Some algebraic surfaces with canonical map of degree 10, 12, 14 NGUYEN BIN

###### Abstract

Surfaces of general type with canonical map of degree $d$ bigger than $8$ have
bounded geometric genus and irregularity. In particular the irregularity is at
most $2$ if $d\geq 10$. In the present paper, the existence of surfaces with
$d=10$ and all possible irregularities, surfaces with $d=12$ and irregularity
$1$ and $2$, and surfaces with $d=14$ and irregularity $0$ and $1$ is proven,
by constructing these surfaces as $\mathbb{Z}_{2}^{3}$-covers of certain
rational surfaces. These results together with the construction by C. Rito of
a surface with $d=12$ and irregularity $0$ show that all the possibilities for
the irregularity in the cases $d=10$, $d=12$ can occur, whilst the existence
of a surface with $d=14$ and irregularity $2$ is still an open problem.

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

##  1 Introduction

The problem of constructing examples of surfaces of general type with
canonical map of high degree has been studied by many authors in the last
decades. 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 $K_{X}$ is the canonical divisor of $X$ and
$p_{g}\left(X\right)=\dim H^{0}\left(X,K_{X}\right)$ is the geometric genus.
In 1979, A. Beauville proved that the degree $d$ of the canonical map is at
most $9$ if the surface has holomorphic Euler-Poincaré characteristic bigger
than $30$ [2, Proposition 4.1]. Later in 1986, G. Xiao showed that the degree
of the canonical map is at most $8$ if the surface has geometric genus bigger
than $132$ [19, Theorem 3]. Only few surfaces with $d$ greater than $8$ have
been known so far, such as: S. L. Tan’s example [18] with $d=9$, U. Persson’s
example [13] with $d=16$. In the last decade, some surfaces with
$d=12,16,20,24,27,32,36$ were constructed by C. Rito [14], [16], [15], [17],
C. Gleissner, R. Pignatelli and C. Rito [3], Ching-Jui Lai and Sai-Kee Yeung
[4], and the author [7], [9].

When the canonical map has degree $d=10,12,14$, by the Bogomolov-Miyaoka-Yau
inequality the irregularity $q$ is at most $2$. Indeed, we have

$\displaystyle d\left(p_{g}-2\right)\leq
d\operatorname{deg}\left(\operatorname{im}\left(\varphi_{\left|K_{X}\right|}\right)\right)\leq
K_{X}^{2}\leq 9\chi\left(\mathcal{O}_{X}\right)=9p_{g}-9q+9.$

This implies that $q\leq 2$ since $p_{g}\geq 3$. In this paper, by using
$\mathbb{Z}_{2}^{3}$-covers of del Pezzo surfaces of degree $5$, we construct
three surfaces with $d=10$ and with all possible irregularities $q=0,1,2$. By
using $\mathbb{Z}_{2}^{3}$-covers of del Pezzo surfaces of degree $6$, we
construct two surfaces with $d=12$ and with $q=1,2$. These two examples
together with C. Rito’s with $d=12$ and with $q=0$ show the existence of
surfaces with $d=12$ for every possibility of the irregularity. By using
$\mathbb{Z}_{2}^{3}$-covers of del Pezzo surfaces of degree $7$, we construct
two surfaces with $d=14$ and with $q=0,1$. We do not have any example with
$d=14$ and with $q=2$. The following theorem is the result of this paper:

######  Theorem 1.

There exist minimal surfaces of general type $X$ satisfying the following

$d$ | $q\left(X\right)$ | $p_{g}\left(X\right)$ | $K_{X}^{2}$ | $\left|K_{X}\right|$
---|---|---|---|---
$10$ | $0$ | $3$ | $10$ | is base point free
$10$ | $1$ | $3$ | $10$ | is base point free
$10$ | $2$ | $3$ | $12$ | has a non-trivial fixed part
$12$ | $1$ | $3$ | $12$ | is base point free
$12$ | $2$ | $3$ | $12$ | is base point free
$14$ | $0$ | $3$ | $14$ | is base point free
$14$ | $1$ | $3$ | $14$ | is base point free

The idea of our constructions is the following: we first construct the
surfaces with $d=14$ and with $q=0,1$ by taking specific
$\mathbb{Z}_{2}^{3}$-covers $X$ of del Pezzo surface $Y_{2}$ of degree $7$
(see Notation 1). In particular, we take the covers to be in such a way that
the following diagram commutes

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}^{3}}$$\scriptstyle{g}$$\scriptstyle{\mathbb{Z}_{2}}$$\scriptstyle{\varphi_{\left|K_{X}\right|}}$$\textstyle{Y_{2}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}^{2}}$$\scriptstyle{\varphi_{\left|K_{Z}\right|}}$$\textstyle{\mathbb{P}^{2}}$

where the intermediate surface $Z$, which is a $\mathbb{Z}_{2}$-quotient of
$X$, is a surface of general type whose only singularities are of type
$A_{1}$. Moreover, we require that:

  1. 1.

the canonical map of $Z$ is of degree $7$,

  2. 2.

the canonical map of $X$ factors through the quotient map
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$.

More precisely, we take the $\mathbb{Z}_{2}^{3}$-covers such that the
$\mathbb{Z}_{2}$-quotient $Z:=X/\left(0,0,1\right)$ is the bidouble cover of
$Y_{2}$ with the building data
$\left\\{D_{1},D_{2},D_{3},L_{1},L_{2},L_{3}\right\\}$ determined as follows:

$\displaystyle D_{1}:=$ $\displaystyle D_{010}+D_{011}\equiv-K_{Y_{2}}$
$\displaystyle D_{2}:=$ $\displaystyle D_{100}+D_{101}\equiv-K_{Y_{2}}$
$\displaystyle D_{3}:=$ $\displaystyle D_{110}+D_{111}\equiv-K_{Y_{2}}$
$\displaystyle L_{1}:=$ $\displaystyle L_{100}\equiv-K_{Y_{2}}$ $\displaystyle
L_{2}:=$ $\displaystyle L_{010}\equiv-K_{Y_{2}}$ $\displaystyle L_{3}:=$
$\displaystyle L_{110}\equiv-K_{Y_{2}}.$

The branch divisors $D_{\sigma}$ are chosen so that the intermediate surface
$Z$ has only singularities of type $A_{1}$ and that the canonical linear
system $\left|K_{Z}\right|$ is base point free. So the canonical map of $Z$ is
of degree $7$.

Furthermore, we take the $\mathbb{Z}_{2}^{3}$-covers with
$h^{0}\left(L_{\chi}+K_{Y}\right)=0$ for all
$\chi\in\left\\{\chi_{001},\chi_{101},\chi_{011},\chi_{111}\right\\}$ so that
the canonical map of $X$ factors through the quotient map
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$.
In fact, by Proposition 5, we have the following decomposition:

$\displaystyle
H^{0}\left(X,K_{X}\right)=H^{0}\left(Y_{2},K_{Y_{2}}\right)\oplus\bigoplus_{\chi\neq\chi_{000}}{H^{0}\left(Y_{2},K_{Y_{2}}+L_{\chi}\right)}.$

We consider the 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}$$\textstyle{\Gamma^{*}}$,
where $\Gamma^{*}$ is the character group of $\Gamma$. We have
$\Gamma^{\perp}=\left\langle\chi_{100},\chi_{010},\chi_{110}\right\rangle$.
The subgroup $\Gamma$ acts trivially on $H^{0}\left(X,K_{X}\right)$ since
$h^{0}\left(L_{\chi}+K_{Y_{2}}\right)=0$ for all $\chi\notin\Gamma^{\perp}$.
So the canonical map $\varphi_{\left|K_{X}\right|}$ is the essential
composition of the quotient map
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z:=X/\Gamma}$
with the canonical map $\varphi_{\left|K_{Z}\right|}$ of $Z$ (see e.g. [12,
Example 2.1], [8, Section 3.2], [10, Section 3.1.2]).

To obtain the other surfaces described in Theorem 1, we modify these
$\mathbb{Z}_{2}^{3}$-covers by imposing triple and quadruple points to the
branch loci.

