Some infinite sequences of canonical covers of degree 2

TL;DR

This paper introduces three new infinite families of algebraic surfaces of general type with canonical maps of degree 2, including one with base points, expanding the understanding of such surfaces.

Contribution

It constructs three novel infinite families of surfaces with canonical maps of degree 2, including one with base points, which was not previously known.

Findings

Three new infinite families of surfaces of general type with degree 2 canonical maps.

One family has a canonical system with base points.

Enhances classification of surfaces with canonical maps.

Abstract

In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

Equations

$$K_X^2$$

$${\chi }_{j_1{j}_2{j}_3}\left(a_1,a_2,a_3\right):=e^{\left(\pi a_1{j}_1\right)i}{e}^{\left(\pi a_2{j}_2\right)i}{e}^{\left(\pi a_3{j}_3\right)i}$$

$$2L_{100}$$

$$2K_X$$

$$K_X^2$$

$$p_g\left(X\right)$$

$$\chi \left({\mathcal{O}}_X\right)$$

$$L_{100}\equiv$$

$$2K_X$$

$$K_X^2=16mn,p_g\left(X\right)=2mn+3,\chi \left({\mathcal{O}}_X\right)=2mn+4,q\left(X\right)=0.$$

$$H^0\left(X,K_X\right)=H^0\left({\mathbb{P}}^1\times {\mathbb{P}}^1,K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\right)\oplus \underset{\chi \ne {\chi }_{000}}{⨁}{H}^0\left({\mathbb{P}}^1\times {\mathbb{P}}^1,K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}+L_{\chi }\right)$$

$$h^0\left(L_{\chi }+K_X\right)=0$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^0\left(K_Z\right)=H^0\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\right)\oplus H^0\left(\left(m-2\right)F+\left(n-2\right)G\right)\oplus H^0\left(mF+nG\right).$$

$$B=\overline{D_{100}}+\overline{D_{101}}+\overline{D_{110}}+\overline{D_{111}}.$$

$$L_{100}\equiv$$

$$2K_{\overline{X}}$$

$$\equiv f^{\ast }\left(\left(2m-1\right)\overline{F}+\left(2n-1\right)\overline{G}+\overline{F_0}+\overline{G_0}\right)$$

$$K_{\overline{X}}^2=16mn-8,p_g\left(\overline{X}\right)=2mn+2,q\left(\overline{X}\right)=0.$$

$$\overline{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^0\left(K_Z\right)=H^0\left({\mathcal{O}}_Y\right)\oplus H^0\left(\left(m-2\right)\overline{F}+\left(n-2\right)\overline{G}\right)\oplus H^0\left(m\overline{F}+n\overline{G}-E\right).$$

$$B=\overline{D_{011}}+\overline{D_{100}}+\overline{D_{101}}+\overline{D_{110}}+\overline{D_{111}},$$

$$L_{100}\equiv$$

$$2K_{\overline{X}}$$

$$\equiv f^{\ast }\left(\left(2m-1\right)\overline{F}+2n\overline{G}+\overline{F_0}\right),$$

$$K_{\overline{X}}^2=16mn-2,p_g\left(\overline{X}\right)=2mn+2,q\left(\overline{X}\right)=0.$$

$$\overline{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^0\left(K_Z\right)=H^0\left({\mathcal{O}}_Y\right)\oplus H^0\left(\left(m-2\right)\overline{F}+\left(n-2\right)\overline{G}+E\right)\oplus H^0\left(m\overline{F}+n\overline{G}-E\right).$$

$$K_Z\equiv g^{\ast }\left(m\overline{F}+n\overline{G}-E\right)+\tilde{E},$$

$$K_{Z_1}\equiv g_1^{\ast }\left(m\overline{F}+n\overline{G}-E\right).$$

$$h^0\left(Z,g^{\ast }\left(m\overline{F}+n\overline{G}-E\right)\right)=2mn+1$$

$$h^0\left(Z,K_Z\right)\ne h^0\left(Z,g^{\ast }\left(m\overline{F}+n\overline{G}-E\right)\right)$$

$$K_{\overline{X}}\equiv h^{\ast }\left(K_Z\right)$$

Full text

††Mathematics Subject Classification (2010): 14J29.††Key words: Surfaces of general type, Canonical maps, Abelian covers.

