---
abstract: 'A real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$ yields a real anti-bicanonical curve $s \cup  A^\prime_1$ (disjoint union) on the $4$-th real Hirzebruch surface ${\mathbb{F}}_4$ where $s$ is the exceptional section of ${\mathbb{F}}_4$ and the real curve $A^\prime_1$ has one real double point. (See Section \[RealK3-311\] below.) We give a criterion (Proposition \[criterion\]) which determines whether the real double point is degenerate or not. One direction of the assertion of Proposition \[criterion\] has already been proved in Lemma 4.6 in the preceding paper [@SaitoSachiko2015]. In this paper we prove the inverse direction.'
address: |
    Department of Mathematics Education, Asahikawa Campus,\
    Hokkaido University of Education, Asahikawa, JAPAN
author:
- Sachiko Saito
title: 'On real anti-bicanonical curves with one double point on the $4$-th real Hirzebruch surface. II'
---

Introduction:  Review of real $2$-elementary K3 surfaces {#real_2-elementary K3}
========================================================

Real $2$-elementary K3 surfaces
-------------------------------

In this paper we mainly discuss K3 surfaces $X$ with a non-symplectic holomorphic involution $\tau$. We often call them [*$2$-elementary K3 surfaces*]{} $(X,\tau)$ ([@Nikulin81], [@AlexeevNikulin2006], [@NikulinSaito05], [@NikulinSaito07], [@SaitoSachiko2015], and e.t.c.). Note that every K3 surface with a non-symplectic holomorphic involution is algebraic. Hence, it has hyperplane sections.

We say that a triple $(X,\tau,\varphi)$ is a [*real*]{} K3 surface with non-symplectic holomorphic involution (or [*real*]{} $2$-elementary K3 surface) if\
  [(1)]{} $(X,\tau)$ is a K3 surface $X$ with a non-symplectic holomorphic involution $\tau$,\
  [(2)]{} $\varphi$ is an anti-holomorphic involution on $X$, and\
  [(3)]{} $\varphi \circ \tau = \tau \circ \varphi$.

For a $2$-elementary K3 surface $(X,\tau)$, let $${H_2}_+(X, {\mathbb{Z}})$$ denote the fixed part of $\tau_* : H_2(X, {\mathbb{Z}}) \to H_2(X, {\mathbb{Z}})$. It is well-known that $H_2(X, {\mathbb{Z}})$ is an even unimodular lattice of signature $(3,19)$. ${H_2}_+(X, {\mathbb{Z}})$ is a primitive hyperbolic $2$-elementary sublattice of $H_2(X, {\mathbb{Z}})$. Note that $${H_2}_+(X, {\mathbb{Z}}) \subset {\mathop\mathrm{Pic}}(X),$$ where ${\mathop\mathrm{Pic}}(X)$ denotes the Picard lattice of $X$.

——————————————————————————————————————————–

[ partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research 25610001 (2013/4 — 2016/3).\
[*2010 AMS Mathematics Subject Classification*]{}:  14J28, 14P25, 14J10.]{}

Let ${\mathbb{L}}_{K3}$ be an even unimodular lattice of signature $(3,19)$ and fix it. Note that the isometry class of ${\mathbb{L}}_{K3}$ is unique. Let $$S \ \ (\subset {\mathbb{L}}_{K3})$$ be a primitive hyperbolic $2$-elementary sublattice of ${\mathbb{L}}_{K3}$.

We set $r(S) := {\mathop\mathrm{rank}}S$.   The non-negative integer $a(S)$ is defined by $S^\ast /S \cong ({\mathbb{Z}}/2{\mathbb{Z}})^{a(S)}$. We define the “parity" $\delta (S)$ of $S$ as follows. $$\delta (S) := \left\{
\begin{array}{cl}
0 &\ \ \ \mbox{if}\ z \cdot \sigma (z) \equiv 0 {\ \mathrm{mod}\ }2 \ \ (\forall z \in {\mathbb{L}}_{K3})\\
1 &\ \ \ \mbox{otherwise,}
\end{array}
\right.$$ where $\sigma : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ is the unique integral involution whose fixed part is $S$.

It is known that the triplet $(r(S),a(S),\delta(S))$ determines the isometry class of the lattice $S$ ([@Nikulin81]). Moreover, if $S$ and $S^\prime$ are isometric primitive hyperbolic $2$-elementary sublattices of the K3 lattice ${\mathbb{L}}_{K3}$, then there exists an ambient automorphism $f$ of ${\mathbb{L}}_{K3}$ such that $f(S^\prime) = S$ ([@AlexeevNikulin2006], [@Nikulin79]).

We fix a half cone $$V^+(S)$$ of the cone $$V(S):= \{ x \in S \otimes {\mathbb{R}}\ |\ x^2 > 0\}.$$

Moreover, we fix a fundamental subdivision $$\Delta(S)=\Delta(S)_+\cup -\Delta(S)_+$$ of all elements with square $-2$ in $S$.

