NOTES ON AUTOMORPHISMS OF SURFACES OF GENERAL TYPE WITH $p_{g}=0$ AND $K^{2}=7$

Abstract

Let $S$ be a smooth minimal complex surface of general type with $p_{g}=0$ and $K^{2}=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^{2}=7$ is isomorphic to $\mathbb{Z}_{2}^{2}$ or $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

1 Introduction

The birational automorphism groups of projective varieties are extensively studied. Nowadays we know that, for a projective variety of general type $X$ over an algebraically closed field of characteristic zero, the number of birational automorphisms of $X$ is bounded by $c_{d}\cdot \text{vol}(X,K_{X})$ , where $c_{d}$ is a constant which only depends on the dimension $d$ of $X$ , and $\text{vol}(X,K_{X})$ is the volume of the canonical divisor $K_{X}$ (cf. [13]). Furthermore, we know that $c_{1}=42$ and $c_{2}=42^{2}$ according to the classical Hurwitz theorem and to Xiao’s theorem (cf. [25] and [26]). However, even in low dimensions, it is usually nontrivial to calculate the automorphism groups of explicit varieties of general type (for example, see [15], [23], [8] and [17]).

We focus on automorphisms of minimal smooth complex surfaces of general type with $p_{g}=0$ and $K^{2}=7$ . Involutions on such surfaces have been studied in [16] and [24]. All the possibilities of the quotient surfaces and the fixed loci of the involutions are listed. In order to find new examples, we have tried to classify such surfaces with commuting involutions in [10] and succeeded in constructing a new family of surfaces in [9]. We briefly recall the main results of [10]. Throughout the article, $S$ denotes a minimal smooth surface of general type with $p_{g}=0$ and $K^{2}=7$ over $\mathbb{C}$ .

Theorem 1.1. [10, Theorem 1.1]

Assume that the automorphism group $\text{Aut}(S)$ contains a subgroup $G=\{1,g_{1},g_{2},g_{3}\}$ , which is isomorphic to $\mathbb{Z}_{2}^{2}$ . Let $R_{g_{i}}$ be the divisorial part of the fixed locus of the involution $g_{i}$ for $i=1,2,3$ . Then the canonical divisor $K_{S}$ is ample and $R_{g_{i}}^{2}=-1$ for $i=1,2,3$ . Moreover, there are only three numerical possibilities for the intersection numbers $(K_{S}R_{g_{1}},K_{S}R_{g_{2}},K_{S}R_{g_{3}})$ : $(a)~(7,5,5)$ , $(b)~(5,5,3)$ and $(c)~(5,3,1)$ . The intersection numbers $(R_{g_{1}}R_{g_{2}},R_{g_{1}}R_{g_{3}},R_{g_{2}}R_{g_{3}})$ have the following values: $(a)~(5,9,7)$ , $(b)~(7,5,1)$ and $(c)~(1,3,1)$ , respectively.

In the above theorem, we adopt the convention that $K_{S}R_{g_{1}}\geqslant K_{S}R_{g_{2}}\geqslant K_{S}R_{g_{3}}$ , $R_{g_{1}}R_{g_{2}}\leqslant R_{g_{1}}R_{g_{3}}$ in case (a) and $R_{g_{1}}R_{g_{3}}\geqslant R_{g_{2}}R_{g_{3}}$ in case (b). Actually, we have completely classified the surfaces in case (a) and case (b) in [10]. But we do not know any example of the surfaces in case (c). One may ask whether there are noncommutative involutions on $S$ . Here we give a negative answer.

Theorem 1.2. If ${\it\alpha}$ is an involution of $S$ , then ${\it\alpha}$ is contained in the center of $\text{Aut}(S)$ .

We prove the above theorem in Section 3. The key step is Theorem 3.1 which shows that any two involutions on $S$ commute. Theorem 3.1 also has the following corollary.

Corollary 1.3. Assume that $(S,G)$ is a pair satisfying the assumption of Theorem 1.1. Then $g_{1},g_{2}$ and $g_{3}$ are exactly all the involutions of $\text{Aut}(S)$ .

The corollary immediately implies that if $\text{Aut}(S)$ contains a nontrivial subgroup which is isomorphic to $\mathbb{Z}_{2}^{r}$ , then $r=1$ or $r=2$ . We remark that there are surfaces of general type with $p_{g}=0$ , $K^{2}=8$ and their automorphism groups contain subgroups which are isomorphic to $\mathbb{Z}_{2}^{3}$  (cf. [19, Example 4.2–4.4]).

As an application, we calculate the automorphism groups of the surfaces in the case (a) of Theorem 1.1. These surfaces are those constructed by Inoue in [14] who found the first examples of surfaces of general type with $p_{g}=0$ and $K^{2}=7$ . They can be described as finite Galois $\mathbb{Z}_{2}^{2}$ -covers of the $4$ -nodal cubic surface (see Example 4.1, which is from [19, Example 4.1]).

Theorem 1.4. Let $S$ be an Inoue surface. Then $\text{Aut}(S)\cong \mathbb{Z}_{2}^{2}$ or $\text{Aut}(S)\cong \mathbb{Z}_{2}\times \mathbb{Z}_{4}$ . If $S$ is a general Inoue surface, then $\text{Aut}(S)\cong \mathbb{Z}_{2}^{2}$ .

Inoue surfaces form a $4$ -dimensional irreducible connected component in the Gieseker moduli space of canonical models of surfaces of general type (cf. [3]). The proof of Theorem 1.4 actually shows that $\text{Aut}(S)\cong \mathbb{Z}_{2}^{2}$ for $S$ outside a $2$ -dimensional irreducible closed subset of this connected component (see Remark 4.3). We also exhibit a $2$ -dimensional family of Inoue surfaces with $\text{Aut}(S)\cong \mathbb{Z}_{2}\times \mathbb{Z}_{4}$ (see Section 5). They are finite Galois $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ -covers of a $5$ -nodal weak Del Pezzo surface of degree two, which is the minimal resolution of one node of the $6$ -nodal Del Pezzo surface of degree two.

2 Preliminaries

2.1 Fixed point formulas and rational curves

Let $X$ be a smooth projective surface over the complex number field. We only consider surfaces with $p_{g}(X)=q(X)=0$ . In this case, $X$ has Picard number ${\it\rho}(X)=10-K_{X}^{2}$ by Noether’s formula and the Hodge decomposition. Also the expotential cohomology sequence gives $\text{Pic}(X)\cong H^{2}(X,\mathbb{Z})$ . Poincaré duality implies that the intersection form on $\text{Num}(X)=\text{Pic}(X)/\text{Pic}(X)_{\text{Tors}}$ is unimodular.

Assume that $X$ has a nontrivial automorphism ${\it\tau}$ . Denote by $\text{Fix}({\it\tau})$ the fixed locus of ${\it\tau}$ . Let $k_{{\it\tau}}$ be the number of isolated fixed points of ${\it\tau}$ and let $R_{{\it\tau}}$ be the divisorial part of $\text{Fix}({\it\tau})$ . Then $R_{{\it\tau}}$ is a disjoint union of irreducible smooth curves. We denote by ${\it\tau}^{\ast }:H^{2}(X,\mathbb{C})\rightarrow H^{2}(X,\mathbb{C})$ the induced linear map on the second singular cohomology group (note that $H^{k}(X,\mathbb{C})=0$ for $k=1,3$ ). The following proposition follows directly from the topological and holomorphic Lefschetz fixed point formulas (cf. [1, p. 567]; see also [11, Lemma 4.2]). The automorphism ${\it\tau}$ is called an involution if it is of order $2$ .

Proposition 2.1. If ${\it\tau}$ is an involution, then $k_{{\it\tau}}=K_{X}R_{{\it\tau}}+4$ and $\text{tr}({\it\tau}^{\ast })=2-R_{{\it\tau}}^{2}$ . If ${\it\tau}$ is of order $3$ , then $k_{{\it\tau}}=r_{1}+r_{2}=\text{tr}({\it\tau}^{\ast })+2+K_{X}R_{{\it\tau}}+R_{{\it\tau}}^{2}$ and $r_{1}+2r_{2}=6+\frac{3}{2}K_{X}R_{{\it\tau}}-\frac{R_{{\it\tau}}^{2}}{2},$ where $r_{j}$ is the number of isolated fixed points of ${\it\tau}$ of type $\frac{1}{3}(1,j)$ for $j=1,2$ .

Throughout this article, we denote by $S$ a smooth minimal complex surface of general type with $p_{g}=0$ and $K_{S}^{2}=7$ . Then ${\it\rho}(S)=3$ and $S$ contains at most one $(-2)$ -curve (this follows from Poincaré duality; cf. [10, Lemma 2.5]). Here an $m$ -curve (for $m\leqslant 0$ ) on a smooth surface stands for an irreducible smooth rational curve with self-intersection number $m$ . We have a similar result for $(-3)$ -curves.

Lemma 2.2. The surface $S$ contains at most one $(-3)$ -curve.

Proof. Assume by contradiction that $S$ contains two distinct $(-3)$ -curves $C_{1}$ and $C_{2}$ . Then the intersection matrix of $K_{S},C_{1}$ and $C_{2}$ has determinant $-7(C_{1}C_{2})^{2}+2C_{1}C_{2}+69$ . Since $K_{S}C_{1}=K_{S}C_{2}=1$ , the algebraic index theorem yields $C_{1}C_{2}\leqslant 3$ . Because the intersection form on $\text{Num}(S)$ is unimodular and ${\it\rho}(S)=3$ , $-7(C_{1}C_{2})^{2}+2C_{1}C_{2}+69$ is a square number. Then we have $C_{1}C_{2}=1$ since $C_{1}C_{2}\geqslant 0$ .

The divisor $K_{S}+C_{1}+C_{2}$ has Zariski decomposition (see [21, Section 3]) $P+N$ with $P:=K_{S}+\frac{1}{2}(C_{1}+C_{2})$ the positive part and $N:=\frac{1}{2}(C_{1}+C_{2})$ the negative part. According to [21, Theorem 1.1], we have $c_{2}(S)-e(C_{1}+C_{2})-\frac{1}{3}P^{2}-\frac{1}{4}N^{2}\geqslant 0$ . Here $e(C_{1}+C_{2})$ denotes the topological Euler number of $C_{1}+C_{2}$ and $c_{2}(S)$ denotes the second Chern number of $S$ . It is clear that $e(C_{1}+C_{2})=3$ and the Noether formula gives $c_{2}(S)=5$ . The inequality above yields $-\frac{5}{12}\geqslant 0$ , a contradiction.◻

Lemma 2.3. (See also the table in [16])

Let ${\it\tau}$ be an involution on $S$ . Then $K_{S}R_{{\it\tau}}\in \{1,3,5,7\}$ and $R_{{\it\tau}}^{2}=\pm 1$ . If $R_{{\it\tau}}^{2}=1$ , then $K_{S}$ is ample and $R_{{\it\tau}}$ is irreducible with $K_{S}R_{{\it\tau}}=3$ .

