Construction of hyperbolic Horikawa surfaces

TL;DR

This paper constructs new examples of Brody hyperbolic surfaces, specifically Horikawa surfaces and double covers of Hirzebruch surfaces, expanding the known classes of hyperbolic algebraic surfaces.

Contribution

It introduces explicit constructions of Brody hyperbolic surfaces as double covers with specific branch loci, including the first such examples of Horikawa surfaces.

Findings

Constructed a Brody hyperbolic Horikawa surface as a double cover of P^2.

Built Brody hyperbolic double covers of Hirzebruch surfaces with minimal branch degrees.

Expanded the catalog of known hyperbolic algebraic surfaces.

Abstract

We construct a Brody hyperbolic Horikawa surface that is a double cover of $\mathbb{P}^2$ branched along a smooth curve of degree $10$. We also construct Brody hyperbolic double covers of Hirzebruch surfaces with branch loci of the lowest possible bidegree.

Equations

$$\varphi (\mathbb{C})\cap S^{\circ }\subset \underset{n\to \infty }{\lim }{\varphi }_n(\mathbb{C})\cap \mathrm{Supp}(S),$$

$$\mathbb{F}_{N}\setminus\big{(}(\cup_{i=1}^{a+b}C_{i})\setminus S\big{)}\subset\mathbb{F}_{N}\setminus\big{(}((\cup_{i=1}^{a_{\min}}C_{i})\cup(\cup_{j=a+1}^{a+b_{\min}}C_{j}))\setminus S\big{)}$$

$$\mathrm{deg}(v):=\#\{e\in E:v\in e\},\mathrm{rdeg}(v):=\#\{w\in V:\{v,w\}\in r(E)\}.$$

$$\Gamma_{0}\setminus\Gamma_{0}^{*}\supset\{x_{1},\cdots,x_{l}\}\cup\big{(}\cup_{\{i,j\}\neq\{1,2\}}(C_{i}\cap C_{j})\big{)}=:V.$$

$$(-K_{\tilde{X}}\cdot \tilde{C})=({\sigma }^{\ast }(-K_X)\cdot \tilde{C})-(E\cdot \tilde{C})=(-K_X\cdot C)-4\ge 4.$$

$$\#\mathrm{Sing}({\Gamma }_{t_{m-1}}^{(m-1)})$$

$$\ge \sum _{1\le i<j\le m}(C_i\cdot C_j)-(m-1)\ge m+1\ge 6.$$

$$Y_0^{(0)}$$

$$Y_0^{(i)}$$

Full text

1. Introduction

A complex algebraic variety $X$ is said to be Brody hyperbolic if there are no non-constant holomorphic maps from $\mathbb{C}$ to $X$. Thanks to Brody Lemma [Bro78], we know that a proper Brody hyperbolic variety is Kobayashi hyperbolic, i.e. its Kobayashi pseudometric is non-degenerate. In [Lan86], Lang conjectured that a complex projective variety $X$ is Brody hyperbolic if every subvariety of $X$ is of general type. More generally, Green, Griffiths [GG79] and Lang [Lan86] proposed the following conjecture:

Conjecture 1.1** (Green-Griffiths-Lang).**

If a complex projective variety $X$ is of general type, then there exists a proper Zariski closed subset $Z\subsetneq X$ such that any non-constant holomorphic map $f:\mathbb{C}\to X$ will satisfy $f(\mathbb{C})\subset Z$.

It is easy to see that Lang’s conjecture follows from the Green-Griffiths-Lang conjecture by a Noetherian induction argument. Even in the case of surfaces, these conjectures are still open. Based on works of Bogomolov [Bog77] and Lu-Yau [LY90], McQuillan [McQ98] showed that Conjecture 1.1 is true for minimal surfaces of general type with $c_{1}^{2}>c_{2}$. Demailly and El Goul [DEG00] proved Conjecture 1.1 for some surfaces with $13c_{1}^{2}>9c_{2}$. In principle, minimal surfaces of general type with minimal $s_{2}=c_{1}^{2}-c_{2}$ should be the most difficult case for these conjectures. For example, a very general quintic surface in $\mathbb{P}^{3}$ ($c_{1}^{2}=5$, $c_{2}=55$) does not contain any rational or elliptic curve by a result of Xu [Xu94], but we do not have a single example of quintic surface that is Brody hyperbolic.

Recall that the Chern numbers of minimal surfaces of general type satisfy the Noether inequality $c_{2}\leq 5c_{1}^{2}+36$. In the extreme case, a surface that reaches the equality $c_{2}=5c_{1}^{2}+36$ if $c_{1}^{2}$ is even and $c_{2}=5c_{1}^{2}+30$ otherwise is called a Horikawa surface. A Horikawa surface with even $c_{1}^{2}$ is classified to be either a double cover of $\mathbb{P}^{2}$ or of a Hirzebruch surface (see [Hor76]). For instance, a double cover of $\mathbb{P}^{2}$ branched along a smooth curve of degree $10$ is Horikawa. Using orbifold techniques, Roulleau and Rousseau [RR13] showed that a very general member of this class of Horikawa surfaces is algebraic hyperbolic (in particular it has no rational or elliptic curve). Hence a very general member of this class of surfaces is expected to be Brody hyperbolic according to Conjecture 1.1.

Our first main result shows that there exists a Horikawa surface in this class that is Brody hyperbolic. This gives an analytic generalization of Roulleau-Rousseau’s result (in particular implies [RR13, Theorem 3.2]) and also provides evidence supporting Conjecture 1.1.

Theorem 1.2**.**

Let $d$ be an even integer. Then there exists a smooth plane curve $D$ of degree $d$ such that the double cover of $\mathbb{P}^{2}$ branched along $D$ is Brody hyperbolic if and only if $d\geq 10$.

We remark that here that some Brody hyperbolic double covers of $\mathbb{P}^{2}$ have been constructed in [Liu16, Theorem 5] with branch loci of minimal degree $30$.

For an integer $N\geq 0$, let $\mathbb{F}_{N}$ be the $N$-th Hirzebruch surface. The surface $\mathbb{F}_{N}$ has a natural fibration $\mathbb{F}_{N}\to\mathbb{P}^{1}$. Denote by $F$ a fiber, and by $T$ a section of the fibration such that $(T^{2})=N$. Any divisor $D$ on $\mathbb{F}_{N}$ is linearly equivalent to $aF+bT$ for integers $a$ and $b$, and we say that $D$ is of bidegree $(a,b)$.

In [RR13], Roulleau and Rousseau also showed that a very general Horikawa surface that is a double cover of $\mathbb{F}_{N}$ branched along a curve of bidegree $(6,6)$ does not contain a rational curve. In general it will contain an elliptic curve, so it cannot be Brody hyperbolic. In the next theorem, we construct smooth curves of the lowest possible bidegrees in $\mathbb{F}_{N}$ along which the double covers of $\mathbb{F}_{N}$ are Brody hyperbolic.

Theorem 1.3**.**

Let $a,b$ be even integers. Then there exists a smooth curve $D\subset\mathbb{F}_{N}$ in the linear system $|aF+bT|$ such that the double cover of $\mathbb{F}_{N}$ branched along $D$ is Brody hyperbolic if and only if one of the following is true:

• •

$N=0$* and $a,b\geq 8$;*

• •

$N\geq 1$, $a\geq 6$ and $b\geq 8$.

The “only if” parts of Theorem 1.2 and 1.3 are somewhat easy which follow by showing the existence of a rational or elliptic curve on the double cover when the branch locus has a smaller (bi)degree.

Our strategy to prove the “if” parts of Theorem 1.2 and 1.3 is by using a degeneration process consisting of three steps. Denote by $X$ the base surface $\mathbb{P}^{2}$ or $\mathbb{F}_{N}$. In step 1, we degenerate the branch locus $D$ to a non-reduced double curve $2C$ where $C$ is smooth. As a result, the double cover degenerates to a union of two copies of $X$ glued along $C$. Using stability of intersections of entire curves, it suffices to show that both $X\setminus(C\setminus D)$ and $C$ are Brody hyperbolic. In step 2, we degenerate $C$ into a line arrangement $\cup_{i}C_{i}$. By a variant of Zaidenberg’s method [Zai89], it suffices to show that $X\setminus((\cup_{i}C_{i})\setminus D)$ is Brody hyperbolic. By classical results, we know that for $X=\mathbb{P}^{2}$ or $\mathbb{F}_{N}$, $X\setminus(\cup_{i}C_{i})$ is Brody hyperbolic. In step 3, we apply Zaidenberg-Duval’s method [Zai89, SZ02, Duv04, Duv17] of degenerating $D$ into line arrangements in order to deduce hyperbolicity of $X\setminus((\cup_{i}C_{i})\setminus D)$ from hyperbolicity of $X\setminus(\cup_{i}C_{i})$ which is known by classical results.