This is equivalent to fixing a fundamental (closed) chamber (see [@NikulinSaito05]) $${\mathcal{M}}\ \ \ \ (\subset V^+(S))$$ for the group $W^{(-2)}(S)$ generated by reflections in all elements with square $(-2)$ in $S$.

Note that ${\mathcal{M}}$ and $\Delta(S)_+$ define each other by the condition ${\mathcal{M}}\cdot \Delta(S)_+ \ge 0$.

Let $\theta$ be an integral involution of $S$.

\[real\_2-elementary K3\_S\_theta\] We say that $(X,\tau,\varphi)$ is a real $2$-elementary K3 surface [*of type $(S,\theta)$*]{} if there exists an isometry (so-called “marking" later) $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$ and the following diagram commutes: $$\begin{CD}
{H_2}_+(X, {\mathbb{Z}}) @> {\alpha}>>S\\
@V{\varphi_*}VV @VV{\theta}V\\
{H_2}_+(X, {\mathbb{Z}}) @> {\alpha}>>S .
\end{CD}$$

\[marked\_real\_K3\] We define that a [*marked real $2$-elementary K3 surface of type $(S,\theta)$*]{} is a pair $$((X,\tau,\varphi),\ \alpha)$$ of a real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $(S,\theta)$ (Definition \[real\_2-elementary K3\_S\_theta\] above) and an isometry, which is called [*marking*]{}, $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that

-   $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$,\

-   $\alpha \circ \varphi_* = \theta \circ \alpha \ \ \ \text{on} \ {H_2}_+(X, {\mathbb{Z}})$,\

-   $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$, where $\alpha_{{\mathbb{R}}}$ stands for the real extension of $\alpha$, and\

-   the set $\alpha^{-1}(\Delta(S)_+)$ contains only effective classes of $X$.

Note that ([@NikulinSaito05]) for any $(X,\tau)$, we can take $\alpha$ such that $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$.

Integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$
----------------------------------------------------------------

Let $S$ be a hyperbolic $2$-elementary sublattice of ${\mathbb{L}}_{K3}$ and $\theta : S \to S$ be an integral involution (as above).

Let $\psi : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ be an integral involution of the lattice ${\mathbb{L}}_{K3}$ such that the following diagram commutes: $$\begin{array}{rcl}
         S           & \subset &  {\mathbb{L}}_{K3}          \\
\theta \ \downarrow  &         &  \downarrow \ \psi \\
         S           & \subset &  {\mathbb{L}}_{K3} .
\end{array}$$ We call such a pair $({\mathbb{L}}_{K3},\psi)$ (or $\psi$ itself) an [*integral involution of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$*]{}.

Let $((X,\tau,\varphi),\ \alpha)$ be a marked real $2$-elementary K3 surface of type $(S,\theta)$ as above. If we set $$\psi := \alpha \circ \varphi_* \circ \alpha^{-1} : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3},$$ then we have $\psi(S) = S$, and $\psi (x) = \theta (x)$ for every $x \in S$. Hence, $({\mathbb{L}}_{K3},\psi)$ is an integral involution of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$.

We call the integral involution $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ [*the associated integral involution*]{} with a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ if the following diagram commutes: $$\begin{CD}
H_2(X, {\mathbb{Z}}) @>{\alpha}>>{\mathbb{L}}_{K3} \\
@V{\varphi_*}VV @VV{\psi}V\\
H_2(X, {\mathbb{Z}}) @>{\alpha}>>{\mathbb{L}}_{K3} .
\end{CD}$$

Let $\Delta(S,L)^{(-4)}$ be the set of all elements $\delta_1$ in $S$ such that $\delta_1^2=-4$ and there exists $\delta_2\in S^\perp_L$ such that $(\delta_2)^2=-4$ and $\delta=(\delta_1+\delta_2)/2\in L$. Let $W^{(-4)}(S,L)$ be the subgroup of $O(S)$ generated by reflections in all elements in $\Delta(S,L)^{(-4)}$, and $W^{(-4)}(S,L)_{\mathcal{M}}$ be the stabilizer subgroup of ${\mathcal{M}}$ in $W^{(-4)}(S,L)$. We define the subgroup $G$ to be generated by reflections $s_{\delta_1}$ in all elements $\delta_1\in \Delta(S,L)^{(-4)}$ which are contained either in $S_+$ or in $S_-$ and satisfy $(s_{\delta_1})_{{\mathbb{R}}}({\mathcal{M}})={\mathcal{M}}$, where $s_{\delta_1}$ denotes the reflection at the orthogonal hyperplane ${\delta_1}^{\perp}$ on $S$, $(s_{\delta_1})_{{\mathbb{R}}}$ stands for the real extension of $s_{\delta_1}$, and we set $S_{\pm} := \{ x\in S \,|\, \theta(x) = \pm x \}$. Then $G$ is a subgroup of $W^{(-4)}(S,L)_{\mathcal{M}}$.