Proof. For $R_{{\it\tau}}^{2}=\pm 1$ , see the proof of [4, Proposition 3.6]. According to [2, Lemma 3.2 and Proposition 3.3(v)], $k_{{\it\tau}}$ is an odd integer and $k_{{\it\tau}}\leqslant 11$ . So $K_{S}R_{{\it\tau}}\in \{1,3,5,7\}$ by Proposition 2.1.

Assume that $R_{{\it\tau}}^{2}=1$ . If $K_{S}$ is not ample, then $S$ has a unique $(-2)$ -curve $C$ . The intersection number matrix of $K_{S},R_{{\it\tau}}$ and $C$ has determinant $-14+2(K_{S}R_{{\it\tau}})^{2}-7(R_{{\it\tau}}C)^{2}$ . The determinant equals $0$ , for, otherwise, the Chern classes of $K_{S},R_{{\it\tau}}$ and $C$ form a basis of $H^{2}(S,\mathbb{C})$ and they are ${\it\tau}^{\ast }$ -invariant, a contradiction to $\text{tr}({\it\tau}^{\ast })=2-R_{{\it\tau}}^{2}=1$ by Proposition 2.1. It follows that $K_{S}R_{{\it\tau}}=7$ and $(R_{{\it\tau}}C)^{2}=12$ . This is impossible. So $K_{S}$ is ample.

The algebraic index theorem gives $(K_{S}R_{{\it\tau}})^{2}\geqslant K_{S}^{2}R_{{\it\tau}}^{2}=7$ and thus $K_{S}R_{{\it\tau}}\in \{3,5,7\}$ . Let ${\it\pi}_{{\it\tau}}:S\rightarrow {\rm\Sigma}_{{\it\tau}}:=S/\langle {\it\tau}\rangle$ be the quotient morphism. We have $K_{S}={\it\pi}_{{\it\tau}}^{\ast }(K_{{\rm\Sigma}_{{\it\tau}}})+R_{{\it\tau}}$ .

If $K_{S}R_{{\it\tau}}=5$ , then $k_{{\it\tau}}=K_{S}R_{{\it\tau}}+4=9$ and $K_{{\rm\Sigma}_{{\it\tau}}}^{2}=\frac{1}{2}(K_{S}-R_{{\it\tau}})^{2}=-1$ . So ${\rm\Sigma}$ has $9$ nodes. The minimal resolution $W_{{\it\tau}}$ of ${\rm\Sigma}_{{\it\tau}}$ has Picard number $11$ and it contains $9$ disjoint $(-2)$ -curves. If ${\rm\Sigma}_{{\it\tau}}$ has Kodaira dimension ${\it\kappa}({\rm\Sigma}_{{\it\tau}})\geqslant 0$ , by [11, Proposition 4.1], $W_{{\it\tau}}$ is minimal. This contradicts $K_{W_{{\it\tau}}}^{2}=-1$ . So ${\it\kappa}({\rm\Sigma}_{{\it\tau}})=-\infty$ and $W_{{\it\tau}}$ is a rational surface. This contradicts [11, Theorem 3.3]. Hence $K_{S}R_{{\it\tau}}\not =5$ .

In the same manner we see that $K_{S}R_{{\it\tau}}\not =7$ (see also [20]). So $K_{S}R_{{\it\tau}}=3$ . Because $K_{S}$ is ample and $R_{{\it\tau}}$ is a disjoint union of smooth irreducible curves, the algebraic index theorem shows that $R_{{\it\tau}}$ is irreducible.◻

2.2 Abelian covers

We briefly recall some facts from the theory of abelian covers from [22]. Assume that ${\it\pi}:X\rightarrow Y$ is a finite abelian cover between projective varieties with $X$ normal and $Y$ smooth. Let $\mathfrak{S}$ be the Galois group of ${\it\pi}$ and let $\mathfrak{S}^{\ast }$ be the group of characters of $\mathfrak{S}$ . Then the action of $\mathfrak{S}$ induces a splitting: ${\it\pi}_{\ast }({\mathcal{O}}_{X})=\bigoplus _{{\it\chi}\in \mathfrak{S}^{\ast }}{\mathcal{L}}_{{\it\chi}}^{-1}$ , where ${\mathcal{L}}_{{\it\chi}}\in \text{Pic}(Y)$ and ${\mathcal{L}}_{1}={\mathcal{O}}_{X}$ . For each nontrivial cyclic subgroup $\mathfrak{C}$ of $\mathfrak{S}$ and each generator ${\it\psi}\in \mathfrak{C}^{\ast }$ , there is a unique effective divisor $D_{\mathfrak{C},{\it\psi}}$ of $Y$ associated to the pair $(\mathfrak{C},{\it\psi})$ . The cover ${\it\pi}$ is determined by ${\mathcal{L}}_{{\it\chi}}$ and $D_{\mathfrak{C},{\it\psi}}$ with some specified relations (cf. [22, Theorem 2.1]). We mainly apply this theory when $\mathfrak{S}\cong \mathbb{Z}_{2}\times \mathbb{Z}_{4}$ or $\mathfrak{S}\cong \mathbb{Z}_{2}^{2}$ .

We set up some notation and conventions. Denote by $H=\langle g_{1}\rangle \times \langle g\rangle$ a group isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ , where $g_{1},g$ are generators of $H$ , $g_{1}$ is of order $2$ and $g$ is of order $4$ . Denote by $H^{\ast }=\langle {\it\chi}\rangle \times \langle {\it\rho}\rangle$ the group of characters of $H$ , where ${\it\chi}(g_{1})=-1,{\it\rho}(g)=\text{i}$ and ${\it\chi}(g)={\it\rho}(g_{1})=1$ . The group $H$ contains a unique subgroup $G=\{1,g_{1},g_{2},g_{3}\}$ which is isomorphic to $\mathbb{Z}_{2}^{2}$ , where $g_{2}=g^{2}$ and $g_{3}=g_{1}g_{2}$ . Denote by ${\it\chi}_{i}\in G^{\ast }$ the nontrivial character orthogonal to $g_{i}$ for $i=1,2,3$ .

When $\mathfrak{S}=G$ , we simply set ${\mathcal{L}}_{i}:={\mathcal{L}}_{{\it\chi}_{i}}$ and ${\rm\Delta}_{i}:=D_{\langle g_{i}\rangle ,{\it\psi}}$ , where ${\it\psi}$ is the unique nontrivial character of $\langle g_{i}\rangle$ . Similarly, when $\mathfrak{S}=H$ , we set $D_{i}:=D_{\langle g_{i}\rangle ,{\it\psi}}$ for $1\not ={\it\psi}\in \langle g_{i}\rangle ^{\ast }$ . For the cyclic group $\langle g\rangle \cong \mathbb{Z}_{4}$ , we set $D_{g,\pm \text{i}}:=D_{\langle g\rangle ,{\it\psi}}$ for ${\it\psi}\in \langle g\rangle ^{\ast }$ with ${\it\psi}(g)=\pm \text{i}$ . We adopt a similar convention for the cyclic group $\langle g_{1}g\rangle \cong \mathbb{Z}_{4}$ .

In what follows, the indices $i\in \{1,2,3\}$ should be understood as residue classes modulo $3$ . Also linear equivalence and numerical equivalence between divisors are denoted by $\equiv$ and $\stackrel{num}{{\sim}}$ , respectively.

Proposition 2.4. (cf. [6], [22, Theorem 2.1 and Corollary 3.1])

Let ${\it\pi}:X\rightarrow Y$ be a finite abelian cover between projective varieties. Assume that $X$ is normal and $Y$ is smooth.

1. (a) If the Galois group of ${\it\pi}$ is $G$ , then ${\it\pi}$ is determined by the following data: Divisors ${\rm\Delta}_{i}$ and ${\mathcal{L}}_{i}$ ( $i=1,2,3$ ) such that

   (2.1) $$\begin{eqnarray}\displaystyle 2{\mathcal{L}}_{i}\equiv {\rm\Delta}_{i+1}+{\rm\Delta}_{i+2},\qquad {\mathcal{L}}_{i}+{\rm\Delta}_{i}\equiv {\mathcal{L}}_{i+1}+{\mathcal{L}}_{i+2}\quad \text{for }i=1,2,3, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
   and moreover the divisors ${\rm\Delta}_{i}$ are effective and ${\rm\Delta}:={\rm\Delta}_{1}+{\rm\Delta}_{2}+{\rm\Delta}_{3}$ is reduced.

2. (b) If the Galois group of ${\it\pi}$ is $H$ , then ${\it\pi}$ is determined by the following reduced data (see [22, Proposition 2.1]): Divisors ${\mathcal{L}}_{{\it\chi}},{\mathcal{L}}_{{\it\rho}},D_{1},D_{2},D_{3}$ , $D_{g,\text{i}},D_{g,-\text{i}},D_{g_{1}g,\text{i}}$ and $D_{g_{1}g,-\text{i}}$ such that

   (2.2) $$\begin{eqnarray}\displaystyle 2{\mathcal{L}}_{{\it\chi}} & \equiv & \displaystyle D_{1}+D_{3}+D_{g_{1}g,\text{i}}+D_{g_{1}g,-\text{i}},\nonumber\\ \displaystyle 4{\mathcal{L}}_{{\it\rho}} & \equiv & \displaystyle 2D_{2}+2D_{3}+D_{g,\text{i}}+3D_{g,-\text{i}}+D_{g_{1}g,\text{i}}+3D_{g_{1}g,-\text{i}},\end{eqnarray}$$
   and moreover the divisors $D_{1},D_{2},D_{3},D_{g,\text{i}},D_{g,-\text{i}},D_{g_{1}g,\text{i}}$ and $D_{g_{1}g,-\text{i}}$ are effective and
   $$\begin{eqnarray}\displaystyle D:=D_{1}+D_{2}+D_{3}+D_{g,\text{i}}+D_{g,-\text{i}}+D_{g_{1}g,\text{i}}+D_{g_{1}g,-\text{i}} & & \displaystyle \nonumber\end{eqnarray}$$
   is reduced.

3 Two involutions commute

We first deduce Corollary 1.3 and Theorem 1.2 from the following theorem.

Theorem 3.1. Let $S$ be a smooth minimal surface of general type with $p_{g}(S)=0$ and $K_{S}^{2}=7$ . Assume that $\text{Aut}(S)$ contains two distinct involutions ${\it\alpha}$ and ${\it\beta}$ . Then ${\it\alpha}{\it\beta}={\it\beta}{\it\alpha}$ .

Proof of Corollary 1.3.