Historical remark. Note that Duval [Duv04] constructed a Brody hyperbolic sextic surface in $\mathbb{P}^{3}$ by nicely adopting Zaidenberg’s method [Zai89], together with the hyperbolic non-percolation introduced in [SZ02]. In this paper, we follow precisely Duval’s approach [Duv04] which was further developed in [Huy16]. Thus we will use the term Zaidenberg-Duval’s method for this approach in our presentation.

The paper is organized as follows. In Section 2, we recall Zaidenberg-Duval’s method [Zai89, SZ02, Duv04, Duv17] in constructing a smooth curve $D$ satisfying the hyperbolicity of $X\setminus((\cup_{i}C_{i})\setminus D)$. We recall results in [Huy16, Section 4] in the surface case in Lemma 2.2, and we apply this lemma to $\mathbb{P}^{2}$ and $\mathbb{F}_{N}$ in Corollary 2.3 and 2.4. In order to deform $\cup_{i}C_{i}$ into a smooth curve $C$ preserving the hyperbolicity of $X\setminus(C\setminus D)$, we apply Zaidenberg’s method [Zai89] in Section 4 (see Lemma 4.2). Starting with a log smooth projective surface pair $(X,D)$ and a set of rational curves $\{C_{i}\}$ with $X\setminus((\cup_{i}C_{i})\setminus D)$ being Brody hyperbolic, we introduce the concept of admissible deformation (see Definition 4.1) in order to preserve the hyperbolicity of $X\setminus(C\setminus D)$ under deformation. Using the technique of smoothing of rational trees in the deformation process (e.g. [Kol96, II.7]), we are able to translate an admissible deformation of rational curves into an admissible contraction of their dual graphs (see Lemma 4.4). In Section 3, we study dual graphs that can be admissibly contracted into singletons. Using these results, we construct a smoothing $C$ of $\cup_{i}C_{i}$ preserving the hyperbolicity of $X\setminus(C\setminus\Delta)$ under certain assumptions on the dual graph of $\cup_{i}C_{i}$ (see Lemma 4.8). Applying this lemma to $X=\mathbb{P}^{2}$ or $\mathbb{F}_{N}$ gives smooth curves $C$ and $D$ with certain (bi)degrees such that $X\setminus(C\setminus D)$ is Brody hyperbolic. In Section 5, we prove Theorem 1.2 and 1.3. As an application of Theorem 1.2, we give new examples of Brody hyperbolic surfaces in $\mathbb{P}^{3}$ of minimal degree $10$ that are cyclic covers of $\mathbb{P}^{2}$ under linear projections (see Theorem 5.3). This also improves [Liu16, Theorem 25]. We mention that a Brody hyperbolic Horikawa surface of even $c_{1}^{2}$ has to be a double cover of $\mathbb{P}^{2}$ branched along a degree $10$ curve (see Remark 5.5).

Notation

Throughout this paper, we work over the complex numbers $\mathbb{C}$. For a subset $U$ of a projective variety $X$, we say that $U$ is Brody hyperbolic if any non-constant holomorphic map $\phi:\mathbb{C}\to X$ satisfies $\phi(\mathbb{C})\not\subset U$. A divisor $D$ on a smooth surface $X$ is normal crossing if $D$ is reduced and has only nodal singularities. Moreover, a normal crossing divisor $D$ is said to be simple normal crossing if all irreducible components of $D$ are smooth. We say that a surface pair $(X,D)$ is log smooth if $X$ is a smooth surface and $D$ is a simple normal crossing divisor on $X$. A reduced projective curve is stable (in the sense of Deligne-Mumford) if it has only nodal singularities and its dualizing sheaf is ample.

Acknowledgement

I would like to thank Xavier Roulleau and Erwan Rousseau for fruitful discussions. I wish to thank Dinh Tuan Huynh, János Kollár and Ziquan Zhuang for helpful comments and suggestions, and Christian Liedtke for his interest. I also wish to thank the anonymous referees for their careful work. The author was partially supported by NSF Grant DMS-1362960.

2. Zaidenberg-Duval’s method

We first recall the following known facts from complex analysis whose proof is a simple application of the classical Hurwitz Theorem. (See also [Kob98, 3.6.11], [Duv17, Stability of intersections] or [Huy16, Section 3.1].)

Lemma 2.1** (Stability of intersections).**

Let $X$ be a normal proper complex analytic space. Let $S$ be an effective Weil divisor in $X$, i.e. $S$ is a sum of closed analytic subvarieties of codimension $1$. Suppose that a sequence of entire curves $(\phi_{n})$ in $X$ converges to an entire curve $\phi$. If $\phi(\mathbb{C})\not\subset\mathrm{Supp}(S)$, then

[TABLE]

where $S^{\circ}:=\{x\in\mathrm{Supp}(S)\mid S\textrm{ is }\mathbb{Q}\textrm{-Cartier in a neighborhood of }x\}$.

The following lemma was proved in [Huy16, Section 4] (see also [Duv17, Lemma]).

Lemma 2.2**.**

Let $X$ be a smooth projective surface. Let $\{C_{i}\}_{i=1}^{m}$ be a set of irreducible curves on $X$ such that $(X,\sum_{i=1}^{m}C_{i})$ is log smooth. Let $L$ be a globally generated line bundle on $X$. Assume the following holds:

1. (a)

$X\setminus(\cup_{i=1}^{m}C_{i})$* is Brody hyperbolic;* 2. (b)

For any $i$, $\cup_{j\neq i}C_{j}$ is a stable curve; 3. (c)

For any $i$, there exists an effective Cartier divisor $H_{i}\in|L|$ such that $\mathrm{Supp}(H_{i})=\cup_{j\neq i}C_{j}$.

Then there exists a smooth curve $S\in|L|$ such that $(X,S+\sum_{i=1}^{m}C_{i})$ is log smooth and X\setminus\big{(}(\cup_{i=1}^{m}C_{i})\setminus S\big{)} is Brody hyperbolic.

Proof.

See [Huy16, Section 4]. ∎

The following corollary was proved in [Huy16, Section 4] using Lemma 2.2.

Corollary 2.3** ([Huy16, Section 4]).**

Let $\{C_{i}\}_{i=1}^{m}$ be a set of lines in general position in $\mathbb{P}^{2}$ with $m\geq 5$. Let $d\geq 4$ be an integer. Then there exists a smooth plane curve $S$ of degree $d$ such that $(\mathbb{P}^{2},S+\sum_{i=1}^{m}C_{i})$ is log smooth and \mathbb{P}^{2}\setminus\big{(}(\cup_{i=1}^{m}C_{i})\setminus S\big{)} is Brody hyperbolic.

Corollary 2.4**.**

Let $\{C_{i}\}_{i=1}^{a+b}$ be a set of curves in $\mathbb{F}_{N}$. Assume that $C_{i}$ is a general curve in $|F|$ for any $i\leq a$; $C_{j}$ is a general curve in $|T|$ for any $j>a$. Then there exists a smooth curve $S\in|cF+dT|$ in $\mathbb{F}_{N}$ such that $(\mathbb{F}_{N},S+\sum_{i=1}^{a+b}C_{i})$ is log smooth and \mathbb{F}_{N}\setminus\big{(}(\cup_{i=1}^{a+b}C_{i})\setminus S\big{)} is Brody hyperbolic if one of the following is true:

• •

$N=0$* and $a,b,c,d\geq 4$;*

• •

$N\geq 1$, $a,c\geq 3$ and $b,d\geq 4$.

Proof.

Firstly, let us consider special cases where $a,b$ achieve their minima, i.e. $N=0$, $a=b=4$ or $N\geq 1$, $a=3$, $b=4$. Since both linear systems $|F|$ and $|T|$ are base point free, for a general choice of $\{C_{i}\}_{i=1}^{a+b}$ the pair $(\mathbb{F}_{N},\sum_{i=1}^{a+b}C_{i})$ is log smooth. Let $L:=\mathcal{O}_{\mathbb{F}_{N}}(cF+dT)$ be a line bundle on $\mathbb{F}_{N}$. Then we only need to show that the assumptions (a), (b) and (c) of Lemma 2.2 are fulfilled for $(\mathbb{F}_{N},\sum_{i=1}^{a+b}C_{i})$ and $L$.

If $N=0$ and $a=b=4$, then $\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $\{C_{i}\}_{i=1}^{8}$ consists of $4$ vertical lines and $4$ horizontal lines in general position. It is clear that $\mathbb{F}_{0}\setminus\cup_{i=1}^{8}C_{i}\cong(\mathbb{P}^{1}\setminus\{4\textrm{ points}\})\times(\mathbb{P}^{1}\setminus\{4\textrm{ points}\})$ is Brody hyperbolic, so (a) is satisfied. For (b), each $C_{j}$ intersects four $C_{k}$’s with $k\neq j$. So for any $i\neq j$, $C_{j}$ intersects with at least three $C_{k}$’s with $k\not\in\{i,j\}$. Since $C_{i}\cap C_{j}\cap C_{k}=\emptyset$, $\cup_{j\neq i}C_{j}$ is stable, hence (b) is satisfied. Since $c,d\geq 4$, $C_{i}\sim F$ for $1\leq i\leq 4$ and $C_{j}\sim T$ for $5\leq j\leq 8$, it is easy to see that (c) is also satisfied.

