A new family of surfaces of general type with $K^2=7$ and $p_g=0$

Abstract

We construct a new family of smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$. We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.

1 Introduction

Minimal surfaces of general type with $p_g=0$ have been constructed and studied since the 1930’s (cf. [7] and [12]). These surfaces have invariants $p_g=q=0$ and $1 \le K^2 \le 9$. For each value of $K^2$, except the case $K^2=7$, there exists quite a list of examples. However, to the best of the author’s knowledge, there is only one known family of minimal surfaces of general type with $K^2=7$ and $p_g=0$ (cf. [4, Tables 1–3]). This family of surfaces is due to M. Inoue (cf. [13]). We will show that the surfaces with $K^2=7$ constructed in [18] are in fact Inoue surfaces (see Sect. 6).

Inoue surfaces with $K^2=7$ are quotients of complete intersections inside the product of four elliptic curves by a group isomorphic to $\mathbb{Z }_2^5$ acting freely (cf. [13]). Alternatively, Inoue surfaces can be realized as finite $\mathbb{Z }_2^2$-covers of the $4$-nodal cubic surface (cf. [16, Example 4.1]). We refer to the recent article [5], where the authors use both constructions to study the deformations of Inoue surfaces.

The bicanonical morphism of an Inoue surface has degree $2$ and is composed with exactly one involution of the Galois group $\mathbb{Z }_2^2$. The bicanonical morphism of a smooth minimal surface of general type with $K^2=7$ and $p_g=0$ has degree either $1$ or $2$ (cf. [16] and [17]). Involutions on surfaces of general type with $K^2=7$ and $p_g=0$ are studied in [15] and [18]. Lists of numerical possibilities are given in these articles. However, no new example is constructed (see Sect. 6). It is also claimed in a pre-version of [15] that three quotients of an Inoue surface by the involutions are rational surfaces. However, we point out that one of the quotients is birational to an Enriques surface (see Sect. 6).

In this article, we construct a family of surfaces with $K^2=7$ and $p_g=0$, as finite $\mathbb{Z }_2^2$-covers of certain singular Del Pezzo surfaces of degree one. These surfaces have ample canonical divisor. For a surface $S$ in our family, the bicanonical morphism of $S$ is not composed with any involution of the Galois group $\mathbb{Z }_2^2$. Using the results of [17], we show that the bicanonical morphism of $S$ is birational. So the family is indeed new.

We show that three quotients of $S$ by the involutions have Kodaira dimensions $-\infty , 0, 1$, respectively, realizing some numerical possibilities on the lists of [15] and [18]. Applying the results of [3], we prove that $S$ satisfies Bloch’s conjecture.

Notation and conventions We make the convention that the indices $i \in \{1,2,3\}$ should be understood as residue classes modulo $3$. We denote by $g_1,g_2,g_3$ the nonzero elements of the group $G:=\mathbb{Z }_2^2$ and by $\chi _i \in G^*$ the nontrivial character orthogonal to $g_i$ for $i=1, 2, 3$. Linear equivalence is denoted by $\equiv$. The rest of the notation is standard in algebraic geometry.

2 Certain weak Del Pezzo surfaces of degree one

In this section, we construct a family of weak Del Pezzo surfaces of degree one as blowup of $\mathbb P ^2$ at eight points. We use $(x_1:x_2:x_3)$ as the homogeneous coordinates for $\mathbb P ^2$. Let $p_1=(1:0:0), p_2=(0:1:0), p_3=(0:0:1),\,p_0=(1:1:1)$ and let $p_j^{\prime }$ be the infinitely near point over $p_j$ corresponding to the line $\overline{p_jp_0}$ for $j=1, 2, 3$. We state a lemma on conics passing through some of these points (Fig. 1).

Lemma 2.1

For each $i=1, 2, 3$, there is a unique conic $c_i$ passing through the points $p_i,\,p_{i+1},\,p_{i+1}^{\prime },\,p_{i+2}$ and $p_{i+2}^{\prime }$. Its defining polynomial is $x_i(x_{i+1}+x_{i+2})-x_{i+1}x_{i+2}=0$. Moreover, $c_i$ does not pass through the point $p_i^{\prime }$.

Let $\sigma :W \rightarrow \mathbb P ^2$ be the blowup at the eight points $p_0, p_1, p_1^{\prime }, p_2, p_2^{\prime }, p_3, p_3^{\prime }$ and $p$, where the eighth point $p$ satisfies the Zariski open conditions:

• (I) $p \not \in \cup _{i=1}^3\{\overline{p_0p_i}:x_{i+1}=x_{i+2}\} \cup _{i=1}^3 \{ \overline{p_{i+1}p_{i+2}}: x_i=0\};$

• (II) $p \not \in c_1 \cup c_2 \cup c_3$.

Fig. 1

Configurations of $p_0, p_1,\ldots , p_3^{\prime }$ and $p$

Denote by $E_j$ (respectively $E_j^{\prime }$, $E$) the total transform of the point $p_j$ (respectively $p_j^{\prime }$, $p$) and by $L$ the pullback of a general line by $\sigma$. Then

$$\begin{aligned} Pic(W)=\mathbb{Z }L\oplus \mathbb{Z }E_0\oplus \oplus _{j=1}^3 (\mathbb{Z }E_j\oplus \mathbb{Z }E_j^{\prime })\oplus \mathbb{Z }E \end{aligned}$$

and $-K_W \equiv 3L-E_0-\sum \nolimits _{j=1}^{3}(E_j+E_j^{\prime })-E$.

The surface $W$ is a weak Del Pezzo surface of degree $1$, i.e., $-K_W$ is nef and big and $K_W^2=1$. This follows from the fact that any four of the eight points $p_0, p_1, \ldots , p_3^{\prime }, p$ are not collinear (cf. [11, Proposition 8.1.7]). The following proposition describes the $(-2)$-curves on $W$.

