---
abstract: 'We establish a formula for the Gromov-Witten-Welschinger invariants of ${\mathbb{C}P}^3$ with mixed real and conjugate point constraints. The method is based on a suggestion by J. Kollár that, considering pencils of quadrics, some real and complex enumerative invariants of ${\mathbb{C}P}^3$ could be computed in terms of enumerative invariants of ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$ and of elliptic curves.'
address:
- 'Erwan Brugallé, CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France'
- 'Penka Georgieva, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 5, France'
author:
- Erwan Brugallé
- Penka Georgieva
bibliography:
- 'Biblio.bib'
title: 'Pencils of quadrics and Gromov-Witten-Welschinger invariants of ${\mathbb{C}P}^3$'
---

[^1]

Invariant signed counts of real rational curves with point constraints in real surfaces and in many real threefolds were first defined by J.-Y. Welschinger in [@Wel1] and [@Wel2], respectively. In the case of surfaces, various methods for computation of these invariants are established [@Mik1; @Sh8; @Wel4; @Br6b; @HorSol12; @IKS13; @KhaRas13; @Bru14]. In the case of threefolds, methods for computation are developed only in the extremal cases: when all point constraints are real [@Br7], when the number of real point constraints is minimal [@Wel4; @GeorZin13], and in the case of no constraints [@Sol08]. In this paper, we establish a relation between the GWW invariants of ${\mathbb{C}P}^3$ and the GWW invariants of ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$, which allows the computation of the former in terms of the latter. To our knowledge, Theorem \[thm:main\] provides the first systematic explicit computation of the Welschinger invariants of ${\mathbb{C}P}^3$, and more generally of a real algebraic variety of dimension 3, when mixed real and conjugate point constraints are used.\
Throughout the text, we equip ${\mathbb{C}P}^3$ and ${\mathbb{C}P}^1$ with their standard real structure defined by the complex conjugation. We equip ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$ with the product of the standard real structure of ${\mathbb{C}P}^1$, in particular its real part is ${\mathbb{R}P}^1\times{\mathbb{R}P}^1$.\
Given $d\in{\mathbb{Z}}_{>0}$ and ${\underline{x}}$ a generic real configuration of $2d$ points in ${\mathbb{C}P}^3$, with $2l$ non-real points, we denote by $W_{{\mathbb{R}}P^3}(d,l)$ the corresponding Welschinger invariant counting with a sign the real rational curves of degree $d$ in ${\mathbb{C}P}^3$ passing through all points in ${\underline{x}}$. Similarly, given $a,b\in{\mathbb{Z}}_{\ge 0}$ and ${\underline{x}}$ a generic real configuration of $2(a+b)-1$ points in ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$, with $2l$ non-real points, we denote by $W_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}((a,b),l)$ the Welschinger invariant counting with a sign the real rational curves of bidegree $(a,b)$ in ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$ passing through all points in ${\underline{x}}$.

\[thm:main\] For every odd positive integer $d$ and $l\in\{0,\ldots, d-1\}$, one has $$W_{{\mathbb{R}}P^3}(d,l)=\sum_{\substack{a+b=d\\ 0\le a<b}} (-1)^a (d-2a) W_{{\mathbb{R}}P^1\times{\mathbb{R}}P^1}\left((a,b),l\right).$$

By symmetry reasons, G. Mikhalkin proved that $W_{{\mathbb{R}}P^3}(d,l)=0$ for even $d$ (see also Remark \[rem:vanishing\]). The invariants $W_{{\mathbb{R}}P^3}(d,d)$ have been computed in [@GeorZin13].

We also establish a corresponding relation between the Gromov-Witten invariants of ${\mathbb{C}P}^3$ and ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$. We denote by $GW_{{\mathbb{C}}P^3}(d)$ the count of degree $d$ rational curves in ${\mathbb{C}P}^3$ passing through $2d$ generic points and by $GW_{{\mathbb{C}}P^1\times{\mathbb{C}}P^1}\left(a,b\right)$ the count of rational curves of bidegree $(a,b)$ in ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$ passing through $2(a+b)-1$ generic points.

\[thm:kollar\] For every positive integer $d$, one has $$GW_{{\mathbb{C}}P^3}(d)=\sum_{\substack{a+b=d\\ 0\le a<b}} (d-2a)^2 GW_{{\mathbb{C}}P^1\times{\mathbb{C}}P^1}\left(a,b\right).$$

The idea of the proof begins with a result of J. Kollár, [@Kol14 Proposition 3], which can be summarised as follows. A non-degenerate elliptic curve $C_0$ of degree 4 in ${\mathbb{C}P}^3$ generates a pencil of quadrics ${\mathcal{Q}}$ with base locus $C_0$. If the configuration ${\underline{x}}$ of $2d$ points is contained in $C_0$, every curve in degree $d$, passing through all points in ${\underline{x}}$, is contained in a non-singular quadric of ${\mathcal{Q}}$, where it is of bidegree $(a,b)$, with $a\ne b$. Furthermore, all such quadrics can be recovered by a computation in the Jacobian of $C_0$ and in particular there are exactly $(d-2a)^2$ of them. To complete the proof of Theorem \[thm:kollar\] from [@Kol14 Proposition 3], an additional transversality argument is needed: one has to show that a configuration ${\underline{x}}$ contained in $C_0$ can be chosen so that the number of the degree $d$ curves passing through ${\underline{x}}$ is indeed the corresponding Gromov-Witten invariant of ${\mathbb{C}P}^3$. We prove that this is indeed the case in Section \[sec:proof\]. The proof of Theorem \[thm:main\] further requires a comparison of the signs of the curves which enter in the definitions of the Welschinger invariants of ${\mathbb{C}P}^3$ and ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$. This is accomplished in Section \[sec:quadric\].\
Theorems \[thm:main\] and \[thm:kollar\] relate the enumerative geometry of ${\mathbb{C}P}^3$ on one hand and of ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$ and of elliptic curves on the other hand. Both theorems can certainly be generalised to ${\mathbb{C}P}^3$ blown up in a small number of points. It would be interesting to understand further generalisations.\
The paper is organised as follows. We start by recalling the definition of the Welschinger invariants of ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$ and ${\mathbb{C}P}^3$ in Section \[sec:def welsch\]. In Section \[sec:elliptic pencil\] we review some standard facts concerning elliptic curves and pencils of quadrics in ${\mathbb{C}P}^3$. In Section \[sec:quadric\] we investigate properties of rational curves in a smooth quadric of ${\mathbb{C}P}^3$ and establish the comparison of their two Welschinger signs. We combine results from Sections \[sec:elliptic pencil\] and \[sec:quadric\] in Section \[sec:proof\] to provide a transversality argument that allows us to deduce Theorem \[thm:kollar\] from [@Kol14 Proposition 3] and we prove Theorem \[thm:main\]. We end this paper with explicit computations of Welschinger invariants of ${\mathbb{C}P}^3$ together with further qualitative results and comments about them.

We are grateful to B. Bertrand, I. Itenberg, and G. Mikhalkin for useful discussions. We also wish to thank the anonymous referee for his valuable comments on the first version of this text.

Welschinger invariants {#sec:def welsch}
======================

Welschinger invariants of ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$
----------------------------------------------------------------