Let $({\mathbb{L}}_{K3},\psi_1)$ and $({\mathbb{L}}_{K3},\psi_2)$ be two integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$. We define that an [*isometry with respect to the group*]{} $G$ from $({\mathbb{L}}_{K3},\psi_1)$ to $({\mathbb{L}}_{K3},\psi_2)$ is an isometry $f: {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ such that $f(S)=S$, ${f|}_S \in G$, and the following diagram commutes: $$\begin{CD}
{\mathbb{L}}_{K3} @>{f}>>{\mathbb{L}}_{K3} \\
@VV\psi_1 V@V\psi_2 VV\\
{\mathbb{L}}_{K3} @>{f}>>{\mathbb{L}}_{K3} .
\end{CD}$$ We say that two integral involutions $({\mathbb{L}}_{K3},\psi_1)$ and $({\mathbb{L}}_{K3},\psi_2)$ of type $(S,\theta)$ are [*isometric with respect to the group $G$*]{} if there exists an isometry with respect to the group $G$ from $({\mathbb{L}}_{K3},\psi_1)$ to $({\mathbb{L}}_{K3},\psi_2)$. By an [*automorphism*]{} of an integral involution $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ [*with respect to the group $G$*]{} we mean an isometry with respect to the group $G$ from $({\mathbb{L}}_{K3},\psi)$ to itself. Namely, an isometry $f: {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ which satisfies that $\psi \circ f = f \circ \psi,\ f(S)=S \ \text{and} \  {f|}_S \in G$.

\[analytic-iso\] We say that two marked real $2$-elementary K3 surfaces $((X,\tau,\varphi),\ \alpha)$ and $((X^\prime,\tau^\prime,\varphi^\prime),\alpha^\prime)$ of type $(S,\theta)$ are [*analytically isomorphic with respect to the group*]{} $G$ if there exists an analytic isomorphism $f : X \to X^\prime$ such that $f \circ \tau = \tau^\prime \circ f$, $f \circ \varphi = \varphi^\prime \circ f$ and $\alpha^\prime \circ f_* \circ \alpha^{-1}|S \in G $.

Period domains {#period domains section}
--------------

Now let us [**fix**]{} an integral involution $$({\mathbb{L}}_{K3},\psi)$$ of type $(S,\theta)$ throughout this subsection. We follow the formulations of period domains of marked real $2$-elementary K3 surfaces (see Itenberg [@Itenberg92] and Nikulin-Saito [@NikulinSaito05]).

We set $$\Omega_\psi := 
\{ \omega \ (\in {\mathbb{L}}_{K3}\otimes {\mathbb{C}}) \ |\ \omega \cdot \omega =0, \ \omega \cdot \overline{\omega} >0, \ \omega \cdot S =0, \ 
\psi_{{\mathbb{C}}}(\omega)=\overline{\omega} \}/{\mathbb{R}}^{\times} .$$

Let $((X,\tau,\varphi),\ \alpha)$ be a marked real $2$-elementary K3 surface of type $(S,\theta)$ satisfying $$\alpha \circ \varphi_* \circ \alpha^{-1} = \psi ,
\footnote{All marked real $2$-elementary K3 surfaces 
whose associated integral involutions are \underline{isometric} to $({\mathbb{L}}_{K3},\psi)$ with respect to $G$ 
satisfy $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$ if we change their markings appropriately 
(see \cite{SaitoSachiko2015}).}$$ and let $H \ \ (\subset H_2(X,{\mathbb{C}}))$ be the Poincare dual of $H^{2,0}(X)$. The $1$-dimensional subspace $\alpha_{{\mathbb{C}}}(H)$ of ${\mathbb{L}}_{K3}\otimes {\mathbb{C}}$ is regarded as an element of $\Omega_\psi$.

We call $\alpha_{{\mathbb{C}}}(H)$ the [*period*]{} of a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$.

We say that a point $[\omega] \ (\in \Omega_\psi)$ is [*equivalent*]{} to a point $[\omega^\prime] \ (\in \Omega_\psi)$ if $[\omega^\prime] = f_{{\mathbb{C}}}([\omega])$ for an automorphism $f$ of $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ with respect to the group $G$.

\[equivalence-lemma\] If a point $[\omega] \ (\in \Omega_\psi)$ is equivalent to $[\omega^\prime] \ (\in \Omega_\psi)$ and $[\omega]$ is the period of some marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$, then $[\omega^\prime]$ is also the period of a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha^\prime)$ of type $(S,\theta)$ satisfying $(\alpha^\prime) \circ \varphi_* \circ (\alpha^\prime)^{-1} = \psi$ where $\alpha^\prime$ is another marking of $(X,\tau,\varphi)$.

Using the global Torelli theorem, if two periods are equivalent, then corresponding marked real $2$-elementary K3 surfaces are analytically isomorphic (see Definition \[analytic-iso\]). The converse is also true.