Proposition 2.2

The surface $W$ has exactly six $(-2)$-curves. Their divisor classes are as follows:

$$\begin{aligned} C_j \equiv L-E_0-E_j-E_j^{\prime },\ C_j^{\prime }\equiv E_j-E_j^{\prime } \text{ for } j=1,2, 3. \end{aligned}$$

(2.1)

Proof

Assume that $C \subset W$ is a $(-2)$-curve and its divisor class is $xL-a_0E_0-\sum _{j=1}^3 (a_jE_j+a_j^{\prime }E_j^{\prime })-aE$. If $\sigma (C)$ is a point, then $C$ is one of the curves $C_1^{\prime }, C_2^{\prime }, C_3^{\prime }$. If $c:=\sigma (C)$ is a curve, then $c$ is an irreducible curve of degree $x$ having multiplicity at least $a_0$ (respectively $a_1, \ldots , a$) at the point $p_0$ (respectively $p_1, \ldots , p$). Therefore $a_0, \ldots , a$ are nonnegative integers. If $x=1$, then $C$ is one of the curves $C_1, C_2, C_3$. It suffices to exclude the case $x \ge 2$. Since $C^2=-2$ and $K_WC=0$,

$$\begin{aligned} x^2+2=a_0^2+\sum \limits _{j=1}^3\left( a_j^2+a_j^{\prime 2}\right) +a^2,\qquad \qquad 3x=a_0+\sum \limits _{j=1}^3\left( a_j+a_j^{\prime }\right) +a. \end{aligned}$$

By Cauchy’s inequality, $9x^2 \le (x^2+2) \cdot 8$ and thus $x \le 4$.

If $x=4$, then equality holds and $a_0=\cdots =a=2$. It follows that $CC_1^{\prime }=x-a_0-a_1-a_1^{\prime }=-2$. Then $C=C_1^{\prime }$, a contradiction. Hence $x \not = 4$.

If $x=2$, then $c$ is an irreducible smooth conic and thus $a_0, \cdots , a \in \{0, 1\}$. Moreover, $a_0+\sum _{j=1}^3(a_j+a_j^{\prime })+a=a_0^2+\sum _{j=1}^3(a_j^2+a_j^{\prime 2})+a^2=6$. Therefore exactly six of $a_0, \cdots , a$ are $1$. This contradicts Lemma 2.1 or condition (II). Hence $x \not =2$.

If $x=3$, then $c$ is an irreducible cubic and thus $a_0, \ldots , a \in \{0, 1, 2\}$. Moreover, $a_0+\sum _{j=1}^3(a_j+a_j^{\prime })+a=9$ and $a(a-1)+\sum _{j=1}^3(a_j(a_j-1)+a_j^{\prime }(a_j^{\prime }-1))+a(a-1)=2$. Therefore exactly one of $a_0 ,\ldots , a$ is $2$ and the others are $1$. If $a_0=2$ or $a_j=2$ or $a_j^{\prime }=2$, then $CC_j=-1$. This contradicts the fact that $C$ is irreducible. So $a=2$ and thus $C \equiv -K_W-E$. We exclude this case by the following lemma.$\square$

Lemma 2.3

The linear system $|-K_W-E|$ is empty.

Proof

Assume by contradiction that $|-K_W-E| \not = \emptyset$. Then an element of $|-K_W-E|$ corresponds to a cubic $c$ on $\mathbb P ^2$ passing through $p_0, p_1, p_2, p_3, p_1^{\prime }, p_2^{\prime }, p_3^{\prime }$ and having $p$ as a singularity. Let $F(x_1,x_2,x_3)$ be the defining polynomial of $c$. Since $c$ passes through $p_1, p_2, p_3$, $F$ does not contain the terms $x_1^3, x_2^3, x_3^3$. Since $\overline{p_0p_j}:x_{j+1}=x_{j+2}$ is the tangent line to $c$ at the point $p_j$, the coefficient of the term $x_j^2x_{j+1}$ is the negative of the term $x_j^2x_{j+2}$. We may assume that

$$\begin{aligned} F(x_1,x_2,x_3)=Ax_1^2(x_2-x_3)+Bx_2^2(x_3-x_1)+Cx_3^2(x_2-x_1)+Dx_1x_2x_3, \end{aligned}$$

where $A, B, C, D \in \mathbb C$. Since $c$ contains $p_0=(1:1:1)$, $D=0$. Assume that $p=(1:\alpha :\beta )$, where $\alpha \not =0, 1$, $\beta \not = 0, 1$ and $\alpha \not =\beta$ (see condition (I)). The singularity $p$ of $c$ imposes the following conditions on $F$:

$$\begin{aligned} (\alpha -\beta )A+\alpha ^2(\beta -1)B+\beta ^2(\alpha -1)C=0,\\ A+2\alpha (\beta -1)B+\beta ^2C=0,\\ -A+\alpha ^2B+2\beta (\alpha -1)C=0. \end{aligned}$$

The coefficient matrix has determinant $2\alpha \beta (\alpha -1)(\beta -1)(\alpha -\beta )$, which is nonzero, we have $A=B=C=0$. Hence $|-K_W-E| = \emptyset$.$\square$

The $(-2)$-curves $C_1, C_1^{\prime }, C_2, C_2^{\prime }, C_3, C_3^{\prime }$ are pairwise disjoint. Let $\eta :W \rightarrow \Sigma$ be the morphism contracting these curves. By Proposition 2.2, $\Sigma$ has six nodes and $-K_\Sigma$ is ample.

Now we turn to study the $(-1)$-curves on $W$. Denote by $\Gamma$ the strict transform of the line $\overline{p_0p}$. Then $\Gamma$ is a $(-1)$-curve and $\Gamma \equiv L-E_0-E$. Note that the pencil of lines on $\mathbb P ^2$ passing through $p_0$ induces a the fibration $g :W \rightarrow \mathbb P ^1$. Denote by $F$ a general fiber of $g$. Then $F \equiv L-E_0$. Moreover, $g$ has exactly four singular fibers:

for $j=1, 2, 3$. Here numbers on nodes represent the multiplicities.