On the contrary, suppose that ${\it\alpha}$ is an involution of $\text{Aut}(S)$ other than $g_{1},g_{2},g_{3}$ . Theorem 3.1 implies $\langle {\it\alpha},g_{1},g_{2}\rangle \cong \mathbb{Z}_{2}^{3}$ . Therefore, there are seven involutions ${\it\theta}$ in $\langle {\it\alpha},g_{1},g_{2}\rangle$ and seven numbers $K_{S}R_{{\it\theta}}\in \{1,3,5,7\}$ by Lemma 2.3. Since any two involutions generate a subgroup of type $\mathbb{Z}_{2}^{2}$ , by Theorem 1.1, we conclude that each of the three numbers $1,3$ and $7$ occurs at most once. Furthermore, Theorem 1.1(a) also implies that if $7$ occurs, then the other six numbers are all equal to $5$ . Hence there are at least five involutions ${\it\theta}$ for which $K_{S}R_{{\it\theta}}=5$ . There must be a subgroup of type $\mathbb{Z}_{2}^{2}$ containing three of these five, a contradiction to Theorem 1.1.◻

Proof of Theorem 1.2.

We may assume that $\text{Aut}(S)$ contains at least two involutions. These two involutions generate a subgroup $G\cong \mathbb{Z}_{2}^{2}$ by Theorem 3.1. We still denote by $g_{1},g_{2}$ and $g_{3}$ the involutions of $G$ . Let ${\it\tau}$ be any automorphism of $S$ . Corollary 1.3 gives ${\it\tau}G{\it\tau}^{-1}=G$ . Since ${\it\tau}(R_{g_{i}})=R_{{\it\tau}g_{i}{\it\tau}^{-1}}$ , we have $K_{S}R_{g_{i}}=K_{S}R_{{\it\tau}g_{i}{\it\tau}^{-1}}$ and $R_{g_{i}}R_{g_{i+1}}=R_{{\it\tau}g_{i}{\it\tau}^{-1}}R_{{\it\tau}g_{i+1}{\it\tau}^{-1}}$ for $i=1,2,3$ . From this observation and Theorem 1.1, we conclude that ${\it\tau}g_{i}{\it\tau}^{-1}=g_{i}$ for $i=1,2,3$ and complete the proof.◻

The remaining of this section is devoted to prove Theorem 3.1. We assume by contradiction that ${\it\alpha}{\it\beta}\not ={\it\beta}{\it\alpha}$ . We will deduce a contradiction through a sequence of lemmas and propositions. We use the same notation as Section 2. Recall that $\text{tr}({\it\alpha}^{\ast })=2-R_{{\it\alpha}}^{2}$ , $R_{{\it\alpha}}^{2}=\pm 1$ and $k_{{\it\alpha}}=K_{S}R_{{\it\alpha}}+4$ (see Proposition 2.1 and Lemma 2.3).

Lemma 3.2. The order of ${\it\alpha}{\it\beta}$ is an odd integer.

Proof. Assume by contradiction that the order of ${\it\alpha}{\it\beta}$ is $2k$ and $k\geqslant 2$ . Let ${\it\gamma}:=({\it\alpha}{\it\beta})^{k}=({\it\beta}{\it\alpha})^{k}$ . Then ${\it\gamma}$ is an involution and ${\it\gamma}{\it\alpha}=({\it\alpha}{\it\beta})^{k}{\it\alpha}={\it\alpha}({\it\beta}{\it\alpha})^{k}={\it\alpha}{\it\gamma}$ . Therefore, $\langle {\it\gamma},{\it\alpha}\rangle \cong \mathbb{Z}_{2}^{2}$ . Then $R_{{\it\alpha}}^{2}=R_{{\it\gamma}}^{2}=-1$ by Theorem 1.1. Similarly, ${\it\gamma}{\it\beta}={\it\beta}{\it\gamma}$ and $R_{{\it\beta}}^{2}=R_{{\it\gamma}}^{2}=-1$ . So $\text{tr}({\it\alpha}^{\ast })=\text{tr}({\it\beta}^{\ast })=3$ .

Let ${\it\iota}:={\it\alpha}{\it\beta}{\it\alpha}$ . Note that ${\it\alpha},{\it\beta}$ and ${\it\iota}$ are three distinct involutions in $\text{Aut}(S)$ and

(3.1) $$\begin{eqnarray}\displaystyle {\it\alpha}(R_{{\it\iota}})=R_{{\it\beta}},\qquad {\it\alpha}(R_{{\it\beta}})=R_{{\it\iota}},\qquad {\it\alpha}(R_{{\it\alpha}})=R_{{\it\alpha}}. & & \displaystyle\end{eqnarray}$$

Recall that $\dim H^{2}(S,\mathbb{C})={\it\rho}(S)=3$ . Now $c_{1}(R_{{\it\alpha}}),c_{1}(R_{{\it\beta}})$ and $c_{1}(R_{{\it\iota}})$ are not a basis of $H^{2}(S,\mathbb{C})$ , for otherwise, (3.1) implies $\text{tr}({\it\alpha}^{\ast })=1$ , which is a contradiction to $\text{tr}({\it\alpha}^{\ast })=3$ .

So the intersection number matrix of $R_{{\it\alpha}},R_{{\it\beta}}$ and $R_{{\it\iota}}$ has determinant zero. That is $2x^{2}y+2x^{2}+y^{2}-1=0$ , where $x:=R_{{\it\alpha}}R_{{\it\iota}}=R_{{\it\alpha}}R_{{\it\beta}}$ (see (3.1)) and $y:=R_{{\it\beta}}R_{{\it\iota}}$ . Observe that $y\geqslant 0$ ; otherwise, $R_{{\it\beta}}=R_{{\it\iota}}$ and ${\it\beta}={\it\iota}$ , a contradiction to ${\it\alpha}{\it\beta}\not ={\it\beta}{\it\alpha}$ . It follows that $x=0,y=1$ and the nontrivial linear relation among $c_{1}(R_{{\it\alpha}}),c_{1}(R_{{\it\beta}})$ and $c_{1}(R_{{\it\iota}})$ is $c_{1}(R_{{\it\beta}})+c_{1}(R_{{\it\iota}})=0$ . This contradicts the fact that the divisor $R_{{\it\beta}}+R_{{\it\iota}}$ is strictly effective. Hence the order of ${\it\alpha}{\it\beta}$ is an odd integer.◻

Recall that our aim is to deduce a contradiction from the assumption ${\it\alpha}{\it\beta}\not ={\it\beta}{\it\alpha}$ . According to the previous lemma, from now on, we may assume that the order $r$ of ${\it\alpha}{\it\beta}$ is an odd prime. In fact, if $r=p(2t+1)$ for some prime $p\geqslant 3$ and some integer $t>0$ , then ${\it\alpha}^{\prime }:=({\it\alpha}{\it\beta})^{t}{\it\alpha}$ and ${\it\beta}^{\prime }:=({\it\beta}{\it\alpha})^{t}{\it\beta}$ are involutions and the order of ${\it\alpha}^{\prime }{\it\beta}^{\prime }$ is $p$ . In particular, we have ${\it\alpha}^{\prime }{\it\beta}^{\prime }\not ={\it\beta}^{\prime }{\it\alpha}^{\prime }$ . We may replace ${\it\alpha},{\it\beta}$ by ${\it\alpha}^{\prime },{\it\beta}^{\prime }$ and continue our discussion.

The subgroup $\langle {\it\alpha},{\it\beta}\rangle$ of $\text{Aut}(S)$ is isomorphic to the dihedral group of order $2r$ . Let $D_{r}$ denote this subgroup. Since $r$ is a prime, all the involutions in $D_{r}$ are pairwise conjugate and $D_{r}$ has exactly one nontrivial normal subgroup $\langle {\it\alpha}{\it\beta}\rangle$ , which is the commutator subgroup. Any irreducible linear representation of $D_{r}$ has dimension at most two, and any irreducible $2$ -dimensional representation of $D_{r}$ is isomorphic to the matrix representation given by

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}0 & 1\\ 1 & 0\end{array}\right)\quad \text{and}\quad \left(\begin{array}{@{}cc@{}}c & 0\\ 0 & c^{-1}\end{array}\right)\quad \text{for some}~c\not =1~\text{and}~c^{r}=1.\end{eqnarray}$$

Lemma 3.3. With the same assumption as above, we have

1. (a) the canonical $K_{S}$ is ample;

2. (b) the curves $R_{{\it\alpha}}$ and $R_{{\it\beta}}$ generate a pencil $|F|$ of curves with $F^{2}=1$ and $K_{S}F=3$ , and $|F|$ has a simple base point $p$ ;

3. (c) the group $D_{r}$ acts faithfully on $|F|$ .

Proof. Set $k:=\frac{1}{2}(r-1)$ and set ${\it\gamma}:={\it\alpha}({\it\beta}{\it\alpha})^{k}={\it\beta}({\it\alpha}{\it\beta})^{k}$ . Then ${\it\alpha},{\it\beta}$ and ${\it\gamma}$ are three distinct involutions and they are pairwise conjugate. Therefore, $K_{S}R_{{\it\alpha}}=K_{S}R_{{\it\beta}}=K_{S}R_{{\it\gamma}}$ and $R_{{\it\alpha}}^{2}=R_{{\it\beta}}^{2}=R_{{\it\gamma}}^{2}$ . Since ${\it\gamma}{\it\alpha}={\it\beta}{\it\gamma}$ and ${\it\gamma}{\it\beta}={\it\alpha}{\it\gamma}$ ,

(3.2) $$\begin{eqnarray}\displaystyle {\it\gamma}(R_{{\it\alpha}})=R_{{\it\beta}},\qquad {\it\gamma}(R_{{\it\beta}})=R_{{\it\alpha}},\qquad {\it\gamma}(R_{{\it\gamma}})=R_{{\it\gamma}}. & & \displaystyle\end{eqnarray}$$

We claim that $R_{{\it\gamma}}^{2}=R_{{\it\alpha}}^{2}=R_{{\it\beta}}^{2}=1$ . Otherwise, as in the proof of Lemma 3.2, we could deduce a contradiction by calculating the determinant of the intersection number matrix of $R_{{\it\alpha}},R_{{\it\beta}}$ and $R_{{\it\gamma}}$ and by calculating $\text{tr}({\it\gamma}^{\ast })$ .

Then (a) follows from Lemma 2.3. Lemma 2.3 also gives $K_{S}R_{{\it\alpha}}=K_{S}R_{{\it\beta}}=3$ . The algebraic index theorem implies $(R_{{\it\alpha}}+R_{{\it\beta}})^{2}\leqslant \frac{6^{2}}{7}$ and thus $R_{{\it\alpha}}R_{{\it\beta}}\leqslant 1$ . Since $R_{{\it\alpha}}^{2}=R_{{\it\beta}}^{2}=1$ , the equality holds and $R_{{\it\alpha}}\stackrel{num}{{\sim}}R_{{\it\beta}}$ . Similarly, we have $R_{{\it\gamma}}\stackrel{num}{{\sim}}R_{{\it\alpha}}.$

Let $p$ be the unique intersection point of $R_{{\it\alpha}}$ and $R_{{\it\gamma}}$ . Then (3.2) implies that $R_{{\it\alpha}},R_{{\it\beta}}$ and $R_{{\it\gamma}}$ pairwise intersect transversely at the point $p$ . Recall that $\text{Pic}(S)\cong H^{2}(S,\mathbb{Z})$ and $\text{Num}(S)=\text{Pic}(S)/\text{Pic}(S)_{\text{Tors}}$ . Let $m$ be the smallest positive integer such that $mR_{{\it\alpha}}\equiv mR_{{\it\gamma}}\equiv mR_{{\it\beta}}$ . Let ${\it\varepsilon}:\tilde{S}\rightarrow S$ be the blowup at $p$ , let $E$ be the exceptional curve and let $\tilde{R_{{\it\alpha}}}$ be the strict transform of $R_{{\it\alpha}}$ , and so forth. Then $|m\tilde{R_{{\it\alpha}}}|$ induces a fibration $f:\tilde{S}\rightarrow \mathbb{P}^{1}$ and $m\tilde{R_{{\it\gamma}}},m\tilde{R_{{\it\alpha}}}$ and $m\tilde{R_{{\it\beta}}}$ are fibers of $f$ .