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

###  2.1 Notation and conventions

Throughout this paper all surfaces are projective algebraic over the complex
numbers. Linear equivalence of divisors is denoted by $\equiv$. A character
$\chi$ of the group $\mathbb{Z}_{2}^{3}$ is a homomorphism from
$\mathbb{Z}_{2}^{3}$ to $\mathbb{C}^{*}$, the multiplicative group of the non-
zero complex numbers. We also use the following notation of del Pezzo
surfaces:

######  Notation 1.

We denote by $Y_{2}$ the blow-up of $\mathbb{P}^{2}$ at two distinct points
$P_{1}$, $P_{2}$, by $l$ the pull-back of a general line in $\mathbb{P}^{2}$,
by $e_{1}$, $e_{2}$ the exceptional divisors corresponding to $P_{1}$,
$P_{2}$, respectively, by $f_{1}$, $f_{2}$ the strict transforms of a general
line through $P_{1}$, $P_{2}$, respectively and by $h_{12}$ the strict
transform of the line $P_{1}P_{2}$. The anti-canonical class $-K_{Y_{2}}\equiv
f_{1}+f_{2}+l$ is very ample and the linear system $\left|-K_{Y_{2}}\right|$
embeds $Y_{2}$ as a smooth del Pezzo surface of degree $7$ in
$\mathbb{P}^{7}$.

######  Notation 2.

We denote by $Y_{3}$ the blow-up of $\mathbb{P}^{2}$ at three distinct non-
collinear points $P_{1}$, $P_{2}$, $P_{3}$. Let us denote by $l$ the pull-back
of a general line in $\mathbb{P}^{2}$, by $e_{1}$, $e_{2}$, $e_{3}$ the
exceptional divisors corresponding to $P_{1}$, $P_{2}$, $P_{3}$, respectively,
by $f_{1}$, $f_{2}$, $f_{3}$ the strict transforms of a general line through
$P_{1}$, $P_{2}$, $P_{3}$, respectively and by $h_{12}$, $h_{23}$, $h_{31}$
the strict transforms of the lines $P_{1}P_{2}$, $P_{2}P_{3}$, $P_{3}P_{1}$,
respectively. The anti-canonical class $-K_{Y_{3}}\equiv f_{1}+f_{2}+f_{3}$ is
very ample and the linear system $\left|-K_{Y_{3}}\right|$ embeds $Y_{3}$ as a
smooth del Pezzo surface of degree $6$ in $\mathbb{P}^{6}$.

######  Notation 3.

We denote by $Y_{4}$ the blow-up of $\mathbb{P}^{2}$ at four points in general
position $P_{1},P_{2},P_{3},P_{4}$. Let us denote by $l$ the pull-back of a
general line in $\mathbb{P}^{2}$, by $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$ the
exceptional divisors corresponding to $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$,
respectively, by $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ the strict transforms of a
general line through $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$, respectively and by
$h_{ij}$ the strict transforms of the line $P_{i}P_{j}$, for all
$i,j\in\left\\{1,2,3,4\right\\}$, respectively. The anti-canonical class

$\displaystyle-K_{Y_{4}}$ $\displaystyle\equiv f_{1}+f_{2}+f_{3}-e_{4}$
$\displaystyle\equiv f_{1}+f_{2}+f_{4}-e_{3}$ $\displaystyle\equiv
f_{1}+f_{3}+f_{4}-e_{2}$ $\displaystyle\equiv f_{2}+f_{3}+f_{4}-e_{1}$

is very ample and the linear system $\left|-K_{Y_{4}}\right|$ embeds $Y_{4}$
as a smooth del Pezzo surface of degree $5$ in $\mathbb{P}^{5}$.

###  2.2 $\mathbb{Z}_{2}^{3}$-coverings

An abelian cover of a smooth surface $Y$ with finite abelian group $G$ is a
finite map
$\textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$
together with a faithful action of $G$ on $X$ such that $f$ exhibits $Y$ as
the quotient of $X$ via $G$.

The construction of abelian covers was studied by R. Pardini in [11]. For
detail about the building data of abelian covers and their notations, we refer
the reader to Section 1 and Section 2 of R. Pardini’s work. For the sake of
completeness, we recall some facts on $\mathbb{Z}_{2}^{3}$-covers, in a form
which is convenient for our later constructions.

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

$\displaystyle\chi_{j_{1}j_{2}j_{3}}\left(a_{1},a_{2},a_{3}\right):=e^{\left(\pi
a_{1}j_{1}\right)\sqrt{-1}}e^{\left(\pi
a_{2}j_{2}\right)\sqrt{-1}}e^{\left(\pi a_{3}j_{3}\right)\sqrt{-1}}$

for all $j_{1},j_{2},j_{3},a_{1},a_{2},a_{3}\in\mathbb{Z}_{2}$. A
$\mathbb{Z}_{2}^{3}$-cover
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$
can be determined a collection of non-trivial divisors $L_{\chi}$ labelled by
characters of $\mathbb{Z}_{2}^{3}$ and effective divisors $D_{\sigma}$
labelled elements of $\mathbb{Z}_{2}^{3}$ of the surface $Y$. More precisely,
from [11, Theorem 2.1] we can define $\mathbb{Z}_{2}^{3}$-covers as follows:

######  Proposition 4.

Given $Y$ a smooth projective surface with no non-trivial $2$-torsion, let
$L_{\chi}$ be divisors of $Y$ such that $L_{\chi}\not\equiv\mathcal{O}_{Y}$
for all non-trivial 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 total 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

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $2L_{010}$
$\equiv$ $D_{010}$ $+D_{011}$ $+D_{110}$ $+D_{111}$ $2L_{001}$ $\equiv
D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$ $2L_{110}$ $\equiv$ $D_{010}$
$+D_{011}$ $+D_{100}$ $+D_{101}$ $2L_{101}$ $\equiv D_{001}$ $+D_{011}$
$+D_{100}$ $+D_{110}$ $2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$
$+D_{110}$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$ $+D_{100}$ $+D_{111}$.

The condition $L_{\chi}\not\equiv\mathcal{O}_{Y}$ for all non-trivial
characters $\chi$ assures that the surface $X$ is irreducible.

By [11, Theorem 3.1] if each branch component $D_{\sigma}$ is smooth and the
total branch locus $B$ is a simple normal crossings divisor, the surface $X$
is smooth.

Also from [11, Lemma 4.2, Proposition 4.2] we have:

######  Proposition 5.

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 the building data
$\left\\{L_{\chi},D_{\sigma}\right\\}_{\chi,\sigma}$, the surface $X$
satisfies the following:

$\displaystyle 2K_{X}$ $\displaystyle\equiv
f^{*}\left(2K_{Y}+\sum\limits_{\sigma\neq 0}{D_{\sigma}}\right);$
$\displaystyle f_{*}\mathcal{O}_{X}$
$\displaystyle=\mathcal{O}_{Y}\oplus\bigoplus\limits_{\chi\neq\chi_{000}}L_{\chi}^{-1}.$

This implies that

$\displaystyle H^{0}\left(X,K_{X}\right)$
$\displaystyle=H^{0}\left(Y,K_{Y}\right)\oplus\bigoplus_{\chi\neq\chi_{000}}{H^{0}\left(Y,K_{Y}+L_{\chi}\right)};$
$\displaystyle K^{2}_{X}$ $\displaystyle=2\left(2K_{Y}+\sum\limits_{\sigma\neq
0}{D_{\sigma}}\right)^{2};$ $\displaystyle p_{g}\left(X\right)$
$\displaystyle=p_{g}\left(Y\right)+\sum\limits_{\chi\neq\chi_{000}}{h^{0}\left(L_{\chi}+K_{Y}\right)};$
$\displaystyle\chi\left(\mathcal{O}_{X}\right)$
$\displaystyle=8\chi\left(\mathcal{O}_{Y}\right)+\sum\limits_{\chi\neq\chi_{000}}{\frac{1}{2}L_{\chi}\left(L_{\chi}+K_{Y}\right)}.$

From [11, Proposition 4.1,c 2.1], the generators of the canonical linear
system $\left|K_{X}\right|$ are obtained as follows:

######  Proposition 6.

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 canonical linear
system $\left|K_{X}\right|$ is generated by

$\displaystyle
f^{*}\left(\left|K_{Y}+L_{\chi}\right|\right)+\sum\limits_{\chi\left(\sigma\right)=1}{R_{\sigma}},\hskip
5.0pt\forall\chi\in J$

where
$J:=\left\\{\chi{{}^{\prime}}:\left|K_{Y}+L_{\chi{{}^{\prime}}}\right|\neq\emptyset\right\\}$
and $R_{\sigma}$ is the reduced divisor supported on
$f^{*}\left(D_{\sigma}\right)$.