The domain $\Omega_\psi$ has two connected components which are interchanged by $-\psi$. Since $-\psi$ is an automorphism of $({\mathbb{L}}_{K3},\psi)$ with respect to the group $G$, by Lemma \[equivalence-lemma\], it is enough to investigate the quotient space $$\Omega_\psi /-\psi .$$

Now we set $${\mathbb{L}}^\psi := \{ x \in {\mathbb{L}}_{K3}\ |\ \psi (x) = x \},\ \ {\mathbb{L}}_\psi := \{ x \in {\mathbb{L}}_{K3}\ |\ \psi (x) = - x \} .$$

We restrict ourselves to the case $$S \subset {\mathbb{L}}_\psi,\ \ \text{i.e.,}\ \ \theta = - {\mathrm{id}},$$ where “${\mathrm{id}}$" stands for the identity map on $S$. We set $${\mathbb{L}}_{-,S} := {\mathbb{L}}_\psi \cap S^{\perp}.$$

For $[\omega] \in \Omega_\psi$  ($\omega \in {\mathbb{L}}_{K3} \otimes {\mathbb{C}}$), we consider the orthogonal decomposition $\omega = \omega_+ + i\,\omega_- \ \ (\omega_+ \in {\mathbb{L}}^\psi \otimes {\mathbb{R}},\ \omega_- \in {\mathbb{L}}_\psi \otimes {\mathbb{R}})$. Then we have $\omega_- \in {\mathbb{L}}_{-,S}\otimes {\mathbb{R}}$ and $\omega_+^2 = \omega_-^2 > 0$. Hence, both ${\mathbb{L}}^\psi$ and ${\mathbb{L}}_{-,S}$ are hyperbolic lattices. We set $V({\mathbb{L}}^\psi):= \{ x \in {\mathbb{L}}^\psi \otimes {\mathbb{R}}\ |\ x^2 > 0\}$. $V({\mathbb{L}}^\psi)$ has two connected components. Let $V^+({\mathbb{L}}^\psi)$ be one of those (half cone). Let ${\mathcal{L}}_+$ denote the set of all rays (half lines) through ${\mathbf{0}}$ in $V^+({\mathbb{L}}^\psi)$. (${\mathcal{L}}_+$ is called the [*hyperbolic*]{} (or [*Lobachevsky*]{}) space obtained from ${\mathbb{L}}^\psi$. ) We define the hyperbolic space ${\mathcal{L}}_{-,S}$ obtained from ${\mathbb{L}}_{-,S}$ in the same way.

Then we have the following identification: $$\label{domain-quotient}
\Omega_\psi /-\psi \ = \ {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S}\ \ \ \ (\mbox{a direct product}).$$

(${\mathcal{D}}{\mathbb{R}}$)-nondegenerate marked real $2$-elementary K3 surfaces
----------------------------------------------------------------------------------

We say that an element $x (\neq 0) \in {\mathop\mathrm{Pic}}(X)\otimes {\mathbb{R}}$ is [*nef*]{} (for $X$) if $x \cdot C \ge 0$ for every effective curve $C$ in ${\mathop\mathrm{Pic}}(X)$.

\[da-degenerate\] [**(i)**]{} We say that a marked $2$-elementary K3 surface $((X,\tau),\ \alpha)$ of type $S$ is [*(${\mathcal{D}}$)-degenerate*]{} if there exists an element $x_0 \in \alpha_{{\mathbb{R}}}^{-1}({\mathcal{M}})$ which is [**not**]{} nef. Namely, $x_0 \cdot C < 0$ for an effective curve $C$ in ${\mathop\mathrm{Pic}}(X)$. This condition is equivalent (see [@Nikulin86], [@AlexeevNikulin2006]) to the existence of an irreducible $(-2)$-curve on the quotient surface $Y:=X/\tau$. And this condition is also equivalent to the existence of an element $\delta \in {\mathop\mathrm{Pic}}(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in \alpha^{-1}(S), \ \ \delta_2\in \alpha^{-1}(S)^\perp_{{\mathop\mathrm{Pic}}(X)}, \ \ \text{and} \ \ \delta_1^2=\delta_2^2 = -4$. [**(ii)**]{} We say that a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ is [*(${\mathcal{D}}{\mathbb{R}}$)-degenerate*]{} if there exists a “real" element $x_0 \in \alpha_{{\mathbb{R}}}^{-1}(S_- \cap {\mathcal{M}})$ which is [**not**]{} nef, where we set $S_{\pm} := \{ x\in S \,|\, \theta(x) = \pm x \}$. This condition is equivalent to the existence of an element $\delta \in {\mathop\mathrm{Pic}}(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in \alpha^{-1}(S),\ \  \delta_2\in \alpha^{-1}(S)^\perp_{{\mathop\mathrm{Pic}}(X)}\ \ \text{and} \ \ \delta_1^2=\delta_2^2=-4,$ and $\delta_1$ is orthogonal to an element $x \in \alpha_{{\mathbb{R}}}^{-1}(S_-\cap \text{int}({\mathcal{M}}))$, where $\text{int}({\mathcal{M}})$ denote the interior part of ${\mathcal{M}}$, i.e., the polyhedron ${\mathcal{M}}$ without its faces.

Considering associated integral involutions, we have:

\[theorem2005moduli\] The natural map gives a bijective correspondence between the connected components of the period domain of (${\mathcal{D}}{\mathbb{R}}$)-nondegenerate marked real $2$-elementary K3 surfaces of type $(S,\theta)$ and the isometry classes with respect to $G$ of integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$ such that the fixed part ${\mathbb{L}}^\psi$ of $\psi$ is hyperbolic.

