    = & \sum_{0\leq a<\frac{d}{2}}(-1)^{a+\frac{1}{2}(k+k^2)} (d-2a) W_{(\mathbb{R} P^1)^2_1} ((a,d-a;k),l).
\end{align*}
Since the genus-$0$ Welschinger invariants $ W_{(\mathbb{R} P^1)^2_1} ((a,d-a;k),l)$ vanish when $a \notin [0,d]$, the Formula (\ref{eq-Welschinger.formula.dim.7}) follows.
\end{proof}

\begin{remark}  Directly from Corollary \ref{coro-Theorem2-W} and the fact that $$|D.S|=|(a,d-a;k).(1,-1;0)|=| d-2a|\equiv d \mod 2,$$
one deduces that the invariants $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$ vanish if $d$ is even for any $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^3)_1}$ over $(\mathbb{R} P^3)_1$.
\end{remark}
\begin{proposition} \label{prop-application.deg7}
 Suppose that $\mathbb{R} E$ is disconnected and that the hyperplane class and the exceptional divisor class are on its pointed-component. If $d$ is even and if the configuration $\underline{x}_d$ has an odd number of real points on the non-pointed component of $\mathbb{R} E$, then there is no real irreducible rational curve of class $(d;k)$  in  $\mathbb{C} P^3 \sharp \overline{\mathbb{C} P^3}$ passing through $\underline{x}_d$.

\end{proposition}
\begin{proof}
This is a consequence of Theorem \ref{Theorem3-W}     and the following remarks.
    \begin{itemize}
        \item The base locus of the pencil of surfaces in the linear system $|-\frac{1}{2}K_{\mathbb{C} P^3\sharp \overline{\mathbb{C} P^3}}|$ is an elliptic curve $E$ of homology class
        $$\frac{1}{4}K_{\mathbb{C} P^3\sharp \overline{\mathbb{C} P^3}}^2=(4;1)\in H_2(\mathbb{C} P^3\sharp \overline{\mathbb{C} P^3};\mathbb{Z}).$$
        Moreover, $E$ is of homology class
        $$(2,2;1)\in  H_2(\mathbb{C} P^1\times \mathbb{C} P^1 \sharp \overline{\mathbb{C} P^2};\mathbb{Z}).$$
        \item By the intersection form on $\mathbb{C} P^3 \sharp \overline{\mathbb{C} P^3}$ and the surjective morphism $\psi$, one has the hyperplane section is of class
        $$\tilde{H}=(1,1;0) \in H_2(\mathbb{C} P^1\times \mathbb{C} P^1 \sharp \overline{\mathbb{C} P^2};\mathbb{Z}) \cong Pic(\mathbb{C} P^1\times \mathbb{C} P^1 \sharp \overline{\mathbb{C} P^2}).$$
        Moreover, one has
        $\tilde{H}=L+T(L)=L_1+L_2$.
        \item  Let $\tilde{h}:=\tilde{H}|_E \in Pic_4(E)$ be the hyperplane class on $E$. One can precise the divisor  $ \phi(\overline{L},\overline{D})$ whose degree is $|D.E+(D.S)(L.E)|=|4(d-a)-k|$ as
        \begin{align*}
           \phi(\overline{L},\overline{D})&=(a,d-a;k)|_E-(d-2a) (1,0;0)|_E\\
           &=(d-a)\tilde{h}-k\hat{E}|_E.
        \end{align*}
        Hence, the position of the divisor $\phi(\overline{L},\overline{D})$ in $\mathbb{R} E$ depends on that of the hyperplane class $\tilde{h}$ and the exceptional divisor class $\hat{E}|_E$ on $E$. Moreover, if $\underline{x}_d$ has an odd number of real points on the non-pointed component $\mathbb{S}^1_1$ of $\mathbb{R} E$, then $[\underline{x}_d]\in \mathbb{S}^1_1$. Thus, by  Corollary \ref{coro-degree.case.real}.c), the statement follows.
    \end{itemize}

\end{proof}
  \subsubsection{Real threefold product of the projective line} \label{Subsect-exam.deg.6.W}

   We have seen in Section \ref{sect-applications.GW} that non-singular surfaces $\Sigma\in |-\frac{1}{2}K_{\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1}|$ are non-singular  del Pezzo surfaces of degree $6$. In particular, they are  non-singular tri-degree $(1,1,1)$-surfaces  in $\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1$. Moreover, they isomorphic to the blown-ups of non-singular quadric surfaces in $\mathbb{C} P^3$ at two distinct points, i.e. $\Sigma\cong \mathbb{C}P^1\times \mathbb{C}P^1 \sharp 2 \overline\mathbb{C}P^2$.

   To simply the notations, in this part we denote by $$(\mathbb{C} P^1)^3:=\mathbb{C}P^1\times \mathbb{C}P^1\times \mathbb{C}P^1$$

   and by
   $$(\mathbb{C} P^1)^2_2:=\mathbb{C}P^1\times \mathbb{C}P^1\sharp 2\overline{\mathbb{C} P^2}.$$

   We recall the following identifications (see Section \ref{Subsect-exam.deg.6.GW})
\begin{itemize}
    \item[$\bullet$] $H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})=<(L_1,L_2;E_1,E_2)>$ with  $\mathbb{Z}^4$ and a homology class in $H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})$ of the form $a L_1+b L_2-\alpha E_1 - \beta E_2$ is written as $(a,b;\alpha,\beta)$. Note that $(a,b;\alpha,\beta)$ is an effective class if $(a,b)\in \mathbb{N}^2$ and $(\alpha,\beta)\in \mathbb{Z}^2$ such that if $a=b=0$ then both $\alpha$ and $\beta$ are negative; otherwise both $\alpha$ and $\beta$ are non-negative.
    \item[$\bullet$] $H_2((\mathbb{C} P^1)^3;\mathbb{Z})=<(M_1,M_2,M_3)>$ with  $\mathbb{Z}^3$ and a homology class in $ H_2((\mathbb{C} P^1)^3;\mathbb{Z})$ of the form $a_1 M_1 +a_2 M_2+a_3 M_3$ is written as $(a_1,a_2,a_3)$. Note that a class $(a_1,a_2,a_3)$ is effective if $a_i\in \mathbb{N}$ for all $i\in \{1,2,3\}$ and at least one $a_i>0$.
\end{itemize}

Moreover, the inclusion of $(\mathbb{C} P^1)^2_2$ in $(\mathbb{C} P^1)^3$ implies the surjective map
    \begin{align*}
        \psi:H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})&\longrightarrow H_2((\mathbb{C} P^1)^3;\mathbb{Z})\\
       (a,b;\alpha,\beta)&\longmapsto (a,b,a+b-\alpha-\beta).
    \end{align*}

      We have $\psi (0,0;\alpha,-\alpha)=(0,0,0)$ then $\ker(\psi)=\mathbb{Z}(0,0;1,-1)$.