For the explanation of this proposition, we refer the reader to [3, Page 3] or
[6, Remark 3.16].

##  3 Constructions of the regular surfaces with $d=10,14$

In this section, we present the constructions of the regular surfaces with
$d=10,14$ listed in Theorem 1. We first construct a surface with $d=14$ and
$q=0$ by using a $\mathbb{Z}_{2}^{3}$-cover of a del Pezzo surface of degree
$7$. To obtain a surface with $d=10$ and $q=0$, we impose two distinct triple
points in the total branch locus and resolve the singularities.

###  3.1 A surface with $d=14$ and $q=0$.

Let $Y_{2}$ be a del Pezzo surface of degree $7$ (see Notation 1). We consider
the following smooth divisors:

$\displaystyle D_{010}:=$ $\displaystyle f_{11}+f_{12}$ $\displaystyle
D_{011}:=$ $\displaystyle f_{21}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle f_{22}$ $\displaystyle D_{101}:=$ $\displaystyle
C_{1}\in\left|l+f_{1}\right|$ $\displaystyle D_{110}:=$ $\displaystyle
C_{2}\in\left|l+f_{1}\right|$ $\displaystyle D_{111}:=$ $\displaystyle
f_{23},$

and $D_{001}=0$ where $f_{11},f_{12}\in\left|f_{1}\right|$ and
$f_{21},f_{22},f_{23}\in\left|f_{2}\right|$ are distinct divisors of $Y_{2}$
such that no more than two of these divisors $D_{\sigma}$ go through the same
point. We consider the following non-trivial divisors of $Y_{2}$:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
$L_{001}:=$ |  | $f_{2}$ | $+l$
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
$L_{101}:=$ |  | $f_{2}$ | $+l$
$L_{011}:=$ | $2f_{1}$ |  | $+l$
$L_{111}:=$ | $f_{1}$ |  $+f_{2}$. |

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2l$ $2L_{010}$ $\equiv$ $D_{010}$ $+D_{011}$ $+D_{110}$
$+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2l$ $2L_{001}$ $\equiv D_{001}$
$+D_{011}$ $+D_{101}$ $+D_{111}$ $\equiv$ $2f_{2}$ $+2l$ $2L_{110}$ $\equiv$
$D_{010}$ $+D_{011}$ $+D_{100}$ $+D_{101}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2l$
$2L_{101}$ $\equiv D_{001}$ $+D_{011}$ $+D_{100}$ $+D_{110}$ $\equiv$ $2f_{2}$
$+2l$ $2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$ $+D_{110}$ $\equiv$
$4f_{1}$ $+2l$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$ $+D_{100}$ $+D_{111}$
$\equiv$ $2f_{1}$ $+2f_{2}$.

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}}$.
Because each branch component $D_{\sigma}$ is smooth and the total branch
locus $B$ is a normal crossings divisor, the surface $X$ is smooth. Moreover,
by Proposition 5, the surface $X$ satisfies the following:

$\displaystyle 2K_{X}$ $\displaystyle\equiv g^{*}\left(f_{1}+f_{2}+l\right).$

We notice that a surface is of general type and minimal if the canonical
divisor is big and nef (see e.g. [5, Section 2]). Since the divisor $2K_{X}$
is the pull-back of a nef and big divisor, the canonical divisor $K_{X}$ is
nef and big. Thus, the surface $X$ is of general type and minimal.
Furthermore, from Proposition 5, the surface $X$ possesses the following
invariants:

$\displaystyle K_{X}^{2}=$ $\displaystyle 2\left(f_{1}+f_{2}+l\right)^{2}=14,$
$\displaystyle p_{g}\left(X\right)=$ $\displaystyle
p_{g}\left(Y_{2}\right)+\sum\limits_{\chi\neq\chi_{000}}{h^{0}\left(L_{\chi}+K_{Y_{2}}\right)}=h^{0}\left(\mathcal{O}_{Y_{2}}\right)+h^{0}\left(\mathcal{O}_{Y_{2}}\right)+h^{0}\left(\mathcal{O}_{Y_{2}}\right)=3,$
$\displaystyle\chi\left(\mathcal{O}_{X}\right)=$ $\displaystyle
8\chi\left(\mathcal{O}_{Y_{2}}\right)+\sum\limits_{\chi\neq\chi_{000}}{\frac{1}{2}L_{\chi}\left(L_{\chi}+K_{Y_{2}}\right)}=4,$
$\displaystyle q\left(X\right)=$ $\displaystyle
1+p_{g}\left(X\right)-\chi\left(\mathcal{O}_{X}\right)=0.$

We show that the canonical map $\varphi_{\left|K_{X}\right|}$ has degree $14$.
By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{f}_{11}+\overline{f}_{12}+\overline{f}_{21}+\overline{e}_{1},\overline{f}_{22}+\overline{C}_{1},\overline{C}_{2}+\overline{f}_{23},$

where $\overline{C}_{1}:=g^{*}\left(C_{1}\right)_{\text{red}}$,
$\overline{C}_{2}:=g^{*}\left(C_{2}\right)_{\text{red}}$,
$\overline{e}_{1}:=g^{*}\left(e_{1}\right)_{\text{red}}$ and
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$. Because the three
divisors
$\overline{f}_{11}+\overline{f}_{12}+\overline{f}_{21}+\overline{e}_{1}$,
$\overline{f}_{22}+\overline{C}_{1}$, $\overline{C}_{2}+\overline{f}_{23}$
have no common intersection, the linear system $\left|K_{X}\right|$ is base
point free. This together with $K_{X}^{2}=14>0$ implies that the linear system
$\left|K_{X}\right|$ is not composed with a pencil. Thus, the canonical image
is $\mathbb{P}^{2}$ and the canonical map is of degree $14$. Therefore we
obtain the surface in the sixth row of Theorem 1.

######  Remark 1.

The surface $X$ has two pencils of genus $5$ corresponding to the fibres
$f_{1},f_{2}$.

###  3.2 A surface with $d=10$ and $q=0$.

The main idea to construct a surface with $d=10$ and $q=0$ is as follows: we
take the branch locus

$\displaystyle
D_{010}\in\left|f_{11}+f_{12}\right|,D_{011}\in\left|f_{21}+e_{1}\right|,D_{100}\in\left|f_{22}\right|,D_{101}\in\left|l+f_{1}\right|,D_{110}\in\left|l+f_{1}\right|,D_{111}\in\left|f_{23}\right|,$

as Section 3.1 except that now we require that the three branch components
$D_{011}$, $D_{101}$, $D_{110}$ go through the same point $P_{3}$ and that the
three branch components $D_{010}$, $D_{101}$, $D_{111}$ go through the same
point $P_{4}$. The $\mathbb{Z}^{3}_{2}$-cover ramified on this branch locus
has singularities. Resolving the singularities, we obtain a surface with
$d=10$ and $q=0$. For the sake of simplicity, we present the construction by
resolving the singularities of the branch locus.