Real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$ {#RealK3-311}
===================================================================

Now we fix a sublattice $S$ of the K3 lattice ${\mathbb{L}}_{K3}$ with the invariants $$(r(S),a(S),\delta(S)) = (3,1,1).$$ We consider $2$-elementary K3 surfaces of type $S \ (\cong (3,1,1))$ ([@NikulinSaito07],[@SaitoSachiko2015]). We quote the following results from Alexeev and Nikulin [@AlexeevNikulin2006]. See also [@NikulinSaito07] and [@SaitoSachiko2015].

Let $(X,\tau)$ be a $2$-elementary K3 surface of type $S \cong (3,1,1)$. Let $A := X^{\tau}$ be the fixed point set (nonsingular complex curve) of $\tau$. Then we have $$A = A_0 \cup A_1\ \ \mbox{(disjoint union)},$$ where $A_0$ is a nonsingular rational curve ($\cong {\mathbb{P}}^1$), and  $A_1$ is a nonsingular curve of genus $9$.

$(X,\tau)$ has a structure of an elliptic pencil $|E+F|$, and $\tau$ is the inversion map of the group structure of the elliptic pencil with the zero section $A_0$. The unique reducible fiber $E+F$ having the following properties:

(i)

:   $E$ is a nonsingular rational curve ($\cong {\mathbb{P}}^1$) and $E\cdot A_0=1$.

(ii)

:   $E\cdot F=2$,  $F^2=-2$,  $F\cdot A_0=0$,  and   $F$ is either\
    a nonsingular rational curve (“[*type IIa case*]{}"), or\
    the union of two nonsingular rational curves $F^\prime$ and $F^{\prime\prime}$ which are conjugate by $\tau$, $F^\prime \cdot F^{\prime\prime} = 1$ (“[*type IIb case*]{}").

(iii)

:   The classes $[A_0]$, $[E]$ and $[F]$ generate the lattice ${H_2}_+(X, {\mathbb{Z}}) \ (\cong S)$. Moreover, $A_1 \cdot E = 1,\ \ A_1 \cdot F = 2$. The Gram matrix of the lattice ${H_2}_+(X, {\mathbb{Z}})$ with respect to the basis $[E],\ [F]$ and $[A_0]$ is $$\begin{array}{crrr}
            & {[E]}   & {[F]} & {[A_0]} \\
    {[E]}   &   -2    &   2   &   1     \\
    {[F]}   &    2    &  -2   &   0     \\
    {[A_0]} &    1    &   0   &  -2 .
    \end{array}$$

Then we have an orthogonal decomposition $${H_2}_+(X, {\mathbb{Z}}) = {\mathbb{Z}}([A_0],\ [E]+[F]) \oplus {\mathbb{Z}}([F]) ,$$ where the subgroups ${\mathbb{Z}}([A_0],\ [E]+[F])$ and ${\mathbb{Z}}([F])$ are isometric to the hyperbolic plane and $\langle -2 \rangle$ respectively.

We now consider the quotient surface $Y := X/\tau$ (so-called “DPN surface" ([@NikulinSaito05])) and let $\pi : X \to Y$ be the quotient map. We define the curves on $Y$ as follows: $$e:=\pi(E) \ \ \ \text{and} \ \ \ f:= \pi(F) .$$ If $F$ is the union of two nonsingular rational curves $F^\prime$ and $F^{\prime\prime}$ which are conjugate by $\tau$ and $F^\prime \cdot F^{\prime\prime} = 1$, then we have $f = \pi(F) = \pi(F^\prime \cup F^{\prime\prime}) = \pi(F^\prime) = \pi(F^{\prime\prime}).$

We use the same symbols $A_0$ and $A_1$ for their images in $Y$ by $\pi$. Then, the Picard group ${\mathop\mathrm{Pic}}(Y)$ of $Y$ is generated by the classes $[e]$, $[f]$ and $[A_0]$. The Gram matrix of ${\mathop\mathrm{Pic}}(Y)$ with respect to the basis $[e],\ [f]$ and $[A_0]$ is $$\begin{array}{crrr}
        &  {[e]}  &  {[f]}  & {[A_0]} \\
{[e]}   &    -1   &    1    &    1    \\
{[f]}   &     1   &   -1    &    0    \\
{[A_0]} &     1   &    0    &   -4 .
\end{array}$$

We have $A_1\cdot e =  1 \ \ \ \mbox{and} \ \ \ A_1\cdot f = 2.$

We next contract the exceptional curve $f = \pi(F)$ to a point. Then we get a blow up $${\mathrm{bl}}: Y \to {\mathbb{F}}_4,$$ where ${\mathbb{F}}_4$ is the $4$-th Hirzebruch surface. (See [@NikulinSaito07].)