For every $D=(a,b;\alpha,\beta)\in H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})$, one has the following topological invariants:

    $$k_D=(2,2;1,1).(a,b;\alpha,\beta)-1\equiv  \alpha + \beta +1 \mod 2;$$
    and
    \begin{align*}
        g(D)&=\frac{1}{2}\Big((2,2;1,1).(a,b;\alpha,\beta)+(a,b;\alpha,\beta)^2+2\Big)\\
        &=a+b+ab+ \frac{1}{2}(\alpha + \alpha^2 ) + \frac{1}{2}(\beta+\beta^2)+1 \mod 2.
    \end{align*}

 We consider two real structures on $(\mathbb{C} P^1)^3$: the \textit{standard} one, denoted by $\tau_{1,1}$ with  the fixed locus $\mathbb{R}P^1\times \mathbb{R}P^1\times \mathbb{R}P^1$, and the \textit{twisted} one, denoted  by $\tilde{\tau}_{1,1}$ with the fixed locus $\mathbb{S}^2\times \mathbb{R} P^1$.
\begin{remark}
    Since we are working on the real $(1,1,1)$-surfaces on $(\mathbb{C} P^1)^3$ such that $S$ is $\tau_{1,1}$-anti-invariant and $\tilde{\tau}_{1,1}$-anti-invariant respectively,  two blowing up points of the non-singular quadric are real.
\end{remark}
Thus, these two real structures restricted to $(\mathbb{C} P^1)_2^2$ gives the associating real parts are the surfaces $\mathbb{R} P^1\times \mathbb{R} P^1 \sharp 2 {\mathbb{R} P^2}$ and $\mathbb{S}^2 \sharp 2 {\mathbb{R} P^2}$ respectively.
 \subsubsection*{ \underline{Case 1:} The standard real structure}

To simply the notations, we denote by $$(\mathbb{R} P^1)^3:=\mathbb{R} P^1\times \mathbb{R} P^1 \times \mathbb{R} P^1$$

and by
$$(\mathbb{R} P^1)^2_2:=\mathbb{R} P^1\times \mathbb{R} P^1 \sharp 2 {\mathbb{R} P^2}.$$

 We can choose a basis for the group $H_2^{-\tau_{1,1} }((\mathbb{C} P^1)^3;\mathbb{Z})$ as the basis for the group $H_2((\mathbb{C} P^1)^3;\mathbb{Z})$, i.e.
    $$(M_1,M_2,M_3)=([\mathbb{C} P^1\times \{p_1\}\times \{p_2\}],[\{q_1\}\times \mathbb{C} P^1\times \{q_2\}],[\{r_1\}\times \{r_2\}\times \mathbb{C} P^1]),$$
    where $p_i,q_i,r_i \in \mathbb{C} P^1$;
and a basis for the group  $H_1((\mathbb{R} P^1)^3;\mathbb{Z}_2)$ as
 $$(\rho M_1,\rho M_2,\rho M_3)=([\mathbb{R} P^1\times \{p_{1\mathbb{R}}\}\times \{p_{2\mathbb{R}}\}],[\{q_{1\mathbb{R}}\}\times \mathbb{R} P^1\times \{q_{2\mathbb{R}}\}],[\{r_{1\mathbb{R}}\}\times \{r_{2\mathbb{R}}\}\times \mathbb{R} P^1]),$$
    where $p_{i,\mathbb{R}},q_{i,\mathbb{R}},r_{i,\mathbb{R}} \in \mathbb{R} P^1$ ($i\in \{1,2\}$).

Suppose that two blowing up points are $x_1\neq x_2\in \mathbb{R} P^1\times \mathbb{R} P^1$.

We can choose a basis for the group $H_2^{-\tau_{1,1}}((\mathbb{C} P^1)_2^2;\mathbb{Z})$ as the basis for the group $H_2((\mathbb{C} P^1)_2^2;\mathbb{Z})$, i.e.
\begin{itemize}
    \item[$\bullet$]  $L_1=[\mathbb{C} P^1\times \{p\}]$,
     \item[$\bullet$] $L_2=[\{q\}\times \mathbb{C} P^1]$,
      \item[$\bullet$] $E_1$ is the exceptional divisor class over $x_1$,
       \item[$\bullet$] $E_2$ is the exceptional divisor class over $x_2$,
\end{itemize}
where $p,q \in \mathbb{C} P^1$,
and a basis for the group  $H_1((\mathbb{R} P^1)_2^2;\mathbb{Z}_2)$ as
\begin{itemize}
    \item[$\bullet$]  $\rho L_1=[\mathbb{R} P^1\times \{p_\mathbb{R}\}]$,
     \item[$\bullet$] $\rho L_2=[\{q_\mathbb{R}\}\times \mathbb{R} P^1]$,
      \item[$\bullet$] $\rho E_1$ is the real part of the exceptional divisor class over $x_1$,
      \item[$\bullet$] $\rho E_2$ is the real part of the exceptional divisor class over $x_2$,
\end{itemize}
where $p_\mathbb{R} , q_\mathbb{R} \in \mathbb{R} P^1.$

   We choose the $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^1)^3}$ over $(\mathbb{R} P^1)^3$ such that $sp_{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}(f(\mathbb{C} P^1))=+1$ where $f:(\mathbb{C} P^1,\tau_1)\to ((\mathbb{C} P^1)^3,\tau_{1,1})$ is a real immersion and $$f_*[\mathbb{C} P^1]\in \{(0,0,1),(0,1,0),(0,0,1)\}\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z}).$$
 The formula induced from Theorem \ref{Theorem2-W} in this case is given as in the following theorem.
\begin{theorem} \label{Theorem-W.in.deg.6.1}

For every triplet of non-negative integers $(a,b,c)$ such that $a+b+c$ is odd, and every integer $l$ is such that $0\leq l \leq \frac{1}{2}( a+b+c-1)$, one has
\begin{equation} \label{Eq:W.in.deg.6.1}
    W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)
    =\sum_{0\leq\alpha<\frac{|a+b-c|}{2}}(-1)^{ \alpha +\frac{a+b-c-1}{2}} (a+b-c-2\alpha) W_{(\mathbb{R} P^1)^2_2} ((a,b;\alpha,a+b-c-\alpha),l)
\end{equation}
\end{theorem}
\begin{proof}
Let $D=(a,b;\alpha,a+b-c-\alpha)\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z})$.
One has

   $$k_D\equiv a+b+c+1 \mod 2;$$

  and
        \begin{align*}
        g(D)&=a+b+ab+
    \alpha^2+ \alpha(a+b-c)+\frac{1}{2}(a+b-c)(a+b-c-1)+1\\
        & \equiv    a+b+1+ab+\frac{1}{2}(a+b-c-1)\mod 2, \quad \text{ if } a+b+c=1 \mod 2.
    \end{align*}

We consider a real  balanced immersion $f:(\mathbb{C} P^1,\tau_1) \to ((\mathbb{C} P^1)^3,\tau_{1,1})$ such that $Im(f)\subset (\mathbb{C} P^1)_2^2$ and $f_*[\mathbb{C} P^1]\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z})$.