We denote by $Y_{4}$ the blow-up of $Y_{2}$ at two distinct points $P_{3}$ and
$P_{4}$. The surface $Y_{4}$ is a del Pezzo surface of degree $5$ (see
Notation 3). We consider the following smooth divisors:

$\displaystyle D_{010}:=$ $\displaystyle f_{11}+h_{14}$ $\displaystyle
D_{011}:=$ $\displaystyle h_{23}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle f_{21}$ $\displaystyle D_{101}:=$ $\displaystyle
C_{1}\in\left|f_{1}+h_{34}\right|$ $\displaystyle D_{110}:=$ $\displaystyle
C_{2}\in\left|f_{1}+f_{3}\right|$ $\displaystyle D_{111}:=$ $\displaystyle
h_{24}$

and $D_{001}=0$, where $f_{11}\in\left|f_{1}\right|$,
$f_{21}\in\left|f_{2}\right|$ are divisors of $Y_{4}$ such that no more than
two of these divisors $D_{\sigma}$ go through the same point. We consider the
following divisors:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$ |
---|---|---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$ |
$L_{001}:=$ |  | $f_{2}$ | $+f_{3}$ | $-e_{4}$ |
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$ |
$L_{101}:=$ |  | $f_{2}$ | $+f_{3}$ |  |
$L_{011}:=$ | $2f_{1}$ |  | $+f_{3}$ | $-e_{4}$ |
$L_{111}:=$ | $f_{1}$ | $+f_{2}$ |  | $-e_{4}.$ |

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{010}$ $\equiv$ $D_{010}$
$+D_{011}$ $+D_{110}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$
$-2e_{4}$ $2L_{001}$ $\equiv D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$
$\equiv$ $2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{110}$ $\equiv$ $D_{010}$ $+D_{011}$
$+D_{100}$ $+D_{101}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$
$2L_{101}$ $\equiv D_{001}$ $+D_{011}$ $+D_{100}$ $+D_{110}$ $\equiv$ $2f_{2}$
$+2f_{3}$ $2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$ $+D_{110}$
$\equiv$ $4f_{1}$ $+2f_{3}$ $-2e_{4}$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$
$+D_{100}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $-2e_{4}.$

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{4}}$.
Because each branch component $D_{\sigma}$ is smooth and the total branch
locus is a normal crossings divisor, the surface $X$ is smooth. By Proposition
5, we get

$\displaystyle 2K_{X}$ $\displaystyle\equiv
g^{*}\left(f_{1}+f_{2}+h_{34}\right)$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. Thus, the surface $X$ is of general
type and minimal. Furthermore, from Proposition 5, the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=10,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=4,q\left(X\right)=0.$

We show that the canonical map $\varphi_{\left|K_{X}\right|}$ has degree $10$.
By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{f}_{11}+\overline{h}_{14}+\overline{h}_{23}+\overline{e}_{1},\overline{f}_{21}+\overline{C}_{1},\overline{C}_{2}+\overline{h}_{24},$

where $\overline{C}_{i}:=g^{*}\left(C_{i}\right)_{\text{red}}$,
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$,
$\overline{e}_{1}:=g^{*}\left(e_{1}\right)_{\text{red}}$ and
$\overline{h}_{ij}:=g^{*}\left(h_{ij}\right)_{\text{red}}$. Because the three
divisors
$\overline{f}_{11}+\overline{h}_{14}+\overline{h}_{23}+\overline{e}_{1}$,
$\overline{f}_{21}+\overline{C}_{1}$, $\overline{C}_{2}+\overline{h}_{24}$
have no common intersection, the linear system $\left|K_{X}\right|$ is base
point free. This together with $K_{X}^{2}=10>0$ implies that the linear system
$\left|K_{X}\right|$ is not composed with a pencil. Thus, the canonical image
is $\mathbb{P}^{2}$ and the canonical map is of degree $10$. Therefore we
obtain the surface in the first row of Theorem 1.

######  Remark 2.

The surface $X$ has four pencils of genus $5$ corresponding to the fibres
$f_{1},f_{2},f_{3},f_{4}$.

######  Remark 3.

Taking the $\mathbb{Z}_{2}^{3}$-cover of $Y_{2}$ ramified on the above branch
locus with two triple points $P_{3},P_{4}$, we would obtain a surface with
four singular points of type $\frac{1}{4}\left(1,1\right)$ (cf. [1, Table 1,
Section 3.3]). The preimage of each $P_{i}$ is two $\left(-4\right)$-curves.
The surface $X$ is the minimal resolution of this singular surface.

In fact, we consider the following commutative diagram

$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}^{3}_{2}}$$\scriptstyle{\overline{g}}$$\textstyle{Y_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{\mathbb{Z}^{3}_{2}}$$\scriptstyle{g}$$\textstyle{Y_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$

where the map $\overline{g}$ is the $\mathbb{Z}_{2}^{3}$-cover of $Y_{2}$
ramified on the above branch locus with two triple points $P_{3},P_{4}$ and
the map $\pi$ is the blow-up at $P_{3}$, $P_{4}$. Each divisor
$g^{*}\left(\pi\left(P_{i}\right)\right)$ are two $\left(-4\right)$-curves.
The map $c$ is the contraction of these four $\left(-4\right)$-curves.

##  4 Constructions of the irregular surfaces with $d=10,12,14$

In this section, we present the constructions of the surfaces with
$d=10,12,14$ and $q=1,2$ listed in Theorem 1. We first construct a surface
with $d=14$ and $q=1$ by using a $\mathbb{Z}_{2}^{3}$-cover of a del Pezzo
surface of degree $7$. To obtain the other surfaces described in Theorem 1, we
impose triple or quadruple points in the total branch locus and resolve the
singularities.

###  4.1 A surface with $d=14$ and $q=1$

Let $Y_{2}$ be a del Pezzo surface of degree $7$ (see Notation 1). We consider
the following smooth divisors:

$\displaystyle D_{010}:=$ $\displaystyle f_{11}+f_{12}$ $\displaystyle
D_{011}:=$ $\displaystyle f_{21}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle l_{1}+e_{2}$ $\displaystyle D_{101}:=$ $\displaystyle
f_{22}+h_{12}$ $\displaystyle D_{110}:=$ $\displaystyle
C_{2}\in\left|l+f_{1}\right|$ $\displaystyle D_{111}:=$ $\displaystyle
f_{23},$

and $D_{001}=0$, where $f_{11},f_{12}\in\left|f_{1}\right|$,
$f_{21},f_{22},f_{23}\in\left|f_{2}\right|$ and $l_{1}\in\left|l\right|$ are
distinct divisors in $Y_{2}$ such that no more than two of these divisors
$D_{\sigma}$ go through the same point. We consider the following non-trivial
divisors in $Y_{2}$:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
$L_{001}:=$ |  | $2f_{2}$ |
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+l$
$L_{101}:=$ |  |  | $2l$
$L_{011}:=$ | $2f_{1}$ | $+f_{2}$ |
$L_{111}:=$ | $f_{1}$ |  |  $+l$.

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2l$ $2L_{010}$ $\equiv$ $D_{010}$ $+D_{011}$ $+D_{110}$
$+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2l$ $2L_{001}$ $\equiv D_{001}$
$+D_{011}$ $+D_{101}$ $+D_{111}$ $\equiv$ $4f_{2}$ $2L_{110}$ $\equiv$
$D_{010}$ $+D_{011}$ $+D_{100}$ $+D_{101}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2l$
$2L_{101}$ $\equiv D_{001}$ $+D_{011}$ $+D_{100}$ $+D_{110}$ $\equiv$ $4l$
$2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$ $+D_{110}$ $\equiv$ $4f_{1}$
$+2f_{2}$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$ $+D_{100}$ $+D_{111}$
$\equiv$ $2f_{1}$ $+2l$.

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}}$.
Because each branch component $D_{\sigma}$ is smooth and the total branch
locus is a normal crossings divisor, the surface $X$ is smooth. Furthermore,
by Proposition 5, we get

$\displaystyle 2K_{X}$ $\displaystyle\equiv g^{*}\left(f_{1}+f_{2}+l\right)$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. Thus, the surface $X$ is of general
type and minimal. Moreover, Proposition 5 shows that the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=14,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=3,q\left(X\right)=1.$

We prove that the canonical map $\varphi_{\left|K_{X}\right|}$ is of degree
$14$. By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by
the three following divisors:

$\displaystyle\overline{f}_{11}+\overline{f}_{12}+\overline{f}_{21}+\overline{e}_{1},\overline{l}_{1}+\overline{e}_{2}+\overline{f}_{22}+\overline{h}_{12},\overline{C}_{2}+\overline{f}_{23},$

where $\overline{l}_{1}:=g^{*}\left(l_{1}\right)_{\text{red}}$,
$\overline{C}_{2}:=g^{*}\left(C_{2}\right)_{\text{red}}$,
$\overline{e}_{i}:=g^{*}\left(e_{i}\right)_{\text{red}}$,
$\overline{h}_{12}:=g^{*}\left(f_{12}\right)_{\text{red}}$ and
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$. Because the three
divisors
$\overline{f}_{11}+\overline{f}_{12}+\overline{f}_{21}+\overline{e}_{1}$,
$\overline{l}_{1}+\overline{e}_{2}+\overline{f}_{22}+\overline{h}_{12}$,
$\overline{C}_{2}+\overline{f}_{23}$ have no common intersection, the linear
system $\left|K_{X}\right|$ is base point free. This together with
$K_{X}^{2}=14>0$ implies that the linear system $\left|K_{X}\right|$ is not
composed with a pencil. Thus, the canonical image is $\mathbb{P}^{2}$ and the
canonical map is of degree $14$. Therefore we obtain the surface in the last
row of Theorem 1.