The fibration $f$ has $E$ as a $m$ -section. If $m\geqslant 2$ , we easily obtain a contradiction by applying the Hurwitz formula for $f|_{E}:E\rightarrow \mathbb{P}^{1}$ . Therefore, $m=1$ , $R_{{\it\alpha}}\equiv R_{{\it\gamma}}\equiv R_{{\it\beta}}$ and $h^{0}(S,{\mathcal{O}}_{S}(R_{{\it\alpha}}))=2$ . And (b) is proved.

For (c), first note that $p$ is a fixed point of $D_{r}$ . So $D_{r}$ acts faithfully on the tangent space $\text{T}_{p}S$ of $S$ to the point $p$ . According to the discussion before the lemma, this action is irreducible. Because $r$ is an odd number, the corresponding action of $D_{r}$ on $\mathbb{P}(\text{T}_{p}S)$ is faithful. Since $F^{2}=1$ , $p$ is a smooth point of $F$ and thus $\text{T}_{p}F$ is a $1$ -dimensional linear subspace of $\text{T}_{p}S$ for any $F\in |F|$ . From this, we conclude that $D_{r}$ acts faithfully on $|F|$ .◻

Because $D_{r}$ acts faithfully on $|F|\cong \mathbb{P}^{1}$ , every automorphism has exactly two invariant curves in $|F|$ . For every involution ${\it\theta}\in D_{r}$ , one of the two ${\it\theta}$ -invariant curves in $|F|$ is $R_{{\it\theta}}$ . Denote the other one by $F_{{\it\theta}}$ . Then $F_{{\it\theta}}$ contains the seven isolated fixed points of ${\it\theta}$ . Denote by $F_{0}$ one of the two ${\it\alpha}{\it\beta}$ -invariant curves in $|F|$ . Then the other one is ${\it\alpha}(F_{0})(={\it\beta}(F_{0}))$ and $\text{Fix}({\it\alpha}{\it\beta})\subseteq F_{0}\cup {\it\alpha}(F_{0})$ . We shall show that $F_{0}$ is not $2$ -connected. But first we need the following lemma about the action of $D_{r}$ on the singular cohomology group.

Lemma 3.4. The automorphism ${\it\alpha}{\it\beta}$ acts trivially on $H^{2}(S,\mathbb{C})$ . In particular, the quotient surface $S/D_{r}$ has Picard number $2$ .

Proof. We have seen that ${\it\alpha},{\it\beta}$ and thus $D_{r}$ act trivially on the $2$ -dimensional linear subspace generated by $c_{1}(K_{S})$ and $c_{1}(F)$ . Because $H^{2}(S,\mathbb{C})$ is $3$ -dimensional and ${\it\alpha}{\it\beta}$ is contained in the kernel of any $1$ -dimensional representation of $D_{r}$ , ${\it\alpha}{\it\beta}$ acts trivially on $H^{2}(S,\mathbb{C})$ . Hence the invariant subspace of $H^{2}(S,\mathbb{C})$ for the $D_{r}$ -action is $2$ -dimensional and $S/D_{r}$ has Picard number $2$ .◻

We analyze the members of the pencil $|F|$ which are not $2$ -connected. This will help us to determine the base locus of the linear system $|K_{S}+F|$ in the proof of Proposition 3.6 and to find a basis of $\text{Num}(S)$ . We continue to use the fact that $S$ has Picard number $3$ .

Lemma 3.5. Assume that $|F|$ contains a curve which is not $2$ -connected. Then

1. (a) the curves in $|F|$ which are not $2$ -connected are exactly the ${\it\alpha}{\it\beta}$ -invariant curves $F_{0}$ and ${\it\alpha}(F_{0})$ ;

2. (b) $F_{0}=A+B$ , where $A$ and $B$ are irreducible curves, and $K_{S}A=2,K_{S}B=1,A^{2}=0,B^{2}=-1$ and $AB=1$ . Moreover, $A$ contains the base point $p$ of $|F|$ .

Proof. Assume that $A+B\in |F|$ , $A>0$ , $B>0$ and $AB\leqslant 1$ . Because $K_{S}$ is ample and $K_{S}F=3$ , we may assume $K_{S}A=2$ and $K_{S}B=1$ . Then $B$ is irreducible. The algebraic index theorem and the adjunction formula imply $A^{2}\leqslant 0$ and $B^{2}\leqslant -1$ . In particular, by Lemma 3.4, ${\it\alpha}{\it\beta}(B).B=B^{2}<0$ and thus ${\it\alpha}{\it\beta}(B)=B$ . Hence $A+B$ is one of the ${\it\alpha}{\it\beta}$ -invariant curves $F_{0}$ and ${\it\alpha}(F_{0})$ .

Because $A^{2}+B^{2}=F^{2}-2AB\geqslant -1$ , the argument above yields $A^{2}=0$ , $B^{2}=-1$ and $AB=1$ . Then $FA=1$ and $FB=0$ . So the simple base point $p$ of $|F|$ belongs to $A$ . It remains to show that $A$ is irreducible.

Assume $A=A_{1}+A_{2}$ with $A_{1}>0$ and $A_{2}>0$ . Because $K_{S}A=2$ and $K_{S}$ is ample, $K_{S}A_{1}=K_{S}A_{2}=1$ and both $A_{1}$ and $A_{2}$ are irreducible. We now calculate the intersection numbers $A_{1}^{2},A_{2}^{2}$ and $A_{1}A_{2}$ . For $i=1,2$ , the algebraic index theorem and the adjunction formula imply $A_{i}^{2}\in \{-1,-3\}$ . Note that $A_{i}^{2}=-3$ if and only if $A_{i}$ is a $(-3)$ -curve. Since ${\it\alpha}(F_{0})\cap F_{0}$ consists of one point $p$ , ${\it\alpha}(A_{i})\not =A_{i}$ for $i=1,2$ . We conclude that $A_{1}^{2}=A_{2}^{2}=-1$ by Lemma 2.2. Then $A_{1}A_{2}=\frac{1}{2}(A^{2}-A_{1}^{2}-A_{2}^{2})=1$ .

Because $FA=1$ , we may assume $FA_{1}=1$ and $FA_{2}=0$ . It follows that $p\in A_{1}$ , $p\not \in A_{2}$ and $A_{2}B=A_{2}(F-A_{1}-A-2)=0$ . Since $p\not \in A_{2}\cup B$ , $S$ contains four disjoint curves $B,{\it\alpha}(B),A_{2}$ and ${\it\alpha}(A_{2})$ , all with self-intersection number $(-1)$ , a contradiction to ${\it\rho}(S)=3$ . Therefore $A$ is irreducible.◻

The following proposition determines the order of the automorphism ${\it\alpha}{\it\beta}$ .

Proposition 3.6. The pencil $|F|$ contains a curve which is not $2$ -connected and the automorphism ${\it\alpha}{\it\beta}$ is of order $3$ .

Proof. Let $F$ be any curve in $|F|$ . The long exact sequence of cohomology groups associated to the exact sequence $0\rightarrow {\mathcal{O}}_{S}(K_{S})\rightarrow {\mathcal{O}}_{S}(K_{S}+F)\rightarrow {\mathcal{O}}_{F}(K_{F})\rightarrow 0$ shows that $h^{0}(S,{\mathcal{O}}_{S}(K_{S}+F))=h^{0}(F,{\mathcal{O}}_{F}(K_{F}))=p_{a}(F)=3$ and the trace of $|K_{S}+F|$ on $F$ is complete. Thus $|K_{S}+F|$ defines a rational map $h:S{\dashrightarrow}\mathbb{P}^{2}$ and $h$ is defined on $F$ whenever $|K_{F}|$ is base point free. In particular, $h$ is defined on the smooth curve $R_{{\it\alpha}}~(\in |F|)$ and $h(R_{{\it\alpha}})$ is the canonical image of $R_{{\it\alpha}}$ . The same statement holds by replacing ${\it\alpha}$ by ${\it\beta}$ .

Because there is a $D_{r}$ -linearization on ${\mathcal{O}}_{S}(K_{S}+F)$ , the rational map $h$ is $D_{r}$ -equivariant. Therefore, $h(R_{{\it\alpha}})$ is contained in the fixed locus of the action of ${\it\alpha}$ on $\mathbb{P}^{2}$ . Note that an involution on $\mathbb{P}^{2}$ has a line and a point as the fixed locus. It follows that ${\it\alpha}$ acts trivially on $\mathbb{P}^{2}$ because $h(R_{{\it\alpha}})$ is a conic curve or a quartic curve. Similarly, ${\it\beta}$ and thus $D_{r}$ act trivially on $\mathbb{P}^{2}$ . Therefore, $h:S{\dashrightarrow}\mathbb{P}^{2}$ factors through the quotient morphism $S\rightarrow S/D_{r}$ .

Note that $K_{S}$ is ample, $F$ is nef and $(K_{S}+F)^{2}=14$ . First assume that $h$ is a morphism. Then it is finite and it has degree $14$ . We thus get $|D_{r}|=\deg h$ and $r=7$ . It follows that the induced morphism $h^{\prime }:S/D_{r}\rightarrow \mathbb{P}^{2}$ is an isomorphism. So the invariant linear subspace of $H^{2}(S,\mathbb{C})$ for the $D_{r}$ -action is isomorphic to $H^{2}(\mathbb{P}^{2},\mathbb{C})$ , which is $1$ -dimensional. This contradicts Lemma 3.4 and thus $h$ is not a morphism.

We now analyze the base locus of $h$ . If $F$ is $2$ -connected, $|K_{F}|$ is base point free by [5, Theorem 3.3] and $h$ is defined on $F$ . Hence the base locus of $|K_{S}+F|$ is contained in the curves of $|F|$ , which are not $2$ -connected. According to Lemma 3.5, $|F|$ contains exactly two such curves $F_{0}=A+B$ and ${\it\alpha}(F_{0})$ . Similar arguments as above show that the trace of $|K_{S}+A|$ (respectively $|K_{S}+B|$ ) on $A$ (respectively $B$ ) is complete. Since $p_{a}(A)=2$ and $p_{a}(B)=1$ , $|K_{A}|$ and $|K_{B}|$ are base point free by [5, Theorem 3.3]. Because $|K_{S}+F|\supseteq |K_{S}+A|+B,|K_{S}+B|+A$ , $|K_{S}+F|$ has exactly two base points $q:=A\cap B$ and ${\it\alpha}(q)={\it\alpha}(A)\cap {\it\alpha}(B)$ .