Remark that the homology class $(0,0,1)\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$ has two effective classes in its preimages via the surjective homomorphism $\psi:H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})\to H_2((\mathbb{C} P^1)^3;\mathbb{Z})$, they are classes $(0,0;0,-1)$ and $(0,0;-1,0)$ in $H_2^{-\tau_{1,1}}( (\mathbb{C} P^1)^2_2;\mathbb{Z})$.

\begin{lemma} \label{lemma2}
Let $L= (0,0;0,-1)\in H_2^{-\tau_{1,1}}( (\mathbb{C} P^1)^2_2;\mathbb{Z})$.

Suppose that $\mathbb{R} f^*\mathcal{N}_{L/(\mathbb{C} P^1)^2_2}=E^+$. Then, the function $\epsilon: H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})\to \mathbb{Z}_2$ (see Definition \ref{def-epsilon.function}) satisfies $\epsilon(a,b;\alpha,\beta)=0$ if and only if $\alpha > \beta$.
\end{lemma}

\begin{proof}
By definition, one has $\epsilon(L)=k_L+1 \equiv 0+0+1+1\equiv 0 \mod 2$.

Recall that $S\in \mathbb{Z}(0,0;1,-1)=\ker\big(H_2((\mathbb{C} P^1)_2^2;\mathbb{Z})\to H_2((\mathbb{C} P^1)^3;\mathbb{Z})\big)$. By an easy calculation, one has

$$(D.S)(L.S) = \alpha - \beta.$$
Thus, the lemma follows by Proposition \ref{prop-function.epsilon2}.
\end{proof}

Moreover, the restricted $Pin^-_2$-structure  on $O(T (\mathbb{R} P^1)^2_2))$ makes  $s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;0,-1)=0$, and so $s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;-1,0)=1$.
Since the spinor states of a  curve of class either $(1,0,0)$  or $(0,1,0)$ or $(0,0,1)$ in $(\mathbb{C} P^1)^3$ are identified, i.e. $$sp_{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}(0,0,1)=sp_{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}(1,0,0)=sp_{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}(0,1,0),$$ one deduces that

$$s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(1,0;1,0)\equiv s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,1;1,0)\equiv 0.$$
Hence,
$$s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(1,0;0,0)\equiv s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(1,0;1,0) +s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;-1,0) \equiv 1,$$
and
$$s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,1;0,0) \equiv s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,1;1,0) +s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;-1,0)\equiv 1.$$

Thus,

\begin{align*}
    &s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(a,b;\alpha,\beta)\\
    \equiv & a s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(1,0;0,0)+bs^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,1;0,0) \\
    & -\alpha s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;-1,0)-\beta  s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(0,0;0,-1)+ab \mod 2\\
    \equiv & a+b+ab+\alpha \mod 2.
\end{align*}

Therefore,
\begin{itemize}
    \item if $D=(a,b;\alpha,\beta)$  such that $\alpha>\beta$ and  $\alpha + \beta= a+b-c \equiv 1 \mod 2$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(\rho D)}=(-1)^{0+(a+b+1+ab+\frac{1}{2}(a+b-c-1))+(a+b+ab+\alpha)};$$
by simplifying, this sign becomes $(-1)^{\alpha+1+\frac{1}{2}(a+b-c-1)};$
    \item  if $D=(a,b;\alpha,\beta)$ such that $\alpha<\beta$ and $\alpha + \beta= a+b-c \equiv 1 \mod 2$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{(\mathbb{R} P^1)^3|(\mathbb{R} P^1)^2_2}}(\rho D)}=(-1)^{1+(a+b+1+ab+\frac{1}{2}(a+b-c-1))+(a+b+ab+\alpha)};$$
by simplifying,  this sign becomes $(-1)^{\alpha+\frac{1}{2}(a+b-c-1)}.$
\end{itemize}

In conclusion, if $a+b +c= 1 \mod 2$ then
\begin{align*}
    & W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)\\
    &=\frac{1}{2}\Big(\sum_{0\leq \alpha<\frac{|a+b-c|}{2}}(-1)^{\alpha +\frac{a+b-c-1}{2}} (a+b-c-2\alpha) W_{(\mathbb{R} P^1)^2_2} ((a,b;\alpha,a+b-c-\alpha),l)\\ +&\sum_{|a+b-c|\geq \alpha>\frac{|a+b-c|}{2}}(-1)^{\alpha +1+\frac{a+b-c-1}{2}} (-(a+b-c-2\alpha)) W_{(\mathbb{R} P^1)^2_2} ((a,b;\alpha,a+b-c-\alpha),l) \Big)\\
    &=\sum_{0\leq \alpha<\frac{|a+b-c|}{2}}(-1)^{\alpha +\frac{a+b-c-1}{2}} (a+b-c-2\alpha) W_{(\mathbb{R} P^1)^2_2} ((a,b;\alpha,a+b-c-\alpha),l).
\end{align*}

Since the genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^1)^2_2} ((a,b;\alpha,a+b-c-\alpha),l)$ vanish when $\alpha \notin [0,|a+b-c|]$,
the Formula (\ref{Eq:W.in.deg.6.1}) follows.

\end{proof}

We call a \textit{ $(x,y,z)$-surface class on elliptic curve $E$} the restriction of  a surface in $(\mathbb{C} P^1)^3$ of \textit{tri-degree} $(x,y,z)$ to $E$.
For example, the $(0,0,1)$-surface class on $E$ is $h_3:=(1,1;1,1)|_E\in Pic_2(E)$. In other words, one can write
$$h_3=L_1{|_E}+L_2{|_E}-E_1{|_E}-E_2{|_E}.$$

Note that $$L_1{|_E},L_2{|_E} \in Pic_2(E), \quad E_1{|_E},E_2{|_E}\in Pic_1(E),$$

and
$$E_1{|_E}+E_2{|_E}=h_{1,1}-h_3,$$
where $h_{a,b}$ is the $(a,b,0)$-surface class on $E$.
\begin{remark}
Directly from Corollary \ref{coro-Theorem2-W} and the fact that $$|D.S|=|(a,b;\alpha,a+b-c-\alpha).(0,0;1,-1)|=|a+b-c-2\alpha|\equiv a+b+c \mod 2,$$

one deduces that the invariants $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$ vanish if $a+b+c$ is even for any $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^1)^3}$ over $(\mathbb{R} P^1)^3$.
\end{remark}
\begin{proposition} \label{prop-application.deg6.1}
Suppose that $\mathbb{R} E$ is disconnected. Suppose also that the  $(0,0,1)$-surface class,  the  $(1,1,0)$-surface class and the  $(b,a,0)$-surface class ($a,b\in \mathbb{N}^*$) are on the pointed-component of $\mathbb{R} E$.
If $a+b+c$ is even and if the real configuration $\underline{x}_d$ of $k_d=a+b+c$ points in $E$     has an odd number of real points on each component of $\mathbb{R} E$, then there is no real rational curve of class $(a,b,c)\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$ passing through $\underline{x}_d$.