We set $s:={\mathrm{bl}}(A_0),\ A^\prime_1 := {\mathrm{bl}}(A_1),\ c:={\mathrm{bl}}(e)$. Then $s$ is the exceptional section of ${\mathbb{F}}_4$ with $s^2 = -4$, and $c$ is a fiber of the fibration ${\mathbb{F}}_4\to s$ with $c^2=0$.

We have $${\mathrm{bl}}(A) = {\mathrm{bl}}(A_0) + \, {\mathrm{bl}}(A_1) =  s +  A^\prime_1 \ \ \in |-2K_{{\mathbb{F}}_4}|.$$ Namely, ${\mathrm{bl}}(A)$ is . Since $s \cdot A^\prime_1 = 0$, $A^\prime_1$ does not intersect the section $s$. Since $-2K_{{\mathbb{F}}_4} \sim 12c+4s$, we have $A^\prime_1 \ \ \in |12c+3s|$.

For any real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $(S,\theta)$ with $S \cong (3,1,1)$, we have $$\theta  =  - {\mathrm{id}},$$ and $$G = \{ {\mathrm{id}}\}.$$ It is known that all real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ are (${\mathcal{D}}{\mathbb{R}}$)-non-degenerate.

\[the real double point\] $F$ is a nonsingular rational curve if and only if $A_1$ intersects $f$ in two distinct points, and $F$ is a union of two nonsingular rational curves if and only if $A_1$ touches $f$.

If $A_1$ intersects with $f$ at two distinct points, then they are real points or non-real conjugate points. In the former case the real double point of $A^\prime_1$ is a real node, and in the latter case it is a real isolated point. Anyway the real double point of $A^\prime_1$ is nondegenerate. If $A_1$ touches to $f$ in $Y$, then the real double point is a real cusp (a degenerate double point).

Since all real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ are (${\mathcal{D}}{\mathbb{R}}$)-non-degenerate, by Theorem \[theorem2005moduli\], the connected components of the period domain of marked real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ and the isometry classes with respect to $G = \{ {\mathrm{id}}\}$ of integral involutions of ${\mathbb{L}}_{K3}$ of type $((3,1,1), - {\mathrm{id}})$ such that the fixed part ${\mathbb{L}}^\psi$ of $\psi$ is hyperbolic are in bijective correspondence.

However, as is written in Remark 8 in Subsection 2.4 in [@NikulinSaito07], it is possible that an isometry class of integral involutions of ${\mathbb{L}}_{K3}$ of type $((3,1,1), - {\mathrm{id}})$ corresponds to [**both type IIa case and type IIb case**]{}. In other words, such isometry classes with respect to $G$ (in the sense of Theorem \[theorem2005moduli\]) cannot distinguish [**degenerate and nondegenerate double points**]{} of the curves $A^\prime_1$ on ${\mathbb{F}}_4$ (Recall Remark \[the real double point\]).

Therefore, we would like to determine whether the real double point of the curve $A^\prime_1$ on ${\mathbb{F}}_4$ is degenerate or not. See also Lemma 4.6 and Problem 1 in [@SaitoSachiko2015]. In order to do this, we define more strict markings of real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$. Compare the following Definition \[marked311\] to the Definition \[marked\_real\_K3\] above.

Let $
{\mathcal{E}},\ \ {\mathcal{F}},\ \ \mbox{and}\ \ {\mathcal{A}}$ be generators of $S$ with the Gram matrix $$\begin{array}{crrr}
       & {{\mathcal{E}}}  & {{\mathcal{F}}} & {{\mathcal{A}}}  \\
{{\mathcal{E}}}  &   -2   &   2   &     1 \ \\
{{\mathcal{F}}}  &    2   &  -2   &     0 \ \\
{{\mathcal{A}}}  &    1   &   0   &    -2
\end{array}$$ where ${\mathcal{M}}\cdot {\mathcal{E}}\ge 0$,  ${\mathcal{M}}\cdot {\mathcal{F}}\ge 0$, and   ${\mathcal{M}}\cdot {\mathcal{A}}\ge 0$.

And we set $${\mathbb{U}}:= {\mathbb{Z}}({\mathcal{A}},\ {\mathcal{E}}+ {\mathcal{F}}).$$ ${\mathbb{U}}$ is isometric to the hyperbolic plane, and $$S = {\mathbb{U}}\oplus {\mathbb{Z}}({\mathcal{F}})$$ is an orthogonal decomposition of $S$.

\[marked311\] We define that a [*marked real $2$-elementary K3 surfaces $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$*]{} is a pair of a real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $((3,1,1),- {\mathrm{id}})$ (Definition \[real\_2-elementary K3\_S\_theta\]) and a marking (isometry) $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that

-   $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$,

-   $\alpha \circ \varphi_* = \theta \circ \alpha \ \text{on} \ {H_2}_+(X, {\mathbb{Z}})$,

-   $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$,

-   the set $\alpha^{-1}(\Delta(S)_+)$ contains only classes of effective curves of $X$,

-   $\alpha([A_0]) = {\mathcal{A}}$, $\alpha([E]) = {\mathcal{E}}$, and $\alpha([F])={\mathcal{F}}$.

Note that every real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $((3,1,1),- {\mathrm{id}})$ has such a marking $\alpha$.