If $N\geq 1$, $a=3$ and $b=4$, then the natural fibration $\pi:\mathbb{F}_{N}\to\mathbb{P}^{1}$ maps $C_{1}$, $C_{2}$, $C_{3}$ to three distinct points in $\mathbb{P}^{1}$. It is clear that $(F\cdot C_{i})=(F\cdot T)=1$ for $i=4,\cdots,7$. Hence for a general choice of $\{C_{i}\}_{i=4}^{7}$, the set $F\cap(\cup_{i=4}^{7}C_{i})$ has at least three points for any fiber $F$ of $\pi$. Since $\mathbb{P}^{1}\setminus\{3\textrm{ points}\}$ is Brody hyperbolic, the fiber and the base of $\pi:\mathbb{F}_{N}\setminus\cup_{i=1}^{7}C_{i}\to\mathbb{P}^{1}\setminus\{\pi(C_{1}),\pi(C_{2}),\pi(C_{3})\}$ are Brody hyperbolic. Hence (a) is satisfied. For (b), each $C_{i}$ with $1\leq i\leq 3$ intersects each $C_{k}$ with $4\leq k\leq 7$. Since $(T^{2})=N\geq 1$, each $C_{i}$ with $4\leq i\leq 7$ intersects each $C_{k}$ with $k\neq i$. As a result, each $C_{i}$ intersects with at least four $C_{k}$’s with $k\neq i$. So (b) is satisfied by the same reason as in the last paragraph. Since $c\geq 3$, $d\geq 4$, $C_{i}\sim F$ for $1\leq i\leq 3$, and $C_{i}\sim T$ for $4\leq i\leq 7$, it is easy to see that (c) is also satisfied.

Up to now we have shown the corollary for cases where $a,b$ achieve their minima. More precisely, under the assumptions of $N,a,b,c,d$, for general choices of $\{C_{i}\}_{i=1}^{a_{\min}}$ and $\{C_{j}\}_{j=a+1}^{a+b_{\min}}$ there exists a smooth curve $S\in|cF+dT|$ in $\mathbb{F}_{N}$, such that $(\mathbb{F}_{N},S+\sum_{i=1}^{a_{\min}}C_{i}+\sum_{j=a+1}^{a+b_{\min}}C_{j})$ is log smooth and \mathbb{F}_{N}\setminus\big{(}((\cup_{i=1}^{a_{\min}}C_{i})\cup(\cup_{j=a+1}^{a+b_{\min}}C_{j}))\setminus S\big{)} is Brody hyperbolic. If one of $a,b$ is strictly bigger than its minimum, then

[TABLE]

where the latter set is Brody hyperbolic. Hence \mathbb{F}_{N}\setminus\big{(}(\cup_{i=1}^{a+b}C_{i})\setminus S\big{)} is Brody hyperbolic. Besides, since $(\mathbb{F}_{N},S+\sum_{i=1}^{a_{\min}}C_{i}+\sum_{j=a+1}^{a+b_{\min}}C_{j})$ is log smooth, for general choices of $\{C_{i}\}_{i=a_{\min}+1}^{a}$ and $\{C_{j}\}_{j=a+b_{\min}+1}^{a+b}$ we also have that $(X,S+\sum_{i=1}^{a+b}C_{i})$ is log smooth. This finishes the proof. ∎

3. Admissible contractions of multigraphs

Definition 3.1**.**

1. (a)

A vertex-weighted multigraph $G$ is an ordered quadruple $(V,E,r,\mathrm{wt})$ such that

• •

$V$ is a finite set of vertices;

• •

$E$ is a finite set of edges;

• •

$r:E\to\{\{v,w\}:v,w\in V,v\neq w\}$ assigns each edge an unordered pair of endpoint vertices;

• •

$\mathrm{wt}:V\to\mathbb{Z}$ assigns to each vertex an integer as its weight. 2. (b)

For a vertex $v\in V$, we define the degree (respectively reduced degree) of $v$ to be its number of incident edges (respectively adjacent vertices). More precisely,

[TABLE] 3. (c)

Let $G_{1},G_{2}$ be two vertex-weighted multigraphs. We say that $G_{1}$ is a submultigraph of $G_{2}$ if there exists injective maps $\phi:V_{1}\to V_{2}$ and $\psi:E_{1}\to E_{2}$, such that $r_{2}\circ\psi=\phi\circ r_{1}$ and $\mathrm{wt}_{1}\leq\mathrm{wt}_{2}\circ\phi$. If, moreover, $\phi$ is bijective, then we say that $G_{1}$ is a spanning submultigraph of $G_{2}$. 4. (d)

A vertex-weighted multigraph is completely multipartite if there does not exist a triple of vertices $\{v_{1},v_{2},v_{3}\}$ such that both $\{v_{1},v_{2}\}$ and $\{v_{1},v_{3}\}$ are non-adjacent, but $\{v_{2},v_{3}\}$ is adjacent.

Definition 3.2**.**

1. (a)

Let $G,G^{\prime}$ be two vertex-weighted multigraphs. We say that $G^{\prime}$ is a contraction of $G$ with respect to a pair of adjacent vertices $\{v,w\}$ in $G$ if there exist maps $\phi:V\to V^{\prime}$ and $\psi:E\setminus r^{-1}(\{v,w\})\to E^{\prime}$ such that

• •

$\phi(v)=\phi(w)$, and $\phi$ induces a bijection between $V\setminus\{v,w\}$ and $V^{\prime}\setminus\{\phi(v)\}$;

• •

$\psi$ is bijective, and $r^{\prime}\circ\psi=\phi\circ r$ as maps from $E\setminus r^{-1}(\{v,w\})$;

• •

$\mathrm{wt}^{\prime}(\phi(v))=\mathrm{wt}(v)+\mathrm{wt}(w)$, and $\mathrm{wt}^{\prime}\circ\phi=\mathrm{wt}$ as maps from $V\setminus\{v,w\}$ to $\mathbb{Z}$. 2. (b)

A contraction $G^{\prime}$ of $G$ with respect to $\{v,w\}$ is said to be admissible if there exists a non-negative integer $l<\#r^{-1}(v,w)$ such that the following conditions hold:

• •

For each vertex $x$ other than $v$ and $w$, $\deg(x)\geq 3$;

• •

$\mathrm{wt}(v)\geq l+1$ and $\mathrm{wt}(w)\geq l+2$;

• •

$\deg(v)-\#r^{-1}(\{v,w\})+l\geq 3$ and $\deg(w)-\#r^{-1}(\{v,w\})+l\geq 3$. 3. (c)

A vertex-weighted multigraph $G$ is said to be admissibly contractible if there exists a sequence of vertex-weighted multigraphs $(G_{i})_{i=0}^{k}$ such that $G_{0}=G$, $G_{k}$ is a singleton, and $G_{i}$ is an admissible contraction of $G_{i-1}$ for each $1\leq i\leq k$.

Example 3.3**.**

We give an illustration of an admissible contraction of vertex-weighted multigraphs.

$G:=$

3$$v_{1}$$3$$v_{2}$$3$$v_{3}

$\xrightarrow{l=1}$

3$$w_{2}$$6$$w_{1}$$*

$=:G^{\prime}$

Here $V_{G}=\{v_{1},v_{2},v_{3}\}$, $V_{G^{\prime}}=\{w_{1},w_{2}\}$, $\mathrm{wt}_{G}(v_{i})=3$ for any $1\leq i\leq 3$, $\mathrm{wt}_{G^{\prime}}(w_{1})=6$ and $\mathrm{wt}_{G^{\prime}}(w_{2})=3$. Each vertex is represented as a circle in the picture. The name of each vertex is marked outside the circle, and its weight is marked inside the circle. Each edge connecting two vertices is represented as an arc connecting two circles.

In the illustration above, we see that $H$ is a contraction of $G$ with respect to $\{v_{1},v_{2}\}$, where $\phi$ is given by $\phi(v_{1})=\phi(v_{2})=w_{1}$ and $\phi(v_{3})=w_{2}$. Each contraction is represented as an arrow. The two merging vertices of $G$ are represented as yellow filled circles, and we mark $*$ outside circles representing their images under $\phi$. If a contraction is admissible, we mark the corresponding value of $l$ above the arrow. It is easy to verify that in the picture above, $G^{\prime}$ is an admissible contraction of $G$ with respect to $\{v_{1},v_{2}\}$.

The following lemma follows easily from the definitions.

Lemma 3.4**.**

Let $G$ be a vertex-weighted multigraph. Let $H$ be a spanning submultigraph of $G$. If $H$ is admissibly contractible, then so is $G$.