Starting from the two $(-1)$-curves $\Gamma$ and $E$, we will find two more $(-1)$-curves. We first need some properties of the linear system $|-2K_W|$.

Proposition 2.4

([11, Theorem 8.3.2]) The linear system $|-2K_W|$ defines a morphism $\phi :W \rightarrow \mathbb P ^3$. It factors as a birational morphism $\eta :W \rightarrow \Sigma$ contracting exactly the six $(-2)$-curves and a finite morphism $q :\Sigma \rightarrow Q$ of degree $2$, where $Q$ is a quadric cone.

See [11, Theorem 8.3.2] for a general statement on weak Del Pezzo surfaces of any degree and for a proof.

Proposition 2.5

• (1) The linear system $|-2K_W-\Gamma |$ consists of a single $(-1)$-curve. Denote this $(-1)$-curve by $\tilde{B}_2$. Then $\tilde{B}_2\Gamma =3$ and $\tilde{B}_2E=1$.

• (2) The linear system $|-2K_W-E|$ consists of a single $(-1)$-curve. Denote this $(-1)$-curve by $\tilde{B}_3$. Then $\tilde{B}_3\Gamma =1$ and $\tilde{B}_3E=3$.

• (3) The curve $\Gamma +E+\tilde{B}_2+\tilde{B}_3$ has only nodes.

Proof

• (1) There is an exact sequence

  $$\begin{aligned} 0 \rightarrow \mathcal O _W(-2K_W-\Gamma ) \rightarrow \mathcal O _W(-2K_W) \rightarrow \mathcal O _{\Gamma }(-2K_W) \rightarrow 0. \end{aligned}$$

  Since $-2K_W\Gamma =2$, by Proposition 2.4, $h^0(W, \mathcal O _W(-2K_W-\Gamma )) \ge 1$. Now assume that $A \in |-2K_W-\Gamma |$. Then $A^2=-1$ and $K_WA=-1$. Since $-K_W$ is nef and big, we may assume that $A = A_1+A_2$, where $A_1$ is an irreducible curve with $-K_WA_1=1$ and $Supp(A_2)$ is contained in the union of the $(-2)$-curves. Since $\Gamma$ is disjoint from the $(-2)$-curves, we have $A_1A_2+A_2^2=(-2K_W-\Gamma )A_2=0$. Since $(A_1+A_2)^2=A^2=-1$, it follows that $A_1^2+A_1A_2=-1$. In particular, $A_1^2 \le -1$. By the adjunction formula, $A_1$ is a $(-1)$-curve and thus $A_1A_2=A_2^2=0$. Hence $A_2=0$ and $A=A_1$ is a $(-1)$-curve. Hence $|-2K_W-\Gamma |$ consists of a $(-1)$-curve $\tilde{B}_2$. Moreover, $\tilde{B}_2\Gamma =(-2K_W-\Gamma )\Gamma =3$ and $\tilde{B}_2E=(-2K_W-\Gamma )E=1$.

• (2) The proof is similar to (1).

• (3) Recall that $\Gamma +E$ is disjoint from the $(-2)$-curves, since they are contained in different fibers of $g$. It follows that $\tilde{B}_2 \equiv -2K_W-\Gamma$ and $\tilde{B}_3 \equiv -2K_W-E$ are also disjoint from the $(-2)$-curves. Note that $\tilde{B}_2\tilde{B}_3=(-2K_W-\Gamma )(-2K_W-E)=1$ and $\tilde{B}_2E=\tilde{B}_3\Gamma =\Gamma E=1$. It suffices to prove that

  • (a) $\Gamma$ (respectively $E$) intersects $\tilde{B}_2+\tilde{B}_3$ transversely.

  • (b) $\tilde{B}_2$ (respectively $\tilde{B}_3$) intersects $\Gamma +E$ transversely.

  For (a), let $M:=\tilde{B}_2+\tilde{B}_3$. Then $|M|$ induces a genus $0$ fibration $h :W \rightarrow \mathbb P ^1$. Since $MC_j=MC_j^{\prime }=0$ for $j=1, 2, 3$, the six $(-2)$-curves are contained in the singular fibers of $h$. We claim that $h$ has exactly four singular fibers: $\tilde{B}_2+\tilde{B}_3$ and $M_j (\ j=1,2,3):$

  

  where the $(-2)$-curves are $C_1, \ldots , C_3^{\prime }$, and $\Theta _j$ is a $(-1)$-curve for $j=1, 2, 3$. Actually, since $-K_W$ is nef, any irreducible component $A$ in a singular fiber is either a $(-2)$-curve or a $(-1)$-curve. Since $-K_WM=-K_W(\tilde{B}_2+\tilde{B}_3)=2$, any singular fiber contains either one $(-1)$-curve with multiplicity $2$, or two $(-1)$-curves with each multiplicity $1$. Since the $(-2)$-curves of $W$ are pairwise disjoint, any singular fiber is of one of the following types:

  

  Each fiber of the first two types contributes $2$ to the Picard number $\rho (W)$. Note that $W$ has six $(-2)$-curves and $\rho (W)=9$. By considering how the $(-2)$-curves are distributed along the singular fibers, we see that there are four singluar fibers, one of which is $\tilde{B}_2+\tilde{B}_3$, the other three are of the first type. Our claim is proved. Since $M\Gamma =(-4K_W-\Gamma -E)\Gamma =4,\, h|_\Gamma :\Gamma \rightarrow \mathbb P ^1$ is of degree $4$. Since $\Gamma$ is disjoint from $C_j$ and $C_j^{\prime }$, $\Gamma \Theta _j=\frac{1}{2} \Gamma M=2$. Thus $M_j$ induces ramification points of $h|_\Gamma$. The Riemann-Hurwitz formula implies that $h|_\Gamma$ does not have any other ramification points than those on $M_1, M_2, M_3$. In particular, $\Gamma$ intersects the fiber $\tilde{B}_2+\tilde{B}_3$ transversely. A similar argument shows that $E$ intersects $\tilde{B}_2+\tilde{B}_3$ transversely. For (b), we return to the fibration $g :W \rightarrow \mathbb P ^1$. Note that the singular fibers of $g$ are of the same types of those of $h$. Also note that $\tilde{B}_2F=\tilde{B}_2(\Gamma +E)=4$ and $\tilde{B}_3F=\tilde{B}_3(\Gamma +E)=4$. A similar argument as the proof of (a) shows that $\tilde{B}_2$ (respectively $\tilde{B}_3$) intersects $\Gamma +E$ transversely.