Therefore, $h$ is a finite morphism outside the base locus and $\deg h=(K_{S}+F)^{2}-2=12$ . Since $h$ factors through $S/D_{r}$ , we have $|D_{r}|=6$ and $r=3$ .◻

According to Lemma 3.5 and Proposition 3.6, the ${\it\alpha}{\it\beta}$ -invariant curve $F_{0}$ has $A$ and $B$ as irreducible components with $AB=1$ . It is easy to check that $F,A$ and ${\it\alpha}(A)$ generate $\text{Num}(S)$ and $K_{S}\stackrel{num}{{\sim}}F+A+{\it\alpha}(A)$ . We shall show that $K_{S}$ is indeed linearly equivalent to $F+A+{\it\alpha}(A)$ , hence contradict $p_{g}(S)=0$ and complete the proof of Theorem 3.1. For this purpose, we turn to the quotient surface $S/D_{3}$ and analyze $\text{Fix}({\it\alpha}{\it\beta})$ .

Proposition 3.7. The automorphism ${\it\alpha}{\it\beta}$ has $B\cup {\it\alpha}(B)$ as the divisorial part of the fixed locus and it has five isolated fixed points $p,q_{1},q_{2},{\it\alpha}(q_{1})$ and ${\it\alpha}(q_{2})$ , where $q_{1}$ and $q_{2}$ are contained in $A$ . Each isolated fixed point of ${\it\alpha}{\it\beta}$ is of type $\frac{1}{3}(1,2)$ .

Proof. We have seen that $F_{0}$ and ${\it\alpha}(F_{0})$ are ${\it\alpha}{\it\beta}$ -invariant and $\text{Fix}({\it\alpha}{\it\beta})\subseteq F_{0}\cup {\it\alpha}(F_{0})$ . Moreover, the curves $A,{\it\alpha}(A),B$ and ${\it\alpha}(B)$ are ${\it\alpha}{\it\beta}$ -invariant. Also note that a point $q$ is a fixed point (respectively an isolated fixed point) of ${\it\alpha}{\it\beta}$ if and only if so is the point ${\it\alpha}(q)(={\it\beta}(q))$ .

We claim that neither $A$ nor ${\it\alpha}(A)$ is contained in $\text{Fix}({\it\alpha}{\it\beta})$ . Otherwise, both $A$ and ${\it\alpha}(A)$ are contained in $\text{Fix}({\it\alpha}{\it\beta})$ . Since $A\cap {\it\alpha}(A)=p$ , this contradicts the fact that the divisorial part of ${\it\alpha}{\it\beta}$ is a disjoint union of smooth curves. The claim is proved.

Now assume by contradiction that $B$ is not contained in $\text{Fix}({\it\alpha}{\it\beta})$ . Then nor is ${\it\alpha}(B)$ and $\text{Fix}({\it\alpha}{\it\beta})$ consists of isolated fixed points. Then $\text{Fix}({\it\alpha}{\it\beta})$ has five fixed points by Proposition 2.1 and Lemma 3.4. Three of these points are $p,q:=A\cap B$ and ${\it\alpha}(q)$ . Denote the other two by $p_{1}(\in F_{0})$ and by ${\it\alpha}(p_{1})$ . We must have $p_{1}\in B$ . Otherwise, the nontrivial automorphism ${\it\alpha}{\it\beta}|_{B}$ has exactly one fixed point $q$ , which is a smooth point of $B$ since $AB=1$ . This is impossible because $p_{a}(B)=1$ . Therefore, $p_{1}\in B$ . It follows that ${\it\alpha}{\it\beta}|_{A}$ has exactly two fixed points $p$ and $q$ , which are smooth points of $A$ . Note that $A$ has at most two singular points since $p_{a}(A)=2$ . Because the singular locus of $A$ is ${\it\alpha}{\it\beta}$ -invariant and ${\it\alpha}{\it\beta}|_{A}$ has order $3$ , we conclude that $A$ is indeed smooth. However, the Hurwitz formula shows that ${\it\alpha}{\it\beta}|_{A}$ has either one or four fixed points, a contradiction.

So $B$ and ${\it\alpha}(B)$ are contained in $\text{Fix}({\it\alpha}{\it\beta})$ . In particular, $B$ and ${\it\alpha}(B)$ are smooth curves. Then $\text{Fix}({\it\alpha}{\it\beta})\setminus \{B\cup {\it\alpha}(B)\}$ consists of five isolated fixed points and each fixed point is of type $\frac{1}{3}(1,2)$ by Proposition 2.1 and Lemma 3.4. These points must be contained in $A\cup {\it\alpha}(A)$ .◻

Now we are able to describe the quotient map ${\it\pi}:S\rightarrow Y:=S/D_{3}$ , where $D_{3}=\{1,{\it\alpha},{\it\beta},{\it\gamma},{\it\alpha}{\it\beta},{\it\beta}{\it\alpha}\}$ and ${\it\gamma}:={\it\alpha}{\it\beta}{\it\alpha}={\it\beta}{\it\alpha}{\it\beta}$ . The divisorial parts and isolated fixed points of cyclic subgroups of $D_{3}$ are as follows (see the discussion before Lemma 3.4 and Proposition 3.7):

$$\begin{eqnarray}\displaystyle \begin{array}{@{}rrr@{}}\hspace{-1.3pt}\text{cyclic subgroups} & \text{divisorial part} & \text{isolated fixed points}\\ \hspace{-1.3pt}\langle {\it\alpha}\rangle ~(\text{resp.}~\langle {\it\beta}\rangle ,\langle {\it\gamma}\rangle )\cong \mathbb{Z}_{2} & R_{{\it\alpha}}~(\text{resp.}~R_{{\it\beta}},R_{{\it\gamma}}) & \text{7 points on}~F_{{\it\alpha}}~(\text{resp.}~F_{{\it\beta}},F_{{\it\gamma}})\\ \hspace{-1.3pt}\langle {\it\alpha}{\it\beta}\rangle \cong \mathbb{Z}_{3} & B,{\it\alpha}(B) & p,q_{1},q_{2},{\it\alpha}(q_{1}),{\it\alpha}(q_{2})\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

Note that $p$ is the unique point with the stabilizer $D_{3}$ . From the action of $D_{3}$ on the tangent space $\text{T}_{p}S$ (see the proof of Lemma 3.3(c)), it is easily seen that ${\it\pi}(p)$ is a smooth point of $Y$ . We conclude that $Y$ has seven nodes and two $A_{2}$ -singularities ${\it\pi}(q_{1})$ and ${\it\pi}(q_{2})$ . In particular, $Y$ is Gorenstein. The ramification formula gives

(3.3) $$\begin{eqnarray}\displaystyle K_{S}={\it\pi}^{\ast }K_{Y}+R_{{\it\alpha}}+R_{{\it\beta}}+R_{{\it\gamma}}+2B+2{\it\alpha}(B)\equiv {\it\pi}^{\ast }K_{Y}+3F+2B+2{\it\alpha}(B) & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and thus $K_{Y}^{2}=\frac{1}{6}(K_{S}-3F-2B-2{\it\alpha}(B))^{2}=-3$ .

Let $B^{\prime }={\it\pi}(B)$ . Then $B^{\prime }$ is contained in the smooth locus of $Y$ . Note that $B^{\prime }$ is a smooth elliptic curve and ${\it\pi}^{\ast }B^{\prime }=3B+3{\it\alpha}(B)$ . So ${B^{\prime }}^{2}=-3$ and $K_{Y}B^{\prime }=3$ . Since $-K_{Y}B^{\prime }=K_{Y}^{2}={B^{\prime }}^{2}$ , $-K_{Y}\stackrel{num}{{\sim}}B^{\prime }$ by Lemma 3.4. This implies that $H^{0}(mK_{Y})=0$ for $m\geqslant 1$ . As the quotient of $S$ , $Y$ has irregularity $q(Y)=0$ . Therefore, $Y$ is a rational surface. Note that linear equivalence and numerical equivalence between divisors are the same on a smooth rational surface. Since $Y$ contains only rational double points and $B^{\prime }$ is contained in the smooth locus of $Y$ , we have $-K_{Y}\equiv B^{\prime }$ indeed. Then by (3.3),

$$\begin{eqnarray}\displaystyle K_{S} & \equiv & \displaystyle {\it\pi}^{\ast }(-B^{\prime })+3F+2B+2{\it\alpha}(B)\nonumber\\ \displaystyle & \equiv & \displaystyle (-3B-3{\it\alpha}(B))+3F+2B+2{\it\alpha}(B)\nonumber\\ \displaystyle & \equiv & \displaystyle F+A+{\it\alpha}(A).\nonumber\end{eqnarray}$$

We obtain a contradiction to $p_{g}(S)=0$ and complete the proof of Theorem 3.1.

4 Inoue surfaces

As mentioned in the introduction, Inoue surfaces have been the first examples of surfaces of general type with $p_{g}=0$ and $K^{2}=7$ (cf. [14]). Here we describe them as finite Galois $\mathbb{Z}_{2}^{2}$ -covers of the $4$ -nodal cubic surface, following [19, Example 4.1]. At the end of this section, we prove Theorem 1.4.

Example 4.1. Let ${\it\sigma}:W\rightarrow \mathbb{P}^{2}$ be the blowup of the six vertexes $p_{1},p_{2},p_{3},p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime }$ of a complete quadrilateral on $\mathbb{P}^{2}$ (see Figure 1). Denote by $E_{i}$ (respectively $E_{i}^{\prime }$ ) the exceptional curve of $W$ over $p_{i}$ (respectively $p_{i}^{\prime }$ ) and denote by $L$ the pullback of a general line by ${\it\sigma}$ . Then $\text{Pic}(W)=\mathbb{Z}L\oplus \bigoplus _{i=1}^{3}(\mathbb{Z}E_{i}\oplus \mathbb{Z}E_{i}^{\prime })$ .

Figure 1. Configurations of the points $p_{1},\ldots ,p_{3}^{\prime }$ .

The surface $W$ has four disjoint $(-2)$ -curves. They are the proper transforms of the four sides of the quadrilateral and their divisor classes are

$$\begin{eqnarray}\displaystyle Z_{i}\equiv L-E_{i}-E_{i+1}^{\prime }-E_{i+2}^{\prime },\qquad Z\equiv L-E_{1}-E_{2}-E_{3}. & & \displaystyle \nonumber\end{eqnarray}$$

Let ${\it\eta}:W\rightarrow {\rm\Sigma}$ be the morphism contracting there curves. Then ${\rm\Sigma}$ is the $4$ -nodal cubic surface.