Given $a,b\in{\mathbb{Z}}_{\ge 0}$ and ${\underline{x}}$ a generic real configuration of $2(a+b)-1$ points in ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$, let ${\mathbb{R}}{\mathcal{C}}'({\underline{x}})$ be the set of all real rational curves of bidegree $(a,b)$ in ${\mathbb{C}P}^1\times {\mathbb{C}P}^1$ passing through all points in ${\underline{x}}$. One can associate a sign $(-1)^{s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(C)}$ to each real curve $C$ in ${\mathbb{R}}{\mathcal{C}}'({\underline{x}})$ such that the sum $$W_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}((a,b),l)=\sum_{C\in {\mathbb{R}}{\mathcal{C}}'({\underline{x}})}(-1)^{s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(C)}$$ only depends on $a$, $b$, and the number $l$ of pairs of complex conjugated points in ${\underline{x}}$; see [@Wel1]. Since ${\underline{x}}$ is generic, every curve $C\in {\mathbb{R}}{\mathcal{C}}'({\underline{x}})$ is nodal. A real node of $C$ is either the intersection of two real branches of $C$, or the intersection of two complex conjugated branches. The node is called hyperbolic in the former and elliptic in the latter case. The number $s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(C)$ is defined to be the number of elliptic real nodes of $C$.\
The parity of $s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(C)$ can be interpreted in terms of a spin structure on ${\mathbb{R}P}^1\times{\mathbb{R}P}^1$ as follows. Since a curve $C\in {\mathbb{R}}{\mathcal{C}}'({\underline{x}})$ is nodal, it has a parametrisation $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^1\times {\mathbb{C}P}^1$ which is a real algebraic immersion. A choice of a trivialisation of $T{\mathbb{R}P}^1$ induces a trivialisation $$\phi_0: T({\mathbb{R}P}^1\times{\mathbb{R}P}^1)\to {\mathbb{R}P}^1\times{\mathbb{R}P}^1\times {\mathbb{R}}^2.$$ The canonical orientation and scalar product on ${\mathbb{R}}^2$ induce, via $\phi_0$, an orientation and a Riemannian metric on $T({\mathbb{R}P}^1\times{\mathbb{R}P}^1)$. In their turn, the latter induce a trivialisation $\phi$ and a Riemannian metric on the ${\mathbb{R}}$-vector bundle $f_{|{\mathbb{R}P}^1}^*T({\mathbb{R}P}^1\times{\mathbb{R}P}^1)$. The tangent bundle $T{\mathbb{R}P}^1$ is naturally a subbundle of $f_{|{\mathbb{R}P}^1}^*T({\mathbb{R}P}^1\times{\mathbb{R}P}^1)$; let ${\mathcal{N}}$ be its orthogonal subbundle. Since $w_1(T{\mathbb{R}P}^1)=0$, there exist a nowhere vanishing smooth section $\sigma_T:{\mathbb{R}P}^1\to T{\mathbb{R}P}^1$. Let $\sigma_{\mathcal{N}}:{\mathbb{R}P}^1\to{\mathcal{N}}$ be a section such that $(\sigma_T,\sigma_{\mathcal{N}})$ is a positive basis of $f_{|{\mathbb{R}P}^1}^*T({\mathbb{R}P}^1\times{\mathbb{R}P}^1)$ and $N$ be the parity of the number of times the basis $\phi\circ (\sigma_T,\sigma_{\mathcal{N}})$ rotates around the canonical basis of ${\mathbb{R}}^2$ as one goes around ${\mathbb{R}P}^1$. In other words, the number $N$ is the parity of the degree of the Gau[ß]{} map of $f({\mathbb{R}P}^1)$. Note that it does not depend on the choice of the trivialisation of $T{\mathbb{R}P}^1$.

\[lem:w2\] One has $s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(f({\mathbb{C}P}^1))=N \mod 2$.

  -----------------------------------------------------------------------------------------------------
   ![Smoothing of $f({\mathbb{R}P}^1)$[]{data-label="fig:node"}](Figures/Node.pdf "fig:"){width="8cm"}
  -----------------------------------------------------------------------------------------------------

We equip $f({\mathbb{R}P}^1)$ with the orientation induced by $\sigma_T$. By smoothing each node of $f({\mathbb{R}P}^1)$ as depicted in Figure \[fig:node\], we obtain a collection $\gamma$ of $n$ disjoint oriented circles embedded in ${\mathbb{R}P}^1\times{\mathbb{R}P}^1$. The sum of the Gau[ß]{} index of all components of $\gamma$ is the Gau[ß]{} index of $f({\mathbb{R}P}^1)$ and we have $$n=1+\kappa \mod 2,$$ where $\kappa$ is the number of hyperbolic nodes of $f({\mathbb{C}P}^1)$. Every embedded circle realising the zero class in $H_1({\mathbb{R}P}^1\times{\mathbb{R}P}^1;{\mathbb{Z}})$, has Gau[ß]{} index $\pm1$. Every embedded circle, which is non-trivial in the homology of $ {\mathbb{R}P}^1\times{\mathbb{R}P}^1$, is isotopic to a closed geodesic. In particular, it has Gau[ß]{} index 0 and represents a class $(p,q)\!\in\!H_1({\mathbb{R}P}^1\times{\mathbb{R}P}^1;{\mathbb{Z}})$, with $p$ and $q$ relatively prime. Furthermore, since the components of the set $\gamma$ do not intersect, all non-trivial components must represent (up to orientation) the same class $(p,q) \!\in\! H_1({\mathbb{R}P}^1\times{\mathbb{R}P}^1;{\mathbb{Z}})$ and the class of $f({\mathbb{R}}P^1)$ can be written as $(mp,mq)$ for some $m\!\in\!{\mathbb{Z}}$. Since the number $N$ is equal to the parity of the number of homologically trivial components in $\gamma$, i.e. $n-m$ mod 2, we have

  ----------------- ------------
  $N=\kappa +1+m$   $ \mod 2$.
  ----------------- ------------

If $f({\mathbb{C}}P^1)$ has bidegree $(a,b)$, by the adjunction formula it has exactly $(a-1)(b-1)$ nodes, $\kappa$ of which are hyperbolic. Thus,

  ------------------------------------------------------------------------------------- ------------
  $N=(a-1)(b-1) +1+m + s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(f({\mathbb{C}P}^1))$   $ \mod 2$.
  ------------------------------------------------------------------------------------- ------------

Since $(mp,mq)=(a,b)\mod 2$, we have that $m$ is even if and only if both $a$ and $b$ are even and so $$s_{{\mathbb{R}P}^1\times {\mathbb{R}P}^1}(f({\mathbb{C}P}^1))=N\mod 2,$$ as announced.

Welschinger invariants of ${\mathbb{C}P}^3$ {#sec:w3}
-------------------------------------------

Given $d\in{\mathbb{Z}}_{\ge 0}$ and ${\underline{x}}$ a generic real configuration of $2d$ points in ${\mathbb{C}P}^3$, let ${\mathbb{R}}{\mathcal{C}}({\underline{x}})$ be the set of all real rational curves of degree $d$ in ${\mathbb{C}P}^3$ passing through all points in ${\underline{x}}$. One can again associate a sign $(-1)^{s_{{\mathbb{R}P}^3}(C)}$ to each real curve $C$ in ${\mathbb{R}}{\mathcal{C}}({\underline{x}})$ such that the sum