1 Introduction

Let $X$ be a minimal smooth complex surface of general type and denote by \textstyle{\varphi_{\left|K_{X}\right|}:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}\left(X\right)-1}} the canonical map of $X$, where $p_{g}\left(X\right)=\dim\left(H^{0}\left(X,K_{X}\right)\right)$ is the geometric genus and $\left|K_{X}\right|$ is the canonical system of $X$. The study of the degree of the canonical map for surfaces of general type has a long history and was inspired by the following well-known work of A. Beauville [2].

Theorem 1**.**

[2, Theorem 3.1] If the image $\Sigma:=\operatorname{im}\left(\varphi_{\left|K_{X}\right|}\right)$ is a surface, then either

$p_{g}\left(\Sigma\right)=0,$* or* 2. 2.

$\Sigma$* is a canonical surface (in particular, $p_{g}\left(\Sigma\right)=p_{g}\left(X\right)$).*

Moreover, in case $\left(1\right),$ $d:=\operatorname{deg}\left(\varphi_{\left|K_{X}\right|}\right)\leq 36,$ and in case $\left(2\right)$, $d\leq 9$.

In the same paper [2], A. Beauville also proved that in the second case if $\chi(\mathcal{O}_{X})\geq 14$ the degree of the canonical map is less than or equal to $3$.

Only a few families of surfaces with arbitrarily high geometric genus in the second case are known. The first construction due to A. Beauville (see [3] or [6]) gives an infinite series of surfaces with $d=2$ and irregularity $q=2$. Later, M.M. Lopes and R. Pardini proved that these are almost the only examples satisfying $K^{2}=6p_{g}-14$ and $q\geq 2$ (see [6]). By means of a construction analogous to that of A. Beauville’s, C. Ciliberto, R. Pardini and F. Tovena created two more series of surfaces: one with $d=2$ and $q=2$, and another with $d=2$ and $q=3$ (see [4]). A few years later they found three more unlimited families of surfaces with $d=2$ and $q=0$ (see [5]). These six families of surfaces all satisfy $K^{2}<7p_{g}$. Furthermore, these surfaces all possess canonical linear systems which are base point free.

As far as we know, all the known examples of the second case of Theorem 1 have base point free canonical system. In this note, we construct three new families of surfaces satisfying the second case of Theorem 1 and for one of the families the canonical system has base points. These three infinite families of surfaces all satisfy either $K^{2}=8p_{g}-24$ or $K^{2}=8p_{g}-18$. The following theorem is the main result of this note:

Theorem 2**.**

Let $m,n$ be integer numbers such that $m,n\geq 2$. Then there exist minimal surfaces of general type $X$ whose canonical map is of degree 2 onto a surface of general type $\Sigma$ with the following invariants

[TABLE]

The idea of our constructions is the following.

We construct the surfaces in the first row of Theorem 2 by taking a specific $\mathbb{Z}_{2}^{3}-$cover $X$ of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. In particular, we take the cover to be in such a way that one of the $\mathbb{Z}_{2}-$quotients $Z$ of $X$ is a surface of general type whose only singularities are nodes (i.e. points of type $A_{1}$). Moreover, we require that:

$p_{g}\left(Z\right)=p_{g}\left(X\right)$, 2. 2.

the canonical map $\varphi_{\left|K_{Z}\right|}$ is birational.

To obtain the other surfaces described in Theorem 2, we modify this $\mathbb{Z}_{2}^{3}-$cover by imposing an ordinary quadruple point on the branch locus in such a way that the two conditions above hold.