Let ${\rm\Gamma}_{1},{\rm\Gamma}_{2}$ and ${\rm\Gamma}_{3}$ be the proper transforms of the three diagonals of the quadrilateral, that is, ${\rm\Gamma}_{i}\equiv L-E_{i}-E_{i}^{\prime }$ for $i=1,2,3$ . Note that they are exactly the $(-1)$ -curves which are disjoint from any $(-2)$ -curve. For each $i=1,2,3,$ $W$ has a pencil of rational curves $|F_{i}|:=|2L-E_{i+1}-E_{i+2}-E_{i+1}^{\prime }-E_{i+2}^{\prime }|$ . Observe that $-K_{W}\equiv {\rm\Gamma}_{1}+{\rm\Gamma}_{2}+{\rm\Gamma}_{3}\equiv {\rm\Gamma}_{i}+F_{i}$ for $i=1,2,3$ .

We define three effective divisors on $W$

(4.1) $$\begin{eqnarray}\displaystyle & & \displaystyle {\rm\Delta}_{1}:={\rm\Gamma}_{1}+F_{2}+Z_{1}+Z_{3},\qquad {\rm\Delta}_{2}:={\rm\Gamma}_{2}+F_{3},\nonumber\\ \displaystyle & & \displaystyle {\rm\Delta}_{3}:={\rm\Gamma}_{3}+F_{1}+F_{1}^{\prime }+Z_{2}+Z.\end{eqnarray}$$

We require that $F_{i}$ ( $i=1,2,3$ ) and $F_{1}^{\prime }$ are smooth $0$ -curves such that the divisor ${\rm\Delta}:={\rm\Delta}_{1}+{\rm\Delta}_{2}+{\rm\Delta}_{3}$ has only nodes. It is directly shown that there are divisors ${\mathcal{L}}_{1},{\mathcal{L}}_{2}$ and ${\mathcal{L}}_{3}$ satisfying (2.1) in Proposition 2.4. Then there is a smooth finite $G$ -cover $\overline{{\it\pi}}:V\rightarrow W$ branched on the divisors ${\rm\Delta}_{1},{\rm\Delta}_{2}$ and ${\rm\Delta}_{3}$ . The (set theoretic) inverse image of a $(-2)$ -curve under $\overline{{\it\pi}}$ is a disjoint union of two $(-1)$ -curves. Let ${\it\varepsilon}:V\rightarrow S$ be the blowdown of these eight $(-1)$ -curves. Then there is a finite $G$ -cover ${\it\pi}:S\rightarrow {\rm\Sigma}$ such that the following diagram (4.2) commutes.

(4.2)

The surface $S$ is a smooth minimal surface of general type with $p_{g}(S)=0$ and $K_{S}^{2}=7$ . It is called an Inoue surface. When the curves $F_{1},F_{1}^{\prime },F_{2}$ and $F_{3}$ vary, we obtain a $4$ -dimensional family of Inoue surfaces.

Lemma 4.2. Let $W$ be as in Example 4.1.

1. (a) Let ${\it\alpha}$ be an automorphism on $W$ . If the induced map ${\it\alpha}^{\ast }:H^{2}(W,\mathbb{C})\rightarrow H^{2}(W,\mathbb{C})$ is the identity, then ${\it\alpha}=\text{Id}_{W}$ .

2. (b) Let ${\it\alpha}_{\mathbb{P}^{2}}$ be the involution on $\mathbb{P}^{2}$ such that ${\it\alpha}_{\mathbb{P}^{2}}(p_{k})=p_{k}^{\prime }$ for $k=1,3$ . It induces an involution ${\it\alpha}_{0}$ on $W$ . Then $\text{Fix}({\it\alpha}_{0})$ consists of the $(-1)$ -curve ${\rm\Gamma}_{2}$ and three isolated fixed points ${\rm\Gamma}_{1}\cap {\rm\Gamma}_{3}$ , $E_{2}\cap F_{3}^{\ast }$ and $E_{2}^{\prime }\cap F_{3}^{\ast }$ , where $F_{3}^{\ast }$ is the unique smooth ${\it\alpha}_{0}$ -invariant curve in the pencil $|F_{3}|=|2L-E_{1}-E_{1}^{\prime }-E_{2}-E_{2}^{\prime }|$ .

Proof. For (a), the assumption implies that the $(-1)$ -curves $E_{i}$ and $E_{i}^{\prime }$ ( $i=1,2,3$ ) are ${\it\alpha}$ -invariant. So ${\it\alpha}$ comes from an automorphism on $\mathbb{P}^{2}$ which has $p_{1},\ldots ,p_{3}^{\prime }$ as fixed points and thus it is the identity morphism.

For (b), note that $\text{Fix}({\it\alpha}_{\mathbb{P}^{2}})=\overline{p_{2}p_{2}^{\prime }}\cup \{p_{13}:=\overline{p_{1}p_{1}^{\prime }}\cap \overline{p_{3}p_{3}^{\prime }}\}$ and ${\it\sigma}(\text{Fix}({\it\alpha}_{0}))=\text{Fix}({\it\alpha}_{\mathbb{P}^{2}})$ . Because ${\it\sigma}^{-1}(\overline{p_{2}p_{2}^{\prime }})=E_{2}\cup E_{2}^{\prime }\cup {\rm\Gamma}_{2}$ and the divisorial part of ${\it\alpha}_{0}$ is smooth, $\text{Fix}({\it\alpha}_{0})$ has ${\rm\Gamma}_{2}$ as the divisorial part. Then ${\it\alpha}_{0}$ has three isolated fixed points by Proposition 2.1. The point ${\it\sigma}^{-1}(p_{13})={\rm\Gamma}_{1}\cap {\rm\Gamma}_{3}$ is an isolated fixed point of ${\it\alpha}_{0}$ . Note that ${\it\alpha}_{0}$ induces a nontrivial action on the pencil $|F_{3}|$ . So $|F_{3}|$ contains exactly two ${\it\alpha}_{0}$ -invariant curves ${\rm\Gamma}_{1}+{\rm\Gamma}_{2}$ and $F_{3}^{\ast }$ . Since $E_{2}$ and $E_{2}^{\prime }$ are also ${\it\alpha}_{0}$ -invariant, the intersection points $E_{2}\cap F_{3}^{\ast }$ and $E_{2}^{\prime }\cap F_{3}^{\ast }$ are isolated fixed points of ${\it\alpha}_{0}$ .◻

Proof of Theorem 1.4.

Let ${\it\tau}\in \text{Aut}(S)$ . By Theorem 1.2, $\text{Fix}(g_{i})$ is ${\it\tau}$ -invariant for $i=1,2,3$ and ${\it\tau}$ induces an automorphism ${\it\alpha}_{{\rm\Sigma}}$ on the quotient surface ${\rm\Sigma}=S/G$ . So the branch locus ${\it\pi}(\text{Fix}(g_{i}))$ (for $i=1,2,3$ ) of ${\it\pi}:S\rightarrow {\rm\Sigma}$ is ${\it\alpha}_{{\rm\Sigma}}$ -invariant.

Assume that ${\it\tau}\not \in G$ , that is, ${\it\alpha}_{{\rm\Sigma}}\not =\text{Id}_{{\rm\Sigma}}$ . The automorphism ${\it\alpha}_{{\rm\Sigma}}$ lifts to the minimal resolution $W$ of ${\rm\Sigma}$ . Denote by ${\it\alpha}$ the induced automorphism on $W$ . Then ${\rm\Delta}_{1},{\rm\Delta}_{2},{\rm\Delta}_{3}$ (see the diagram (4.2)) are ${\it\alpha}$ -invariant because ${\rm\Delta}_{i}$ is the inverse image of ${\it\pi}(\text{Fix}(g_{i}))$ under the morphism ${\it\eta}:W\rightarrow {\rm\Sigma}$ . These divisors are given by (4.1). It follows that the $(-1)$ -curves ${\rm\Gamma}_{1},{\rm\Gamma}_{2},{\rm\Gamma}_{3}$ , the $0$ -curves $F_{2},F_{3}$ and the curves $F_{1}+F_{1}^{\prime },Z_{1}+Z_{3},Z_{2}+Z$ are ${\it\alpha}$ -invariant. Note that the Chern classes of ${\rm\Gamma}_{1},{\rm\Gamma}_{2},{\rm\Gamma}_{3}$ and the Chern classes of $Z_{1},Z_{2},Z_{3},Z$ generate $H^{2}(S,\mathbb{C})$ . The argument above implies that $({\it\alpha}^{2})^{\ast }=({\it\alpha}^{\ast })^{2}$ is the identity morphism. Then ${\it\alpha}$ is an involution by Lemma 4.2.

Since $F_{i}\equiv {\rm\Gamma}_{i+1}+{\rm\Gamma}_{i+2}$ , the fibration $f_{i}:W\rightarrow \mathbb{P}^{1}$ induced by $|F_{i}|$ is ${\it\alpha}$ -equivalent for $i=1,2,3$ . Note that $f_{2}$ has three singular fibers ${\rm\Gamma}_{1}+{\rm\Gamma}_{3},Z_{1}+2E_{2}^{\prime }+Z_{3}$ and $Z_{2}+2E_{2}+Z$ . According to the discussion above, these three fibers are ${\it\alpha}$ -invariant. Because any nontrivial automorphism on $\mathbb{P}^{1}$ has at most two fixed points, ${\it\alpha}$ respects the fibration $f_{2}$ , that is, $f_{2}=f_{2}{\it\alpha}$ . In particular, $E_{2}$ and $E_{2}^{\prime }$ are ${\it\alpha}$ -invariant.

Note that $f_{1}$ has three singular fibers ${\rm\Gamma}_{2}+{\rm\Gamma}_{3},Z_{1}+2E_{1}+Z$ and $Z_{2}+2E_{1}^{\prime }+Z_{3}$ . If $f_{1}=f_{1}{\it\alpha}$ , then all the $(-2)$ -curves $Z_{1},Z_{2},Z_{3}$ and $Z$ are ${\it\alpha}$ -invariant since ${\it\alpha}$ also respects $f_{2}$ . Then ${\it\alpha}^{\ast }$ is the identity morphism and so is ${\it\alpha}$ by Lemma 4.2, a contradiction to our assumption. So ${\it\alpha}$ induces a nontrivial action on $|F_{1}|\cong \mathbb{P}^{1}$ . Since the singular ${\rm\Gamma}_{2}+{\rm\Gamma}_{3}$ is ${\it\alpha}$ -invariant, ${\it\alpha}$ must permute the other two singular fibers of $f_{1}$ . Hence ${\it\alpha}(E_{1})=E_{1}^{\prime }$ and ${\it\alpha}(E_{1}^{\prime })=E_{1}$ . Similarly, by considering the action of ${\it\alpha}$ on $|F_{3}|$ , we see that ${\it\alpha}(E_{3})=E_{3}^{\prime }$ and ${\it\alpha}(E_{3}^{\prime })=E_{3}$ .