\end{proposition}
\begin{proof}
This is a consequence of Theorem \ref{Theorem3-W}     and the following remarks.
    \begin{itemize}
        \item The base locus of the pencil of surfaces in the linear system $|-\frac{1}{2}K_{(\mathbb{C} P^1)^3}|$ is an elliptic curve $E$ of homology class
        $$\frac{1}{4}K_{(\mathbb{C} P^1)^3}^2=(2,2,2)\in H_2^{-{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z}).$$
        Moreover, $E$ is of homology class
        $$(2,2;1,1)\in  H_2^{-{\tau}_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z}).$$
        \item By the intersection forms of $(\mathbb{C} P^1)^3$ and of $(\mathbb{C} P^1)^2_2$ described as above,  one has  the class $$L+T(L)=E_1+E_2=(0,0;-1,-1)\in H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})\cong Pic((\mathbb{C} P^1)^2_2).$$
        Taking its restriction to $E$,  we get the divisor
        $$\phi(\overline{L})=h_{1,1}-h_3 \in Pic_2(E)$$
        which is determined by the $(0,0,1)$-surface class  and the  $(1,1,0)$-surface class on $E$.
        \item Moreover, if $0\leq \alpha\leq \frac{1}{2}(a+b-c)$ then the degree $|D.E-|D.S|(L.E)|= 2(c+\alpha)$  divisor $\phi(\overline{L},\overline{D})$ in this case is
        \begin{align*}
            \phi(\overline{L},\overline{D})&=(a,b;a+b-c-\alpha,\alpha)|_E-(a+b-c-2\alpha)(0,0;0,-1)|_E\\
           & =(a,b;0,0)|_E-(a+b-c-\alpha) (0,0;-1,-1)|_E\\
           & = h_{b,a}-(a+b-c-\alpha)(h_{1,1}-h_3)
        \end{align*}

        Hence, Equation (\ref{eq-solutions.in.PicE}) becomes
\begin{equation}
     (a+b-c-2\alpha)l=\underline{x}_{d}-h_{b,a}+(a+b-c-\alpha)(h_{1,1}-h_3).
\end{equation}

    \end{itemize}

    Thus, by hypothesis and by Corollary \ref{coro-degree.case.real}.c), the statement follows.
\end{proof}

  \subsubsection*{\underline{Case 2:} The twisted real structure}

   We can choose a basis for the group $H_2^{-\tilde{\tau}_{1,1} }((\mathbb{C} P^1)^3;\mathbb{Z})$ as
    $$(M_1+M_2,M_3)=([\mathbb{C} P^1\times \{p_1\}\times \{p_2\}]+[\{q_1\}\times \mathbb{C} P^1\times \{q_2\}],[\{r_1\}\times \{r_2\}\times \mathbb{C} P^1])$$
    where $p_i,q_i,r_i \in \mathbb{C} P^1$.

    Hence, the group $H_2^{-\tilde{\tau}_{1,1} }((\mathbb{C} P^1)^3;\mathbb{Z})$  can be identified with $\mathbb{Z}^2$.

Note that the group $H_1(\mathbb{S}^2\times \mathbb{R} P^1;\mathbb{Z}_2)$ has a generator $\rho M_3:=[\{r_{\mathbb{R}}\}\times \mathbb{R} P^1],$
    where $r_{\mathbb{R}} \in \mathbb{S}^2$.

Note also that two blowing up points are $x_1\neq x_2\in \mathbb{S}^2$.

We can choose a basis for the group $H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)_2^2;\mathbb{Z})$ as
\begin{itemize}
    \item[$\bullet$]  $L_1+L_2=[\mathbb{C} P^1\times \{p\}]+[\{q\}\times \mathbb{C} P^1]$,
      \item[$\bullet$] $E_1$ is the exceptional divisor class over $x_1$,
       \item[$\bullet$] $E_2$ is the exceptional divisor class over $x_2$,
\end{itemize}
where $p,q \in \mathbb{C} P^1$.

Hence, the group $H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^2_2 ;\mathbb{Z})$  can be identified with $\mathbb{Z}^3$.

Note that the group $H_1(\mathbb{S}^2\sharp 2 {\mathbb{R} P^2};\mathbb{Z}_2) $ has two generators given as
\begin{itemize}
      \item[$\bullet$] $\rho E_1$ is the real part of the exceptional divisor class over $x_1$,
      \item[$\bullet$] $\rho E_2$ is the real part of the exceptional divisor class over $x_2$,
\end{itemize}

One can write $H_1(\mathbb{S}^2\sharp 2 {\mathbb{R} P^2};\mathbb{Z}_2) = <(0;-1,0),(0;0,-1)> $.

  We choose the $Spin_3$-structure $\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1}$ over $\mathbb{S}^2 \times \mathbb{R} P^1$ such that $sp_{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}(f(\mathbb{C} P^1))=+1$ where $f:(\mathbb{C} P^1,\tau_1)\to ((\mathbb{C} P^1)^3,\tilde{\tau}_{1,1})$ is a real immersion and $f_*[\mathbb{C} P^1]=(0;1)\in H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$.

The formula induced from Theorem \ref{Theorem2-W} in this case is given in the following theorem.
\begin{theorem} \label{Theorem-W.in.deg.6.2}
For every non-negative integer $a$, every odd positive integer $c$ and $0\leq l \leq \frac{1}{2}(2a+c-1)$, one has
\begin{equation}\label{Eq:W.in.deg.6.2}
    W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)=\sum_{0\leq\alpha<\frac{|2a-c|}{2}}(-1)^{ \alpha +\frac{c+1}{2}} (2a-c-2\alpha) W_{\mathbb{S}^2\sharp 2 {\mathbb{R} P^2}} ((a;\alpha,2a-c-\alpha),l).
\end{equation}

\end{theorem}

\begin{proof}
Similar proof as the one of Theorem \ref{Theorem-W.in.deg.6.1} can be applied. We emphasize some distinguished points coming from the changing of real structures on $(\mathbb{C} P^1)^3$ as follows.