2 $\mathbb{Z}_{2}^{3}-$coverings

The construction of abelian covers was studied by R. Pardini in [7]. For details about the building data of abelian covers we refer the reader to Section 1 and Section 2 of R. Pardini’s work ([7]).

We will denote by $\chi_{j_{1}j_{2}j_{3}}$ the character of $\mathbb{Z}_{2}^{3}$ defined by

[TABLE]

for all $j_{1},j_{2},j_{3},a_{1},a_{2},a_{3}\in\mathbb{Z}_{2}$.

From [7, Theorem 2.1] we can define $\mathbb{Z}_{2}^{3}-$covers as follows:

Proposition 1**.**

Given $Y$ a smooth projective surface, let $L_{\chi}$ be divisors of $Y$ such that $L_{\chi}\not\equiv\mathcal{O}_{Y}$ for all nontrivial characters $\chi$ of $\mathbb{Z}_{2}^{3}$ and let $D_{\sigma}$ be effective divisors of $Y$ for all $\sigma\in\mathbb{Z}_{2}^{3}\setminus\left\{\left(0,0,0\right)\right\}$ such that the branch divisor $B:=\sum\limits_{\sigma\neq 0}{D_{\sigma}}$ is reduced. Then $\left\{L_{\chi},D_{\sigma}\right\}_{\chi,\sigma}$ is the building data of a $\mathbb{Z}_{2}^{3}-$cover \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y} if and only if

[TABLE]

By [7, Theorem 3.1] if each $D_{\sigma}$ is smooth and $B$ is a simple normal crossings divisor, then the surface $X$ is smooth.

Also from [7, Lemma 4.2, Proposition 4.2] we have:

Proposition 2**.**

If  $Y$ is a smooth surface and \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y} is a smooth $\mathbb{Z}_{2}^{3}-$cover with building data $\left\{L_{\chi},D_{\sigma}\right\}_{\chi,\sigma}$, the surface $X$ satisfies the following:

[TABLE]

3 Construction

3.1 Construction and computation of invariants

We will denote by $F=\left\{0\right\}\times\mathbb{P}^{1}$ and $G=\mathbb{P}^{1}\times\left\{0\right\}$ the generators of $\operatorname{Pic}\left(\mathbb{P}^{1}\times\mathbb{P}^{1}\right)$. Let $D_{100}$, $D_{101}$ $\in\left|2F+2G\right|$, $D_{110}\in\left|2mF\right|$ (with $m\geq 2$) and $D_{111}$ $\in\left|2nG\right|$ (with $n\geq 2$) be smooth divisors of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ intersecting transversally and such that no more than two of these divisors go through the same point.

Let \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} be the $\mathbb{Z}^{3}_{2}-$cover with branch locus $B=\sum\limits_{\sigma\neq 0}{D_{\sigma}}$, where $D_{\sigma}=0$ for the other $D_{\sigma}$. The building data is as follows:

[TABLE]

The surface $X$ is smooth and satisfies

[TABLE]

This implies that the surface $X$ is minimal and of general type. Furthermore, from Proposition 2, the surface $X$ has the following invariants:

[TABLE]

3.2 The canonical map and the canonical image

In this section, we show that the degree $d$ of the canonical map of $X$ is $2$ and that the canonical image $\Sigma$ is a surface of general type. We have the $\mathbb{Z}_{2}^{3}-$equivariant decomposition

[TABLE]

where the group $\mathbb{Z}_{2}^{3}$ acts on $H^{0}\left(\mathbb{P}^{1}\times\mathbb{P}^{1},K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}+L_{\chi}\right)$ via the character $\chi$ (see [7, Proposition 4.1c]).