$$W_{{\mathbb{R}}P^3}(d,l)=\sum_{C\in {\mathbb{R}}{\mathcal{C}}({\underline{x}})}(-1)^{s_{{\mathbb{R}P}^3}(C)}$$ only depends on $d$ and the number $l$ of pairs of complex conjugated points in ${\underline{x}}$; see [@Wel2]. We now recall the definition of $s_{{\mathbb{R}P}^3}(C)$.\
We fix once and for all an orientation on ${\mathbb{R}P}^3$. The projective space ${\mathbb{R}P}^3$ is spin and we may choose a trivialisation $\phi_0:T{\mathbb{R}P}^3\to {\mathbb{R}P}^3\times {\mathbb{R}}^3$ of its tangent bundle compatible with the chosen orientation. The pullback by $\phi_0$ of the canonical Euclidean scalar product on ${\mathbb{R}}^3$ provides a Riemannian metric on ${\mathbb{R}P}^3$. Let $f:{\mathbb{C}P}^1\to{\mathbb{C}P}^3$ be a real algebraic immersion. The trivialisation of $T{\mathbb{R}P}^3$ induces a trivialisation and a Riemannian metric on the ${\mathbb{R}}$-vector bundle $f_{|{\mathbb{R}P}^1}^*T{\mathbb{R}P}^3$. The tangent bundle $T{\mathbb{R}P}^1$ is naturally a ${\mathbb{R}}$-subbundle of $f_{|{\mathbb{R}P}^1}^*T{\mathbb{R}P}^3$; let ${\mathcal{N}}_{\mathbb{R}}$ be its orthogonal ${\mathbb{R}}$-subbundle. Choose an orientation of ${\mathbb{R}P}^1$ and a positive orthonormal section $\sigma_T$ of $T{\mathbb{R}P}^1$. Given a line ${\mathbb{R}}$-subbundle $E$ of ${\mathcal{N}}_{\mathbb{R}}$ with a non-vanishing smooth section $\sigma_E$, such that $(\sigma_T,\sigma_E)$ is an orthonormal section of $T{\mathbb{R}P}^1\oplus E$, there exists a unique choice of a section $\sigma_{\mathcal{N}}$ of ${\mathcal{N}}_{\mathbb{R}}$, such that $(\sigma_T,\sigma_E,\sigma_{\mathcal{N}})$ is a positive orthonormal section of $f_{|{\mathbb{R}P}^1}^*T{\mathbb{R}P}^3$. Combining with the trivialisation $\phi_0$ of the latter bundle, this section defines a loop in $SO_3({\mathbb{R}})$. We define $$s(E)\in \{0,1\}$$ to be the number realised by this loop in $\pi_1(SO_3({\mathbb{R}}))=\{0,1\}$. Note that $s(E)$ only depends on the isotopy class of $E$ as an ${\mathbb{R}}$-subbundle of ${\mathcal{N}}_{\mathbb{R}}$ and on the homotopy class of the restriction of $\phi_0$ to $T{\mathbb{R}}P^3_{|f({\mathbb{R}}P^1)}$.\
Let $d$ be the degree of $f({\mathbb{C}P}^1)$ in ${\mathbb{C}P}^3$. The quotient ${\mathcal{N}}_{\mathbb{C}}$ of $f^*T{\mathbb{C}P}^3$ by $T{\mathbb{C}P}^1$ is a holomorphic vector bundle over ${\mathbb{C}P}^1$ with first Chern class $4d-2$. Suppose that the curve $f({\mathbb{C}P}^1)$ is *balanced*, i.e. ${\mathcal{N}}_{\mathbb{C}}$ is isomorphic to the holomorphic bundle $\mathcal O_{{\mathbb{C}P}^1}(2d-1)\oplus \mathcal O_{{\mathbb{C}P}^1}(2d-1)$. In this case, there is a consistent way of choosing a line ${\mathbb{R}}$-subbundle $L(f)$ of the ${\mathbb{R}}$-bundle ${\mathcal{N}}_{\mathbb{R}}$ as follows. Holomorphic line subbundles of ${\mathcal{N}}_{\mathbb{C}}$ are in one-to-one correspondence with rational functions $F:{\mathbb{C}P}^1\to{\mathbb{C}P}^1$: a fiber of such subbundle over the point $u$ has equation $w=F(u)z$, where $(u,z,w)\subset {\mathbb{C}}^3$ are local coordinates on ${\mathcal{N}}_{\mathbb{C}}$ such that both $(u,z)$ and $(u,w)$ are local coordinates on $\mathcal
O_{{\mathbb{C}P}^1}(2d-1)$. Moreover, a holomorphic line subbundle of ${\mathcal{N}}_{\mathbb{C}}$ defined by the rational function $F$ has degree $2d-1-\deg F$ and is real if and only if $F$ is real. In particular, up to real isotopy, there exists a unique holomorphic real line subbundle of ${\mathcal{N}}_{\mathbb{C}}$ of degree $2d-1$ and two holomorphic real line subbundles of ${\mathcal{N}}_{\mathbb{C}}$ of degree $2d-2$, depending on whether $F_{|{\mathbb{R}P}^1}$ is orientation preserving or not. Phrased differently, the two (real isotopy classes of) holomorphic real line subbundles of ${\mathcal{N}}_{\mathbb{C}}$ of degree $2d-2$ are characterised by the direction in which the real part of a fiber rotates in ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$. The Riemannian metric on ${\mathbb{R}P}^3$ identifies the ${\mathbb{R}}$-bundles ${\mathcal{N}}_{\mathbb{R}}$ and ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$. In particular, the section $\sigma_T$ of $T{\mathbb{R}P}^1$ together with the orientation of ${\mathbb{R}P}^3$ induce an orientation on the bundle ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$: given $u\in
{\mathbb{R}P}^1$, a positive basis of the fiber at $u$ of $f_{|{\mathbb{R}P}^1}^*T{\mathbb{R}P}^3$ is formed by $(\sigma_T(u), v_1,v_2)$ with $(v_1,v_2)$ a positive basis of ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$. We denote by $L(f)$ (resp. $\overline L(f)$) the isotopy class of the real part of the degree $2d-2$ line subbundle of ${\mathcal{N}}_{\mathbb{C}}$ whose real fibers rotate positively (resp. negatively) in local holomorphic coordinates on ${\mathcal{N}}_{\mathbb{C}}$ defining a real holomorphic splitting ${\mathcal{N}}_{\mathbb{C}}=\mathcal
O_{{\mathbb{C}P}^1}(2d-1)\oplus\mathcal O_{{\mathbb{C}P}^1}(2d-1)$. Since $L(f )$ and $\overline L(f )$ differ by exactly one full rotation, which is a generator of $\pi_1(SO_2({\mathbb{R}}))$, we have $s(L(f )) \ne s(\overline L(f ))$.\
For a generic real configuration ${\underline{x}}$ of points in ${\mathbb{R}P}^3$, every curve $C$ in ${\mathbb{R}}{\mathcal{C}}({\underline{x}})$ is parametrised by a balanced immersion $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ and the number $s_{{\mathbb{R}P}^3}(C)$ is defined as $$s_{{\mathbb{R}P}^3}(C)=s(L(f)).$$ The number $s_{{\mathbb{R}P}^3}(C)$ is independent of the parametrisation but in general depends on the choice of a trivialisation $\phi_0$. In the remaining of this note, we always assume that $\phi_0$ is chosen so that $s_{{\mathbb{R}P}^3}(D)=0$ for a line $D$ in ${\mathbb{C}P}^3$. [^2]

Elliptic curves and pencil of quadrics in ${\mathbb{C}P}^3$ {#sec:elliptic pencil}
===========================================================

In this section we recall some known facts about the Picard group of complex and real elliptic curves, their torsion points, and their relation with pencils of quadrics in ${\mathbb{C}P}^3$.

Torsion points of complex and real elliptic curves {#sec:elliptic}
--------------------------------------------------

Let $C_0$ be a complex elliptic curve. Recall that a choice of $p_0\in C_0$ induces an isomorphism $$\begin{array}{cccc}
\psi:&C_0 &\longrightarrow & {\text{Pic}}_0(C_0)
\\ & p &\longmapsto & [p]-[p_0],
\end{array}$$ which induces in its turn a group structure on $C_0$. Geometrically, writing $C_0$ as the quotient of ${\mathbb{C}}$ by a full rank lattice $\Lambda$ for which $p_0$ is the orbit of $0$, the group structure induced by $\psi$ on $C_0$ is simply the group structure inherited from $({\mathbb{C}},+)$ by the quotient map. This description allows to easily describe torsion points of order $d$ on $C_0$: if $\Lambda=u{\mathbb{Z}}+ v{\mathbb{Z}}$, with $u$ and $v$ two complex numbers, then the set of solutions of $$\label{equ:torsion}
dp=0$$ is a group of order $d^2$ isomorphic to ${\mathbb{Z}}/d{\mathbb{Z}}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$ and generated by $\frac{1}{d}u$ and $\frac{1}{d}v$.\
Analogously, the map $\psi$ induces the following series of isomorphisms $\Psi_d$ with $d\in{\mathbb{Z}}$: $$\begin{array}{cccc}
\Psi_d:&{\text{Pic}}_d(C_0) &\longrightarrow & C_0
\\ & \sum_{i=1}^d \left[p_i\right] &\longmapsto & \psi^{-1}\left(\sum_{i=1}^dp_i - dp_0\right)
\end{array}$$ satisfying $$\Psi_d(E)+\Psi_{d'}(E')= \Psi_{d+d'}(E+E').$$

Suppose now that $C_0$ is real, with ${\mathbb{R}}C_0\ne \emptyset$, and that $p_0\in{\mathbb{R}}C_0$. If ${\mathbb{R}}C_0$ is not connected, the connected component of ${\mathbb{R}}C_0$ containing $p_0$ is called the *pointed component* of ${\mathbb{R}}C_0$. The real structure on $C_0$ induces a real structure on ${\text{Pic}}_d(C_0)$ for every $d\in{\mathbb{Z}}$ and the maps $\Psi_d$ are all real maps (see [@GroHar81]). Recall also (see [@Nat90]) that $C_0$ can be expressed as ${\mathbb{C}}/\Lambda$ with the real structure inherited by the complex conjugation on ${\mathbb{C}}$, where $\Lambda$ has one of the following forms:

-   $\Lambda=u{\mathbb{Z}}+ iv {\mathbb{Z}}$ with $u$ and $v$ two real numbers. In this case ${\mathbb{R}}C_0$ has two connected components: ${\mathbb{R}}/u{\mathbb{Z}}$ and $({\mathbb{R}}+\frac{iv}{2})/u{\mathbb{Z}}$ (see Figure \[fig:real ell\]a). When $d$ is even, both connected components of ${\mathbb{R}}C_0$ contain exactly $d$ solutions of Equation $(\ref{equ:torsion})$. When $d$ is odd, Equation $(\ref{equ:torsion})$ has exactly $d$ real solutions, all located on the pointed component of ${\mathbb{R}}C_0$.

-   $\Lambda= u{\mathbb{Z}}+ \overline u {\mathbb{Z}}$ with $u$ a complex number. In this case ${\mathbb{R}}C_0={\mathbb{R}}/(u+\overline u){\mathbb{Z}}$ is connected (see Figure \[fig:real ell\]b). Equation $(\ref{equ:torsion})$ has exactly $d$ real solutions for any $d$.

[ccc]{} ![Uniformisation of real elliptic curves with a non-empty real part. The points represent the solutions of $3p=0$.[]{data-label="fig:real ell"}](Figures/Elliptic2cc.pdf "fig:"){height="3.5cm"} (5, 2)[$u$]{} (-135, 95)[$iv$]{} (5, 52)[${\mathbb{R}}+\frac{iv}{2}$]{} & & ![Uniformisation of real elliptic curves with a non-empty real part. The points represent the solutions of $3p=0$.[]{data-label="fig:real ell"}](Figures/Elliptic1cc.pdf "fig:"){height="3.5cm"} (-80, 95)[$u$]{} (-80, 2)[$\overline u$]{} (5, 52)[${\mathbb{R}}$]{}\
\
a) A maximal real elliptic curve&& b) A real elliptic curve with a connected real part

In this paper, all considered elliptic curves are assumed to be equipped with a distinguished point $p_0$, which is real if the curve is real. In particular, we always identify ${\text{Pic}}_d(C_0)$ with $C_0$ via the map $\Psi_d$.