Let $D=(a;\alpha,2a-c-\alpha)\in H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z})$.
One has
$$k_D\equiv c+1 \mod 2;$$
and
\begin{align*}
    g(D)&= 2a+a^2+\alpha^2+(2a-c)\alpha +\frac{1}{2}(2a-c)(2a-c-1)+1\\
    &\equiv \alpha^2+ c\alpha+1+\frac{1}{2}c(c+1) \mod 2\\
    & \equiv \frac{1}{2}(c-1)\mod 2, \quad \text{ if } c\equiv 1 \mod 2.
\end{align*}

We consider a real balanced immersion $f:(\mathbb{C} P^1,\tau_1) \to ((\mathbb{C} P^1)^3,\tilde{\tau}_{1,1})$ such that $Im(f)\subset (\mathbb{C} P^1)^2_2$ and $f_*[\mathbb{C} P^1]\in H_2^{-\tilde{\tau}_{1,1}}( (\mathbb{C} P^1)^2_2;\mathbb{Z})$.

Remark that the homology class $(0;1)\in H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$ has two effective classes in its preimages via the surjective homomorphism $\psi:H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})\to H_2((\mathbb{C} P^1)^3;\mathbb{Z})$, they are classes $(0;0,-1)$ and $(0;-1,0)$ in $H_2^{-\tilde{\tau}_{1,1}}( (\mathbb{C} P^1)^2_2;\mathbb{Z})$.

Lemma \ref{lemma2} implies directly the following lemma.
\begin{lemma}
Let $L= (0;0,-1)\in H_2^{-\tilde{\tau}_{1,1}}( (\mathbb{C} P^1)^2_2;\mathbb{Z})$.

Suppose that $\mathbb{R} f^*\mathcal{N}_{L/(\mathbb{C} P^1)^2_2}=E^+$. Then, the function $\epsilon: H_2((\mathbb{C} P^1)^2_2;\mathbb{Z})\to \mathbb{Z}_2$ (see Definition \ref{def-epsilon.function}) satisfies $\epsilon(a;\alpha,\beta)=0$ if and only if $\alpha > \beta$.
\end{lemma}

Moreover, the restricted $Pin^-_2$-structure  on $O(T \mathbb{S}^2 \times \mathbb{R} P^1)$ makes  $s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(0;0,-1)=0$, and so $s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(0;-1,0)=1$.

Since $H_1(\mathbb{S}^2\sharp 2 {\mathbb{R} P^2};\mathbb{Z}_2)=<(0;-1,0),(0;0,-1)>$, one has

\begin{align*}
    &s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(0;-\alpha,-\beta)\\
    \equiv & \alpha s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(0;-1,0)+\beta  s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(0;0,-1) \mod 2\\
    \equiv &  \alpha  \mod 2.
\end{align*}

Therefore,
\begin{itemize}
    \item if $D=(a;\alpha,\beta)$ such that $\alpha>\beta$ and $\alpha +\beta =2a-c \equiv 1 \mod 2$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(\rho D)}=(-1)^{0+\frac{1}{2}(c-1)+\alpha};$$
    \item  if $D=(a;\alpha,\beta)$ such that $\alpha<\beta$ and $\alpha +\beta =2a-c \equiv 1 \mod 2$ then
$$(-1)^{\epsilon(D)+g(D)+s^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1|\mathbb{S}^2 \times \mathbb{R} P^1}}(\rho D)}=(-1)^{1+\frac{1}{2}(c-1)+\alpha }.$$
\end{itemize}

In conclusion, if $c= 1 \mod 2$ then
\begin{align*}
    & W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)\\
    &=\frac{1}{2}\Big(\sum_{0\leq \alpha<\frac{|2a-c|}{2}}(-1)^{\alpha +\frac{c+1}{2}} (2a-c-2\alpha) W_{\mathbb{S}^2\sharp 2 {\mathbb{R} P^2}} ((a;\alpha,2a-c-\alpha),l)\\ +&\sum_{|2a-c|\geq \alpha>\frac{|2a-c|}{2}}(-1)^{\alpha +1+\frac{c+1}{2}} (-(2a-c-2\alpha)) W_{\mathbb{S}^2\sharp 2 {\mathbb{R} P^2}} ((a;\alpha,2a-c-\alpha),l) \Big)\\
    &=\sum_{0\leq \alpha<\frac{|2a-c|}{2}}(-1)^{\alpha +\frac{c+1}{2}} (2a-c-2\alpha) W_{\mathbb{S}^2\sharp 2 {\mathbb{R} P^2}} ((a;\alpha,2a-c-\alpha),l).
\end{align*}

Since the genus-$0$ Welschinger invariants $W_{\mathbb{S}^2\sharp 2 {\mathbb{R} P^2}} ((a;\alpha,2a-c-\alpha),l)$ vanish when $\alpha \notin [0,|2a-c|]$,
the Formula (\ref{Eq:W.in.deg.6.2}) follows.

\end{proof}

Note that in this case, we have

\begin{enumerate}
    \item The $(0,0,1)$-surface class on $E$:  $$h_3:=(1;1,1)|_E=(L_1+L_2){|_E}-E_1{|_E}-E_2{|_E}\in Pic_2(E).$$
\item The $(a,a,0)$-surface class on $E$:
$$h_{a,a}:=(a;0,0)|_E=a(L_1+L_2){|_E}\in Pic_{4a}(E).$$
\end{enumerate}

\begin{remark}
Directly from Corollary \ref{coro-Theorem2-W} and the fact that $$|D.S|=|(a;\alpha,2a-c-\alpha).(0;1,-1)|=|2a-c-2\alpha|\equiv  c \mod 2,$$

one has the invariants $W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)$ vanish if $c$ is even for any $Spin_3$-structure $\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1}$ over $\mathbb{S}^2 \times \mathbb{R} P^1$.
\end{remark}

\begin{proposition} \label{prop-application.deg6.2}
Suppose that $\mathbb{R} E$ is disconnected and that the $(0,0,1)$-surface class and the $(a,a,0)$-surface class ($a\in \mathbb{N}^*$)  are on the pointed-component of $\mathbb{R} E$.     If $c$ is even and if the real configuration $\underline{x}_d$ of $k_d=2a+c$ points in $E$ has an odd number of real points on each component of $\mathbb{R} E$, then there is no real rational curve of class $(a;c)$ in $H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$  passing through $\underline{x}_d$.