We conclude that ${\it\alpha}$ is the involution ${\it\alpha}_{0}$ in Lemma 4.2. We actually prove that if $\text{Aut}(S)\not =G$ , then $\text{Aut}(S)/G\cong \langle {\it\alpha}_{0}\rangle$ , and in the Equation (4.1), the curve $F_{3}$ in ${\rm\Delta}_{2}$ is indeed the curve $F_{3}^{\ast }$ in Lemma 4.2(b) and $F_{1}^{\prime }={\it\alpha}_{0}(F_{1})$ in ${\rm\Delta}_{3}$ . Combining with Theorem 1.2 and Corollary 1.3, we complete the proof of Theorem 1.4.◻

Remark 4.3. When the curves $F_{1}$ and $F_{2}$ vary, the Inoue surfaces corresponding to the branch divisors (4.1) with $F_{3}=F_{3}^{\ast }$ and $F_{1}^{\prime }={\it\alpha}_{0}(F_{1})$ form a $2$ -dimensional irreducible closed subset of the total $4$ -dimensional family of Inoue surfaces. Also Lemma 4.2 shows that $W/\langle {\it\alpha}_{0}\rangle$ has three nodes. Moreover, it contains three $(-2)$ -curves in the smooth locus and these curves are the images of $Z_{1}+Z_{3}$ , $Z_{2}+Z$ and ${\rm\Gamma}_{2}$ under the quotient map from $W$ to $W/\langle {\it\alpha}_{0}\rangle$ . This observation motivates us to construct some special Inoue surfaces in the next section.

5 Special Inoue surfaces

We construct a $2$ -dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ . We use the notation in Section 2.2.

Let $q,q_{1},q_{2},q_{3},q_{1}^{\prime },q_{2}^{\prime }$ and $q_{3}^{\prime }$ be seven points on $\mathbb{P}^{2}$ with the configuration as in Figure 2.

Figure 2. Configurations of the points $q,q_{1},\ldots ,q_{3}^{\prime }$ .

Let ${\it\nu}:Y\rightarrow \mathbb{P}^{2}$ be the blowup of these points. Denote by $Q_{i}$ (respectively $Q_{i}^{\prime }$ , $Q$ ) the exceptional curve of $Y$ over $q_{i}$ (respectively $q_{i}^{\prime }$ , $q$ ) and by $L$ the pullback of a general line by ${\it\nu}$ . Then $\text{Pic}(W)=\mathbb{Z}L\oplus \mathbb{Z}Q\oplus (\bigoplus _{i=1}^{3}\mathbb{Z}Q_{i}\oplus \mathbb{Z}Q_{i}^{\prime })$ . The surface $Y$ has six disjoint $(-2)$ -curves and their divisor classes are:

$$\begin{eqnarray}\displaystyle M_{i}=L-Q-Q_{i}-Q_{i}^{\prime },\qquad N_{i}=L-Q_{i}^{\prime }-Q_{i+1}-Q_{i+2}\quad \text{for }i=1,2,3. & & \displaystyle \nonumber\end{eqnarray}$$

Let ${\rm\Lambda}_{i}$ be the proper transform of the line $\overline{q_{i+1}^{\prime }q_{i+2}^{\prime }}$ , that is, ${\rm\Lambda}_{i}\equiv L-Q_{i+1}^{\prime }-Q_{i+2}^{\prime }$ for $i=1,2,3$ .

We describe four base-point-free pencils of rational curves on $Y$ . They are $|{\rm\Phi}|:=|2L-Q-Q_{1}-Q_{2}-Q_{3}|$ and $|{\rm\Phi}_{i}|:=|2L-Q-Q_{i}-Q_{i+1}^{\prime }-Q_{i+2}^{\prime }|$ ( $i=1,2,3$ ). The singular members of $|{\rm\Phi}|$ are $M_{1}+2Q_{1}^{\prime }+N_{1},M_{2}+2Q_{2}^{\prime }+N_{2},M_{3}+2Q_{3}^{\prime }+N_{3}$ and those of $|{\rm\Phi}_{i}|$ (fixed $i$ ) are ${\rm\Lambda}_{i}+M_{i}+Q_{i}^{\prime },M_{i+1}+2Q_{i+1}+N_{i+2},M_{i+2}+2Q_{i+2}+N_{i+1}$ . Also note that ${\rm\Phi}_{i}+N_{i}\equiv -K_{Y}$ for $i=1,2,3$ .

Let ${\it\zeta}:Y\rightarrow {\rm\Upsilon}$ be the morphism contracting the five $(-2)$ -curves $M_{1},M_{2},N_{2},M_{3}$ and $N_{3}$ . Then ${\rm\Upsilon}$ has five nodes and contains a unique $(-2)$ -curve ${\it\zeta}(N_{1})$ in the smooth locus.

Now we define the following effective divisors on $Y$ :

(5.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad \hspace{5.0pt}D_{1}:={\rm\Lambda}_{1}+{\rm\Phi}_{1}+M_{3},\qquad D_{2}:={\rm\Lambda}_{2},\qquad D_{3}:=Q_{1}^{\prime }+{\rm\Phi}+N_{3},\nonumber\\ \displaystyle & & \displaystyle \quad \hspace{5.0pt}D_{g,\text{i}}:=N_{1}+N_{2},\qquad D_{g,-\text{i}}:=M_{2},\qquad D_{g_{1}g,\text{i}}:=0,\qquad D_{g_{1}g,-\text{i}}:=M_{1}.\end{eqnarray}$$

We also define the following divisors:

(5.2) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{{\it\chi}} & := & \displaystyle 4L-2Q-2Q_{1}-Q_{2}-Q_{2}^{\prime }-Q_{3}-2Q_{3}^{\prime },\nonumber\\ \displaystyle {\mathcal{L}}_{{\it\rho}} & := & \displaystyle 4L-2Q-2Q_{1}-Q_{1}^{\prime }-2Q_{2}-Q_{2}^{\prime }-Q_{3}-Q_{3}^{\prime }.\end{eqnarray}$$

We require that ${\rm\Phi}\in |{\rm\Phi}|$ and ${\rm\Phi}_{1}\in |{\rm\Phi}_{1}|$ are smooth curves such that the divisor $D=D_{1}+\ldots +D_{g_{1}g,-\text{i}}$ has only nodes. These divisors satisfy (2.2) in Proposition 2.4. So there is a finite Galois $H$ -cover $\widehat{{\it\pi}}:X\rightarrow Y$ and $X$ is normal.

We use [22, Propositions 3.1 and 3.3] to analyze the singular locus of $X$ .

Lemma 5.1. Let $m:={\rm\Lambda}_{2}\cap M_{2}$ and $n:={\rm\Lambda}_{2}\cap N_{2}$ .

1. (a) The inverse image $\widehat{{\it\pi}}^{-1}(m)$ (resp. $\widehat{{\it\pi}}^{-1}(n)$ ) consists of two points $\widehat{m_{1}}$ and $\widehat{m_{2}}$ (resp. $\widehat{n_{1}}$ and $\widehat{n_{2}}$ ), each of which has stabilizer $\langle g\rangle$ .

2. (b) The points $\widehat{m_{1}},\widehat{m_{2}},\widehat{n_{1}}$ and $\widehat{n_{2}}$ are exactly the singularities of $X$ and they are nodes.

3. (c) The curve $\widehat{{\it\pi}}^{-1}(M_{2})$ is a disjoint union of two irreducible smooth curves $\widehat{M}_{21}$ and $\widehat{M}_{22}$ , and $\widehat{M}_{2j}$ has self-intersection number $(-\frac{1}{2})$ and $\widehat{m}_{j}\in \widehat{M}_{2j}$ for $j=1,2$ . The curve $\widehat{{\it\pi}}^{-1}(N_{2})$ consists of two irreducible smooth curves $\widehat{N}_{21}$ and $\widehat{N}_{22}$ , and $\widehat{N}_{2j}$ has self-intersection number $(-\frac{1}{2})$ and $\widehat{n}_{j}\in \widehat{N}_{2j}$ for $j=1,2$ .

4. (d) The curve $\widehat{{\it\pi}}^{-1}(M_{3})$ is a disjoint union four $(-1)$ -curves and so is $\widehat{{\it\pi}}^{-1}(N_{3})$ .

5. (e) The curve $\widehat{{\it\pi}}^{-1}(M_{1})$ is a $(-1)$ -curve.

Proof. [22, Proposition 3.1] shows that $X$ is smooth outside $\widehat{{\it\pi}}^{-1}(m)$ and $\widehat{{\it\pi}}^{-1}(n)$ . Note that $M_{2}$ intersects only one irreducible component of $D-M_{2}$ ; that is $M_{2}{\rm\Lambda}_{2}=1$ . Because ${\rm\Lambda}_{2}=D_{2}$ , $M_{2}\leqslant D_{g,-\text{i}}$ and $[H:\langle g\rangle ]=2$ , we conclude that $\widehat{{\it\pi}}^{-1}(m)$ consists of two points, each of which has stabilizer $\langle g\rangle$ . These two points are nodes of $X$ according to [22, Proposition 3.3]. For the same reason, we have $\widehat{{\it\pi}}^{-1}(M_{2})=\widehat{M}_{21}\cup \widehat{M}_{22}$ with $\widehat{M}_{21}\cap \widehat{M}_{22}=\emptyset$ and $\widehat{{\it\pi}}|_{\widehat{M}_{2j}}:\widehat{M}_{2j}\rightarrow M_{2}$ is an isomorphism. We also have $\widehat{{\it\pi}}^{\ast }(M_{2})=4\widehat{M}_{21}+4\widehat{M}_{22}$ . Thus (a)–(c) follow from the discussion above. Similar arguments apply to (d) and (e). For (d), just note that $M_{3}({\leqslant}D_{1})$ and $N_{3}({\leqslant}D_{3})$ are connected irreducible components of $D$ . And (e) follows from the observation that $M_{1}(=D_{g_{1}g,-\text{i}})$ intersects exactly two irreducible components of $D-M_{1}$ and $M_{1}D_{1}=M_{1}D_{3}=1$ .◻

Now we explain how to obtain the smooth minimal model of $X$ . On the minimal resolution $\widetilde{X}$ of $X$ , the strict transforms of $\widehat{M}_{21},\widehat{M}_{22},\widehat{N}_{21}$ and $\widehat{N}_{22}$ are $(-1)$ -curves. Each of these $(-1)$ -curves intersects transversely at one point with exactly one of the four $(-2)$ -curves over the nodes of $X$ . So we can contract the four curves $\widehat{M}_{21},\widehat{M}_{22},\widehat{N}_{21}$ and $\widehat{N}_{22}$ of $X$ to smooth points on another surface.

Let ${\it\theta}:X\rightarrow S$ be the morphism contracting the disjoint union of the nine $(-1)$ -curves $\widehat{{\it\pi}}^{-1}(M_{3}),\widehat{{\it\pi}}^{-1}(N_{3}),\widehat{{\it\pi}}^{-1}(M_{1})$ and the four curves $\widehat{M}_{21},\widehat{M}_{22},\widehat{N}_{21}$ and $\widehat{N}_{22}$ . Then there is a smooth $H$ -cover ${\it\pi}:S\rightarrow {\rm\Upsilon}$ such that the outer square of the following diagram (5.3) commutes.