Pencils of quadrics {#sec:pencil}
-------------------

Let ${\mathcal{Q}}$ be a pencil of quadrics in ${\mathbb{C}P}^3$, i.e. a line in the space ${\mathbb{C}P}^{9}$ of quadrics in ${\mathbb{C}P}^3$. We assume that ${\mathcal{Q}}$ is generic enough so that the base locus of ${\mathcal{Q}}$ is a non-degenerate elliptic curve $C_0$ of degree 4 in ${\mathbb{C}P}^3$. Conversely, every non-degenerate elliptic curve $C_0$ of degree 4 in ${\mathbb{C}P}^3$ defines a pencil of quadrics with base locus $C_0$. We denote by $h\in C_0\simeq {\text{Pic}}_4(C_0) $ the hyperplane section class.\
A non-singular quadric in ${\mathbb{C}P}^3$ is isomorphic to ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$ and admits a ruling by two families of lines in ${\mathbb{C}P}^3$; let $D_1$ and $D_2$ be two lines representing the families. Since $C_0$ is of bidegree $(2,2)$ in the non-singular quadrics of ${\mathcal{Q}}$, any such quadric defines two elements $E_i=D_i\cap C_0$ of $ C_0\simeq {\text{Pic}}_2(C_0)$, whose sum is $h$. Conversely, given $E\in{\text{Pic}}_2(C_0)$, the union of all lines in ${\mathbb{C}P}^3$, whose intersection with $C_0$ is a divisor defining $E$ or $h-E$, is a quadric $Q_E$ in ${\mathcal{Q}}$. Hence, the map $$\begin{array}{cccc}
\pi_{{\mathcal{Q}}}:&C_0\simeq {\text{Pic}}_2(C_0) &\longrightarrow & {\mathcal{Q}}\\  & E  &\longmapsto & Q_E
\end{array}$$ is a ramified covering of degree two. The ramification values of $\pi_{{\mathcal{Q}}}$ correspond to singular quadrics in ${\mathcal{Q}}$, which are all quadratic cones. The corresponding critical points of $\pi_{{\mathcal{Q}}}$ are the solutions of the equation $2E=h$ and so any two of them differ by a torsion point of order 2. Thus, ${\mathcal{Q}}$ contains exactly 4 distinct singular quadrics, in accordance with the Riemann-Hurwitz formula.\
Suppose now that $C_0$ is real with ${\mathbb{R}}C_0\ne\emptyset$. In this case, the pencil ${\mathcal{Q}}$ is real and the fiber over a regular value $Q\in{\mathbb{R}}{\mathcal{Q}}$ of $\pi_{\mathcal{Q}}$ consists of two real points (resp. two complex conjugate points) if and only if ${\mathbb{R}}Q$ is homeomorphic to $S^1\times S^1$ (resp. $S^2$). By the description of the real torsion points of $C_0$ given in Section \[sec:elliptic\], we have the following possibilities for the map $\pi_{{\mathcal{Q}}|{\mathbb{R}}C_0}$:

-   ${\mathbb{R}}C_0$ is not connected and $h$ is on the non-pointed component of ${\mathbb{R}}C_0$ (equivalently, both components of ${\mathbb{R}}C_0$ are non-trivial in $\pi_1({\mathbb{R}P}^3)$): none of the four singular fibers of ${\mathcal{Q}}$ are real (see Figure \[fig:real pencil\]a),

-   ${\mathbb{R}}C_0$ is not connected and $h$ is on the pointed component of ${\mathbb{R}}C_0$ (equivalently, both components of ${\mathbb{R}}C_0$ are trivial in $\pi_1({\mathbb{R}P}^3)$): the four singular fibers of ${\mathcal{Q}}$ are real (see Figure \[fig:real pencil\]b),

-   ${\mathbb{R}}C_0$ is connected: exactly two singular fibers of ${\mathcal{Q}}$ are real (see Figure \[fig:real pencil\]c).

[ccccc]{} ![Real pencils of quadrics[]{data-label="fig:real pencil"}](Figures/RealPencil2.pdf "fig:"){height="5.5cm"} (-15, 0)[${\mathbb{R}}{\mathcal{Q}}$]{} (-5, 120)[${\mathbb{R}}C_0$]{} (-80, 65)[$\pi_{\mathcal{Q}}$]{} & & ![Real pencils of quadrics[]{data-label="fig:real pencil"}](Figures/RealPencil1.pdf "fig:"){height="5.5cm"} & &![Real pencils of quadrics[]{data-label="fig:real pencil"}](Figures/RealPencil3.pdf "fig:"){height="5.5cm"}\
\
a) && b)&& c)

Rational curves on a smooth quadric of ${\mathbb{C}P}^3$ {#sec:quadric}
========================================================

In this section we study properties of rational curves in a smooth quadric in ${\mathbb{C}P}^3$ and establish a comparison of the signs of a real curve as an element of the quadric and of the projective space.\
Throughout the section we fix a smooth quadric $Q$ of ${\mathbb{C}P}^3$. In particular, $Q$ is isomorphic to ${\mathbb{C}P}^1\times{\mathbb{C}P}^1$ and we denote by $D_1$ and $D_2$ two distinct intersecting lines in $Q$. Given an algebraic immersion $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ contained in $Q$, there is an exact sequence of holomorphic vector bundles over ${\mathbb{C}P}^1$: $$\label{exseq} 0\to {\mathcal{N}}'_{{\mathbb{C}}} \to {\mathcal{N}}_{\mathbb{C}}\to f^*{\mathcal{N}}_Q\to 0,$$ where ${\mathcal{N}}'_{{\mathbb{C}}}$ is the quotient bundle $f^* TQ/T{\mathbb{C}P}^1$, ${\mathcal{N}}_{{\mathbb{C}}}$ is the quotient bundle $f^* T{\mathbb{C}P}^3/T{\mathbb{C}P}^1$, and ${\mathcal{N}}_Q=T{\mathbb{C}P}^3 |_{Q}/TQ$ is the normal bundle of $Q$ in ${\mathbb{C}P}^3$. When $Q$ and $f$ are real, we also have the corresponding real bundles ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}', {\mathbb{R}}{\mathcal{N}}_{\mathbb{C}},$ and ${\mathbb{R}}{\mathcal{N}}_Q$ fitting in an exact sequence as above.

\[prop:balanced2\] Let $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ be an algebraic immersion such that $f({\mathbb{C}P}^1)$ is contained in $Q$, where it has bidegree $(a,b)$, with $a\ne b$. Then, $f$ is balanced.

We want to prove that the normal bundle ${\mathcal{N}}_{\mathbb{C}}$ is isomorphic to the holomorphic bundle $\mathcal O_{{\mathbb{C}P}^1}(2d-1)\oplus \mathcal
O_{{\mathbb{C}P}^1}(2d-1)$, where $d=a+b$. By the adjunction formula, the line bundle ${\mathcal{N}}'_{\mathbb{C}}$ has degree $2d-2$ and thus $f^*{\mathcal{N}}_Q$ has degree $2d$. Hence, the map $f$ is balanced if and only if the sequence (\[exseq\]) does not split. According to [@GrifHar83 Theorem 4.f.3], whose proof extends to immersions, this sequence splits if and only if $f({\mathbb{C}}P^1)$ is a complete intersection. Since $a\ne b$, this is not the case.\
We briefly recall the main lines of the proof of [@GrifHar83 Theorem 4.f.3]. Suppose that the above exact sequence splits and let $C_0\subset Q$ be an elliptic curve of bidegree $(2,2)$ intersecting $f({\mathbb{C}}P^1)$ transversely. Then, there exists a holomorphic section $\sigma$ of $ {\mathcal{N}}_{\mathbb{C}}$, which vanishes only at the $2d$ points of $f({\mathbb{C}P}^1)\cap
 C_0$. If ${\mathcal{Q}}$ denotes the pencil of quadrics in ${\mathbb{C}P}^3$ with base locus $C_0$, the section $\sigma$ corresponds to a first order deformation $f_{\varepsilon}:{\mathbb{C}P}^1\to Q_{\varepsilon}$ of $f$ in the pencil ${\mathcal{Q}}$. Since $\sigma$ vanishes at the $2d$ points in $f({\mathbb{C}P}^1)\cap C_0$, the divisor class realised in ${\text{Pic}}_{2d}(C_0)$ by $f_{{\varepsilon}}({\mathbb{C}P}^1)\cap  C_0$ is constant. On the other hand, as explained in Section \[sec:pencil\], the quadric $Q_{\varepsilon}$ in ${\mathcal{Q}}$ is determined by the class $E_{1,{\varepsilon}}$ realised by $D_{1,{\varepsilon}}\cap C_0$ in ${\text{Pic}}_2(C_0)$, where $D_{i,{\varepsilon}}$ is the deformation in ${\mathcal{Q}}_{\varepsilon}$ of $D_i$. Since $f_{{\varepsilon}}({\mathbb{C}P}^1)$ realises the class $aD_{1,{\varepsilon}}+bD_{2,{\varepsilon}}= (a-b)D_{1,{\varepsilon}} + bH$ in ${\text{Pic}}({\mathbb{C}P}^1\times{\mathbb{C}P}^1)$, where $H$ is the hyperplane section, $f_{{\varepsilon}}({\mathbb{C}P}^1)\cap C_0$ realises the class $(a-b)E_{1,{\varepsilon}} + bh$ in ${\text{Pic}}_{2d}(C_0)$. However, the class $h$ is constant along the deformation, whereas the class $(a-b)E_{1,{\varepsilon}}$ is not, since $a-b\ne 0$. This is a contradiction.