Proposition 3.5**.**

The following vertex-weighted multigraphs $K_{1}$, $K_{2}$, $K_{3}$ and $K_{4}$ are all admissibly contractible:

$K_{1}=$* * 2$$v_{1}$$2$$v_{2}$$2$$v_{3}$$2$$v_{4}$$2$$v_{5} *   $K_{2}=$ * 2$$v_{1}$$2$$v_{2}$$2$$v_{3}$$2$$v_{4}$$2$$v_{5}$$2$$v_{6} **

$K_{3}=$* * 2$$v_{1}$$2$$v_{7}$$2$$v_{6}$$2$$v_{2}$$2$$v_{3}$$2$$v_{5}$$2$$v_{4} *   $K_{4}=$ * 2$$v_{1}$$2$$v_{2}$$2$$v_{8}$$2$$v_{7}$$2$$v_{3}$$2$$v_{4}$$2$$v_{6}$$2$$v_{5} **

Proof.

For simplicity, we will omit the name of vertices in all pictures. A successive admissible contraction of $K_{1}$ is illustrated as below:

2$$2$$2$$2$$2

$\xrightarrow{l=0}$

4$$*$$2$$2$$2

$\xrightarrow{l=0}$

4$$4$$*$$2

$\xrightarrow{l=1}$

4$$6$$*

$\xrightarrow{l=3}$

10$$*

A successive admissible contraction of $K_{2}$ is illustrated as below:

2$$2$$2$$2$$2$$2

$\xrightarrow{l=0}$

4$$*$$2$$2$$2$$2

$\xrightarrow{l=0}$

4$$4$$*$$2$$2

$\xrightarrow{l=0}$

4$$4$$4$$*

$\xrightarrow{l=0}$

4$$8$$*

$\xrightarrow{l=3}$

12$$*

A successive admissible contraction of $K_{3}$ is illustrated as below:

2$$2$$2$$2$$2$$2$$2

$\xrightarrow{l=0}$

2$$2$$2$$4$$*$$2$$2

$\xrightarrow{l=0}$

2$$2$$2$$4$$4$$*

$=:K_{3}^{\prime}$

It is clear that $K_{1}$ is a spanning submultigraph of $K_{3}^{\prime}$. Since $K_{1}$ is admissibly contractible, so is $K_{3}^{\prime}$. Hence $K_{3}$ is also admissibly contractible.

An admissible contraction of $K_{4}$ is illustrated as below:

2$$2$$2$$2$$2$$2$$2$$2

$\xrightarrow{l=0}$

4$$*$$2$$2$$2$$2$$2$$2

$=:K_{4}^{\prime}$

It is clear that $K_{3}$ is a spanning submultigraph of $K_{4}^{\prime}$. Since $K_{3}$ is admissibly contractible, so is $K_{4}^{\prime}$. Hence $K_{4}$ is also admissibly contractible. ∎

Lemma 3.6**.**

Let $G$ be a vertex-weighted multigraph. Let $H$ be a submultigraph of $G$. Assume the following conditions:

1. (a)

$G$* is completely multipartite;* 2. (b)

$\mathrm{wt}_{G}\geq 2$; 3. (c)

If $\{v_{1},\cdots,v_{s}\}\subset V_{H}$ is a set of mutually non-adjacent vertices of $H$, then $s\leq\#V_{H}-4$.

Then there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ is a spanning submultigraph of $G^{\prime}$.

Proof.

We do induction on $q:=\#(V_{G}\setminus V_{H})$. If $q=0$, then the lemma is proved by taking $G^{\prime}:=G$. Assume that the lemma is proved for $q-1$. Let $w\in V_{G}$ be an arbitrary vertex of $G$. Let $\{v_{1},\cdots,v_{s(w)}\}$ be the set of all vertices in $V_{H}$ that are not adjacent to $w$ in $G$. Since $G$ is completely multipartite, $\{v_{1},\cdots,v_{s(w)}\}$ is a set of mutually non-adjacent vertices of $G$ (hence of $H$). By assumption, we have $s(w)\leq\#V_{H}-4$. This implies that $\mathrm{rdeg}_{G}(w)\geq 4$ for any vertex $w$ of $G$. Let us pick a vertex $w\in V_{G}\setminus V_{H}$, then $w$ is adjacent to a vertex $v\in V_{H}$. Let $G_{1}$ be the contraction of $G$ with respect to $\{v,w\}$. Since $\mathrm{wt}_{G}\geq 2$ and each vertex of $G$ have reduced degree $\geq 4$, $G_{1}$ is an admissble contraction of $G$ when $l=0$. It is clear that $H$ is a also submultigraph of $G_{1}$ with $q-1=\#(V_{G_{1}}\setminus V_{H})$, $\mathrm{wt}_{G_{1}}\geq 2$, and $G_{1}$ is also a completely multipartite. By the inductive hypothesis, there exists a successive admissible contraction $G_{1}^{\prime}$ of $G_{1}$ such that $H$ is a spanning submultigraph of $G_{1}^{\prime}$. The proof is finished by taking $G^{\prime}:=G_{1}^{\prime}$. ∎

Remark 3.7**.**

It is easy to verify that $H=K_{i}$ satisfies assumption (c) of Lemma 3.6 for each $1\leq i\leq 4$.

Lemma 3.8**.**

Let $G$ be a completely multipartite vertex-weighted multigraph. Assume that for any vertex $v$ of $G$ we have $\mathrm{rdeg}(v)\geq 4$ and $\mathrm{wt}(v)\geq 2$. Then $G$ is admissibly contractible.

Proof.

Since $G$ is completely multipartite, there exists a partition of vertices $V=\cup_{i=1}^{k}V_{k}$ such that two vertices are non-adjacent if and only if they belong to the same $V_{i}$. Denote $a_{i}:=\#V_{i}$. For simplicity we may assume that $a_{1}\leq a_{2}\leq\cdots\leq a_{k}$. Then $\mathrm{rdeg}(v)\geq 4$ implies $\sum_{i=1}^{k-1}a_{i}\geq 4$. In particular, $k\geq 2$.

We divide the proof into five cases based on values of $k$ and $a_{1}$. We will use Lemma 3.6 in all cases. Since $G$ satisfies assumptions (a)(b) of Lemma 3.6, we only need to verify assumption (c).

Case 1. $k\geq 5$.

Let us pick $v_{i}\in V_{i}$ for $1\leq i\leq 5$. Let $H$ be the submultigraph of $G$ generated by $\{v_{1},\cdots,v_{5}\}$. Since $\{v_{1},\cdots,v_{5}\}$ are mutually adjacent in $G$, $K_{1}$ is a spanning submultigraph of $H$. Hence $H$ satisfies condition (c). By Lemma 3.6, there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ (hence $K_{1}$) is a spanning submultgraph of $G^{\prime}$. By Proposition 3.5, $K_{1}$ is admissibly contractible, hence $G$ is admissibly contractible.

Case 2. $k=4$.

Since $\sum_{i=1}^{3}a_{i}\geq 4$, we have that $a_{1},a_{2}\geq 1$ and $a_{3},a_{4}\geq 2$. Let us pick $v_{1}\in V_{1}$, $v_{4}\in V_{2}$, $v_{2},v_{5}\in V_{3}$ and $v_{3},v_{6}\in V_{4}$. Let $H$ be the submultigraph of $G$ generated by $\{v_{1},\cdots,v_{6}\}$. It is easy to see that $K_{2}$ is a spanning submultigraph of $H$, hence $H$ satisfies condition (c). By Lemma 3.6, there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ (hence $K_{2}$) is a spanning submultgraph of $G^{\prime}$. By Proposition 3.5, $K_{2}$ is admissibly contractible, hence $G$ is admissibly contractible.

Case 3. $k=3$ and $a_{1}\geq 2$.

We know that $a_{2},a_{3}\geq a_{1}\geq 2$. Let us pick $v_{1},v_{4}\in V_{1}$, $v_{2},v_{5}\in V_{2}$ and $v_{3},v_{6}\in V_{3}$. Let $H$ be the submultigraph of $G$ generated by $\{v_{1},\cdots,v_{6}\}$. It is easy to see that $K_{2}$ is a spanning submultigraph of $H$, hence $H$ satisfies condition (c). By Lemma 3.6, there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ (hence $K_{2}$) is a spanning submultgraph of $G^{\prime}$. By Proposition 3.5, $K_{2}$ is admissibly contractible, hence $G$ is admissibly contractible.

Case 4. $k=3$ and $a_{1}=1$.

Since $a_{1}+a_{2}\geq 4$, we have $a_{2},a_{3}\geq 3$. Let us pick $v_{1}\in V_{1}$, $v_{2},v_{4},v_{6}\in V_{2}$ and $v_{3},v_{5},v_{7}\in V_{3}$. Let $H$ be the submultigraph of $G$ generated by $\{v_{1},\cdots,v_{7}\}$. It is easy to see that $K_{3}$ is a spanning submultigraph of $H$, hence $H$ satisfies condition (c). By Lemma 3.6, there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ (hence $K_{3}$) is a spanning submultgraph of $G^{\prime}$. By Proposition 3.5, $K_{3}$ is admissibly contractible, hence $G$ is admissibly contractible.