For later use, we prove the following lemma.

Lemma 2.6

The linear system $|-2K_W+\Gamma |$ defines a birational morphism $\phi :W \rightarrow \mathbb P ^5$ and $\phi$ contracts exactly the six $(-2)$-curves $C_1, \ldots , C_3^{\prime }$.

Proof

Since $H^1(W, \mathcal O _W(-2K_W))=0$ and $(-2K_W+\Gamma )\Gamma =1$, the trace of $|-2K_W+\Gamma |$ on $\Gamma$ is complete and base point free. By Proposition 2.4, $|-2K_W+\Gamma |$ is base point free. By Ramanujam’s vanishing theorem and Riemann–Roch theorem, $h^0(W, \mathcal O _W(-2K_W\!+\!\Gamma ))\!=\!6$. Since $(-2K_W+\Gamma )^2=7$ and $Im(\phi )$ is nondegenerate, $\phi$ is birational.

Now assume that $C$ is an irreducible curve and $(-2K_W+\Gamma )C=0$. Then $C^2 <0$. Since $-K_W$ is nef, $C$ is either a $(-2)$-curve or a $(-1)$-curve. If $C$ is a $(-1)$-curve, then $\Gamma C=2K_WC=-2$. This contradicts the fact that $\Gamma$ is a $(-1)$-curve. Therefore $C$ is one of the $(-2)$-curves $C_1, \ldots ,C_3^{\prime }$. Hence $\phi$ contracts exactly the $(-2)$-curves $C_1, \ldots , C_3^{\prime }$.$\square$

3 Construction of surfaces of general type

In this section, we construct a family of surfaces of general type as finite $\mathbb{Z }_2^2$-covers of the weak Del Pezzo surface $W$. First, we define three effective divisors on $W$

$$\begin{aligned} \Delta _1:&= F_b+\Gamma +(C_1+C_1^{\prime }+C_2+C_2^{\prime }) \equiv 4L-4E_0-2E_1^{\prime }-2E_2^{\prime }-E,\nonumber \\ \Delta _2:&= \tilde{B}_2+(C_3+C_3^{\prime }) \equiv -2K_W-2E_3^{\prime }+E, \\ \Delta _3:&= \tilde{B}_3 \equiv -2K_W-E.\nonumber \end{aligned}$$

(3.1)

Here we require that

• (A) The curve $F_b$ is a smooth fiber of the fibration $g :W \rightarrow \mathbb P ^1$ (see Sect. 2);

• (B) The divisor $\Delta :=\Delta _1+\Delta _2+\Delta _3$ has only nodes. By Proposition 2.5, $\tilde{B}_2+\tilde{B}_3+\Gamma$ has only nodes. Since $F_b, \Gamma , \tilde{B}_2, \tilde{B}_3$ are disjoint from the $(-2)$-curves $C_1, \ldots , C_3^{\prime }$, (B) is equivalent to the requirement that $F_b$ intersects the curve $\tilde{B}_2+\tilde{B}_3$ transversely. By Bertini’s theorem, this is the case for a general fiber of $g$.

We also define three divisors

$$\begin{aligned} \begin{aligned} \mathcal L _1&=-2K_W-E_3^{\prime },\\ \mathcal L _2&=-K_W+(2L-2E_0-E_1^{\prime }-E_2^{\prime }-E),\\ \mathcal L _3&=-K_W+(2L-2E_0-E_1^{\prime }-E_2^{\prime }-E_3^{\prime }). \end{aligned} \end{aligned}$$

(3.2)

It follows that $2\mathcal L _i \equiv \Delta _{i+1}+\Delta _{i+2},\ \ \mathcal L _i+\Delta _i \equiv \mathcal L _{i+1}+\mathcal L _{i+2}$ for $i=1,2,3$.

By [8, Section 1] or [9, Theorem 2], the data (3.1) and (3.2) define a finite $G$-cover $\tilde{\pi }:V \rightarrow W$. Conditions (A) and (B) imply that $V$ is smooth. By the formulae in [9, Section 2],

$$\begin{aligned} 2K_V&\equiv \tilde{\pi }^*(2K_W+\Delta ) \equiv \tilde{\pi }^*\left( -2K_W+\Gamma +\sum \limits _{j=1}^{3}(C_j+C_j^{\prime })\right) ,\end{aligned}$$

(3.3)

$$\begin{aligned} p_g(V)&= p_g(W)+\sum \limits _{i=1}^{3}h^0(W, \mathcal O _W(K_W+\mathcal L _i)). \end{aligned}$$

(3.4)

Since $C_j$ or $C_j^{\prime }$ (for $j=1,2,3$) is a connected component of $\Delta$, the (set-theoretic) inverse image $\tilde{\pi }^{-1}C_j$ or $\tilde{\pi }^{-1}C_j^{\prime }$ is a disjoint union of two $(-1)$-curves. Let $\varepsilon :V \rightarrow S$ be the blowdown of these twelve $(-1)$-curves. By construction, there is a finite $G$-cover $\pi :S \rightarrow \Sigma$ such that the following diagram commutes:

(3.5)

The discussion above shows that

$$\begin{aligned} 2K_S \equiv \pi ^*(-2K_\Sigma +\gamma ), \end{aligned}$$

(3.6)

where $\gamma =\eta (\Gamma )$ is a $(-1)$-curve contained in the smooth locus of $\Sigma$.