Suppose now that $Q$ is real with a real part homeomorphic to ${\mathbb{R}P}^1\times{\mathbb{R}P}^1$ and that $D_1$ and $D_2$ are also real. Recall that, for a real algebraic immersion $f:{\mathbb{C}}P^1\rightarrow {\mathbb{C}}P^3$, the isotopy classes $L(f)$ and $\overline L(f)$ and the numbers $s(E)$, for an orientable ${\mathbb{R}}$-subbundle $E\subset {\mathbb{R}}\mathcal{N}_{\mathbb{C}}$, have been defined in Section \[sec:w3\].

\[prop:lines\] Let ${\mathbb{R}}\mathcal{N}'_i$ be the real part of the normal bundle of $D_i$ in $Q$, considered as a subbundle of the real part of its normal bundle in ${\mathbb{C}}P^3$. Then, one has $s({\mathbb{R}}\mathcal{N}'_1)\ne s({\mathbb{R}}\mathcal{N}'_2)$.

Given a line $D$ in ${\mathbb{C}P}^3$, the holomorphic line subbundles of degree 1 of its normal bundle in ${\mathbb{C}P}^3$ correspond precisely to the planes of ${\mathbb{C}P}^3$ containing $D$. In particular, the normal bundle ${\mathbb{R}}\mathcal{N}'_i$ realises the isotopy class $L(D_i)$ if and only if it rotates in ${\mathbb{R}P}^3$ around ${\mathbb{R}}D_i$ in the positive direction. Since ${\mathbb{R}}\mathcal{N}'_1$ and ${\mathbb{R}}\mathcal{N}'_2$ do not rotate in the same direction, the result follows.

We will say that $(D_1,D_2)$ is *the positive basis* of $H_2(Q;{\mathbb{Z}})$ if $s({\mathbb{R}}\mathcal{N}'_1)=0$.

\[prop:eps-eps’\] Let $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ be a real algebraic immersion such that $f({\mathbb{C}P}^1)$ is contained in $Q$, where it has a bidegree $(a,b)$ in the positive basis, with $a\ne b$. Then, one has $$s({\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}})= s_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}(f({\mathbb{C}P}^1)) +b.$$

The proof is similar to the proof of Lemma \[lem:w2\]. Equip ${\mathbb{R}P}^1$ with some orientation and smooth each node of $f({\mathbb{R}P}^1)$, as depicted in Figure \[fig:node\], in order to obtain a collection $\gamma$ of oriented circles embedded in ${\mathbb{R}P}^1\times{\mathbb{R}P}^1$. The loop in $\pi_1(SO_3({\mathbb{R}}))$ defined by ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$ is freely homotopic to the product of the loops in $\pi_1(SO_3({\mathbb{R}}))$ defined by $T{\mathbb{R}}Q_{|\gamma_i}/T\gamma_i$ for $\gamma_i$ ranging over elements of $\gamma$. Any loop in $\gamma$ realising the 0 class (resp. the class $p[{\mathbb{R}}D_1] + q[{\mathbb{R}}D_2]$ with gcd$(p,q)$=1) in $H_1({\mathbb{R}P}^1\times{\mathbb{R}P}^1;{\mathbb{Z}})$ defines a non-trivial loop (resp. $q$ times the non-trivial loop) in $\pi_1(SO_3({\mathbb{R}}))$. Now the end of the proof is similar to the end of the proof of Lemma \[lem:w2\].

\[prop:L or not\] Let $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ be a real algebraic immersion such that $f({\mathbb{C}P}^1)$ is contained in $Q$, where it has a bidegree $(a,b)$ in the positive basis, with $a\ne b$. Then, the holomorphic real line subbundle ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$ of ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$ realises the real isotopy class $L(f)$ if and only if $a>b$.

By convention, we have $s_{{\mathbb{R}P}^3}(D)=0$ for a real line $D$ in ${\mathbb{C}P}^3$. Hence, according to Proposition \[prop:lines\], the proposition is true for $(a,b)=(1,0)$ and $(a,b)=(0,1)$.

Recall how the two isotopy classes $L(f)$ and $\overline L(f)$ are characterised: in local holomorphic coordinates on ${\mathcal{N}}_{\mathbb{C}}$ defining a real splitting ${\mathcal{N}}_{\mathbb{C}}=\mathcal
O_{{\mathbb{C}P}^1}(2d-1)\oplus\mathcal O_{{\mathbb{C}P}^1}(2d-1)$, the classes $L(f)$ and $\overline L(f)$ rotate in different directions in ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$. The positive direction of rotation is determined by an orientation of ${\mathbb{R}}P^1$ and of ${\mathbb{R}}P^3$; the class $L(f)$ is by definition the one which rotates in the positive direction. Hence to prove the proposition, it is enough to find a point $u_0\in{\mathbb{R}P}^1$ and a real holomorphic subbundle $E$ of ${\mathcal{N}}_{\mathbb{C}}$ of degree $2d-1$ such that

-   the fibers of $E$ and ${\mathcal{N}}'_{\mathbb{C}}$ over $u_0$ coincide and

-   we can determine the mutual position of the real parts of the fibers of these bundles over a point $u\in{\mathbb{R}P}^1$ in a neighbourhood of $u_0$.

Let $C_0\subset Q$ be a real elliptic curve of bidegree $(2,2)$, with ${\mathbb{R}}C_0\ne 0$, that intersects $f({\mathbb{C}P}^1)$ transversely at some point $p_0=f(u_0)$ with $u_0\in {\mathbb{R}P}^1 $. We denote by ${\mathcal{Q}}$ the real pencil of quadrics defined by $C_0$. In particular, ${\mathcal{Q}}$ is the realisation of a first order real deformation $Q_{\varepsilon}$ of the real quadric $Q$ in ${\mathbb{C}P}^3$. Denoting by $D_{i,{\varepsilon}}$ the deformation of $D_i$ in $Q_{\varepsilon}$ and by $E_{i,{\varepsilon}}\in{\text{Pic}}_2(C_0)$ the class realised by $D_{i,{\varepsilon}}\cap C_0$, recall that $$\label{equ:diff}
\frac{d E_{i,{\varepsilon}}}{d {\varepsilon}}|_{{\varepsilon}=0}\ne 0.$$

Let $f_{\varepsilon}$ be a first order real deformation of $f$ in the pencil ${\mathcal{Q}}$ such that $f_{\varepsilon}({\mathbb{C}P}^1)$ passes through the $2d-1$ points of $f({\mathbb{C}P}^1)\cap
 C_0\setminus\{p_0\}$ for all ${\varepsilon}$. This deformation corresponds to a non-null real holomorphic section $\sigma:{\mathbb{C}P}^1\to {\mathcal{N}}_{\mathbb{C}}$ that vanishes on $f^{-1}\left( C_0\setminus\{p_0\}\right)$. Recall that, since $a\ne b$, the class realised by $f_{\varepsilon}({\mathbb{C}P}^1)\cap C_0$ in ${\text{Pic}}_{2d}(C_0)$ is not constant. In particular, $\sigma(u_0)\ne 0$. Let $E$ be the real holomorphic subbundle of ${\mathcal{N}}_{\mathbb{C}}$ of degree $2d-1$ whose fiber over $u_0$ is the line generated by $\sigma(u_0)$.