We consider the cyclic subgroup $\Gamma:=\left\langle\left(0,0,1\right)\right\rangle$ of $\mathbb{Z}_{2}^{3}$. Let $\Gamma^{\perp}$ denote the kernel of the restriction map \textstyle{\left(\mathbb{Z}_{2}^{3}\right)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{*}} , where $\Gamma^{*}$ is the character group of $\Gamma$. We have $\Gamma^{\perp}=\left\langle\chi_{100},\chi_{010}\right\rangle$. Because

[TABLE]

for all $\chi\notin\Gamma^{\perp}$, the subgroup $\Gamma$ acts trivially on $H^{0}\left(X,K_{X}\right)$. So the canonical map $\varphi_{\left|K_{X}\right|}$ is the composition of the quotient map \textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z:=X/\left\langle 0,0,1\right\rangle} with the canonical map $\varphi_{\left|K_{Z}\right|}$ of $Z$ (see e.g. [8, Example 2.1]). Thus, the following diagram commutes

[TABLE]

The intermediate surface $Z$ is singular but since the singularities are ordinary double points, the formulas for the smooth case hold in this case. Since the intermediate surface $Z$ is the fiber product of the double covers of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ branched on $D_{100}+D_{101}$ and $D_{110}+D_{111}$, respectively, the canonical class $K_{Z}$ is the pullback of $mF+nG$. In addition, the eigenspace decomposition of $H^{0}\left(K_{Z}\right)$ is as follows:

[TABLE]

Because the linear system $\left|K_{Z}\right|$ contains the pullback of the very ample system $\left|mF+nG\right|$, there is a factorization \textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\left|K_{Z}\right|}}$$\textstyle{Z^{{}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} . So if the map $\varphi_{\left|K_{Z}\right|}$ is not birational, then the surface $Z^{{}^{\prime}}$ is birational to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ or to one of the three intermediate quotients $Z/\left\langle g\right\rangle$, where $g\in\mathbb{Z}_{2}^{2}$ is a nonzero element. This does not happen since $H^{0}\left(K_{Z}\right)$ has three nonzero eigenspaces. Thus, the canonical map $\varphi_{\left|K_{Z}\right|}$ is birational. Therefore, the canonical map of $X$ has degree $2$ and the canonical image $\Sigma$ is a surface of general type of degree $8mn$.

3.3 Variations

We now consider variations of the construction of Section 3.1, to obtain the surfaces listed in the second and third row of the table in Theorem 2. In order to achieve this, we impose an ordinary quadruple point on the branch locus and we resolve the singularities.

3.3.1 Variation 1

The first variation of the construction in Section 3.1 consists of allowing all the non zero $D_{\sigma}$ to pass through the same point.

Namely we take a branch locus as before, except that now we require that the four divisors $D_{100}$, $D_{101}$ $\in\left|2F+2G\right|$, $D_{110}\in\left|2mF\right|$ (with $m\geq 2$) and $D_{111}$ $\in\left|2nG\right|$ (with $n\geq 2$) go through the same point $P$ in such a way that the branch locus $B$ has an ordinary quadruple point at $P$. As before we require that $D_{100}$, $D_{101}$, $D_{110}$ and $D_{100}$ are smooth, that all intersections are transversal and that $P$ is the only point where more than two of the $D_{\sigma}$ meet.

Denote by $\overline{Y}$ the blow up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at $P$, by $\overline{F}$, $\overline{G}$ the pullback of general fibers $F$, $G$, by $E$ the exceptional divisor and by $\overline{D_{100}}$, $\overline{D_{101}}$, $\overline{D_{110}}$, $\overline{D_{111}}$ the strict transforms of $D_{100}$, $D_{101}$, $D_{110}$, $D_{111}$, respectively.