Theorem 3.1

$S$ is a smooth minimal surface of general type with $K_S^2=7$ and $p_g(S)=0$. Moreover, $K_S$ is ample.

Proof

By (3.6), $K_S^2=\frac{1}{4} 4(-2K_\Sigma +\gamma )^2=7$. To show that $p_g(S)=p_g(V)=0$, we use (3.4). Since $p_g(W)=0$, it suffices to show that $h^0(W, \mathcal O _W(K_W+\mathcal L _i))=0$ for $i=1, 2, 3$.

By (3.2) and the configurations of the eight points on $\mathbb P ^2$, it is clear that $|K_W+\mathcal L _2|$ and $|K_W+\mathcal L _3|$ are empty. Now assume by contradiction that $|K_W+\mathcal L _1| \not = \emptyset$. Let $D \in |K_W+\mathcal L _1|$. By (3.2) and (2.1), $DC_3=DC_3^{\prime }=-1$ and thus $D \ge C_3+C_3^{\prime }$. Note that $D-C_3-C_3^{\prime } \equiv 2L-E_1-E_1^{\prime }-E_2-E_2^{\prime }-E_3-E$. This contradicts condition (II). Therefore $|K_W+\mathcal L _1|=\emptyset$ and thus $p_g(S)=0$.

Lemma 2.6 implies that $-2K_\Sigma +\gamma$ is ample. Since $\pi$ is a finite morphism, $K_S$ is ample by (3.6). In particular, $S$ is minimal and of general type.$\square$

We have constructed a family of surfaces with a $\mathbb{Z }_2^2$-action, parameterized by a $3$-dimensional open subset $\{(p, F_b)|\ p \in \mathbb P ^2\ \text{ satisfying } \text{ conditions } \text{(I) } \text{ and } \text{(II) },\ F_b\in |F| \text{ satisfying } \text{ conditions } \text{(A) } \text{ and } \text{(B) }\}$ of $\mathbb P ^2 \times \mathbb P ^1$. A natural question arises: is this a new family? To the best of the author’s knowledge, the Inoue surfaces are the only known smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$. So we intend to show that the surfaces constructed here satisfy different properties from the Inoue surfaces.

4 The bicanonical map

It is known that the bicanonical morphism of an Inoue surface has degree $2$ (cf. [16, Example 4.1]). In this section, we prove that our surfaces have birational bicanonical morphism. So they are indeed new surfaces.

Proposition 4.1

For a surface $S$ as in Theorem 3.1, the dimensions of the eigenspaces of $H^0(S, \mathcal O _S(2K_S))$ for the $G$-action are as follows: $\dim H^0(S, \mathcal O _S(2K_S))^{inv}=6,\,\dim H^0(S, \mathcal O _S(2K_S))^{\chi _1}=1,\, \dim H^0(S, \mathcal O _S(2K_S))^{\chi _2}=1$ and $\dim H^0(S, \mathcal O _S(2K_S))^{\chi _3}=0$.

Proof

By the formulae in [9, Section 2],

$$\begin{aligned} \dim H^0(S, \mathcal O _S(2K_S))^{\chi _i}=h^0(W, \mathcal O _W(2K_W+\mathcal L _{i+1}+\mathcal L _{i+2})) \end{aligned}$$

for $i=1, 2, 3$ and $\dim H^0(S, \mathcal O _S(2K_S))^{inv}=h^0(W, \mathcal O _W(2K_W+\Delta ))$. By (3.2) and (2.1),

$$\begin{aligned} 2K_W+\mathcal L _1+\mathcal L _3&\equiv (2L-E_1-E_2-E_3-E_3^{\prime }-E)+\sum \limits _{j=1}^3 (C_j+C_j^{\prime }),\\ 2K_W+\mathcal L _1+\mathcal L _2&\equiv (2L-E_1-E_2-E_3-2E)+\sum \limits _{j=1}^3 (C_j+C_j^{\prime }). \end{aligned}$$

It is immediate that $\dim H^0(S, \mathcal O _S(2K_S))^{\chi _2}= 1$ and $\dim H^0(S, \mathcal O _S(2K_S))^{\chi _3}= 0$.

By (3.2), (3.3) and Lemma 2.6, we have $\dim H^0(S, \mathcal O _S(2K_S))^{inv}=6$. Since $h^0(S, \mathcal O _S(2K_S))=K_S^2+1=8$, it follows that $\dim H^0(S, \mathcal O _S(2K_S))^{\chi _1}=1$.$\square$

Corollary 4.2

The bicanonical morphism $\varphi :S \rightarrow \mathbb P ^7$ is not composed with any involution of the group $G$.

Theorem 4.3

For a surface $S$ as in Theorem 3.1, the bicanonical morphism $\varphi :S \rightarrow \mathbb P ^7$ is birational.

Proof

By [16] and [17], $\varphi$ has degree either $1$ or $2$. By Lemma 2.6, $|-2K_\Sigma +\gamma |$ defines a birational morphism of $\Sigma$. Since $2K_S=\pi ^*(-2K_\Sigma +\gamma )$ and $S$ is a $\mathbb{Z }_2^2$-Galois cover of $\Sigma$, $\varphi$ is birational if and only if $\varphi$ is not composed with any involution of the Galois group. Thus $\varphi$ is a birational by Corollary 4.2.$\square$

5 The intermediate double covers and Bloch’s conjecture

By construction, we see that the automorphism group of the surface $S$ in Theorem 3.1 contains at least three involutions. Involutions on surfaces of general type with $K^2=7$ and $p_g=0$ are studied in [18] and [15]. Lists of numerical possibilities are given in these articles. The surfaces constructed here realize some possibilities of their lists.