$\sigma$ is a section of $E$. Indeed, $\sigma$ induces a holomorphic section of the bundle ${\mathcal{N}}_{\mathbb{C}}/E$ that vanishes at the $2d$ points of $f^{-1}\left(C_0\right)$. Since the latter bundle has degree $2d-1$, this induced section must be the null section i.e. $\sigma(u)\in E$ for all $u\in{\mathbb{C}P}^1$.

the fibers of $E$ and ${\mathcal{N}}'_{\mathbb{C}}$ over $u_0$ coincide. Indeed, the vector $\sigma(u_0)$ corresponds to the preimage of the deformation of $p_0$ in $f_{\varepsilon}({\mathbb{C}P}^1)\cap C_0$. Since this deformation stays in $C_0$ by definition and $C_0$ sits in $Q$, we must have $\sigma(u_0)\in  {\mathcal{N}}'_{\mathbb{C}}$.

the isotopy class realised by ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$ is determined by the direction of the vector $\sigma(u_0)$. Let us denote by $\sigma_Q$ the holomorphic section of $T{\mathbb{C}P}^3/T Q$ corresponding to the deformation ${\mathcal{Q}}$. Since both ${\mathbb{R}P}^3$ and ${\mathbb{R}}Q$ are orientable manifolds, we may also fix a smooth nowhere vanishing section $\lambda_Q$ of the ${\mathbb{R}}$-vector bundle $T{\mathbb{R}P}^3/T{\mathbb{R}}Q$ (see Figure \[fig:L or not\]a for a local picture at $p_0$). The section $\sigma_Q$ vanishes along $C_0$ and we denote by ${\mathbb{R}}Q_+$ the region of ${\mathbb{R}}Q\setminus {\mathbb{R}}C_0$, where $\sigma_Q$ and $\lambda_Q$ have the same direction (see Figure \[fig:L or not\]b). The choices of $\lambda_Q$ and $\sigma_Q$ induce an orientation on the real part of our source curve: we orient ${\mathbb{R}P}^1$ so that $f({\mathbb{R}P}^1)$ points toward ${\mathbb{R}}Q_+$ at $f(u_0)$. The choice of a Riemannian metric on ${\mathbb{R}P}^3$ identifies the ${\mathbb{R}}$-bundle $f^*(T{\mathbb{R}P}^3/T {\mathbb{R}}Q)$ with the orthogonal of ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$ in ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$. With this identification, the vector $\sigma(u)$ for $u\in{\mathbb{R}P}^1$ close enough to $u_0$ decomposes as $$\sigma(u)=  g_1(u)\sigma(u_0)  + g_2(u)\lambda_Q(u),$$ where $g_1$ is a smooth function with $g_1(u_0)=1$ and $g_2$ is a smooth function vanishing at $u_0$ and positive for $u>u_0$ (the local ordering of ${\mathbb{R}P}^1$ at $u_0$ is given by the orientation of ${\mathbb{R}P}^1$). In other words, the choice of $\lambda_Q$ and $\sigma_Q$ determine an orientation of the fiber of ${\mathbb{R}}{\mathcal{N}}_{\mathbb{C}}$ over $u_0$ together with a half-plane $\Pi\subset {\mathbb{R}}{\mathcal{N}}_{{\mathbb{C}}| u_0}\setminus
{\mathbb{R}}{\mathcal{N}}'_{{\mathbb{C}}| u_0}$ containing $\sigma(u)$ when $u>u_0$ (see Figure \[fig:L or not\] c, d). Now clearly, the direction in which ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$ rotates with respect to ${\mathbb{R}}E$ depends only on the direction of $\sigma(u_0)$.

[cc]{} ![Determining the isotopy class of ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$[]{data-label="fig:L or not"}](Figures/Def1.pdf "fig:"){height="4cm"} (-110, 105)[${\mathbb{R}}Q$]{} (5, 70)[${\mathbb{R}}C_0$]{} (-50, 50)[$p_0$]{} (-80, 70)[$\lambda_Q$]{} & ![Determining the isotopy class of ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$[]{data-label="fig:L or not"}](Figures/Def2.pdf "fig:"){height="4cm"} (-150, 85)[${\mathbb{R}}Q_+$]{} (3, 95)[${\mathbb{R}}Q_{{\varepsilon}}$]{}\
a) & b) ${\varepsilon}>0$\
\
![Determining the isotopy class of ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$[]{data-label="fig:L or not"}](Figures/Def3.pdf "fig:"){height="4cm"} (-120, 85)[$f({\mathbb{R}}P^1)$]{} (-70, 55)[$\Pi$]{} (-270, 55)[${\mathbb{R}}{\mathcal{N}}_{{\mathbb{C}}| u_0}$]{} & ![Determining the isotopy class of ${\mathbb{R}}{\mathcal{N}}'_{\mathbb{C}}$[]{data-label="fig:L or not"}](Figures/Def4.pdf "fig:"){height="4cm"} (-90, 40)[$\sigma(u_0)$]{} (-90, 80)[$\sigma(u)$]{} (-40, 40)[$\sigma(u_0)$]{} (-40, 80)[$\sigma(u)$]{} (-100, 100)[$\Pi$]{} (5, 5)[${\mathbb{R}}{\mathcal{N}}_{{\mathbb{C}}| u_0}$]{} (5, 55)[${\mathbb{R}}{\mathcal{N}}'_{{\mathbb{C}}| u_0}$]{}\
\
c) & d)

As in the proof of Proposition \[prop:balanced2\], the deformation $p_{\varepsilon}$ of $p_0$ as an intersection point of $f_{\varepsilon}({\mathbb{C}P}^1)$ and $C_0$ is determined by the condition that $f_{{\varepsilon}}({\mathbb{C}P}^1)\cap C_0$ has to realise the class $(a-b)E_{1,{\varepsilon}} + bh$ in ${\text{Pic}}_{2d}(C_0)$. Since the class $h$ is constant, Inequality $(\ref{equ:diff})$ implies that the direction of the vector $$\frac{d p_{{\varepsilon}}}{d {\varepsilon}}|_{{\varepsilon}=0}$$ is the same as for the line $D_1$ if $a>b$ and opposite if $a<b$. Since the statement of the proposition holds for the classes $(1,0)$ and $(0,1)$, the proof is complete.

\[prop:W2-W3\] Let $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$ be a real algebraic immersion such that $f({\mathbb{C}P}^1)$ is contained in $Q$, where it has a bidegree $(a,b)$ in the positive basis, with $a\ne b$. Then, one has $$s_{{\mathbb{R}P}^3}(f({\mathbb{C}P}^1))= \left\{\begin{array}{ll}
s_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}(f({\mathbb{C}P}^1)) + b+1\,,& \mbox{if $a<b$,}
\\
\\  s_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}(f({\mathbb{C}P}^1)) +b\,,& \mbox{if $a>b$.}

\end{array}\right.$$

Proofs of Theorems \[thm:main\] and \[thm:kollar\] {#sec:proof}
==================================================

In this section we recall in Proposition \[prop:kollar\] the main statement for our purposes from [@Kol14] and we prove transversality results needed to deduce Theorem \[thm:kollar\] from Proposition \[prop:kollar\]. We finish the section with the proof of Theorem \[thm:main\].\
Let $C_0$ be a non-degenerate elliptic curve of degree 4 in ${\mathbb{C}P}^3$ and ${\underline{x}}\subset C_0$ be a configuration of $2d$ distinct points. We denote by ${\mathcal{Q}}$ the pencil of quadrics induced by $C_0$ and by $\overline {\mathcal{C}}({\underline{x}})$ the set of connected algebraic curves of arithmetic genus 0 and degree $d$ in ${\mathbb{C}P}^3$ that contain ${\underline{x}}$. Recall that $h\in{\text{Pic}}_4(C_0)$ denotes the hyperplane section.

\[prop:kollar\] Suppose that the points in ${\underline{x}}$ are in general position in $C_0$. Then, every curve in $\overline {\mathcal{C}}({\underline{x}})$ is irreducible and contained in a quadric of ${\mathcal{Q}}$. Furthermore, the quadrics of ${\mathcal{Q}}$, that contain a curve in $\overline {\mathcal{C}}({\underline{x}})$, are exactly the images under $\pi_{\mathcal{Q}}$ of the solutions $E\in{\text{Pic}}_2(C_0)$ of the equation $$\label{equ:pencil}
(d-2a)E=  (d-a)h -{\underline{x}},$$ with $0\le a<\frac{d}2$.\
If $Q$ is such a quadric and $C$ is a curve in $\overline {\mathcal{C}}({\underline{x}})$, then $C$ is linearly equivalent in $Q$ to $aD_1 + (d-a)D_2$, where $D_1$ (resp. $D_2$) is a line in $Q$ whose intersection with $C_0$ is $E$ (resp. $h-E$). Conversely, any irreducible rational curve $C$ in $Q$, linearly equivalent to $aD_1+ (d-a)D_2$ and containing $2d-1$ of the points in ${\underline{x}}$, is in $\overline {\mathcal{C}}({\underline{x}})$ (i.e. contains ${\underline{x}}$).