Case 5. $k=2$.

Since $a_{1}\geq 4$, we have $a_{1},a_{2}\geq 4$. Let us pick $v_{1},v_{3},v_{5},v_{7}\in V_{1}$ and $v_{2},v_{4},v_{6},v_{8}\in V_{2}$. Let $H$ be the submultigraph of $G$ generated by $\{v_{1},\cdots,v_{8}\}$. It is easy to see that $K_{4}$ is a spanning submultigraph of $H$, hence $H$ satisfies condition (c). By Lemma 3.6, there exists a successive admissible contraction $G^{\prime}$ of $G$ such that $H$ (hence $K_{4}$) is a spanning submultgraph of $G^{\prime}$. By Proposition 3.5, $K_{4}$ is admissibly contractible, hence $G$ is admissibly contractible. ∎

4. Zaidenberg’s method

Definition 4.1**.**

Let $(X,\Delta)$ be a log smooth projective surface pair. Let $C$ be a reduced curve in $X$. Let $\{\Gamma_{t}\}_{t\in\mathbb{D}}$ be a holomorphic flat family of reduced divisors on $X$. Denote by $\Gamma\subset X\times\mathbb{D}$ the development of $\{\Gamma_{t}\}_{t\in\mathbb{D}}$. We say that $\{\Gamma_{t}\}_{t\in\mathbb{D}}$ is an admissible deformation of $C$ if $\Gamma_{0}=C$ and the set $\Gamma_{0}^{*}:=\{x\in\Gamma_{0}\mid\Gamma\textrm{ is locally analytically irreducible at }(x,0)\}$ is Brody hyperbolic. If, moreover, $\Delta+C$ is normal crossing, an admissible deformation $\{\Gamma_{t}\}_{t\in\mathbb{D}}$ of $C$ is nodal if $\Delta+\Gamma_{t}$ is normal crossing for any $t\in\mathbb{D}$. Besides, we say that $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq k}$ is a successive admissible deformation of $C$ if for each $1\leq j\leq k$ there exists $t_{j}\in\mathbb{D}\setminus\{0\}$, such that $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D}}$ is an admissible deformation of $\Gamma_{t_{j-1}}^{(j-1)}$ where $\Gamma_{t_{0}}^{(0)}:=C$.

The following lemma is a generalization of Zaidenberg’s result [Zai89, Lemma-Definition 3.2] to surface pairs.

Lemma 4.2**.**

Let $(X,\Delta)$ be a log smooth projective surface pair. Let $C$ be a reduced curve in $X$ such that $\Delta+C$ is normal crossing. Let $\{\Gamma_{t}\}_{t\in\mathbb{D}}$ be an admissible deformation of $C$. If $X\setminus(C\setminus\Delta)$ is Brody hyperbolic, then $X\setminus(\Gamma_{t}\setminus\Delta)$ is also Brody hyperbolic for any $0<|t|\ll 1$. (Note that $X\setminus(C\setminus\Delta)$ being Brody hyperbolic is the same as saying that $X\setminus C$ has the property of hyperbolic non-percolation through $C\cap\Delta$ according to [SZ02].)

Proof.

The proof is similar to [Zai89, Proof of Lemma-Definition 3.2]. ∎

Lemma 4.3**.**

Let $X$ be a smooth projective rational surface. Let $C_{1}$, $C_{2}$ be two intersecting rational nodal curves such that $C_{1}+C_{2}$ is normal crossing. Assume that $(-K_{X}\cdot C_{1})\geq l+1$ and $(-K_{X}\cdot C_{2})\geq l+2$ for some non-negative integer $l<(C_{1}\cdot C_{2})$. Then for any subset $\{x_{1},\cdots,x_{l}\}\subset C_{1}\cap C_{2}$, there exists a holomorphic flat family of divisors $\{\Gamma_{t}\}$ in $X$ such that $\Gamma_{0}=C_{1}+C_{2}$ and $\Gamma_{t}$ is an irreducible rational nodal curve singular at $x_{i}$ for any $t\neq 0$ and any $1\leq i\leq l$.

Proof.

Denote by $\sigma:\tilde{X}=\mathrm{Bl}_{x_{1},\cdots,x_{l}}X\to X$ the blow up of $X$ at $x_{1},\cdots,x_{l}$. Let $E$ be the reduced exceptional divisor of $\sigma$. Let $\tilde{C}_{1}$ and $\tilde{C_{2}}$ be strict transforms of $C_{1}$ and $C_{2}$ under $\sigma$. It is easy to see that $(-K_{\tilde{X}}\cdot\tilde{C}_{1})\geq 1$, $(-K_{\tilde{X}}\cdot\tilde{C}_{2})\geq 2$ and $(\tilde{C}_{1}\cdot\tilde{C}_{2})>0$. It is clear that both $\tilde{C}_{1}$ and $\tilde{C}_{2}$ are irreducible rational nodal curves intersecting each other transversally. Denote by $f_{i}:\mathbb{P}^{1}\to\tilde{X}$ the normalization of $\tilde{C}_{i}$. Since $f_{i}$ is an immersion, we have an exact sequence $0\to T_{\mathbb{P}^{1}}\to f_{i}^{*}T_{\tilde{X}}\to N_{\tilde{C}_{i}/\tilde{X}}\to 0$, where $\deg N_{\tilde{C}_{i}/\tilde{X}}=(-K_{\tilde{X}}\cdot\tilde{C}_{i})-2\geq i-2$. Hence $f_{1}^{*}T_{\tilde{X}}\otimes\mathcal{O}(1)$ is nef and $f_{2}^{*}T_{\tilde{X}}\otimes\mathcal{O}(1)$ is ample. Denote by $f:\mathbb{P}^{1}\vee\mathbb{P}^{1}\to\tilde{X}$ the gluing morphism of $f_{1}$ and $f_{2}$ at an intersection point of $\tilde{C}_{1}$ and $\tilde{C}_{2}$. Then $H^{1}(\mathbb{P}^{1}\vee\mathbb{P}^{1},f^{*}T_{\tilde{X}})=0$ by [Kol96, II.7.5], so the deformation of $f$ is unobstructed. By [Kol96, I.2.17] there exists a holomorphic flat family of divisors $\{\tilde{\Gamma}_{t}\}_{t\in\mathbb{D}}$ such that $\tilde{\Gamma}_{0}=\tilde{C}_{1}+\tilde{C}_{2}$ and $\tilde{\Gamma}_{t}$ is an irreducible rational nodal curve whenever $t\neq 0$. After a reparametrization of $t$ if necessary we may also assume that $\tilde{\Gamma}_{t}+E$ is normal crossing for each $t$. The lemma is proved by taking $\Gamma_{t}:=\sigma_{*}(\tilde{\Gamma}_{t})$. ∎

Lemma 4.4**.**

Let $(X,\Delta)$ be a log smooth projective surface pair with $X$ rational. Let $C=\sum_{i=1}^{m}C_{i}~{}(m\geq 2)$ be a reduced divisor on $X$ such that each $C_{i}$ is an irreducible nodal rational curve and $\Delta+C$ is normal crossing. Assume

• •

$(C_{1}\cdot C_{2})>0$;

• •

$(C_{i}\cdot(C-C_{i}))\geq 3$* for any $3\leq i\leq m$;*

• •

There exists a non-negative integer $l<(C_{1}\cdot C_{2})$, such that $(-K_{X}\cdot C_{i})\geq l+i$ and $(C_{i}\cdot(C-C_{1}-C_{2}))\geq 3-l$ for any $i\in\{1,2\}$.

Then there exists a nodal admissible deformation $\{\Gamma_{t}\}_{t\in\mathbb{D}}$ of $C$ such that $\Gamma_{t}=A_{t}+\sum_{i=3}^{m}C_{i}$ where $A_{t}$ is an irreducible rational nodal curve whenever $t\neq 0$.

Proof.