Proposition 5.1

Let $S$ be a surface as in Theorem 3.1.

• (1) The involution $g_1$ has $9$ isolated fixed points on $S$ and $S/g_1$ is a rational surface.

• (2) The involution $g_2$ has $9$ isolated fixed points on $S$ and $S/g_2$ is birational to an Enriques surface.

• (3) The involution $g_3$ has $7$ isolated fixed points on $S$. Moreover, $S/g_3$ has Kodaira dimension $1$ and $K_{S/g_3}$ is nef.

Proof

• (1) The $\mathbb{Z }_2^2$-cover $\tilde{\pi }:V \rightarrow W$ of (3.5) factors through the intermediate double cover $\tilde{\pi }_1 :V_1 \rightarrow W$. The covering data associated to $\tilde{\pi }_1$ is $\Delta _2+\Delta _3 \equiv 2\mathcal L _1$. Note that $V_1$ has exactly one node, lying over the node of $\tilde{B}_2+\tilde{B}_3$. The inverse image $\tilde{\pi }_1^{-1}C_3$ or $\tilde{\pi }_1^{-1}C_3^{\prime }$ is a $(-1)$-curve, while the inverse image $\tilde{\pi }_1^{-1}C_k$ or $\tilde{\pi }_2^{-1}C_k^{\prime }$ ($k=1, 2$) is a disjoint union of two $(-2)$-curves. Contracting all these curves, we obtain the quotient surface $S/g_1$. By construction, $S/g_1$ has exactly $9$ nodes (the images of the node of $V_1$ and the $8$ $(-2)$-curves $\tilde{\pi }_1^{-1}C_k, \tilde{\pi }_1^{-1}C_k^{\prime }$ ($k=1, 2$)). Hence $g_1$ has $9$ isolated fixed points on $S$. In the proof of Proposition 2.4 (3), we show that $|M|=|\tilde{B}_2+\tilde{B}_3|$ gives a genus $0$ fibration $h :W \rightarrow \mathbb P ^1$. For a general $M$, we have $M(\Delta _2+\Delta _3)=0$. So the pullback of $M$ by $\tilde{\pi }_1$ is a disjoint union of two smooth rational curves. Applying Stein factorization to the morphism $h \circ \tilde{\pi }:V_1 \rightarrow \mathbb P ^1$, we conclude that $V_1$ has a genus $0$ fibration. Since $V_1$ is a quotient of $V$, $q(V_1)=0$. Hence $V_1$ is a rational surface and so is $S/g_1$.

• (2) Consider the intermediate double cover $\tilde{\pi }_2 :V_2 \rightarrow W$ associated to the data $\Delta _1+\Delta _3 \equiv 2\mathcal L _2$. Then $V_2$ has exactly $5$ nodes, lying over the $5$ nodes of $F_b+\Gamma +\tilde{B}_3$. Contracting the inverse image $\tilde{\pi }_2^{-1}(C_j)$ and $\tilde{\pi }_2^{-1}(C_j^{\prime })$ ($j=1, 2, 3$), we obtain $S/g_2$. It has $9$ nodes (the images of the $5$ nodes and the $4$ ($-2$)-curves $\tilde{\pi }_2^{-1}C_3, \tilde{\pi }_2^{-1}C_3^{\prime }$). Hence $g_2$ has $9$ isolated fixed points on $S$. Since $V_2$ is a quotient of $V$, $p_g(V_2)=q(V_2)=0$. To show that $V_2$ is birational to an Enriques surface, it suffices to show that $P_{2m+1}(V_2)=0$ and $P_{2m}(V_2)=1$ for $m \ge 1$. Note that $K_{V_2}=\tilde{\pi }_2^*(K_W+\mathcal L _2)$. Therefore,

  $$\begin{aligned} P_{2m}(V_2)=h^0(W, \mathcal O _W(2mK_W+(2m-1)\mathcal L _2)+h^0(W, \mathcal O _W(2mK_W+2m\mathcal L _2)) \end{aligned}$$

  By (3.2),

  $$\begin{aligned} 2mK_W+2m\mathcal L _2 \equiv 2m\Gamma +m(C_1+C_1^{\prime }+C_2+C_2^{\prime }). \end{aligned}$$

  (5.1)

  It is immediate that $h^0(W, \mathcal O _W(2mK_W+2m\mathcal L _2))=1$. By (3.2), it is clear that $|2K_W+\mathcal L _2| = \emptyset$. For $m \ge 2$, by (5.1),

  $$\begin{aligned} 2mK_W+(2m-1)\mathcal L _2 \equiv 2(m-1)\Gamma +(m-1)(C_1+C_1^{\prime }+C_2+C_2^{\prime })+(2K_W+\mathcal L _2). \end{aligned}$$

  Note that $\Gamma (2K_W+\mathcal L _2)=-2$, $C_k(2K_W+\mathcal L _2)=C_k^{\prime }(2K_W+\mathcal L _2)=-1 (k=1, 2)$. Similarly to the proof of Theorem 3.1, it is easy to show $|2mK_W+(2m-1)\mathcal L _2|=\emptyset$. Hence $P_{2m}(V_2)=1$. A similar argument using (5.1) shows that $P_{2m+1}(V_2)=0$ for $m \ge 1$. Hence $V_2$ is birational to an Enriques surface.

• (3) Consider the intermediate double cover $\tilde{\pi }_3 :V_3 \rightarrow W$ associated to the data $\Delta _1+\Delta _2 \equiv 2\mathcal L _3$. Then $V_3$ has exactly $7$ nodes, lying over the $7$ nodes of the curve $F_b+\Gamma +\tilde{B}_2$. The inverse image $\tilde{\pi }_3^{-1}C_j$ or $\tilde{\pi }_3^{-1}C_j^{\prime }$ ($j=1, 2, 3$) is a $(-1)$-curve. Contracting these $(-1)$-curves, we obtain $S/g_3$, which has $7$ nodes. Thus $g_3$ has $7$ isolated fixed points on $S$. By construction, there are double covers $\pi _3 :S/g_3 \rightarrow \Sigma$ and $p_3 :S \rightarrow S/g_3$ such that the following diagram commutes.