Any two solutions of Equation $(\ref{equ:pencil})$ differ by a torsion point of order $d-2a$; in particular, Equation $(\ref{equ:pencil})$ has exactly $(d-2a)^2$ solutions in $C_0$. Let $\mathcal
M^*_{0,2d}({\mathbb{C}P}^3,d)$ be the space of stable maps $f\!:\!({\mathbb{C}P}^1,x_1,\ldots,x_{2d})\to{\mathbb{C}P}^3$ from ${\mathbb{C}P}^1$ with $2d$ marked points $x_1,\ldots,x_{2d}$, considered up to reparametrisation, whose image has degree $d$. The evaluation map $ev$ is defined as $$\begin{array}{cccc}
ev:& \mathcal M^*_{0,2d}({\mathbb{C}P}^3,d) &\longrightarrow & ({\mathbb{C}P}^3)^{2d}
\\  & f &\longmapsto & (f(x_1),\ldots,f(x_{2d})).
\end{array}$$

To complete the proof of Theorem \[thm:kollar\], it remains to prove that one can choose ${\underline{x}}\subset C_0$ so that the map $ev$ is regular at every curve in $\overline
{\mathcal{C}}({\underline{x}})$ and that each quadric of ${\mathcal{Q}}$, solution to Equation $(\ref{equ:pencil})$, contains exactly $GW_{{\mathbb{C}}P^1\times{\mathbb{C}}P^1}\left(a,d-a\right)$ elements of $\overline
{\mathcal{C}}({\underline{x}})$. This is done in the next two propositions. Denote by $$V_{n}\subset C_0^{n}\subset ({\mathbb{C}P}^3)^{n}$$ the set of configurations of $n$ distinct points on $C_0$.

\[prop:generic quadric\] Let $Q$ be a quadric in ${\mathcal{Q}}$. Given an integer $a\in\{0,\ldots, d\}$ and ${\underline{y}}$ in $V_{2d-1}$, we denote by $\overline {\mathcal{C}}'_{Q,a}({\underline{y}})$ the set of stable maps $f:(C,x_1,\ldots,x_{2d-1})\to Q$, with $C$ a connected nodal curve of arithmetic genus 0, such that $f(C)$ has bidegree $(a,d-a)$ and $f(\{x_1,\ldots,x_{2d-1}\})={\underline{y}}\in V_{2d-1}$. Then, there exists a dense open subset $U_{2d-1}\subset V_{2d-1}$ such that for every choice of $a$ and ${\underline{y}}\in U_{2d-1}$, and for every stable map $f:(C,x_1,\ldots,x_{2d-1})\to Q$ in $\overline {\mathcal{C}}'_{Q,a}({\underline{y}})$, one has

-   $C$ is non-singular,

-   $f$ is an immersion.

In particular, for every configuration ${\underline{y}}\in U_{2d-1}$, the quadric $Q$ contains exactly $ GW_{{\mathbb{C}}P^1\times{\mathbb{C}}P^1}(a,d-a)$ rational curves of bidegree $(a,d-a)$ and passing through ${\underline{y}}$.

Let $\mathcal V_a$ be the set of irreducible nodal rational curves in $Q$ of bidegree $(a,d-a)$. This is a quasiprojective subvariety of dimension $2d-1$ of the linear system $|aD_1 + (d-a)D_2|$ and we denote by $\overline{\mathcal V}_a$ its Zariski closure. We also denote by $\mathcal U_a$ the linear system on $C_0$ defined by the restriction to $C_0$ of the divisor $aD_1 + (d-a)D_2$. Since $C_0$ has bidegree $(2,2)$ in $Q$, the linear system $\mathcal
U_a$ has degree $2d$. By the Riemann-Roch theorem, every element of $\mathcal U_a$ is determined by $2d-1$ of its points in $C_0$. In particular, every element ${\underline{y}}$ of $V_{2d-1}$ induces an element $[{\underline{y}}]$ of $ \mathcal U_a $. Since $C_0$ cannot be a component of a curve in $\overline{\mathcal V}_a$, the map $$\begin{array}{cccc}
\phi: & \overline{\mathcal V}_a&\longrightarrow & \mathcal U_a 
\\ & C_1 &\longmapsto & C_1\cap C_0
\end{array}$$ is well defined, generically finite, and $\dim \phi(\overline{\mathcal
  V}_a\setminus\mathcal V_a)\le 2d-2$. Hence if ${\underline{y}}\in V_{2d-1}$ is so that $[{\underline{y}}]\notin\phi(\overline{\mathcal
  V}_a\setminus\mathcal V_a)$, then for every element $f:(C,x_1,\ldots,x_{2d-1})\to Q$ of $\overline {\mathcal{C}}'_{Q,a}({\underline{y}})$, the curve $f(C)$ must be a nodal irreducible rational curve. In other words, we have $C={\mathbb{C}P}^1$ and $f$ is an immersion.

Let $\mathcal
M^*_{0,2d-1}(Q,(a,d-a))$ be the space of stable maps $f:({\mathbb{C}P}^1,x_1,\ldots,x_{2d-1})\to Q$ from ${\mathbb{C}P}^1$ with $2d-1$ marked points, considered up to reparametrisation, whose image has bidegree $(a,d-a)$. Since $Q\simeq {\mathbb{C}P}^1\times{\mathbb{C}P}^1$ is convex, the proof of [[@Wel2 Lemma 1.3]]{} implies that a point $f$ in $\mathcal M^*_{0,2d-1}(Q,(a,d-a))$ is regular for the corresponding evaluation map if and only if it is an immersion. This completes the proof.

\[prop:balanced\] Regular values of $ev$ contained in $V_{2d}$ form a dense open subset $U_{2d}\subset V_{2d}$. In particular, if ${\underline{x}}\in U_{2d}$, then the set $\overline{\mathcal{C}}({\underline{x}})$ contains exactly $GW_{{\mathbb{C}P}^3}(d)$ elements.

By [[@Wel2 Lemma 1.3]]{}, an element of $\mathcal
M^*_{0,2d}({\mathbb{C}P}^3,d)$ is a regular point of $ev$ if and only if it is a balanced immersion. Let ${\underline{x}}$ be a configuration of $2d$ points on $C_0$ for which the conclusions of Proposition \[prop:kollar\] hold. Note that the quadrics, solutions to Equation $(\ref{equ:pencil})$, do not change if ${\underline{x}}$ is replaced by a configuration of points linearly equivalent to ${\underline{x}}$ in $C_0$. By the Riemann-Roch theorem, every effective divisor linearly equivalent to ${\underline{x}}$ in $C_0$ is determined by $2d-1$ points in $C_0$. Hence, according to Proposition \[prop:generic quadric\], if ${\underline{x}}$ is chosen generically among all configurations of $2d$ points in $C_0$ linearly equivalent to a given generic divisor of degree $2d$, all elements in $\overline {\mathcal{C}}({\underline{x}})$ will be immersions. Thus, we are left to show that every element $f:({\mathbb{C}P}^1,x_1,\ldots,x_{2d})\to{\mathbb{C}P}^3$ of $\overline {\mathcal{C}}({\underline{x}})$ is balanced, which follows from Proposition \[prop:balanced2\].

[***Proof of Theorem \[thm:kollar\].***]{} Let $U_{2d}$ be as in Proposition \[prop:balanced\]. Choose ${\underline{x}}\in U_{2d}$ and ${\underline{y}}\subset{\underline{x}}$ a set of $2d-1$ points in ${\underline{x}}$. We denote by $\mathcal W_a $ the set of quadrics in ${\mathcal{Q}}$ corresponding to a solution of Equation $(\ref{equ:pencil})$. As explained above, this set contains $(d-2a)^2$ elements. By Proposition \[prop:kollar\], we know that the equality $$|\overline {\mathcal{C}}({\underline{x}})|=\sum_{0\leq a<b} \ \sum_{Q\in \mathcal W_a}|\overline
 {\mathcal{C}}'_{Q,a}({\underline{y}})|$$ holds. By Proposition \[prop:balanced\], we have $|\overline {\mathcal{C}}({\underline{x}})|=GW_{{\mathbb{C}P}^3}(d)$. Since ${\underline{x}}$ is a regular value of the map $ev$, every map in $\overline {\mathcal{C}}({\underline{x}})$ is an immersion $f:{\mathbb{C}P}^1\to {\mathbb{C}P}^3$. Hence by Proposition \[prop:generic quadric\], we have $|\overline
{\mathcal{C}}'_{Q,a}({\underline{y}})|=GW_{{\mathbb{C}P}^1\times{\mathbb{C}P}^1}(a,b)$.  $\hfill\qed$\
[***Proof of Theorem \[thm:main\].***]{} Suppose now that $C_0$ is real, with ${\mathbb{R}}C_0\ne 0$, and ${\underline{x}}\in U_{2d}$ is a real configuration of $2d$ points, containing at least one real point.