Let us pick $l$ distinct points $x_{1},\cdots,x_{l}$ in $C_{1}\cap C_{2}$. By Lemma 4.3, there exists a holomorphic flat family $\{A_{t}\}_{t\in\mathbb{D}}$ of reduced divisors on $X$ such that $A_{0}=C_{1}+C_{2}$ and $A_{t}$ is an irreducible rational nodal curve singular at $x_{1},\cdots,x_{l}$ for any $t\in\mathbb{D}\setminus\{0\}$. By Bertini’s theorem, after a reparametrization of $t$ we may assume that $\Delta+A_{t}+\sum_{i=3}^{m}C_{i}$ is normal crossing for any $t\in\mathbb{D}$. Let $\Gamma_{t}:=A_{t}+\sum_{i=3}^{m}C_{i}$, then it suffices to show that $\Gamma_{0}^{*}$ is hyperbolic. As a divisor in $X\times\mathbb{D}$, $\Gamma=\mathcal{A}+\sum_{i=3}^{m}\mathcal{C}_{i}$, where $\mathcal{A}$ is the development of $\{A_{t}\}_{t\in\mathbb{D}}$ and $\mathcal{C}_{i}:=C_{i}\times\mathbb{D}$. Thus $\Gamma$ is (analytically) reducible at $(x,0)$ if $x\in C_{i}\cap C_{j}$ for some $\{i,j\}\neq\{1,2\}$. By Lemma 4.3, we know that $\mathcal{A}$ is analytically reducible at $(x_{1},0),\cdots,(x_{l},0)$. Thus we have

[TABLE]

Since $\Gamma_{0}=\sum_{i=1}^{m}C_{i}$, we only need to show that $C_{i}\setminus V$ is hyperbolic for any $1\leq i\leq k$. For each $i\in\{1,2\}$, \#C_{i}\cap V=\#\big{(}\{x_{1},\cdots,x_{l}\}\cup(\cup_{j\geq 3}(C_{i}\cap C_{j}))\big{)}=l+(C_{i}\cdot(C-C_{1}-C_{2}))\geq 3. For each $i\geq 3$, \#C_{i}\cap V=\big{(}C_{i}\cdot(C-C_{i})\big{)}\geq 3. Hence $C_{i}\setminus V$ is hyperbolic for each $1\leq i\leq k$. The lemma is proved.∎

Definition 4.5**.**

Let $X$ be a smooth projective surface. Let $C=\sum_{i=1}^{m}C_{i}$ be a reduced normal crossing divisor on $X$. The dual graph $\mathcal{D}(C):=(V,E,r,\mathrm{wt})$ of $C$ is a vertex-weighted multigraph defined as follows:

• •

$V:=\{v_{1},\cdots,v_{m}\}$;

• •

$E:=\cup_{1\leq i<j\leq m}(C_{i}\cap C_{j})$;

• •

For each $p\in E$, $r(p):=\{v_{i},v_{j}\}$ where $\{i,j\}$ is the unique unordered pair with $p\in C_{i}\cap C_{j}$;

• •

For each $v_{i}\in V$, $\mathrm{wt}(v_{i}):=(-K_{X}\cdot C_{i})$.

Lemma 4.6**.**

Let $(X,\Delta)$ be a log smooth projective surface pair with $X$ rational. Let $C=\sum_{i=1}^{m}C_{i}$ be a reduced divisor such that $C+\Delta$ is normal crossing, and each $C_{i}$ is an irreducible rational curve. If the dual graph $\mathcal{D}(C)$ is admissibly contractible, then there exists a successive nodal admissible deformation $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq m-1}$ such that $\Gamma_{t_{m-1}}^{(m-1)}$ is an irreducible rational nodal curve. If, moreover, $X\setminus(C\setminus\Delta)$ is Brody hyperbolic, then $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq m-1}$ can be chosen so that $X\setminus(\Gamma_{t}^{(j)}\setminus\Delta)$ is Brody hyperbolic for any $t\in\mathbb{D}$ and any $1\leq j\leq m-1$.

Proof.

The successive nodal admissible deformation $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq m-1}$ can be constructed inductively by a successive admissible contraction of the dual graph $\mathcal{D}(C)$ using Lemma 4.4. The hyperbolicity part follows from Lemma 4.2 and taking reparametrizations of $t$ if necessary. ∎

Lemma 4.7**.**

Let $(X,\Delta)$ be a log smooth projective surface pair with $X$ rational. Let $C$ be an irreducible rational nodal curve in $X$ such that $\Delta+C$ is normal crossing. If $(-K_{X}\cdot C)\geq 8$ and $\#\mathrm{Sing}(C)\geq 4$, then there exists a successive nodal admissible deformation $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq 2}$ of $C$ such that $\Gamma_{t_{2}}^{(2)}$ is an irreducible smooth hyperbolic curve. If, moreover, $X\setminus(C\setminus\Delta)$ is Brody hyperbolic, then $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq 2}$ can be chosen so that $X\setminus(\Gamma_{t}^{(j)}\setminus\Delta)$ is Brody hyperbolic for any $t\in\mathbb{D}$ and any $1\leq j\leq 2$.

Proof.

Let us pick two nodes $p_{1},p_{2}$ of $C$. Denote by $\sigma:\tilde{X}=\mathrm{Bl}_{p_{1},p_{2}}X\to X$ the blow up of $X$ at $p_{1},p_{2}$. Let $E=E_{1}+E_{2}$ be the reduced exceptional divisor of $\sigma$. Let $\tilde{C}\subset\tilde{X}$ be the strict transform of $C$ under $\sigma$. We claim that $\tilde{C}$ is base point free in $\tilde{X}$.

Since $\tilde{X}$ is rational, we have $H^{1}(\tilde{X},\mathcal{O}_{\tilde{X}})=0$. Thus the claim is equivalent to saying that $\mathcal{O}_{\tilde{C}}(\tilde{C})$ is globally generated. Since $(-K_{X}\cdot C)\geq 8$, we have

[TABLE]

By adjunction we have $(-K_{\tilde{X}}\cdot\tilde{C})=(\tilde{C}^{2})-2\#\mathrm{Sing}(\tilde{C})+2$, so we have $\deg\nu^{*}\mathcal{O}_{\tilde{C}}(\tilde{C})=(\tilde{C}^{2})\geq 2\#\mathrm{Sing}(\tilde{C})+2$, where $\nu:\mathbb{P}^{1}\to\tilde{C}$ is the normalization of $\tilde{C}$. Hence the global sections of $\nu^{*}\mathcal{O}_{\tilde{C}}(\tilde{C}))$ separate any $2\#\mathrm{Sing}(\tilde{C})+1$ points on $\mathbb{P}^{1}$. In particular, this implies that $\mathcal{O}_{\tilde{C}}(\tilde{C})$ is globally generated.

Now we have shown that $\tilde{C}$ is base point free on $\tilde{X}$. By Bertini’s theorem, there exists a holomorphic flat family of irreducible divisors $\{\tilde{\Gamma}_{t}^{(1)}\}_{t\in\mathbb{D}}$ on $\tilde{X}$ such that $\tilde{\Gamma}_{0}^{(1)}=\tilde{C}$ and $(\tilde{X},\tilde{\Gamma}_{t}^{(1)}+E+\sigma^{*}\Delta)$ is log smooth for any $t\in\mathbb{D}\setminus\{0\}$. Let $\Gamma_{t}^{(1)}:=\sigma_{*}\tilde{\Gamma}_{t}^{(1)}$. Since $\tilde{\Gamma}_{0}^{(1)}=\tilde{C}$ intersects $E_{i}$ transversally at two points for any $i\in\{1,2\}$, it is clear that $\tilde{\Gamma}$ has two analytic branches intersecting $E_{i}\times\{0\}$ in different points. Thus $\Gamma^{(1)}$ has two analytic branches at $(p_{i},0)$ for each $i\in\{1,2\}$ which implies that $\Gamma_{0}^{(1),*}\subset C\setminus\{p_{1},p_{2}\}$ is hyperbolic. Besides, $(\tilde{X},\tilde{\Gamma}_{t}^{(1)}+E+\sigma^{*}\Delta)$ being log smooth implies that $\Gamma_{t}^{(1)}$ is nodal at $p_{1},p_{2}$, smooth elsewhere and intersects transversally with $\Delta$ for any $t\in\mathbb{D}\setminus\{0\}$. Hence $\{\Gamma_{t}^{(1)}\}_{t\in\mathbb{D}}$ is a nodal admissible deformation of $C$ with $\Delta+\Gamma_{t}^{(1)}$ being normal crossing for each $t\in\mathbb{D}$.