  

  By (3.2), $2K_{V_3} \equiv \tilde{\pi }_3^*(2K_W+2\mathcal L _3) \equiv \tilde{\pi }_3^*(L-E_0+C_1+C_2+C_3+C_1^{\prime }+C_2^{\prime }+C_3^{\prime })$. In Sect. 2, we show that $|L-E_0|$ gives a genus $0$ fibration $g :W \rightarrow \mathbb P ^1$, and the $(-2)$-curves $C_1, \ldots , C_3^{\prime }$ are contained in the fibers of $g$. This induces a fibration on $g^{\prime } :\Sigma \rightarrow \mathbb P ^1$. Denote the general fiber of $g^{\prime }$ by $F^{\prime }$. From the diagram, $2K_{S/g_3} \equiv \pi _3^*(F^{\prime })$. Thus $K_{S/g_3}$ is nef, $K_{S/g_3}^2=0$ and $S/g_3$ has Kodaira dimension $1$.$\square$

Remark 5.1

We remark that (2) (respectively (3)) realize some numerical possibilities of case a) (respectively case b)) of [18, Theorem 4.1]. (1), (2) and (3) realize respectively the following possible cases in the list of [15]:

• (1) $k=9, K_W^2=-2$, $W$ is a rational surface and $B_0=\begin{array}{c}{\Gamma _0}\\ (3,0)\end{array} + \begin{array}{c}{\Gamma _1} \\ (1,-2)\end{array}$.

• (2) $k=9, K_W^2=-2$, $W$ is birational to an Enriques surface and $B_0=\begin{array}{c}{\Gamma _0}\\ (3,-2)\end{array}$.

• (3) $k=7, K_W^2=0$, $W$ is minimal proper elliptic and $B_0=\begin{array}{c}{\Gamma _0}\\ (2,-2)\end{array}$.

These cases are different from those of the Inoue surfaces. See [15, Section 5] and Sect. 6.

Recently, it is shown in [3] that the Bloch’s conjecture ([6]) holds for Inoue surfaces with $K^2=7$ and $p_g=0$, by using the method of “enough automorphisms” ([14] and [1]). We observe that the results of [3] apply to our surfaces.

Theorem 5.2

Let $S$ be a surface as in Theorem 3.1. Then $S$ satisfies the Bloch’s conjecture, i.e., the kernel $T(S)$ of the natural morphism $A_0^0(S) \rightarrow Alb(S)$ is trivial. In particular, $A_0^0(S)=0$.

Proof

The first statement follows directly from [3, Proposition 1.3,Corollary 1.5] and Proposition 5.1. Since $Alb(S)$ is trivial in our case, Bloch’s conjecture says that $A_0^0(S)=0$.

6 Remarks on related topics

In a pre-version of [15], it is claimed that the three quotients of an Inoue surface by the involutions are rational. The claim turns out to be wrong. We will point out that one of the quotients is birational to an Enriques surface. In [18], a family of surfaces of general type with $K^2=7$ is constructed as bidouble planes. However, here we show that this family consists of Inoue surfaces.

We first stick to the same notation of [18, Section 4.2] (Fig. 2).

Fig. 2

Configurations of $p, p_1, p_2, p_3, p_1^{\prime }$ and $p_2^{\prime }$

Let $p, p_1, p_2, p_3$ be four points in general position of $\mathbb{P }^2$, and let $p_k^{\prime }$ ($k=1, 2$) be the infinitely near point of $p_k$ corresponding to the line $\overline{p_kp}$. Denote by $T_j$ ( $j=1, 2, 3$) the line $\overline{p_jp}$ and by $T_4$ a general line passing through $p$. Denote by $C_1, C_2$ two distinct smooth conics passing through $p_1, p_2, p_1^{\prime }, p_2^{\prime }$. Denote by $L$ a quintic passing through $p$, having $p_k$ ($k=1, 2$) as a $(2, 2)$-singularity and having $p_3$ as an ordinary triple point (See the last paragraph in [18, Subsection 4.2.1]).

We claim that $L$ is a union of a conic $C$ and a cubic $\Gamma$, where $C$ is the conic passing through $p_1, p_2, p_1^{\prime }, p_2^{\prime }, p_3$, and $\Gamma$ is a cubic passing through $p, p_1, p_2, p_1^{\prime }, p_2^{\prime }$ and having $p_3$ as an ordinary double point. Note that $LC=11$. The claim follows from Bézout’s Theorem.

In [18], it is claimed that the smooth minimal model of the bidouble plane associated to the following branch divisors is a surface of general type with $K^2=7$ and $p_g=0:$

$$\begin{aligned} D_1=L=C+\Gamma ,\ D_2=T_1+C_1+C_2,\ D_3=T_2+T_3+T_4. \end{aligned}$$

(6.1)

We explain how to find the smooth minimal model of the bidouble plane and show that this is indeed an Inoue surface with $K^2=7$.

Let $\sigma :\widetilde{Y}\rightarrow \mathbb P ^2$ be the blowup at the six points $p, p_1, p_2, p_3, p_1^{\prime }$ and $p_2^{\prime }$. Denote by $\hat{L}$ the pullback of a general line of $\mathbb P ^2$ and by $E$ (respectively $E_j,\, E_k^{\prime }$) the total transform of $p$ (respectively $p_j(\ j=1, 2, 3),\, p_k^{\prime }(\ k=1, 2)$). We also denote by $\widetilde{T}_1$ the strict transform of $T_1$, and similarly for other curves.

The surface $\widetilde{Y}$ is the minimal resolution of the $4$-nodal cubic surface $Y$. We explain some geometry of $\widetilde{Y}$.