\end{proposition}
\begin{proof}
This is a consequence of Theorem \ref{Theorem3-W}, Proposition \ref{prop-application.deg6.1}  and the following remarks.
    \begin{itemize}
        \item The base locus of the pencil of surfaces in the linear system $|-\frac{1}{2}K_{(\mathbb{C} P^1)^3}|$ is an elliptic curve $E$ of homology class
        $$\frac{1}{4}K_{(\mathbb{C} P^1)^3}^2=(2;2)\in H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z}).$$
        Moreover, $E$ is of homology class
        $$(2;1,1)\in  H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z}).$$

             \item By the intersection form on $H_4((\mathbb{C} P^1)^3;\mathbb{Z})$,  we deduce that  the class $$L+T(L)=E_1+E_2=(0;-1,-1)\in H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^2_2;\mathbb{Z})\cong Pic((\mathbb{C} P^1)^2_2).$$
        Taking its restriction to $E$,  we get the divisor
        $$\phi(\overline{L})=h_{1,1}-h_3 \in Pic_2(E)$$
        which is formed by the $(0,0,1)$-surface class  and the  $(1,1,0)$-surface class on $E$.
        \item Moreover, if $0\leq \alpha\leq \frac{1}{2}(2a-c)$ then the degree $|D.E-|D.S|(L.E)|= 2(c+\alpha)$  divisor $\phi(\overline{L},\overline{D})$ in this case is
        \begin{align*}
            \phi(\overline{L},\overline{D})&=(a;2a-c-\alpha,\alpha)|_E-(2a-c-2\alpha)(0;0,-1)|_E\\
           & =(a;0,0)|_E-(2a-c-\alpha) (0;-1,-1)|_E\\
           & = h_{a,a}-(2a-c-\alpha)(h_{1,1}-h_3)
        \end{align*}
        Hence, Equation (\ref{eq-solutions.in.PicE}) becomes
\begin{equation}
     (2a-c-2\alpha)l=\underline{x}_{d}-h_{a,a}+(2a-c-\alpha)(h_{1,1}-h_3).
\end{equation}

    \end{itemize}

Thus, by hypothesis and by Corollary \ref{coro-degree.case.real}.c), the statement follows.
\end{proof}

In the next section, we will show some precise values of the Formula (\ref{eq-Welschinger.formula.dim.7}),  Formula (\ref{Eq:W.in.deg.6.1}) and Formula (\ref{Eq:W.in.deg.6.2}) in Table \ref{Tab:W.in.deg.7}, Table \ref{Tab:W.in.deg.6.1} and Table \ref{Tab:W.in.deg.6.2} respectively.

\subsection{Computations and further comments}
\begin{remark}
The explicit computations of genus-$0$ Welschinger invariants in real del Pezzo surfaces has been studied by E. Brugall\'e and G. Mikhalkin by using floor diagrams in tropical geometry (\cite{brugalle2007enumeration});
I. Itenberg, V. Kharlamov, E. Shustin by using another  tropical techniques
\cite{itenberg2013welschinger}); X. Chen and A. Zinger by using WDVV-type relations (\cite{chen2021wdvv}).
\end{remark}
We list the first values of the following Welschinger invariants

\begin{itemize}
\item $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$,
\item  $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$ and
\item $W_{\mathbb{S}^2\times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2\times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2\times \mathbb{R} P^1}}((a;c),l)$
\end{itemize}
 in the Tables \ref{Tab:W.in.deg.7}, \ref{Tab:W.in.deg.6.1} and \ref{Tab:W.in.deg.6.2} respectively. These computations have been made using \textit{Maple program}.

\subsubsection{Real three-dimensional del Pezzo varieties of degree $7$}
We provide in Table \ref{Tab:W.in.deg.7} the values of the genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$ for small values of $d$.

Note that $0\leq l \leq d-\frac{1}{2}(2d-1-k)$ and $k\leq d$.
Note also that
$W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$ vanishes if $d$ is even.

 Table \ref{Tab:W.in.deg.7} shows computations of  genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$ for the $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^3)_1}$ over $(\mathbb{R} P^3)_1$  such that the spinor state $sp_{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}(f(\mathbb{C} P^1))=+1$ where $$f_*[\mathbb{C} P^1]=(1;0)\in H_2^{-\tau}(\mathbb{C} P^3\sharp \overline{\mathbb{C} P^3};\mathbb{Z}).$$

 Note that in this table, $0 \leq k\leq d\leq 9$.

The following observations can be proven by geometrical explanations or deduced directly from Theorem \ref{Theorem-W.in.deg.7}.
\begin{enumerate}
    \item For every $Spin_3$-structure on $(\mathbb{R} P^3)_1$, for every $d\in \mathbb{N}^*$ and  $l \in \{0,\ldots, \dfrac{1}{2}(2d-1-k)\}$, one has
    $$W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;0),l)=-W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;1),l).$$
    \item  For every $Spin_3$-structure on $(\mathbb{R} P^3)_1$,  for every $d\in \mathbb{N}^*$, $k\in \{\dfrac{1}{2}(d+1),\ldots,d\}$ and for every integer $l$ is such that $l \in \{0,\ldots, \dfrac{1}{2}(3k+1)\}$, one has
    $$W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)=0.$$
    \item  For every $Spin_3$-structure on $(\mathbb{R} P^3)_1$,  for every $k\in \mathbb{N}$ and  $l \in \{0,\ldots, \dfrac{1}{2}(3k+1)\}$, one has
    $$W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((2k+1;k),l)=(-1)^{\frac{1}{2}(k^2-k)}W_{\mathbb{R} P^2}(k+1,l).$$
\end{enumerate}

\subsubsection{Real threefold product of the projective line with the standard real structure}

One has
$$ W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)= W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((\sigma(a),\sigma(b),\sigma(c)),l),$$
where $\sigma$ is a permutation of the set $\{a,b,c\}$.
Without lose of generality, we can suppose that $a\geq b \geq c$ when computing of $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$.

 As a direct consequence of Proposition  \ref{Prop-vanishing.GW} and the remark after Theorem \ref{Theorem-W.in.deg.6.1},
 one has the following proposition.

\begin{proposition}\label{Prop-vanishing.W.6.1}
Suppose that $a\geq b \geq c >1$. Then, for every $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^1)^3}$ over $(\mathbb{R} P^1)^3$, one has
$W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l) = 0$ if one of the following cases holds:
    \begin{itemize}
        \item[$\bullet$] $a> b+c$,
        \item[$\bullet$] $a+b+c$ is even.

    \end{itemize}
\end{proposition}

     Table \ref{Tab:W.in.deg.6.1} shows computations of  genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$ for the $Spin_3$-structure $\mathfrak{s}_{(\mathbb{R} P^1)^3}$ over $(\mathbb{R} P^1)^3$  such that the spinor state $sp_{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}(f(\mathbb{C} P^1))=+1$ where $$f_*[\mathbb{C} P^1]\in \{(0,0,1),(0,1,0),(0,0,1)\}\in H_2^{-\tau_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z}).$$

     Note that in this table, $7\geq a\geq b\geq c\geq 0$ and $a+b+c$ increases up to $15$. Note also that we show only the non-vanishing genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$, i.e. the cases where $a\leq b+c$ and $a+b+c$ odd.