######  Remark 4.

The surface $X$ has one pencil of genus $5$ corresponding to the fibres
$f_{1}$. The Albanese pencil of $X$, which comes from the fibration
$\left|f_{2}\right|$, has genus $3$.

###  4.2 A surface with $d=12$ and $q=1$.

The main idea to construct a surface with $d=12$ and $q=1$ is as follows: we
take the branch locus

$\displaystyle
D_{010}\in\left|f_{11}+f_{12}\right|,D_{011}\in\left|f_{21}+e_{1}\right|,D_{100}\in\left|l_{1}+e_{2}\right|,D_{101}\in\left|f_{22}+h_{12}\right|,D_{110}\in\left|l+f_{1}\right|,D_{111}\in\left|f_{23}\right|$

as Section 4.1 except that now we require that the three branch components
components $D_{010}$, $D_{100}$, $D_{110}$ go through the same point $P_{3}$
in such a way that the branch locus contains $P_{3}$ as an ordinary triple
point. The $\mathbb{Z}^{3}_{2}$-cover ramified on this branch locus has
singularities. Resolving the singularities, we obtain a surface with $d=12$
and $q=1$. For the sake of simplicity, we present the construction by
resolving the singularities of the branch locus.

Let us denote by $Y_{3}$ the blow-up of $Y_{2}$ at a point $P_{3}$. The
surface $Y_{3}$ is a del Pezzo surface of degree $6$ (see Notation 2). We
consider the following smooth divisors:

$\displaystyle D_{010}:=$ $\displaystyle f_{11}+h_{13}$ $\displaystyle
D_{011}:=$ $\displaystyle f_{21}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle f_{31}+e_{2}$ $\displaystyle D_{101}:=$ $\displaystyle
f_{22}+h_{12}$ $\displaystyle D_{110}:=$ $\displaystyle
C_{2}\in\left|f_{1}+f_{3}\right|$ $\displaystyle D_{111}:=$ $\displaystyle
f_{23}$

and $D_{001}=0$, where $f_{11}\in\left|f_{1}\right|$,
$f_{21},f_{22},f_{23}\in\left|f_{2}\right|$ and $f_{31}\in\left|f_{3}\right|$
are distinct divisors in $Y_{3}$ such that no more than two of these divisors
$D_{\sigma}$ go through the same point. In addition, we consider the following
non-trivial divisors in $Y_{3}$:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
---|---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
$L_{001}:=$ |  | $2f_{2}$ |  |
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
$L_{101}:=$ | $l$ |  | $+f_{3}$ |
$L_{011}:=$ | $2f_{1}$ | $+f_{2}$ |  | $-e_{3}$
$L_{111}:=$ | $f_{1}$ |  |  $+f_{3}$. |

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2f_{3}$ $2L_{010}$ $\equiv$ $D_{010}$ $+D_{011}$
$+D_{110}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$ $2L_{001}$ $\equiv
D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$ $\equiv$ $4f_{2}$ $2L_{110}$
$\equiv$ $D_{010}$ $+D_{011}$ $+D_{100}$ $+D_{101}$ $\equiv$ $2f_{1}$
$+2f_{2}$ $+2f_{3}$ $2L_{101}$ $\equiv D_{001}$ $+D_{011}$ $+D_{100}$
$+D_{110}$ $\equiv$ $2l$ $+2f_{3}$ $2L_{011}$ $\equiv D_{001}$ $+D_{010}$
$+D_{101}$ $+D_{110}$ $\equiv$ $4f_{1}$ $+2f_{2}$ $-2e_{3}$ $2L_{111}$ $\equiv
D_{001}$ $+D_{010}$ $+D_{100}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{3}$.

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}}$.
Moreover, because each branch component $D_{\sigma}$ is smooth and the total
branch locus is a normal crossings divisor, the surface $X$ is smooth.
Proposition 5 implies that

$\displaystyle 2K_{X}$ $\displaystyle\equiv
g^{*}\left(f_{1}+f_{2}+f_{3}\right).$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. Thus, the surface $X$ is of general
type and minimal. Furthermore, from Proposition 5, the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=12,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=3,q\left(X\right)=1.$

The canonical map $\varphi_{\left|K_{X}\right|}$ has degree $12$. In fact, by
Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{h}_{13}+\overline{f}_{11}+\overline{f}_{21}+\overline{e}_{1},\overline{f}_{31}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{22},\overline{C}_{2}+\overline{f}_{23},$

where $\overline{C}_{2}:=g^{*}\left(C_{2}\right)_{\text{red}}$,
$\overline{h}_{ij}:=g^{*}\left(h_{ij}\right)_{\text{red}}$,
$\overline{e}_{i}:=g^{*}\left(e_{i}\right)_{\text{red}}$ and
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$ for all
$i,j\in\left\\{1,2,3\right\\}$. Because the three divisors
$\overline{h}_{13}+\overline{f}_{11}+\overline{f}_{21}+\overline{e}_{1}$,
$\overline{f}_{31}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{22}$,
$\overline{C}_{2}+\overline{f}_{23}$ have no common intersection, the linear
system $\left|K_{X}\right|$ is base point free. This together with
$K_{X}^{2}=12>0$ implies that the linear system $\left|K_{X}\right|$ is not
composed with a pencil. Thus, the canonical image is $\mathbb{P}^{2}$ and the
canonical map is of degree $12$. Therefore we obtain the surface described in
the fourth row of Theorem 1.

######  Remark 5.

The surface $X$ has two pencils of genus $5$ corresponding to the fibres
$f_{1}$, $f_{3}$. The Albanese pencil of $X$, which comes from the fibration
$\left|f_{2}\right|$, has genus $3$.

######  Remark 6.

Taking the $\mathbb{Z}_{2}^{3}$ cover of $Y_{2}$ ramified on the above branch
locus with a triple point $P_{3}$, we would obtain a surface with two singular
points of type $\frac{1}{4}\left(1,1\right)$ coming from the point $P_{3}$
(cf. [1, Table 1, Section 3.3]). The surface $X$ is the minimal resolution of
this singular surface.

###  4.3 A surface with $d=12$ and $q=2$.

The main idea to construct a surface with $d=12$ and $q=2$ is as follows: we
take the branch locus

$\displaystyle
D_{010}\in\left|f_{11}+f_{12}\right|,D_{011}\in\left|f_{21}+e_{1}\right|,D_{100}\in\left|l_{1}+e_{2}\right|,D_{101}\in\left|f_{22}+h_{12}\right|,D_{110}\in\left|l+f_{1}\right|,D_{111}\in\left|f_{23}\right|$

as Section 4.1 except that now we require that the branch component $D_{110}$
has as unique singular point a node $P_{3}$ and the branch components
$D_{011}$, $D_{100}$ go through the point $P_{3}$. If we take a
$\mathbb{Z}^{3}_{2}$-cover ramified on this branch locus, we obtain a surface
with singularities. The singularities appearing in this case are solved as
follows: since the strict transforms of
$D_{010},D_{011},D_{100},D_{101},D_{110},D_{111}$ of the blow-up at $P_{3}$ do
not satisfy the condition of Proposition 4, we can not take a
$\mathbb{Z}^{3}_{2}$-cover ramified on these strict transforms. To solve this
problem, we add the exceptional divisor $e_{3}$ of the blow-up to $D_{111}$ to
fulfill the condition of Proposition 4. We present the construction in detail
as follows:

We denote by $Y_{3}$ the blow-up of $Y_{2}$ at a point $P_{3}$. The surface
$Y_{3}$ is a del Pezzo surface of degree $6$ (see Notation 2). We consider the
following divisors of $Y_{3}$:

$\displaystyle D_{010}:=$ $\displaystyle f_{11}+f_{12}$ $\displaystyle
D_{011}:=$ $\displaystyle h_{23}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle f_{31}+e_{2}$ $\displaystyle D_{101}:=$ $\displaystyle
f_{21}+h_{12}$ $\displaystyle D_{110}:=$ $\displaystyle f_{32}+h_{13}$
$\displaystyle D_{111}:=$ $\displaystyle f_{22}+e_{3},$

and $D_{001}=0$, where $f_{11},f_{12}\in\left|f_{1}\right|$,
$f_{21},f_{22}\in\left|f_{2}\right|$ and $f_{31},f_{32}\in\left|f_{3}\right|$
are distinct divisors such that no more than two of these divisors
$D_{\sigma}$ go through the same point. We consider the following non-trivial
divisors of $Y_{3}$:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
---|---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
$L_{001}:=$ |  | $2f_{2}$ |  |
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ |
$L_{101}:=$ |  |  | $2f_{3}$ |
$L_{011}:=$ | $2f_{1}$ | $+f_{2}$ |  | $-e_{3}$
$L_{111}:=$ | $f_{1}$ |  |  |  $+l$.