Let \textstyle{\overline{f}:\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{Y}} be the $\mathbb{Z}^{3}_{2}-$cover with branch locus

[TABLE]

The building data is as follows:

[TABLE]

Then the surface $\overline{X}$ is smooth and satisfies

[TABLE]

where $\overline{F_{0}}$, $\overline{G_{0}}$ are the strict transforms of the fibers $F$, $G$ passing through $P$, respectively. Because the canonical divisor $K_{\overline{X}}$ is nef and big, the surface $\overline{X}$ is minimal and of general type. Moreover, the surface $\overline{X}$ has the invariants:

[TABLE]

As in Section 3.2, the canonical map $\varphi_{\left|K_{\overline{X}}\right|}$ is the composition of the quotient map \textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z} with the canonical map $\varphi_{\left|K_{Z}\right|}$ of $Z$, where $Z:=\overline{X}/\left\langle 0,0,1\right\rangle$. Moreover, the following diagram commutes

[TABLE]

The eigenspace decomposition of $H^{0}\left(K_{Z}\right)$ is as follows:

[TABLE]

Similarly to Section 3.2, one checks that $\operatorname{im}\left(\varphi_{\left|K_{\overline{X}}\right|}\right)$ is a surface of general type of degree $8mn-4$ and that $\varphi_{\left|K_{\overline{X}}\right|}$ is a morphism of degree 2. Therefore, we obtain the surfaces described in the second row of Theorem 2.

Remark 1*.*

Taking the $\mathbb{Z}_{2}^{3}$ cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with the above branch locus, we would obtain a singular surface with a Gorenstein elliptic singularity whose minimal resolution is an elliptic curve with self-intersection $-8$, (cf. [1, No. 4.4, Table 1, Section 3.3]). The surface $\overline{X}$ is the minimal resolution of this singular surface and the pullback of the exceptional divisor $E$ is exactly the elliptic curve.

3.3.2 Variation 2

In this second variation of the construction given in Section 3.1 we again impose a quadruple point to the branch locus but in a different way.

We take again a branch locus as before: $D_{100}\in\left|2F+2G\right|$, $D_{101}$ $\in\left|2F+2G\right|$, $D_{110}\in\left|2mF\right|$ (with $m\geq 2$) and $D_{111}$ $\in\left|2nG\right|$ (with $n\geq 2$) , but now we require that $D_{100}$ has as unique singular point a node $P$ and that $D_{101}$ and $D_{110}$ go through the point $P$ in such a way that the total branch $B$ has an ordinary quadruple point. As before we require that $D_{101}$, $D_{110}$ and $D_{100}$ are smooth, that all intersections are transversal and that $P$ is the only point where more than two of the $D_{\sigma}$ meet.

Denote again by $\overline{Y}$ the blow up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at $P$, by $\overline{F}$, $\overline{G}$ the pullback of general fibers $F$, $G$, by $E$ the exceptional divisor and by $\overline{D_{100}}$, $\overline{D_{101}}$, $\overline{D_{110}}$, $\overline{D_{111}}$ the strict transforms of $D_{100}$, $D_{101}$, $D_{110}$, $D_{111}$, respectively.

Let \textstyle{\overline{f}:\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{Y}} be the $\mathbb{Z}^{3}_{2}-$cover with branch locus

[TABLE]

where $\overline{D_{011}}=E$. The building data is as follows:

[TABLE]

Then the surface $\overline{X}$ is smooth and satisfies

[TABLE]

where $\overline{F_{0}}$ is the strict transforms of the fiber $F$ passing through $P$. Because the canonical divisor $K_{\overline{X}}$ is nef and big, the surface $\overline{X}$ is minimal and of general type. Moreover, the surface $\overline{X}$ has the invariants:

[TABLE]

As in Section 3.2, the canonical map $\varphi_{\left|K_{\overline{X}}\right|}$ is the composition of the quotient map \textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Z} with the canonical map $\varphi_{\left|K_{Z}\right|}$ of $Z$, where $Z:=\overline{X}/\left\langle 0,0,1\right\rangle$. Furthermore, the following diagram commutes

[TABLE]

The eigenspace decomposition of $H^{0}\left(K_{Z}\right)$ is as follows:

[TABLE]

Similarly to Section 3.2, one checks that $\operatorname{im}\left(\varphi_{\left|K_{\overline{X}}\right|}\right)$ is a surface of general type and that $\varphi_{\left|K_{\overline{X}}\right|}$ is a map of degree $2$.