 Computations from Table \ref{Tab:W.in.deg.6.1} lead us to make the following conjecture about the positivity of the Welschinger invariants.
 \begin{conjecture}
     Let $\mathfrak{s}_{(\mathbb{R} P^1)^3}$ be the $Spin_3$-structure over $(\mathbb{R} P^1)^3$ as in Theorem \ref{Theorem2-W}.
Then for all triplets $(a,b,c)\in \mathbb{N}^*\times  \mathbb{N}\times \mathbb{N}$ and for all $l \in \{0,\ldots, \frac{1}{2}(a+b+c-1)\}$, one has
     $$W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)\geq  0.$$
 \end{conjecture}

Moreover, the equality happens exactly if either $a+b+c$ is even or $a>b+c$. In other words,  the statement of Proposition \ref{Prop-vanishing.W.6.1} becomes $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l) = 0$ if and only if either $a+b+c \equiv 0 \mod 2$ or $a>b+c$.

\subsubsection{Real threefold product of the projective line with the twisted real structure}
As a direct consequence of Proposition \ref{Prop-vanishing.GW} and the remark after Theorem \ref{Theorem-W.in.deg.6.2},
one has the following proposition.
\begin{proposition}\label{Prop-vanishing.W.6.2}
 For every $Spin_3$-structure $\mathfrak{s}_{\mathbb{S}^2\times\mathbb{R} P^1}$ over $\mathbb{S}^2\times \mathbb{R} P^1$, one has
$ W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)$ vanishes if one of the following cases holds:
    \begin{itemize}
        \item[$\bullet$] $c>2a$,
        \item[$\bullet$] $c$ is even.
            \end{itemize}
\end{proposition}

 Table \ref{Tab:W.in.deg.6.2} shows computations of  genus-$0$ Welschinger invariants $ W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)$ for the $Spin_3$-structure $\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1}$ over $\mathbb{S}^2\times \mathbb{R} P^1$   such that the spinor state $sp_{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}(f(\mathbb{C} P^1))=+1$ where $f_*[\mathbb{C} P^1]=(0;1)$ in $H_2^{-\tilde{\tau}_{1,1}}((\mathbb{C} P^1)^3;\mathbb{Z})$.

 Note that in this table,  $a\leq 5$ and $2a+c$ increases up to $19$. Note also that we show only the genus-$0$ Welschinger invariants $ W_{\mathbb{S}^2 \times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2 \times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2 \times \mathbb{R} P^1}}((a;c),l)$ in the cases where $c<2a$ and $c$ odd (since in these cases, the Welschinger invariants are "almost" non-zeros by Proposition \ref{Prop-vanishing.W.6.2}).

\begin{table}[H]
\centering
     \begin{tabular}{|l||c|c||c|c|c|c||}\hline
\diagbox[innerwidth=1.5cm]{$l$}{$(d;k)$}&
{$(1;0)$}&{$(1;1)$}&{$(3;0)$}&{$(3;1)$}&{$(3;2)$}&
{$(3;3)$}\\
\hline
\hline
0&1 & $-1$&$-1$&$1$&$0$&$0$\\
\hline
1&&&$-1$&1&0&0\\
\hline
2&&&$-1$&1&&\\

\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c|c|c||}\hline
\diagbox[innerwidth=1.5cm]{$l$}{$(d;k)$}&{$(5;0)$}&{$(5;1)$}&{$(5;2)$}&{$(5;3)$}&{$(5;4)$}&{$(5;5)$}\\
\hline
\hline
0&45&$-45$&$-8$&0&0&0\\
\hline
1&$29$ &$-29$&$-6$&0&0&0\\
\hline
2&$17$ &$-17$&$-4$&0&0&0\\
\hline
3&$9$ &$-9$&$-2$&0&&\\
\hline
4&$5$ &$-5$&&&&\\
\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c|c|c|c|c||}\hline
\diagbox[innerwidth=1.5cm]{$l$}{$(d;k)$}&
{$(7;0)$}&{$(7;1)$}&{$(7;2)$}&{$(7;3)$}&{$(7;4)$}&{$(7;5)$}&{$(7;6)$}&{$(7;7)$}\\
\hline
\hline
0&$-14589$& $14589$&$3816$&$-240$&$0$ &$0$&0&0\\
\hline
1& $-6957$ & $6957$ & $1932$ &$-144$&0&$0$&$0$ &0\\
\hline
2& $-3093$ & $3093$ & $912$ &$-80$&0&$0$&$0$ &0\\
\hline
3& $-1269$ & $1269$ & $396$ &$-40$&0&$0$&$0$ &0\\
\hline
4& $-477$  & $477$ & $152$ &$-16$&0&$0$& &\\
\hline
5& $-173$ & $173$ & $44$ &$0$&&& &\\
\hline
6& $-85$ & $85$ & &&&& &\\
\hline
\hline
\end{tabular}

\vspace{1cm}
\centering
     \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c||}\hline
\diagbox[innerwidth=1.5cm]{$l$}{$(d;k)$}&
{$(9;0)$}&{$(9;1)$}&{$(9;2)$}&{$(9;3)$}&{$(9;4)$}&{$(9;5)$}&{$(9;6)$}&{$(9;7)$}&{$(9;8)$}&{$(9;9)$}\\
\hline
\hline
0&$17756793$& $-17756793$&$-5519664$&$603840$&$18264$ &$0$&0&0&0&0\\
\hline
1& $6717465$ & $-6717465$ & $-2155050$ &$256080$&9096&$0$&$0$ &0&0&0\\
\hline
2& $2407365$ & $-2407365$ & $-797604$ &$102912$&4272&$0$&$0$ &0&0&0\\
\hline
3& $812157$ & $-812157$ & $-278046$ &$38880$&1872&$0$&$0$ &0&0&0\\
\hline
4& $256065$  & $-256065$ & $-90392$ &$13568$&744&$0$&0 &0&0&0\\
\hline
5& $75281$ & $-75281$ & $-27058$ &$4208$&248&0&0 &0&&\\
\hline
6& $21165$ & $-21165$ &$-7500$ &1088&64&0& &&&\\
\hline
7& $6165$ & $-6165$ & $-2086$&256&&& &&&\\
\hline
8& $1993$ & $-1993$ & &&&& &&&\\
\hline
\hline
\end{tabular}
\caption{Genus-$0$ Welschinger invariants $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;k),l)$  of real three-dimensional del Pezzo varieties of degree $7$  via Formula (\ref{eq-Welschinger.formula.dim.7})  with $d\leq 9$.}
\label{Tab:W.in.deg.7}
\end{table}