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2f_{3}$ $2L_{010}$ $\equiv$ $D_{010}$ $+D_{011}$
$+D_{110}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$ $2L_{001}$ $\equiv
D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$ $\equiv$ $4f_{2}$ $2L_{110}$
$\equiv$ $D_{010}$ $+D_{011}$ $+D_{100}$ $+D_{101}$ $\equiv$ $2f_{1}$
$+2f_{2}$ $+2f_{3}$ $2L_{101}$ $\equiv D_{001}$ $+D_{011}$ $+D_{100}$
$+D_{110}$ $\equiv$ $4f_{3}$ $2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$
$+D_{110}$ $\equiv$ $4f_{1}$ $+2f_{2}$ $-2e_{3}$ $2L_{111}$ $\equiv D_{001}$
$+D_{010}$ $+D_{100}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2l$.

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}}$.
Moreover, because each branch component $D_{\sigma}$ is smooth and the total
branch locus is a normal crossings divisor, the surface $X$ is smooth.
Furthermore, Proposition 5 implies that

$\displaystyle 2K_{X}$ $\displaystyle\equiv
g^{*}\left(f_{1}+f_{2}+f_{3}\right).$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. Thus, the surface $X$ is of general
type and minimal. Furthermore, from Proposition 5, the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=12,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=2,q\left(X\right)=2.$

We show that the canonical map $\varphi_{\left|K_{X}\right|}$ has degree $12$.
By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{f}_{11}+\overline{f}_{12}+\overline{h}_{23}+\overline{e}_{1},\overline{f}_{31}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{21},\overline{f}_{32}+\overline{h}_{13}+\overline{e}_{3}+\overline{f}_{22},$

where $\overline{h}_{ij}:=g^{*}\left(h_{ij}\right)_{\text{red}}$,
$\overline{e}_{i}:=g^{*}\left(e_{i}\right)_{\text{red}}$ and
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$ for all
$i,j\in\left\\{1,2,3\right\\}$. Because the three divisors
$\overline{f}_{11}+\overline{f}_{12}+\overline{h}_{23}+\overline{e}_{1}$,
$\overline{f}_{31}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{21}$,
$\overline{f}_{32}+\overline{h}_{13}+\overline{e}_{3}+\overline{f}_{22}$ have
no common intersection, the linear system $\left|K_{X}\right|$ is base point
free. This together with $K_{X}^{2}=12>0$ implies that the linear system
$\left|K_{X}\right|$ is not composed with a pencil. Thus, the canonical image
is $\mathbb{P}^{2}$ and the canonical map is of degree $12$. Therefore we
obtain the surface listed in the fifth row of Theorem 1.

######  Remark 7.

Taking the $\mathbb{Z}_{2}^{3}$-cover of $Y_{2}$ ramified on the above branch
locus with a quadruple point $P_{3}$, we would obtain a surface with a
singular point coming from the point $P_{3}$. This point is Gorenstein
elliptic singularity whose minimal resolution are elliptic curves with self-
intersection $-2$ (cf. [1, Table 1, Section 3.3]). The surface $X$ is the
minimal resolution of this singular surface.

######  Remark 8.

The Albanese morphism of this surface is of degree $2$.

In fact, we consider the following commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}^{3}}$$\scriptstyle{g}$$\scriptstyle{\mathbb{Z}_{2}}$$\scriptstyle{g_{1}}$$\textstyle{Y_{3}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}^{2}}$$\scriptstyle{g_{2}}$

where
$\textstyle{g_{2}:Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}}$
is the bidouble cover with the following buiding data:

$D_{1}:=D_{011}$ | $=h_{23}+e_{1}$ | $\equiv l+e_{1}-e_{2}-e_{3}$
---|---|---
$D_{2}:=D_{101}+D_{111}$ | $=f_{21}+h_{12}+f_{22}+e_{3}$ | $\equiv 3l-e_{1}-3e_{2}+e_{3}$
$D_{3}:=D_{100}+D_{110}$ | $=f_{31}+e_{2}+f_{32}+h_{13}$ | $\equiv 3l-e_{1}+e_{2}-3e_{3}$
$L_{1}:=L_{100}$ |  | $\equiv 3l-e_{1}-e_{2}-e_{3}$
$L_{2}:=L_{101}$ |  | $\equiv 2l-2e_{3}$
$L_{3}:=L_{001}$ |  |  $\equiv 2l-2e_{2}$.

We have that $p_{g}\left(Z\right)=1,q\left(Z\right)=2$ and

$\displaystyle 2K_{Z}\equiv g_{2}^{*}\left(h_{23}+e_{1}\right)$

We notice that the surface $Z$ contains $\left(-1\right)$-curves. Let
$\textstyle{c:Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{Z}}$
be the contraction map. The minimal surface $\overline{Z}$ satisfies
$p_{g}\left(\overline{Z}\right)=1,q\left(\overline{Z}\right)=2$ and
$2K_{\overline{Z}}\equiv 0$. So the surface $\overline{Z}$ is an abelian
surface. By the universal property of the Albanese morphism of the surface
$X$, the following diagram commutes:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}}$$\scriptstyle{\alpha}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\overline{Z}}$$\textstyle{Alb\left(X\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Because the surface $X$ is of general type, the Albanese morphism $\alpha$ is
of degree $2$ and $Alb\left(X\right)$ is isomorphic to $\overline{Z}$.

###  4.4 A surface with $d=10$ and $q=1$.

The main idea to construct a surface with $d=10$ and $q=1$ is as follows: we
take the branch locus

$\displaystyle
D_{010}\in\left|f_{11}+f_{12}\right|,D_{011}\in\left|f_{21}+e_{1}\right|,D_{100}\in\left|l_{1}+e_{2}\right|,D_{101}\in\left|f_{22}+h_{12}\right|,D_{110}\in\left|l+f_{1}\right|,D_{111}\in\left|f_{23}\right|$

as Section 4.1 except that now we require that the three branch components
components $D_{010}$, $D_{100}$, $D_{110}$ go through the same two distinct
points $P_{3}$ and $P_{4}$ in such a way that the branch locus contains
$P_{3},P_{4}$ as two ordinary triple points. The $\mathbb{Z}^{3}_{2}$-cover
ramified on this branch locus has singularities. Resolving the singularities,
we obtain a surface with $d=10$ and $q=1$. For the sake of simplicity, we
present the construction by resolving the singularities of the branch locus.