• (1) $\widetilde{Y}$ has exactly four $(-2)$-curves: $\widetilde{T}_1=\hat{L}-E_1-E_1^{\prime }-E$, $\widetilde{T}_2=\hat{L}-E_2-E_2^{\prime }-E$, $N_1=E_1-E_1^{\prime }$ and $N_2=E_2-E_2^{\prime }$. These curves correspond to the four nodes of $Y$.

• (2) $\widetilde{Y}$ contains nine $(-1)$-curves, which correspond to nine lines on the $4$-nodal cubic surface $Y$. Among these curves, there are exactly three, which are disjoint from the $(-2)$-curves: $\widetilde{T}_3=\hat{L}-E_3-E$, $\widetilde{C}=2\hat{L}-E_1-E_1^{\prime }-E_2-E_2^{\prime }-E_3$ and $E_3$. They correspond to three lines on $Y$ which do not pass through any nodes. In particular, they are determined by the $4$-nodal cubic surface $Y$.

• (3) Note that

  $$\begin{aligned} \begin{array}{lll} \widetilde{T}_4\in |\hat{L}-E|, &{}&{}\widetilde{T}_4+\widetilde{C}\equiv -K_{\widetilde{Y}},\\ \widetilde{C}_1,\widetilde{C}_2 \in |2\hat{L}-E_1-E_1^{\prime }-E_2-E_2^{\prime }|,&{}&{} \widetilde{C}_1+\widetilde{T}_3\equiv \widetilde{C}_2+\widetilde{T}_3 \equiv -K_{\widetilde{Y}},\\ \tilde{\Gamma }\in |3\hat{L}-E_1-E_1^{\prime }-E_2-E_2^{\prime }-2E_3-E|,&{}&{} \tilde{\Gamma }+E_3 \equiv -K_{\widetilde{Y}}. \end{array} \end{aligned}$$

  So the divisor classes of $\widetilde{T}_4, \widetilde{C}_1, \widetilde{C}_2$ and $\tilde{\Gamma }$ are also determined by the $4$-nodal cubic surface.

The total transforms of $D_1,D_2,D_3$ on $\widetilde{Y}$ are

$$\begin{aligned} \sigma ^*(D_1)&= \widetilde{C}+\tilde{\Gamma }+2E_1+2E_1^{\prime }+2E_2+2E_2^{\prime }+3E_3+E,\\ \sigma ^*(D_2)&= \widetilde{T}_1+\widetilde{C}_1+\widetilde{C}_2+N_1+2E_1^{\prime }+E+2\left( E_1+E_1^{\prime }+E_2+E_2^{\prime }\right) ,\\ \sigma ^*(D_3)&= \widetilde{T}_2+\widetilde{T}_3+\widetilde{T}_4+N_2+2E_2^{\prime }+E_3+3E. \end{aligned}$$

Applying the normalization procedure in the theory of bidouble covers (cf. [9, Section 2, Remark 3]), we obtain three new divisors:

$$\begin{aligned} \begin{aligned} \tilde{D}_1&=\widetilde{C}+\tilde{\Gamma }, \\ \tilde{D}_2&=\widetilde{T}_1+\widetilde{C}_1+\widetilde{C}_2+N_1+E_3,\\ \tilde{D}_3&=\widetilde{T}_2+\widetilde{T}_3+\widetilde{T}_4+N_2. \end{aligned} \end{aligned}$$

(6.2)

The bidouble cover $\tilde{\pi }:\widetilde{S} \rightarrow \widetilde{Y}$ associated to (6.2) is birational to the bidouble plane associated to (6.1). Using the above explanation of the geometry of $\widetilde{Y}$ and comparing (6.2) with [16, Example 4.1 (I)], we conclude that the smooth minimal model of $\widetilde{S}$ (and thus of the bidouble plane) is an Inoue surface.

Now we point out a mistake in the pre-version of [15]. This observation is due to Carlos Rito. Here we use the notation of [16, Example 4.1], as [15] uses almost the same notation (except denoting the minimal resolution $\Sigma$ of the $4$-nodal cubic surface by $P$). In [15, Section 5, paragraph 4], the authors claim “Also, $H^0(T_2, \mathcal O _{T_2}(2K_{T_2}))=0$ by a similar argument as the case $i=1$”. Here $T_2$ is a double cover of $\Sigma$ associated to the data $D_1+D_3 \equiv 2L_2$. However, we will show that $H^0(T_2, \mathcal O _{T_2}(2K_{T_2}))=1$.

It suffices to show $h^0(\Sigma , \mathcal O _{\Sigma }(2K_{\Sigma }+L_2))=0$ and $h^0(\Sigma , \mathcal O _{\Sigma }(2K_{\Sigma }+2L_2))=1$, where $L_2=6l-2e_1-2e_2-2e_3-2e_4-3e_5-3e_6$ ([16, Example 4.1] (II)). Since $2K_{\Sigma }+L_2=-e_5-e_6$, clearly $h^0(\Sigma , \mathcal O _{\Sigma }(2K_{\Sigma }+L_2))=0$. Note that

$$\begin{aligned} 2K_{\Sigma }+2L_2&\equiv (l-e_1-e_2-e_5)+(l-e_3-e_4-e_5)+(l-e_1-e_4-e_6)\\&+\,(l-e_2-e_3-e_6)+2(l-e_5-e_6). \end{aligned}$$

From the configuration of the six points on $\mathbb P ^2$ ([16, Figure 1]), it is immediate that $h^0(\mathcal O _{\Sigma }, \mathcal O _{\Sigma }(2K_{\Sigma }+2L_2))=1$.

Finally, we remark that $T_2$ is birational to an Enriques surface as described in [18] and it realizes the case $k=9, K_W^2=-2$ and $B_0={\begin{array}{c}\Gamma _0 \\ (3,0)\end{array}}+{\begin{array}{c}\Gamma _1 \\ (1,-2)\end{array}}$ on the list of [15].