Now let us fix an arbitrary $t_{1}\in\mathbb{D}\setminus\{0\}$. Since $p_{a}(\tilde{C})=\#\mathrm{Sing}(\tilde{C})=\#\mathrm{Sing}(C)-2\geq 2$, we know that $\Gamma_{t}^{(1)}$ is hyperbolic for any $t\in\mathbb{D}\setminus\{0\}$. As we argued before in showing the base-point-freeness of $\tilde{C}$, $(-K_{X}\cdot C)\geq 8\geq 4$ also implies that $C$ is base point free on $X$. Hence by Bertini’s theorem there exists a holomorphic flat family of irreducible divisors $\{\Gamma_{t}^{(2)}\}_{t\in\mathbb{D}}$ on $X$ such that $\Gamma_{0}^{(2)}=\Gamma_{t_{1}}^{(1)}$ and $(X,\Gamma_{t}^{(2)}+\Delta)$ is log smooth for any $t\in\mathbb{D}\setminus\{0\}$. Besides, $\Gamma_{0}^{(2),*}\subset\Gamma_{0}^{(2)}=\Gamma_{t_{1}}^{(1)}$ is hyperbolic. Hence $\{\Gamma_{t}^{(2)}\}_{t\in\mathbb{D}}$ is a nodal admissible deformation of $\Gamma_{t_{1}}^{(1)}$ such that $\Delta+\Gamma_{t}^{(2)}$ is normal crossing for any $t\in\mathbb{D}$. Besides, $g(\Gamma_{t}^{(2)})=p_{a}(\Gamma_{0}^{(2)})\geq p_{g}(\Gamma_{0}^{(2)})\geq 2$ for any $t\in\mathbb{D}\setminus\{0\}$, hence $\Gamma_{t}^{(2)}$ is hyperbolic for any $t\in\mathbb{D}\setminus\{0\}$. The lemma is proved by taking arbitrary $t_{2}\neq 0$. ∎

Lemma 4.8**.**

Let $(X,\Delta)$ be a log smooth projective surface pair with $X$ rational. Let $C=\sum_{i=1}^{m}C_{i}$ be a reduced divisor on $X$ such that $C+\Delta$ is normal crossing. Assume that each $C_{i}$ is a base-point-free irreducible rational curve with $(-K_{X}\cdot C_{i})\geq 2$, and it intersects with at least four other $C_{j}$’s. If X\setminus\big{(}(\cup_{i=1}^{m}C_{i})\setminus\Delta\big{)} is Brody hyperbolic, then there exists an irreducible smooth curve $C^{\prime}$ linearly equivalent to $\sum_{i=1}^{m}C_{i}$ such that both $C^{\prime}$ and $X\setminus(C^{\prime}\setminus\Delta)$ are Brody hyperbolic.

Proof.

Let $G:=\mathcal{D}(C)$ be the dual graph of $C$. Since each $C_{i}$ is base-point-free, $G$ is completely multipartite. By assumptions, for each vertex $v$ of $G$ we have $\mathrm{rdeg}(v)\geq 4$ and $\mathrm{wt}(v)\geq 2$. Hence Lemma 3.8 implies that $\mathcal{D}(C)$ is admissibly contractible. By Lemma 4.6, there exists a successive nodal admissible deformation $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},1\leq j\leq m-1}$ of $C$ such that $\Gamma_{t_{m-1}}^{(m-1)}$ is an irreducible rational curve and $X\setminus(\Gamma_{t_{m-1}}^{(m-1)}\setminus\Delta)$ is Brody hyperbolic. Since each $C_{i}$ intersects with at least four other $C_{j}^{\prime}s$, we have $m\geq 5$, hence $(-K_{X}\cdot\Gamma_{t_{m-1}}^{(m-1)})=\sum_{i=1}^{m}(-K_{X}\cdot C_{i})\geq 10$. Since $2\sum_{1\leq i<j\leq m}(C_{i}\cdot C_{j})=\sum_{i=1}^{m}\sum_{j\neq i}(C_{i}\cdot C_{j})\geq 4m$, we have

[TABLE]

By applying Lemma 4.7 to $C:=\Gamma_{t_{m-1}}^{(m-1)}$, we know that there exists a successive nodal admissible deformation $\{\Gamma_{t}^{(j)}\}_{t\in\mathbb{D},m\leq j\leq m+1}$ of $\Gamma_{t_{m-1}}^{(m-1)}$ such that $\Gamma_{t_{m+1}}^{(m+1)}$ is an irreducible smooth hyperbolic curve and $X\setminus(\Gamma_{t_{m+1}}^{(m+1)}\setminus\Delta)$ is Brody hyperbolic. It is clear that $\Gamma_{t_{m+1}}^{(m+1)}$ is numerically equivalent to $C$, hence they are linearly equivalent since $X$ is rational. The lemma is proved by taking $C^{\prime}:=\Gamma_{t_{m+1}}^{(m+1)}$. ∎

The following corollary is a generalization of [Zai89, Theorem 3.1] which says that there exists a smooth plane curve of degree $m$ whose complement is Brody hyperbolic for $m\geq 5$.

Corollary 4.9**.**

Let $m\geq 5$ and $d\geq 4$ be integers. Then there exists smooth plane curves $C$ and $S$ of degree $m$ and $d$ respectively, such that $(\mathbb{P}^{2},S+C)$ is log smooth and \mathbb{P}^{2}\setminus\big{(}C\setminus S\big{)} is Brody hyperbolic.

Proof.

Let $\{C_{i}\}_{i=1}^{m}$ be a set of lines in general position in $\mathbb{P}^{2}$. By Corollary 2.3, there exists a smooth plane curve $S$ of degree $d$ such that $(\mathbb{P}^{2},\sum_{i=1}^{m}C_{i}+S)$ is log smooth and \mathbb{P}^{2}\setminus\big{(}(\cup_{i=1}^{m}C_{i})\setminus S\big{)} is Brody hyperbolic. We know that $(C_{i}^{2})=1$ and each $C_{i}$ intersects all $C_{j}$’s whenever $j\neq i$. Since $m-1\geq 4$, the corollary is proved by applying Lemma 4.8 to $(X,\Delta,C_{i}):=(\mathbb{P}^{2},S,C_{i})$. ∎

The following corollary is related to [IT15, 1.2] where they studied hyperbolic imbeddedness of $\mathbb{F}_{0}\setminus C$.

Corollary 4.10**.**

Let $a,b,c,d$ be integers. Then there exists smooth curves $C$ and $S$ in $\mathbb{F}_{N}$ of bidegree $(a,b)$ and $(c,d)$ respectively, such that $(\mathbb{F}_{N},S+C)$ is log smooth and \mathbb{F}_{N}\setminus\big{(}C\setminus S\big{)} is Brody hyperbolic if one of the following is true:

• •

$N=0$* and $a,b,c,d\geq 4$;*

• •

$N\geq 1$, $a,c\geq 3$ and $b,d\geq 4$.

Proof.

Let $\{C_{i}\}_{i=1}^{a+b}$ be a set of curves in $\mathbb{F}_{N}$, such that $C_{i}$ is a general curve in $|F|$ for any $i\leq a$, and $C_{j}$ is a general curve in $|T|$ for any $j>a$. By Corollary 2.4, there exists a smooth curve $S$ of bidegree $(c,d)$ such that $(\mathbb{F}_{N},\sum_{i=1}^{a+b}C_{i}+S)$ is log smooth and \mathbb{F}_{N}\setminus\big{(}(\cup_{i=1}^{a+b}C_{i})\setminus S\big{)} is Brody hyperbolic. We know that $(C_{i}^{2})=0$ for each $i\leq a$ and $(C_{j}^{2})=N\geq 0$ for each $j>a$. From the proof of Corollary 2.4 we know that $C_{i}$ intersects with at least four $C_{j}$’s for each $1\leq i\leq a+b$. Hence the corollary is proved by applying Lemma 4.8 to $(X,\Delta,C_{i}):=(\mathbb{F}_{N},S,C_{i})$. ∎

5. Proofs

Lemma 5.1**.**

Let $X$ be a smooth projective surface. Let $L$ be a line bundle on $X$. Let $n\geq 2$ be an integer. Assume that there exists irreducible divisors $C\in|L|$ and $S\in|L^{\otimes n}|$ satisfying that $(X,S+C)$ is log smooth, and both $C$ and $X\setminus(C\setminus S)$ are Brody hyperbolic. Then there exists a smooth curve $D\in|L^{\otimes n}|$ such that the degree $n$ cyclic cover of $X$ branched along $D$ is Brody hyperbolic.

Proof.

Let $\{S_{t}\}_{t\in\mathbb{P}^{1}}$ be the linear pencil of divisors on $X$ spanned by $S_{0}:=nC$ and $S_{\infty}:=S$. Then the development of $\{S_{t}\}$ is an effective Cartier divisor $\mathcal{S}$ of $X\times\mathbb{P}^{1}$. Since $S$ and $C$ intersect transversally, it is not hard to check in local charts that $\mathcal{S}$ is smooth away from the finite set $(C\cap S)\times\{0\}$. Let $\pi:\mathcal{Y}\to X\times\mathbb{P}^{1}$ be the degree $n$ cyclic cover of $X\times\mathbb{P}^{1}$ branched along $\mathcal{S}$. Then $\mathcal{Y}$ is smooth away from $\pi^{-1}((C\cap S)\times\{0\})$. From the construction it is clear that each fiber $Y_{t}$ of $pr_{2}\circ\pi:\mathcal{Y}\to\mathbb{P}^{1}$ is a degree $n$ cyclic cover of $X$ branched along $S_{t}$. Since $S_{0}=nC$, $Y_{0}$ is the union of $n$ irreducible components $\{Y_{0,i}\}_{i=1}^{n}$ such that $Y_{0,i}\cap Y_{0,j}=\pi^{-1}(C\times\{0\})$ for any $i\neq j$, and $\pi:(Y_{0,i},\pi^{-1}(C\times\{0\}))\to(X,C)\times\{0\}$ is an isomorphism for any $i$.