Let us denote by $Y_{4}$ the blow-up of $Y_{2}$ at two distinct points $P_{3}$
and $P_{4}$. The surface $Y_{4}$ is a del Pezzo surface of degree $5$ (see
Notation 3). We consider the following smooth divisors of $Y_{4}$:

$\displaystyle D_{010}:=$ $\displaystyle h_{13}+h_{14}$ $\displaystyle
D_{011}:=$ $\displaystyle f_{21}+e_{1}$ $\displaystyle D_{100}:=$
$\displaystyle h_{34}+e_{2}$ $\displaystyle D_{101}:=$ $\displaystyle
h_{12}+f_{22}$ $\displaystyle D_{110}:=$ $\displaystyle
C_{2}\in\left|f_{1}+h_{34}\right|$ $\displaystyle D_{111}:=$ $\displaystyle
f_{23},$

and $D_{001}=0$, where $f_{21},f_{22},f_{23}\in\left|f_{2}\right|$ are
distinct divisors such that no more than two of these divisors $D_{\sigma}$ go
through the same point. We consider the following non-trivial divisors of
$Y_{4}$:

$L_{100}\equiv$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
---|---|---|---|---
$L_{010}\equiv$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
$L_{001}\equiv$ |  | $2f_{2}$ |  |
$L_{110}\equiv$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
$L_{101}\equiv$ |  |  | $f_{3}$ | $+f_{4}$
$L_{011}\equiv$ | $2f_{1}$ | $+f_{2}$ | $-e_{3}$ | $-e_{4}$
$L_{111}\equiv$ | $f_{1}$ |  | $+f_{3}$ | $-e_{4}.$

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{010}$ $\equiv$ $D_{010}$
$+D_{011}$ $+D_{110}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$
$-2e_{4}$ $2L_{001}$ $\equiv D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$
$\equiv$ $4f_{2}$ $2L_{110}$ $\equiv$ $D_{010}$ $+D_{011}$ $+D_{100}$
$+D_{101}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{101}$ $\equiv
D_{001}$ $+D_{011}$ $+D_{100}$ $+D_{110}$ $\equiv$ $2f_{3}$ $+2f_{4}$
$2L_{011}$ $\equiv D_{001}$ $+D_{010}$ $+D_{101}$ $+D_{110}$ $\equiv$ $4f_{1}$
$+2f_{2}$ $-2e_{3}$ $-2e_{4}$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$
$+D_{100}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{3}$ $-2e_{4}.$

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{4}}$.
Moreover, because each branch component $D_{\sigma}$ is smooth and the total
branch locus is a normal crossing divisor, the surface $X$ is smooth. In
addition, Proposition 5 yields that

$\displaystyle 2K_{X}$ $\displaystyle\equiv
g^{*}\left(f_{1}+f_{2}+h_{34}\right).$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. Thus, the surface $X$ is of general
type and minimal. Furthermore, from Proposition 5, the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=10,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=3,q\left(X\right)=1.$

We show that the canonical map $\varphi_{\left|K_{X}\right|}$ has degree $10$.
By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{h}_{13}+\overline{h}_{14}+\overline{f}_{21}+\overline{e}_{1},\overline{h}_{34}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{22},\overline{C}_{2}+\overline{f}_{23},$

where $\overline{C}_{2}:=g^{*}\left(C_{2}\right)_{\text{red}}$,
$\overline{e}_{1}:=g^{*}\left(e_{1}\right)_{\text{red}}$,
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$ and
$\overline{h}_{ij}:=g^{*}\left(h_{ij}\right)_{\text{red}}$. Because the three
divisors
$\overline{h}_{13}+\overline{h}_{14}+\overline{f}_{21}+\overline{e}_{1}$,
$\overline{h}_{34}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{22}$,
$\overline{C}_{2}+\overline{f}_{23}$ have no common intersection, the linear
system $\left|K_{X}\right|$ is base point free. This together with
$K^{2}=10>0$ implies that the linear system $\left|K_{X}\right|$ is not
composed with a pencil. Thus, the canonical image is $\mathbb{P}^{2}$ and the
canonical map is of degree $10$. Therefore we obtain the surface in the second
row of Theorem 1.

######  Remark 9.

The surface $X$ has three pencils of genus $5$ corresponding to the fibre
$f_{1},f_{3},f_{4}$. The Albanese pencil of $X$, which comes from the
fibration $\left|f_{2}\right|$, has genus $3$.

######  Remark 10.

Taking the $\mathbb{Z}_{2}^{3}$-cover of $Y_{2}$ ramified on the above branch
locus with two ordinary triple points $P_{3},P_{4}$, we would obtain a surface
with four points of type $\frac{1}{4}\left(1,1\right)$ coming from $P_{3}$ and
$P_{4}$ (cf. [1, Table 1, Section 3.3]). The surface $X$ is the minimal
resolution of this singular surface.

###  4.5 A surface with $d=10$ and $q=2$.

The main idea to construct a surface with $d=10$ and $q=2$ is as follows: we
take the branch locus

$\displaystyle
D_{010}\in\left|f_{11}+f_{12}\right|,D_{011}\in\left|f_{21}+e_{1}\right|,D_{100}\in\left|l_{1}+e_{2}\right|,D_{101}\in\left|f_{22}+h_{12}\right|,D_{110}\in\left|l+f_{1}\right|,D_{111}\in\left|f_{23}\right|$

as Section 4.1 except that now we require that the branch component $D_{110}$
has as unique singular point a node $P_{3}$ and the branch components
$D_{011}$, $D_{100}$ go through the point $P_{3}$. In addition, we require
that the three branch components $D_{010}$, $D_{100}$, $D_{111}$ go through
the same point $P_{4}$. The $\mathbb{Z}^{3}_{2}$-cover ramified on this branch
locus has singularities. Resolving the singularities, we obtain a surface with
$d=10$ and $q=2$. For the sake of simplicity, we present the construction by
resolving the singularities of the branch locus.

Let us denote by $Y_{4}$ the blow-up of $Y_{2}$ at two distinct points $P_{3}$
and $P_{4}$. $Y_{4}$ is a del Pezzo surface of degree $5$ (see Notation 3). We
consider the following smooth divisors of $Y_{4}$:

$\displaystyle D_{001}:=$ $\displaystyle e_{4}$ $\displaystyle D_{010}:=$
$\displaystyle h_{14}+f_{11}$ $\displaystyle D_{011}:=$ $\displaystyle
h_{23}+e_{1}$ $\displaystyle D_{100}:=$ $\displaystyle h_{34}+e_{2}$
$\displaystyle D_{101}:=$ $\displaystyle h_{12}+f_{21}$ $\displaystyle
D_{110}:=$ $\displaystyle h_{13}+f_{31}$ $\displaystyle D_{111}:=$
$\displaystyle h_{24}+e_{3},$

where $f_{11}\in\left|f_{1}\right|$, $f_{21}\in\left|f_{2}\right|$,
$f_{31}\in\left|f_{3}\right|$ are divisors such that no more than two of these
divisors $D_{\sigma}$ go through the same point. We consider the following
non-trivial divisors of $Y_{4}$:

$L_{100}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
---|---|---|---|---
$L_{010}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
$L_{001}:=$ |  | $2f_{2}$ |  |
$L_{110}:=$ | $f_{1}$ | $+f_{2}$ | $+f_{3}$ | $-e_{4}$
$L_{101}:=$ |  |  | $2f_{3}$ |
$L_{011}:=$ | $2f_{1}$ | $+f_{2}$ | $-e_{3}$ |
$L_{111}:=$ | $f_{1}$ |  |  | $+f_{4}.$

These divisors $D_{\sigma},L_{\chi}$ satisfy the following relations:

$2L_{100}$ $\equiv$ $D_{100}$ $+D_{101}$ $+D_{110}$ $+D_{111}$ $\equiv$
$2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{010}$ $\equiv$ $D_{010}$
$+D_{011}$ $+D_{110}$ $+D_{111}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$
$-2e_{4}$ $2L_{001}$ $\equiv D_{001}$ $+D_{011}$ $+D_{101}$ $+D_{111}$
$\equiv$ $4f_{2}$ $2L_{110}$ $\equiv$ $D_{010}$ $+D_{011}$ $+D_{100}$
$+D_{101}$ $\equiv$ $2f_{1}$ $+2f_{2}$ $+2f_{3}$ $-2e_{4}$ $2L_{101}$ $\equiv
D_{001}$ $+D_{011}$ $+D_{100}$ $+D_{110}$ $\equiv$ $4f_{3}$ $2L_{011}$ $\equiv
D_{001}$ $+D_{010}$ $+D_{101}$ $+D_{110}$ $\equiv$ $4f_{1}$ $+2f_{2}$
$-2e_{3}$ $2L_{111}$ $\equiv D_{001}$ $+D_{010}$ $+D_{100}$ $+D_{111}$
$\equiv$ $2f_{1}$ $+2f_{4}.$