We give a criterion for the unique double point of a real anti-bicanonical curve ${\mathrm{bl}}(A) = s +  A^\prime_1 $ on ${\mathbb{F}}_4$ with one real double point on $A^\prime_1$ to be nondegenerate.

Let us consider and fix an integral involution $$({\mathbb{L}}_{K3},\psi)$$ of type $((3,1,1),- {\mathrm{id}})$ again, and consider the period domain in Subsection \[period domains section\] $$\Omega_\psi /-\psi \ = \ {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S} \ .$$

\[criterion\] Let $[[\omega]]$ be a point [^1] in $\Omega_\psi /-\psi$. Then, the real double point of the curve $A^\prime_1$ on ${\mathbb{F}}_4$ (Remark \[the real double point\]) which corresponds to $((X,\tau,\varphi), \alpha)$ is nondegenerate if and only if there are no ${\mathbf{v}}\ (\neq \pm {\mathcal{F}})$ in ${\mathbb{L}}_{K3}$ satisfying: $${\mathbf{v}}\cdot \omega = 0,\ \ {\mathbf{v}}\cdot {\mathbb{U}}=0,\ \ \text{and}\ \ {\mathbf{v}}^2 = -2.$$

The ($\Longleftarrow$) direction has been proved in Lemma 4.6 of [@SaitoSachiko2015]. Here we prove the ($\Longrightarrow$) direction. Assume that the double point of the real anti-bicanonical curve ${\mathrm{bl}}(A)$ on ${\mathbb{F}}_4$ is nondegenerate. Then $F$ is irreducible ([@NikulinSaito07], [@SaitoSachiko2015]). Suppose that there [**exists**]{} a ${\mathbf{v}}\ (\neq \pm {\mathcal{F}})$ in ${\mathbb{L}}_{K3}$ satisfying that ${\mathbf{v}}\cdot \omega = 0,\ {\mathbf{v}}\cdot {\mathbb{U}}=0$, and ${\mathbf{v}}^2 = -2$. Then $-{\mathbf{v}}$ has the same properties. By Riemann-Roch Theorem, we have $l(\alpha^{-1}({\mathbf{v}}))+l(-\alpha^{-1}({\mathbf{v}})) \geq \alpha^{-1}({\mathbf{v}})^2/2 +2 = 1$. Hence, $\alpha^{-1}({\mathbf{v}})$ or $-\alpha^{-1}({\mathbf{v}})$ is effective. (See [@AlexeevNikulin2006], p.23 or [@BHPV], p.311.) Hence, we may assume $\alpha^{-1}({\mathbf{v}})$ is an effective class. Then, $\tau^*(\alpha^{-1}({\mathbf{v}}))$ is also effective. We have $\tau^*(\alpha^{-1}({\mathbf{v}}))^2 = -2$, $\tau^*(\alpha^{-1}({\mathbf{v}}))\cdot [A_0] = 0,\ \tau^*(\alpha^{-1}({\mathbf{v}}))\cdot ([E]+[F])=0$ and $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}})))\cdot \omega = 0$ since $\tau^*$ is non-symplectic. Hence, $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}})))$ has the same properties as ${\mathbf{v}}$. Since $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}}\in S$, it is contained in ${\mathbb{Z}}({\mathcal{F}})$. There exists an integer $n$ such that $$\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}}= n{\mathcal{F}}.$$ Let us consider a hyperplane section class $h \ \in {\mathop\mathrm{Pic}}X$, which is ample. Since $\tau^*(\alpha^{-1}({\mathbf{v}}))$ and $\alpha^{-1}({\mathbf{v}})$ are effective, we have $\tau^*(\alpha^{-1}({\mathbf{v}}))\cdot h >0$ and $\alpha^{-1}({\mathbf{v}})\cdot h >0$. (Nakai’s criterion) We have $\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}) \neq 0$. Hence, we have $n \neq 0$. Moreover, since $\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}})$ and $[F] (= \alpha^{-1}({\mathcal{F}}))$ are effective, we have . Actually, if $n<0$, then $0=(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}) + (-n)\alpha^{-1}({\mathcal{F}}))\cdot h >0$. This is a contradiction.

If $\tau^*(\alpha^{-1}({\mathbf{v}})) = \alpha^{-1}({\mathbf{v}})$, then $(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}))^2 = (2\alpha^{-1}({\mathbf{v}}))^2 = -8$. On the other hand, $(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}))^2 = (n[F])^2 = -2n^2$. Hence, $n=2$. Thus we have ${\mathbf{v}}= {\mathcal{F}}$. This is a contradiction. Thus we have $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$.