Assume to the contrary that $Y_{t_{n}}$ is not Brody hyperbolic for a sequence of non-zero complex numbers $(t_{n})$ converging to [math]. Let $\phi_{n}:\mathbb{C}\to Y_{t_{n}}$ be the sequence of entire curves. We may assume that $||\phi_{n}^{\prime}(0)||$ tends to infinity after coordinate changes. By Brody Lemma (e.g. [Duv17]), after choosing a subsequence if necessary, there exists a sequence of reparametrizations $r_{n}:\mathbb{D}_{R_{n}}\to\mathbb{D}$ where $\lim_{n\to\infty}R_{n}=+\infty$ such that $(\phi_{n}\circ r_{n})$ converges to an entire curve $\phi_{\infty}:\mathbb{C}\to Y_{0}$ as $n\to\infty$. Notice that $Y_{0,i}$ is Cartier away from $\pi^{-1}((C\cap S)\times\{0\})$, so Lemma 2.1 implies that $\phi_{\infty}(\mathbb{C})$ is contained in at least one of the $(n+1)$ subsets $\{Y_{0}^{(i)}\}_{i=0}^{n}$ of $Y_{0}$, where

[TABLE]

In particular, at least one of the subsets $\{Y_{0}^{(i)}\}_{i=0}^{n}$ is not Brody hyperbolic. Under the projection $\pi$, it is not hard to see that $Y_{0}^{(0)}\cong C$ and $(Y_{0,i},Y_{0}^{(i)})\cong(X,X\setminus(C\setminus S))$ for any $1\leq i\leq n$. Thus $Y_{0}^{(i)}$ is Brody hyperbolic for any $0\leq i\leq n$, we get a contradiction.

As a result, $Y_{t}$ is Brody hyperbolic for any $t\neq 0$ sufficiently small. Since $S_{t}$ is smooth for general $t$, the lemma is proved by choosing $D:=S_{t}$ for $t\neq 0$ sufficiently small. ∎

5.2****Proof of Theorem 1.2.

Let $\pi:Y\to\mathbb{P}^{2}$ be the double cover of $\mathbb{P}^{2}$ branched along $D$.

For the “only if” part, if $d\leq 4$ then $Y$ is a rational surface; if $d=6$ then $Y$ is a K3 surface. In both cases $Y$ is not Brody hyperbolic. If $d=8$, since Brody hyperbolicity is preserved under small deformation, we may deform $D$ a bit to ensure that there exists a bitangent line $\ell$ of $D$ that meets $D$ transversally in four further points. Hence by Riemann-Hurwitz formula, $\pi^{-1}(\ell)$ is an elliptic curve. Thus $Y$ is never Brody hyperbolic when $d\leq 8$.

For the “if” part, Corollary 4.9 implies that there exist plane curves $C$ and $S$ of degree $d/2$ and $d$ respectively, such that $(\mathbb{P}^{2},C+S)$ is log smooth and $\mathbb{P}^{2}\setminus(C\setminus S)$ is Brody hyperbolic. Since $d/2\geq 5$, $C$ is a smooth curve of genus at least $6$, so it is Brody hyperbolic. Thus applying Lemma 5.1 to $(X,L,n,C,S):=(\mathbb{P}^{2},\mathcal{O}(d/2),2,C,S)$ finishes the proof. ∎

The following theorem is an application of Corollary 4.9 and Lemma 5.1. It also improves [Liu16, Theorem 25].

Theorem 5.3**.**

Let $d\geq 10$ be a composite number. Then there exists a smooth Brody hyperbolic surface of degree $d$ in $\mathbb{P}^{3}$ that is a cyclic cover of $\mathbb{P}^{2}$ under some linear projection.

Proof.

By assumption, $d=d_{1}d_{2}$ for some integers $d_{1}\geq 2$, $d_{2}\geq 5$. Corollary 4.9 implies that there exist plane curves $C$ and $S$ of degree $d_{2}$ and $d$ respectively, such that $(\mathbb{P}^{2},C+S)$ is log smooth and $\mathbb{P}^{2}\setminus(C\setminus S)$ is Brody hyperbolic. Since $d_{2}\geq 5$, $C$ is a smooth curve of genus at least $6$, so it is Brody hyperbolic. Applying Lemma 5.1 to $(X,L,n,C,S):=(\mathbb{P}^{2},\mathcal{O}(d_{2}),d_{1},C,S)$ yields that there exists a smooth plane curve $D$ of degree $d$ such that the degree $d_{1}$ cyclic cover $Y$ of $\mathbb{P}^{2}$ branched along $D$ is Brody hyperbolic. Let $W$ be the degree $d$ cyclic cover of $\mathbb{P}^{2}$ branched along $D$, then there is a natural finite surjective morphism $W\to Y$. Since $Y$ is Brody hyperbolic, $W$ is also Brody hyperbolic. ∎

5.4****Proof of Theorem 1.3.

Let $\pi:Y\to\mathbb{F}_{N}$ be the double cover of $\mathbb{F}_{N}$ branched along $D$.

For the “only if” part, assume to the contrary that $b\leq 6$, then $(D\cdot F)=b\leq 6$. Since $\dim|F|=1$, there exists a curve $F_{0}\in|F|$ such that $F_{0}$ is tangent to $D$ at some point. As a result, $\pi^{-1}(F_{0})$ is a double cover of $\mathbb{P}^{1}$ branched along a non-reduced divisor of degree $\leq 6$. This implies that each irreducible component of $\pi^{-1}(F_{0})$ is either a rational curve or an elliptic curve, so $Y$ is not Brody hyperbolic. We get a contradiction. Hence we must have $b\geq 8$. If $N=0$, then $a\geq 8$ by the symmetry between $F$ and $T$. If $N\geq 1$, assume to the contrary that $a\leq 4$, then $(D\cdot(T-NF))=a\leq 4$. Let $T^{\prime}\subset\mathbb{F}_{N}$ be the unique curve with negative self-intersection number, then $T^{\prime}\sim T-NF$. Hence $(D\cdot T^{\prime})\leq 4$. This implies that each irreducible component of $\pi^{-1}(T^{\prime})$ is either a rational curve or an elliptic curve, so $Y$ is not Brody hyperbolic. We get a contradiction. Therefore, the proof of the “only if” part is completed.

For the “if” part, Corollary 4.10 implies that there exist plane curves $C$ and $S$ of bidegree $(a/2,b/2)$ and $(a,b)$ respectively, such that $(\mathbb{F}_{N},C+S)$ is log smooth and $\mathbb{F}_{N}\setminus(C\setminus S)$ is Brody hyperbolic. If $N=0$, then $a,b\geq 8$ implies that $C$ is a smooth curve of genus at least $9$; if $N\geq 1$, then $a\geq 6$ and $b\geq 8$ implies that $C$ is a smooth curve of genus at least $6N+6$. So $C$ is Brody hyperbolic for every $N\geq 0$. Thus applying Lemma 5.1 to $(X,L,n,C,S):=(\mathbb{F}_{N},\mathcal{O}_{\mathbb{F}_{N}}((a/2)F+(b/2)T),2,C,S)$ finishes the proof. ∎

Remark 5.5**.**

1. (a)

According to [Hor76], the canonical model of a Horikawa surface with even $c_{1}^{2}$ is either a double cover of $\mathbb{P}^{2}$ branched along a degree $8$ or $10$ curve, or a minimal resolution of a double cover of $\mathbb{F}_{N}$ branched along a bidegree $(a,6)$ curve where $a$ has finite choices depending on $N$. Hence the “only if” parts of Theorem 1.2 and 1.3 imply that a Brody hyperbolic Horikawa surface with even $c_{1}^{2}$ has to be a double cover of $\mathbb{P}^{2}$ branched along a degree $10$ curve (in fact one only needs to check algebraic hyperbolicity). However, our deformation method cannot be applied to exhibit other Brody quasi-hyperbolic Horikawa surfaces (i.e. satisfying the Green-Griffiths-Lang conjecture). 2. (b)

Smooth quintic surfaces in $\mathbb{P}^{3}$ are natural examples of Horikawa surfaces with odd $c_{1}^{2}$. It was shown by Xu [Xu94] that a very general quintic surface does not contain any rational or elliptic curve. However, no examples of Brody hyperbolic (even algebraic hyperbolic) quintic surfaces are known so far. Notice that the case of a (very) general quintic surface in $\mathbb{P}^{3}$ corresponds to the case $d=2n-1$ in the Kobayashi Conjecture (cf. [Kob70, Kob98]). 3. (c)

Since Brody hyperbolicity is open in the Euclidean topology (see e.g. [Kob98, 3.11.1]), Theorem 1.2 and 1.3 imply that there exist non-empty open subsets of certain moduli spaces of double covers of $\mathbb{P}^{2}$ or $\mathbb{F}_{N}$ that parametrize Brody hyperbolic ones. Besides, we know that Brody hyperbolicity implies algebraic hyperbolicity, and algebraic hyperbolicity is a very generic property in families. Hence Theorem 1.2 gives an alternative proof of [RR13, Theorem 3.2].