Now we show that the canonical linear system $\left|K_{\overline{X}}\right|$ has base points. First, we claim that

[TABLE]

where the map \textstyle{g:Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y} is the $\mathbb{Z}_{2}^{2}-$cover with building data $\left\{L_{100},L_{010},L_{110}\right\}$ and the divisor $\tilde{E}:=\frac{1}{2}g^{*}\left(E\right)$ is an elliptic curve with self-intersection $\tilde{E}^{2}=-1$. In fact, the intermediate surface $Z$ can be obtained by taking the iterated double covers

[TABLE]

where $g_{1}$ is the double cover branched on $\overline{D_{100}}+\overline{D_{101}}+\overline{D_{110}}+\overline{D_{111}}$ and $g_{2}$ is the double cover branched on the pullback $g_{1}^{*}\left(E\right)$ and the nodes coming from the intersections between $\overline{D_{100}}+\overline{D_{101}}$ and $\overline{D_{110}}+\overline{D_{111}}$. So, we get

[TABLE]

Because the elliptic curve $g_{1}^{*}\left(E\right)$ and the nodes are disjoint, we obtain the formula (1).

On the other hand, by the projection formula, we get

[TABLE]

This implies that

[TABLE]

By $(\ref{Condition 1 of fixed point})$ and $(\ref{Condition 2 of fixed point})$, the elliptic curve $\tilde{E}$ is not in the fixed part of $\left|K_{Z}\right|$. On the other hand, because the divisor $\left.K_{Z}\right|_{\tilde{E}}$ on the elliptic curve $\tilde{E}$ has degree $1$, the linear system $\left|\left.K_{Z}\right|_{\tilde{E}}\right|$ has a simple base point. Therefore, the linear system $\left|K_{Z}\right|$ has a simple base point.

The double cover \textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z} ramifies on $8mn+12$ nodes coming from the intersection points between $\overline{D_{100}}$ and $\overline{D_{101}}$, and $\overline{D_{110}}$ and $\overline{D_{111}}$. Because

[TABLE]

and $h^{0}\left(\overline{X},K_{\overline{X}}\right)=h^{0}\left(Z,K_{Z}\right)$, the linear system $\left|K_{\overline{X}}\right|$ possesses two simple base points on the pullback of $\tilde{E}$. Moreover, the image $\operatorname{im}\left(\varphi_{\left|K_{\overline{X}}\right|}\right)$ is a surface of degree $8mn-2$.

Remark 2*.*

Taking the $\mathbb{Z}_{2}^{3}$ cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with the above branch locus, we would obtain a singular surface with a Gorenstein elliptic singularity whose minimal resolution is an elliptic curve with self-intersection $-2$. (cf. [1, No. 4.8, Table 1, Section 3.3]). The surface $\overline{X}$ is the minimal resolution of this singular surface and the inverse image of the exceptional divisor $E$ is exactly the elliptic curve.

Remark 3*.*

In the three above constructions, if $n=1$ or $m=1$, the canonical map $\varphi_{\left|K_{X}\right|}$ is a $4:1$ map onto a rational ruled surface.

In fact, if $n=1$ or $m=1$, then the quotient surface $Z/\left\langle 0,1,0\right\rangle$ has geometric genus [math]. This implies that the canonical map $\varphi_{\left|K_{Z}\right|}$ has degree $2$. Therefore, $\operatorname{deg}\left(\varphi_{\left|K_{X}\right|}\right)=4$.

ACKNOWLEDGMENTS*.*

The author is deeply indebted to Margarida Mendes Lopes for all her help. The author is supported by Fundação para a Ciência e Tecnologia (FCT), Portugal under the framework of the program Lisbon Mathematics PhD (LisMath), Programa de Doutoramento FCT - PD/BD/113632/2015.