Thus by Proposition 4, the divisors $D_{\sigma},L_{\chi}$ define a
$\mathbb{Z}^{3}_{2}$-cover
$\textstyle{g:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{4}}$.
Moreover, because each branch component $D_{\sigma}$ is smooth and the total
branch locus is a normal crossings divisor, the surface $X$ is smooth. In
addition, Proposition 5 implies that

$\displaystyle 2K_{X}$ $\displaystyle\equiv
g^{*}\left(f_{1}+f_{2}+f_{3}\right).$

Since the divisor $2K_{X}$ is the pull-back of a nef and big divisor, the
canonical divisor $K_{X}$ is nef and big. So the surface $X$ is of general
type and minimal. Furthermore, from Proposition 5, the surface $X$ possesses
the following invariants:

$\displaystyle
K_{X}^{2}=12,p_{g}\left(X\right)=3,\chi\left(\mathcal{O}_{X}\right)=2,q\left(X\right)=2.$

We show that the canonical map $\varphi_{\left|K_{X}\right|}$ has degree $10$
and the linear system $\left|K_{X}\right|$ has a non-trivial fixed component.
By Proposition 6, the linear system $\left|K_{X}\right|$ is generated by the
three following divisors:

$\displaystyle\overline{e}_{4}+\overline{h}_{14}+\overline{f}_{11}+\overline{h}_{23}+\overline{e}_{1},\overline{e}_{4}+\overline{h}_{34}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{21},\overline{e}_{4}+\overline{h}_{13}+\overline{f}_{31}+\overline{h}_{24}+\overline{e}_{3},$

where $\overline{C}_{i}:=g^{*}\left(C_{i}\right)_{\text{red}}$,
$\overline{e}_{i}:=g^{*}\left(e_{i}\right)_{\text{red}}$,
$\overline{h}_{ij}:=g^{*}\left(h_{ij}\right)_{\text{red}}$ and
$\overline{f}_{ij}:=g^{*}\left(f_{ij}\right)_{\text{red}}$. Because the
$\left(-2\right)$-curve $\overline{e}_{4}$ is the common part of these three
above divisors, the $\left(-2\right)$-curve $\overline{e}_{4}$ is a fixed
component of $\left|K_{X}\right|$.

On the other hand, since the three divisors
$\overline{h}_{14}+\overline{f}_{11}+\overline{h}_{23}+\overline{e}_{1}$,
$\overline{h}_{34}+\overline{e}_{2}+\overline{h}_{12}+\overline{f}_{21}$,
$\overline{h}_{13}+\overline{f}_{31}+\overline{h}_{24}+\overline{e}_{3}$ have
no common intersection, the linear system $\left|M\right|$ is base point free,
where
$\left|M\right|:=\left|\overline{h}_{14}+\overline{f}_{11}+\overline{h}_{23}+\overline{e}_{1}\right|$.
This together with $M^{2}=10>0$ implies that the linear system
$\left|K_{X}\right|$ is not composed with a pencil. Thus, the canonical image
is $\mathbb{P}^{2}$ and the canonical map is of degree $10$. Therefore we
obtain the surface described in the third row of Theorem 1.

######  Remark 11.

Taking the $\mathbb{Z}_{2}^{3}$-cover of $Y_{1}$ ramified on the above branch
locus with singular points $P_{3},P_{4}$, we would obtain a surface with one
singular point of type $A_{1}$ and one Gorenstein elliptic singular point
whose minimal resolution is an elliptic curve with self-intersection $-2$. The
$A_{1}$ point comes from $P_{4}$ and the Gorenstein elliptic singular point
comes from $P_{3}$. The surface $X$ is the minimal resolution of this singular
surface.

######  Remark 12.

Similarly to the construction of the surface with $d=2,p_{g}=3,q=2$, the
Albanese morphism of this surface is of degree $2$.

######  Remark 13.

Let $H$ be the subgroup generated by $\left(0,0,1\right)$ of the group
$\mathbb{Z}_{2}^{3}$. In the all above constructions, the canonical map
$\varphi_{\left|K_{X}\right|}$ is the composition of the quotient map
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X/H}$
with the canonical map of the quotient surface $X/H$ because
$h^{0}\left(L_{\chi}+K_{Y}\right)=0$ for all
$\chi\in\left\\{\chi_{001},\chi_{101},\chi_{011},\chi_{111}\right\\}$.

## Acknowledgments

The author is deeply indebted to Margarida Mendes Lopes for all her help. The
author was partially supported by Fundação para a Ciência e Tecnologia (FCT),
Portugal through the program Lisbon Mathematics PhD (LisMath) of the
University of Lisbon, scholarship FCT - PD/BD/113632/2015 and project
UID/MAT/04459/2019 of CAMGSD. This paper was finished during the author’s
postdoctoral fellowship at the National Center for Theoretical Sciences
(NCTS), Taiwan, under the grant number MOST 109-2119-M-002-014. The author
would like to thank NCTS for the financial support and kind hospitality.

## References

  * [1] Alexeev, V., and Pardini, R. Non-normal abelian covers.  Compos. Math. 148, 4 (2012), 1051–1084.
  * [2] Beauville, A. L’application canonique pour les surfaces de type général.  Invent. Math. 55, 2 (1979), 121–140.
  * [3] Gleissner, C., Pignatelli, R., and Rito, C. New surfaces with canonical map of high degree.  arXiv e-prints (July 2018), arXiv:1807.11854.
  * [4] Lai, C.-J., and Yeung, S.-K. Examples of surfaces with canonical map of maximal degree.  Taiwanese J. Math. 25, 4 (2021), 699–716.
  * [5] Lopes, M. M., and Pardini, R. The geography of irregular surfaces.  In Current developments in algebraic geometry, vol. 59 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2012, pp. 349–378.
  * [6] Mendes Lopes, M., and Pardini, R. On the degree of the canonical map of a surface of general type.  arXiv e-prints (Mar. 2021), arXiv:2103.01912.
  * [7] Nguyen Bin A new example of an algebraic surface with canonical map of degree 16\.  Arch. Math. (Basel) 113, 4 (2019), 385–390.
  * [8] Nguyen Bin Some infinite sequences of canonical covers of degree 2.  Adv. Geom. 21, 1 (2021), 143–148.
  * [9] Nguyen Bin Some examples of algebraic surfaces with canonical map of degree 20.  C. R. Math. Acad. Sci. Paris 359 (2021), 1145–1153.
  * [10] Nguyen Bin New examples of canonical covers of degree 3.  Math. Nachr. 295, 3 (2022), 450–467.
  * [11] Pardini, R. Abelian covers of algebraic varieties.  J. Reine Angew. Math. 417 (1991), 191–213.
  * [12] Pardini, R. Canonical images of surfaces.  J. Reine Angew. Math. 417 (1991), 215–219.
  * [13] Persson, U. Double coverings and surfaces of general type.  In Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977), vol. 687 of Lecture Notes in Math. Springer, Berlin, 1978, pp. 168–195.
  * [14] Rito, C. New canonical triple covers of surfaces.  Proc. Amer. Math. Soc. 143, 11 (2015), 4647–4653.
  * [15] Rito, C. A surface with canonical map of degree 24.  Internat. J. Math. 28, 6 (2017), 1750041, 10.
  * [16] Rito, C. A surface with $q=2$ and canonical map of degree 16.  Michigan Math. J. 66, 1 (2017), 99–105.
  * [17] Rito, C. Surfaces with canonical map of maximum degree.  J. Algebraic Geom. 31, 1 (2022), 127–135.
  * [18] Tan, S. L. Surfaces whose canonical maps are of odd degrees.  Math. Ann. 292, 1 (1992), 13–29.
  * [19] Xiao, G. Algebraic surfaces with high canonical degree.  Math. Ann. 274, 3 (1986), 473–483.

Current address:

Mathematics Division,

National Center for Theoretical Sciences,

National Taiwan University,

Taiwan.

E-mail address<EMAIL_ADDRESS>

Center for Mathematical Analysis, Geometry and Dynamical Systems

Departamento de Matemática

Instituto Superior Técnico

Universidade de Lisboa

Av. Rovisco Pais

1049-001 Lisboa

Portugal.

E-mail address<EMAIL_ADDRESS>