Let $\alpha^{-1}({\mathbf{v}})=\displaystyle \sum_i m_i {\mathbf{v}}_i$, where $m_i$ are positive integers and ${\mathbf{v}}_i$ are irreducible effective classes. Obviously we have $\alpha({\mathbf{v}}_i) \cdot \omega = 0$ for any $i$. If ${\mathbf{v}}_i = [E]$ or $[F]$, then ${\mathbf{v}}_i \cdot ([E]+[F]) = 0$. And if ${\mathbf{v}}_i \neq [E]$ and $\neq [F]$, then we have ${\mathbf{v}}_i \cdot ([E]+[F]) \geq 0$ for any $i$. Here $E$ is irreducible, and moreover, $F$ is irreducible by the assumption. Since $\alpha^{-1}({\mathbf{v}})\cdot ([E]+[F])=0$, we have ${\mathbf{v}}_i \cdot ([E]+[F]) = 0$ for any $i$. Since $[A_0] \cdot ([E]+[F]) = 1$, we see ${\mathbf{v}}_i \neq [A_0]$ for any $i$. Thus we also have ${\mathbf{v}}_i \cdot [A_0] \geq 0$. Since $\alpha^{-1}({\mathbf{v}})\cdot [A_0]=0$, we have ${\mathbf{v}}_i \cdot [A_0] = 0$ for any $i$.

Thus, we have $\alpha({\mathbf{v}}_i) \cdot {\mathbb{U}}= 0$ for any $i$. Since ${\mathbb{U}}$ is of signature $(1,1)$, we have $({\mathbf{v}}_i)^2 <0$ for any $i$ by the Hodge index theorem. For such ${\mathbf{v}}_i$’s, we have ${\mathbf{v}}_i^2 = -2$, and ${\mathbf{v}}_i$’s are nonsingular rational ($\cong {\mathbb{P}}^1$).

Suppose $\alpha({\mathbf{v}}_i) = {\mathcal{F}}$ for some $i$, say $i=1$. Namely, ${\mathbf{v}}_1 = [F]$. Let ${\mathbf{v}}^\prime := \alpha^{-1}({\mathbf{v}}) - m_1 [F]$. Since $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$ (see above), we have ${\mathbf{v}}^\prime \neq 0$. Thus, there exists a ${\mathbf{v}}_i (\neq \pm [F])$ in $H_2(X,{\mathbb{Z}})$ such that ${\mathbf{v}}_i \cong {\mathbb{P}}^1$ (irreducible), ${\mathbf{v}}_i^2 = -2$, $\alpha({\mathbf{v}}_i) \cdot \omega = 0$, and $\alpha({\mathbf{v}}_i) \cdot {\mathbb{U}}= 0$. Eventually we can choose ${\mathbf{v}}$ such that $\alpha^{-1}({\mathbf{v}})$ is an irreducible class.

Since $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$, $\tau^*(\alpha^{-1}({\mathbf{v}}))$ and $\alpha^{-1}({\mathbf{v}})$ are represented by different irreducible curves respectively. Hence, we have $\tau^*(\alpha^{-1}({\mathbf{v}})) \cdot \alpha^{-1}({\mathbf{v}}) \geq 0$.

Since $(\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}})^2 = (-2) + (-2) + 2(\tau^*(\alpha^{-1}({\mathbf{v}})) \cdot \alpha^{-1}({\mathbf{v}})) = -2n^2$, we have $2-n^2 \geq 0$. Thus, we have $n=1$. Namely, we have $${\mathcal{F}}= {\mathbf{v}}+ \alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))),$$ i.e., $[F] = \alpha^{-1}({\mathbf{v}}) + \tau^*(\alpha^{-1}({\mathbf{v}}))$. Let $F^\prime$ be an irreducible curve representing $\alpha^{-1}({\mathbf{v}})$, and we set $F^{\prime \prime} := \tau^*(F^\prime)$. Thus $F^\prime + F^{\prime \prime}$ represents the class $[F]$. Hence, there exists a marked real $2$-elementary K3 surface corresponding to the period $[\omega]$ which has $E + F^\prime + F^{\prime \prime}$ as the unique reducible fiber of its elliptic fibration. Conversely, every marked real $2$-elementary K3 surface corresponding to the period $[\omega]$ has $E + F^\prime + F^{\prime \prime}$ as the unique reducible fiber of its elliptic fibration. Hence, we have $F= F^\prime + F^{\prime \prime}$. This contradicts the assumption that $F$ is irreducible. This completes the proof of Proposition \[criterion\].

For curves of degree $6$ with one double point on ${\mathbb{R}}P^2$, the corresponding criterion to Proposition \[criterion\] is written on the top of p. 281 in Itenberg’s paper [@Itenberg92].

As is written in [@SaitoSachiko2015], we now get the precise image ($\subset {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S}$) of the period map on the set of all marked real $2$-elementary K3 surfaces $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$ for which the real double points of the curves $A^\prime_1$ on ${\mathbb{F}}_4$ are [**nondegenerate**]{}. Thus we are able to continue the interesting arguments as in [@Itenberg92].

[**Acknowledgments.**]{} The author would like to thank Professor Hisanori Ohashi for his helpful suggestions and comments, Professor Ilia Itenberg for his inspiring papers, and Professor Viacheslav Nikulin for his constant encouragement.

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[^1]: By the surjectivity of the period map of marked K3 surfaces ([@BHPV], p.339), $[\omega]$ is the period of a marked real $2$-elementary K3 surface $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$.