(5.3)

We confirm that $S$ is the smooth minimal model of $X$ by the following proposition.

Proposition 5.2. The surface $S$ is an Inoue surface.

Proof. From [22, Theorem 2.1], we obtain ${\mathcal{L}}_{{\it\rho}^{2}}=2{\mathcal{L}}_{{\it\rho}}-D_{2}-D_{3}-D_{g,-\text{i}}-D_{g_{1}g,-\text{i}}$ . Then $2{\mathcal{L}}_{{\it\rho}^{2}}\equiv D_{g,\text{i}}+D_{g,-\text{i}}+D_{g_{1}g,\text{i}}+D_{g_{1}g,-\text{i}}=M_{1}+N_{1}+M_{2}+N_{2}$ by Proposition 2.4(b) and (5.2). Let $\widehat{{\it\pi}}_{1}:X_{1}\rightarrow Y$ be the double cover branched along the four disjoint $(-2)$ -curves $M_{1},N_{1},M_{2}$ and $N_{2}$ . Note that ${\it\rho}^{2}$ is the unique character of $H^{\ast }$ which is trivial on $G$ . So the Galois group of $\widehat{{\it\pi}}_{1}$ is $H/G$ and the cover $\widehat{{\it\pi}}$ factors through a $G$ -cover $\widehat{{\it\pi}}_{2}:X\rightarrow X_{1}$ .

We have $2K_{X_{1}}=\widehat{{\it\pi}}_{1}^{\ast }(2K_{Y}+M_{1}+N_{1}+M_{2}+N_{2})$ and $K_{X_{1}}^{2}=0$ . The inverse images of $M_{1},N_{1},M_{2}$ and $N_{2}$ under $\widehat{{\it\pi}}_{1}$ are $(-1)$ -curves. Also $\widehat{{\it\pi}}_{1}^{-1}(M_{3})$ is a disjoint union of two $(-2)$ -curves and so is $\widehat{{\it\pi}}_{1}^{-1}(N_{3})$ . Let ${\it\delta}:X_{1}\rightarrow \overline{W}$ be the morphism contracting three $(-1)$ -curves $\widehat{{\it\pi}}_{1}^{-1}M_{1},\widehat{{\it\pi}}_{1}^{-1}M_{2}$ and $\widehat{{\it\pi}}_{1}^{-1}N_{2}$ . Then $\overline{W}$ is a weak Del Pezzo surface of degree three.

Let ${\it\theta}_{2}:X\rightarrow \overline{V}$ be the morphism contracting the curves $\widehat{{\it\pi}}^{-1}(M_{1})$ , $\widehat{{\it\pi}}^{-1}(M_{2})$ and $\widehat{{\it\pi}}^{-1}(N_{2})$ . We obtain a smooth Galois finite $G$ -cover $\overline{{\it\pi}}:\overline{V}\rightarrow \overline{W}$ and a commutative diagram (5.3). The branch locus of $\overline{{\it\pi}}$ is

(5.4) $$\begin{eqnarray}\displaystyle \qquad \overline{{\rm\Delta}_{1}}=\overline{{\rm\Lambda}_{1}}+\overline{{\rm\Phi}_{1}}+\overline{M_{3}},\qquad \overline{{\rm\Delta}_{2}}=\overline{N_{1}}+\overline{{\rm\Lambda}_{2}},\qquad \overline{{\rm\Delta}_{3}}=\overline{Q_{1}^{\prime }}+\overline{{\rm\Phi}}+\overline{N_{3}}. & & \displaystyle\end{eqnarray}$$

Here we denote by $\overline{{\rm\Lambda}_{1}}={\it\delta}\widehat{{\it\pi}}_{1}^{-1}({\rm\Lambda}_{1})$ , and so forth. We claim that

1. (i) $\overline{{\rm\Lambda}_{1}}$ , $\overline{N_{1}}$ and $\overline{Q_{1}^{\prime }}$ are $(-1)$ -curves;

2. (ii) $\overline{{\rm\Phi}_{1}}$ and $\overline{{\rm\Lambda}_{2}}$ are $0$ -curves, and $\overline{{\rm\Phi}}$ is a disjoint union of two $0$ -curves in the same linear system;

3. (iii) $\overline{M_{3}}$ ( $\overline{N_{3}}$ ) is a disjoint union of two $(-2)$ -curves; these two $(-2)$ -curves are disjoint from the $(-1)$ -curves in (i);

4. (iv) $\overline{{\rm\Lambda}_{1}}+\frac{1}{2}\overline{{\rm\Phi}}$ , $\overline{N_{1}}+\overline{{\rm\Phi}_{1}}$ and $\overline{Q_{1}^{\prime }}+\overline{{\rm\Lambda}_{2}}$ and are linearly equivalent to $-K_{\overline{W}}$ .

For example, because the general member of $|{\rm\Phi}|$ is disjoint from $M_{1}+N_{1}+M_{2}+N_{2}$ , the curve $\widehat{{\it\pi}}_{1}^{-1}({\rm\Phi})$ is a disjoint union of two $0$ -curves in the same linear system and so is $\overline{{\rm\Phi}}$ . In particular, $K_{W}\overline{{\rm\Phi}}=-4$ and $\frac{1}{2}\overline{{\rm\Phi}}$ is well defined in $\text{Pic}(\overline{W})$ . For the $(-1)$ -curve ${\rm\Lambda}_{1}$ on $Y$ , since ${\rm\Lambda}_{1}M_{1}={\rm\Lambda}_{1}N_{1}=1$ and ${\rm\Lambda}_{1}M_{2}={\rm\Lambda}_{1}N_{2}=0$ , the curve $\widehat{{\it\pi}}_{1}^{-1}({\rm\Lambda}_{1})$ is a $(-2)$ -curve, and it intersects with $\widehat{{\it\pi}}_{1}^{-1}(M_{1})$ transversely at one point and it is disjoint from $\widehat{{\it\pi}}_{1}^{-1}(M_{2})$ and $\widehat{{\it\pi}}_{1}^{-1}(N_{2})$ . So $\overline{{\rm\Lambda}_{1}}$ is a $(-1)$ -curve. Moreover, we have $\overline{{\rm\Lambda}_{1}}\overline{{\rm\Phi}}=\widehat{{\it\pi}}_{1}^{\ast }({\rm\Lambda}_{1})\widehat{{\it\pi}}_{1}^{\ast }({\rm\Phi})=2{\rm\Lambda}_{1}{\rm\Phi}=4$ . Finally, the algebraic index theorem yields $\overline{{\rm\Lambda}_{1}}+\frac{1}{2}\overline{{\rm\Phi}}\equiv -K_{W}$ . Other statements can be proved in the same manner.

Comparing (5.4) to (4.1), we conclude that $S$ is an Inoue surface.◻

When ${\rm\Phi}$ and ${\rm\Phi}_{1}$ vary, we obtain a $2$ -dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ .

Remark 5.3. We may directly show that $K_{S}$ is ample, $K_{S}^{2}=7$ and $p_{g}(S)=0$ for the surface $S$ in (5.3). According to the proof of [22, Proposition 4.2], we have

$$\begin{eqnarray}\displaystyle 4K_{X} & = & \displaystyle \widehat{{\it\pi}}^{\ast }(4K_{Y}+2D_{1}+2D_{2}+2D_{3}+3D_{g,\text{i}}+3D_{g,-\text{i}}+3D_{g_{1}g,\text{i}}+3D_{g_{1}g,-\text{i}})\nonumber\\ \displaystyle & = & \displaystyle \widehat{{\it\pi}}^{\ast }(-K_{Y}+{\rm\Phi}+{\rm\Phi}_{1})+\widehat{{\it\pi}}^{\ast }(M_{1}+2M_{2}+2N_{2}+2M_{3}+2N_{3}).\nonumber\end{eqnarray}$$

It follows that $4K_{S}={\it\pi}^{\ast }(-K_{{\rm\Upsilon}}+{\it\phi}+{\it\phi}_{1})$ and $K_{S}^{2}=7$ , where $|{\it\phi}|$ and $|{\it\phi}_{1}|$ are base-point-free pencils on ${\rm\Upsilon}$ induced by $|{\rm\Phi}|$ and $|{\rm\Phi}_{1}|$ . The linear system $|-K_{Y}+{\rm\Phi}+{\rm\Phi}_{1}|$ is base point free, and the corresponding morphism contracts exactly the nodal curves $M_{1},M_{2},N_{2},M_{3}$ and $N_{3}$ . Hence $|-K_{{\rm\Upsilon}}+{\it\phi}+{\it\phi}_{1}|$ is ample and so is $K_{S}$ . For each ${\it\psi}\in H^{\ast }$ , we can calculate ${\mathcal{L}}_{{\it\psi}}$ by [22, Theorem 2.1] and then easily show that $H^{0}(Y,{\mathcal{O}}_{Y}(K_{Y}+{\mathcal{L}}_{{\it\psi}}))=0$ . It follows that $p_{g}(S)=p_{g}(X)=0$ by [22, Proposition 4.1].

Remark 5.4. We remark that Theorem 1.4 contributes to the study of the moduli space of the Inoue surfaces. Let ${\mathcal{M}}_{1,7}^{\text{can}}$ be the Gieseker moduli space of canonical models of surfaces of general type with ${\it\chi}({\mathcal{O}})=1$ and $K^{2}=7$ (cf. [12]). Let $S$ be any Inoue surface. Denote by $[S]$ the corresponding point in ${\mathcal{M}}_{1,7}^{\text{can}}$ and by $\text{B}(S)$ be the base of the Kuranishi family of deformations of $S$ . Recall the facts that the tangent space of $\text{B}(S)$ is $H^{1}(S,{\rm\Theta}_{S})$ , where ${\rm\Theta}_{S}$ is the tangent sheaf of $S$ , and that the germ $({\mathcal{M}}_{1,7}^{\text{can}},[S])$ is analytically isomorphic to $\text{B}(S)/\text{Aut}(S)$ . It has been shown in [3] that the group $G$ acts trivially on $H^{1}(S,{\rm\Theta}_{S})$ and $\text{B}(S)$ is smooth of dimension $4$ .

Now assume that $S$ is a special Inoue surface constructed here. We can use the same method as in the proof of [3, Theorem 5.1] to conclude that the invariant subspace of $H^{1}(S,{\rm\Theta}_{S})$ for the $H$ -action has dimension $2$ . Note that $\text{Aut}(S)=H$ and $H/G\cong \mathbb{Z}_{2}$ . Combining the result of [3], we see that $({\mathcal{M}}_{1,7}^{\text{can}},[S])$ is analytically isomorphic to $(\mathbb{C}^{2}\times \text{Spec}\,\mathbb{C}[x,y,z]/(xz-y^{2}),0)$ .