Let $d$ be an odd positive integer and $Q$ be a quadric containing a real map $f\!:\!{\mathbb{C}P}^1\to{\mathbb{C}P}^3\in \overline{\mathcal{C}}({\underline{x}})$. According to Proposition \[prop:kollar\], there exists an integer $a\in\{0,\ldots,\frac{d-1}{2} \}$ such that $f({\mathbb{C}P}^1)$ has bidegree either $(a,d-a)$ or $(d-a,a)$ in the positive basis of $Q$. From Corollary \[prop:W2-W3\], we deduce that $$s_{{\mathbb{R}P}^3}(f({\mathbb{C}P}^1))= \left\{\begin{array}{ll}
s_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}(f({\mathbb{C}P}^1))\,,& \mbox{if $a$ is even,}
\\
\\  s_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}(f({\mathbb{C}P}^1)) +1\,,& \mbox{if $a$ is odd.}

\end{array}\right.$$

Thus, the total contribution to $W_{{\mathbb{R}P}^3}(d,l)$ of elements of $\overline {\mathcal{C}}({\underline{x}})$, whose image is contained in $Q$, is $W_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}((a,d-a),l)$ if $a$ is even and $-W_{{\mathbb{R}P}^1\times{\mathbb{R}P}^1}((a,d-a),l)$ if $a$ odd. Equation $(\ref{equ:pencil})$ always has a real solution and two real solutions differ by a real torsion element of order $d-2a$. Hence, Equation $(\ref{equ:pencil})$ has exactly $d-2a$ real solutions for every $a\in\{0,\ldots,\frac{d-1}{2}\}$. Now the end of the proof is similar to the proof of Theorem \[thm:kollar\]. $\hfill\qed$

\[rem:vanishing\] One can also prove the vanishing of $W_{{\mathbb{R}P}^3}(d,l)$ for $d$ even and $l\le
d-1$ in this way. This is also how J. Kollár exhibited configurations ${\underline{x}}$ for which $ \overline {\mathcal{C}}({\underline{x}})$ contains no real curves: if $h$ is on the pointed component of ${\mathbb{R}}C_0$ and ${\underline{x}}$ is on the non-pointed component, then Equation $(\ref{equ:pencil})$ has no real solution.

Computations and further comments {#sec:compute}
=================================

We provide in Tables \[tab1\] and \[tab2\] the values of $W_{{\mathbb{R}P}^3}(d,l)$ for small values of $d$. The values $W_{{\mathbb{R}P}^3}(d,d)$ are taken from [@GeorZin13]. Our values of $W_{{\mathbb{R}P}^3}(d,0)$ agree with the ones computed in [@Br7] and [@Br8].

$$\begin{array}{|c|r|r|r|r|r|r|r|r|r|}
\hline \mbox{\backslashbox{l}{d}} & 1&3&5&7&9 &11& 13 
\\ \hline
0 & 1 &  -1 & 45& -14589& 17756793 &-58445425017 
&426876362998821
\\\hline 1 & 1 & -1 & 29 & -6957 & 6717465 & -18318948633&
114201657733941
\\\hline 2 & & -1 & 17 & -3093 &2407365  & -5495423913 &
29447853240537
\\\hline 3 & & -1 & 9 & -1269& 812157 & -1571343273 & 7298043143697
\\\hline 4 & & & 5 & -477 &  256065 & -426170217 & 1732456594269
\\\hline 5 &  &  & 5 & -173 &   75281 &  -109136649 & 392521356477
\\\hline 6 &  &  & & -85  & 21165 & -26389305 & 84651531633
\\\hline 7 &  & &  & -85  &6165 & -6109369 &  17390628729
\\\hline 8 &  & &  &  & 1993 & -1401241 & 3432362709
\\\hline 9 &  & &  &  & 1993 & -336441 &  663105669 
\\\hline 10 & & &  &  & & -136457 & 129344841
\\\hline 11 & & &  &  & & -136457 &  27607073
\\\hline 12 & & &  &  & & & 3991693
\\\hline 13 & & &  &  & & & 3991693
\\\hline 
\end{array}$$

$$\begin{array}{|c|r|r|}
\hline \mbox{\backslashbox{l}{d}} &  15 &17 
\\ \hline
0 & -6061743911446054965 & 152244625648721441783409 
\\\hline 1 & -1414422922125979269 & 31497207519483035166897
\\\hline 2 & -319737783634469757 & 6337510847893018140813
\\\hline 3 & -69876860779936989 & 1238195460245786397189
\\\hline 4 & -14727767907263157 & 234469282186353521817
\\\hline 5 &  -2985647746084965 & 42946188374781866313
\\\hline 6 &  -580664589588189 & 7592707791183642453
\\\hline 7 &  -108170761670685 & 1293343577697132477
\\\hline 8 & -19320554509557 &  212071309052944257
\\\hline 9 &  -3327374698245 & 33506171960522913
\\\hline 10 & -558961586685 & 5121214631258589
\\\hline 11 & -93320976413 & 763120829396277
\\\hline 12 &  -16000904949 & 112222758491433
\\\hline 13 &  -2937725541 & 16596074817721
\\\hline 14 & -1580831965 & 2542297019941
\\\hline 15 &  -1580831965&  447392666733
\\\hline 16 &  & -129358296175
\\\hline 17 &  & -129358296175
\\\hline 
\end{array}$$

The tables show that the values of the Welschinger invariants with 2 or 0 real points are the same in the computed degrees. This lead us to make the following conjecture.

\[prop:2-no real\] For every positive integer $d$, one has $$W_{{\mathbb{R}}P^3}(d,d-1)=W_{{\mathbb{R}}P^3}(d,d).$$

From the first values of $W_{{\mathbb{R}P}^3}(d,l)$, it might be tempting to conjecture that $$(-1)^{k} W_{{\mathbb{R}}P^3}(2k+1,l)\ge
(-1)^{k} W_{{\mathbb{R}}P^3}(2k+1,l+1)\ge 0.$$ However, both inequalities turn out not to hold in general, starting in degrees 17 and 19: $$W_{{\mathbb{R}}P^3}(17,16)<0\quad\mbox{and}\quad  
-W_{{\mathbb{R}}P^3}(19,17)=74131154312945<106335656443537 = -W_{{\mathbb{R}}P^3}(19,18).$$ In fact, the sign of $W_{{\mathbb{R}}P^3}$ seems to obey an analogous rule as the one observed in [@Bru14] for the Welschinger invariants of ${\mathbb{R}P}^2$: as $l$ goes from $0$ to $d-1$, the numbers $(-1)^kW_{{\mathbb{R}P}^3}(2k+1,l)$ are first positive and then, starting from some mysterious threshold, have an alternating sign. It would be interesting to investigate what governs the sign of the Welschinger invariants.

We end this paper by establishing a congruence modulo 4 between the Welschinger and the Gromov-Witten invariants, generalising results from [@Br7] and [@GeorZin13].

\[prop:mod W2\] For every positive integers $a$, $b$, and $l\in\{0,\ldots, a+b-1\}$, one has $$GW_{{\mathbb{C}}P^1\times{\mathbb{C}P}^1}(a,b)= W_{{\mathbb{R}}P^1\times{\mathbb{R}P}^1}((a,b),l) \mod 4.$$

Both invariants can be computed via the enumeration of marked floor diagrams (see [@Br6b]) with Newton polygon the rectangle with vertices $$(0,0), \ (a,0), \ (0,b), \ \mbox{and}\ (a,b).$$ Any floor diagram with this Newton polygon has only floors with divergence 0. Hence, every marked floor diagram contributing to $GW_{{\mathbb{C}}P^1\times{\mathbb{C}P}^1}(a,b)$ has only non-negative real multiplicities. Now the result follows from the fact that the absolute value of a real multiplicity of a marked floor diagram is always equal modulo 4 to its complex multiplicity.

For every positive integers $d$ and $l\in\{0,\ldots,d\}$, one has $$GW_{{\mathbb{C}}P^3}(d)= (-1)^{\frac{(d-1)(d-2)}2} W_{{\mathbb{R}}P^3}(d,l) \mod 4.$$

When $d$ is even, this follows from Theorem \[thm:kollar\] and the vanishing of $W_{{\mathbb{R}}P^3}(d,l) $. When $d$ is odd, this follows from Theorem \[thm:main\], Proposition \[prop:mod W2\], and the congruence $(-1)^{u}=2u+1 \mod 4 $.

In [@Br7] it was also shown that the inequality $(-1)^{\frac{(d-1)(d-2)}2} W_{{\mathbb{R}}P^3}(d,0)\ge 0$ holds for all $d$. It is not clear how to deduce this inequality from Theorem \[thm:main\].

[^1]: P.G. is supported by ERC grant STEIN-259118

[^2]: Such a choice is possible since every trivialization of $T{\mathbb{R}}P^3$ over the 2-skeleton extends to the 3-skeleton. The two homotopy classes of trivializations over the 2-skeleton correspond to different values of $s_{{\mathbb{R}}P^3}(D)$.