\begin{table}[H]
\centering
     \begin{tabular}{|l||c||c||c||c|c||}\hline
\diagbox[innerwidth=2cm]{$l$}{$(a,b,c)$}&
{$(1,0,0)$}&{$(1,1,1)$}&{$(2,2,1)$}&{$(3,2,2)$}&{$(3,3,1)$}\\
\hline
\hline
0&$1$& $1$&$1$&$8$&$1$ \\
\hline
1&& $1$&$1$&$6$&$1$ \\
\hline
2&&&$1$&$4$&$1$ \\
\hline
3&&&&$2$&$1$ \\
\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c||c|c|c|c||}\hline
\diagbox[innerwidth=2cm]{$l$}{$(a,b,c)$}&
{$(3,3,3)$}&{$(4,3,2)$}&{$(4,4,1)$}&
{$(4,4,3)$}&{$(5,3,3)$}&{$(5,4,2)$}&{$(5,5,1)$}\\
\hline
\hline
0&$216$&48&1&$3864$& $1086$&$256$&$1$\\
\hline
1&$126$&32&1&$1980$& $606$&$160$&1\\
\hline
2&$68$&20&1&$960$&$318$&$96$&1\\
\hline
3&$34$&12&1&$444$&$158$&$56$&1\\
\hline
4&$16$&8&1&$200$&$78$&$32$&1\\
\hline
5&&&&$92$&$46$&$16$&1\\
\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c|c||}\hline
\diagbox[innerwidth=2cm]{$l$}{$(a,b,c)$}&{$(5,4,4)$}&{$(5,5,3)$}&{$(6,4,3)$}&{$(6,5,2)$}&{$(6,6,1)$}\\
\hline
\hline
0&$174360$ &57608&18424&1280&1\\
\hline
1&$77064$ &27276&9256&768&1\\
\hline
2&$32496$ &12400&4432&448&1\\
\hline
3&$13136$ &5444&2032&256&1\\
\hline
4&5160&2328&904&144&1\\
\hline
5&2040&$988$&408&80&1\\
\hline
6&896 &$448$&224&48&1\\
\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c|c|c|c||}\hline
\diagbox[innerwidth=2cm]{$l$}{$(a,b,c)$}&
{$(5,5,5)$}&{$(6,5,4)$}&{$(6,6,3)$}&{$(7,4,4)$}&{$(7,5,3)$}&{$(7,6,2)$}&{$(7,7,1)$}\\
\hline
\hline
0&15253434&5998848&773808&819200&268575&6144&1\\
\hline
1&5840298&2410496&347202&360896&125855&3584&1\\
\hline
2&2148978&931712&151028&152192&56831&2048&1\\
\hline
3&762450&347520&63958&61568&24831&1152&1\\
\hline
4&262842&125952&26488&24064&10559&640&1\\
\hline
5&89418&45056&10794&9280&4415&352&1\\
\hline
6&31122&16512&4412&3712&1887&192&1\\
\hline
7&12690&6784&1982&1536&991&96&1\\
\hline
\hline
\end{tabular}

\caption{Non-vanishing genus-$0$ Welschinger invariants  $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$ of real three-dimensional del Pezzo varieties of degree $6$ with real part is $\mathbb{R}P^1\times \mathbb{R}P^1\times \mathbb{R}P^1$ via Formula (\ref{Eq:W.in.deg.6.1}) with $a+b+c\leq 15$.}
\label{Tab:W.in.deg.6.1}
\end{table}

 \begin{table}[H]
\centering
     \begin{tabular}{|l||c||c|c||c|c|c||}\hline
\diagbox{$l$}{$(a;c)$}&
{$(1;1)$}&{$(2;1)$}&{$(2;3)$}&{$(3;1)$}&{$(3;3)$}&{$(3;5)$}\\
\hline
\hline
0&$-1$&1&6&$-1$&$-120$&$-576$\\
\hline
1&$-1$&1&4&$-1$&$-62$&$-288$\\
\hline
2&&1&2&$-1$&$-28$&$-128$\\
\hline
3&&&0&$-1$&$-10$&$-48$\\
\hline
4&&&&&$0$&$-16$\\
\hline
5&&&&&&$-16$\\

\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c||}\hline
\diagbox{$l$}{$(a;c)$}&
{$(4;1)$}&{$(4;3)$}&{$(4;5)$}&{$(4;7)$}\\
\hline
\hline
0&1&1692&61704&294336\\
\hline
1&1&768&24048&116352\\
\hline
2&1&324&8592&42624\\
\hline
3&1&128&2760&14208\\
\hline
4&1&44&792&4224\\
\hline
5&&0&224&1152\\
\hline
6&&&96&320\\
\hline
7&&&&$-256$\\
\hline
\hline
\end{tabular}

\vspace{1cm}

\centering
     \begin{tabular}{|l||c|c|c|c|c||}\hline
\diagbox{$l$}{$(a;c)$}&
{$(5;1)$}&{$(5;3)$}&{$(5;5)$}&{$(5;7)$}&{$(5;9)$}\\
\hline
\hline
0&$-1$&$-20760$&$-3792888$&$-97970688$&$-493848576$\\
\hline
1&$-1$&$-8764$&$-1271016$&$-31204224$&$-160966656$\\
\hline
2&$-1$&$-3536$&$-399216$&$-9331200$&$-49582080$\\
\hline
3&$-1$&$-1380$&$-117072$&$-2593536$&$-14303232$\\
\hline
4&$-1$&$-520$&$-32056$&$-663936$&$-3821568$\\
\hline
5&$-1$&$-172$&$-8136$&$-157344$&$-938496$\\
\hline
6&&0&$-1872$&$-35776$&$-215040$\\
\hline
7&&&$-1104$&$-7840$&$-47872$\\
\hline
8&&&&$-4096$&$-10496$\\
\hline
9&&&&&$-26880$\\
\hline
\hline
\end{tabular}

\caption{Genus-$0$ Welschinger invariants $W_{\mathbb{S}^2\times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2\times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2\times \mathbb{R} P^1}}((a;c),l)$ of real three-dimensional del Pezzo varieties of degree $6$ with real part is $\mathbb{S}^2\times \mathbb{R}P^1$  via Formula (\ref{Eq:W.in.deg.6.2}) with $a \leq 5$.}
\label{Tab:W.in.deg.6.2}
\end{table}

\begin{remark}
The values of $W_{(\mathbb{R} P^3)_1}^{\mathfrak{s}_{(\mathbb{R} P^3)_1},\mathfrak{o}_{(\mathbb{R} P^3)_1}}((d;0),l)$ in these computations as shown in Table \ref{Tab:W.in.deg.7} agree with that given by E. Brugall\'e and P. Georgieva (\cite{brugalle2016pencils}).

\end{remark}

\begin{remark}
The values of $W_{(\mathbb{R} P^1)^3}^{\mathfrak{s}_{(\mathbb{R} P^1)^3},\mathfrak{o}_{(\mathbb{R} P^1)^3}}((a,b,c),l)$ and $  W_{\mathbb{S}^2\times \mathbb{R} P^1}^{\mathfrak{s}_{\mathbb{S}^2\times \mathbb{R} P^1},\mathfrak{o}_{\mathbb{S}^2\times \mathbb{R} P^1}}((a;c),l)$ in these computations as shown in Table \ref{Tab:W.in.deg.6.1} and Table \ref{Tab:W.in.deg.6.2} agree with that given by X. Chen and A. Zinger (\cite{chen2021wdvv}, \cite{chen}).

\end{remark}

redpajama_set_name: RedPajamaArXiv