FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

Abstract

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

Introduction

The general structure of smooth noncomplete surfaces $X$ with negative (logarithmic) Kodaira dimension is not fully understood yet. For say smooth quasi-projective surfaces over an algebraically closed field of characteristic zero, it was established by Keel and McKernan [10] that the negativity of the Kodaira dimension is equivalent to the fact that $X$ is generically covered by images of the affine line $\mathbb{A}^{1}$ in the sense that the set of points $x\in X$ with the property that there exists a nonconstant morphism $f:\mathbb{A}^{1}\rightarrow X$ such that $x\in f(\mathbb{A}^{1})$ is dense in $X$ with respect to the Zariski topology. This property, called $\mathbb{A}^{1}$ -uniruledness is equivalent to the existence of an open embedding $X{\hookrightarrow}(\overline{X},B)$ into a complete variety $\overline{X}$ covered by proper rational curves meeting the boundary $B=\overline{X}\setminus X$ in at most one point. In the case where $X$ is smooth and affine, an earlier deep result of Miyanishi–Sugie and Fujita [14] asserts the stronger property that $X$ is $\mathbb{A}^{1}$ -ruled: there exists a Zariski dense open subset $U\subset X$ of the form $U\simeq Z\times \mathbb{A}^{1}$ for a suitable smooth curve $Z$ . Equivalently, $X$ admits a surjective flat morphism $\unicode[STIX]{x1D70C}:X\rightarrow C$ to an open subset $C$ of a smooth projective model $\overline{Z}$ of $Z$ , whose generic fiber is isomorphic to the affine line over the function field of $C$ . Such a morphism $\unicode[STIX]{x1D70C}:X\rightarrow C$ is called an $\mathbb{A}^{1}$ -fibration, and $\unicode[STIX]{x1D70C}$ is said to be of affine type or complete type when the base curve $C$ is affine or complete, respectively.

Smooth $\mathbb{A}^{1}$ -uniruled but not $\mathbb{A}^{1}$ -ruled affine varieties are known to exist in every dimension ${\geqslant}3$  [1]. Many examples of $\mathbb{A}^{1}$ -uniruled affine threefolds can be constructed in the form of flat families $f:X\rightarrow B$ of smooth $\mathbb{A}^{1}$ -ruled affine surfaces parametrized by a smooth base curve $B$ . For instance, the complement $X$ of a smooth cubic surface $S\subset \mathbb{P}_{\mathbb{C}}^{3}$ is the total space of a family $f:X\rightarrow \mathbb{A}^{1}=\text{Spec}(\mathbb{C}[t])$ of $\mathbb{A}^{1}$ -ruled surfaces induced by the restriction of a pencil $\overline{f}:\mathbb{P}^{3}{\dashrightarrow}\mathbb{P}^{1}$ on $\mathbb{P}^{3}$ generated by $S$ and three times a tangent hyperplane $H$ to $S$ whose intersection with $S$ consists of a cuspidal cubic curve. The general fibers of $f$ have negative Kodaira dimension, carrying $\mathbb{A}^{1}$ -fibrations of complete type only, and the failure of $\mathbb{A}^{1}$ -ruledness is intimately related to the fact that the generic fiber $X_{\unicode[STIX]{x1D702}}$ of $f$ , which is a surface defined over the field $K=\mathbb{C}(t)$ , does not admit any $\mathbb{A}^{1}$ -fibration defined over $\mathbb{C}(t)$ . Nevertheless, it was noticed in [3, Theorem 6.1] that one can infer straight from the construction of $f:X\rightarrow \mathbb{A}^{1}$ the existence of a finite base extension $\text{Spec}(L)\rightarrow \text{Spec}(K)$ for which the surface $X_{\unicode[STIX]{x1D702}}\times _{\text{Spec}(K)}\text{Spec}(L)$ carries an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{\unicode[STIX]{x1D702}}\times _{\text{Spec}(K)}\text{Spec}(L)\rightarrow \mathbb{P}_{L}^{1}$ defined over the field $L$ .

A natural question is then to decide whether this phenomenon holds in general for families $f:X\rightarrow B$ of $\mathbb{A}^{1}$ -ruled affine surfaces parameterized by a smooth base curve $B$ , namely, does the existence of $\mathbb{A}^{1}$ -fibrations on the general fibers of $f$ imply the existence of one on the generic fiber of $f$ , possibly after a finite extension of the base $B$ ? A partial positive answer is given by Gurjar et al. [3, Theorem 3.8] under the additional assumption that the general fibers of $f$ carry $\mathbb{A}^{1}$ -fibrations of affine type. The main result in Gurjar et al. [3, Theorem 3.8] is derived from the study of log-deformations of suitable relative normal projective models $\overline{f}:(\overline{X},D)\rightarrow B$ of $X$ over $B$ with appropriate boundaries $D$ . It is established in particular that the structure of the boundary divisor of a well-chosen smooth projective completion of a general closed fiber $X_{s}$ is stable under small deformations, a property which implies in turn, possibly after a finite extension of the base $B$ , the existence of an $\mathbb{A}^{1}$ -fibration of affine type on the generic fiber of $f$ . This log-deformation theoretic approach is also central in the related recent work of Flenner et al. [2] on the classification of normal affine surfaces with $\mathbb{A}^{1}$ -fibrations of affine type up to a certain notion of deformation equivalence, defined for families which admit suitable relative projective models satisfying Kamawata’s axioms of logarithmic deformations of pairs [8]. The fact that the $\mathbb{A}^{1}$ -fibrations under consideration are of affine type plays again a crucial role and, in contrast with the situation considered in [3], the restrictions imposed on the families imply the existence of $\mathbb{A}^{1}$ -fibrations of affine type on their generic fibers.

Our main result (Theorem 7) consists of a generalization of the results in [3] to families $f:X\rightarrow S$ of $\mathbb{A}^{1}$ -ruled surfaces over an arbitrary normal base $S$ , which also includes the case where a general closed fiber $X_{s}$ of $f$ admits $\mathbb{A}^{1}$ -fibrations of complete type only. In particular, we obtain the following positive answer to [3, Conjecture 6.2]:

Theorem.

Let $f:X\rightarrow S$ be a dominant morphism between normal complex algebraic varieties whose general fibers are smooth $\mathbb{A}^{1}$ -ruled affine surfaces. Then there exist a dense open subset $S_{\ast }\subset S$ , a finite étale morphism $T\rightarrow S_{\ast }$ and a normal $T$ -scheme $h:Y\rightarrow T$ such that the induced morphism $f_{T}=\text{p}r_{T}:X_{T}=X\times _{S_{\ast }}T\rightarrow T$ factors as

$$\begin{eqnarray}f_{T}=h\circ \unicode[STIX]{x1D70C}:X_{T}\stackrel{\unicode[STIX]{x1D70C}}{\longrightarrow }Y\stackrel{h}{\longrightarrow }T,\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}:X_{T}\rightarrow Y$ is an $\mathbb{A}^{1}$ -fibration.

In contrast with the log-deformation theoretic strategy used in [3], which involves the study of certain Hilbert schemes of rational curves on well-chosen relative normal projective models $\overline{f}:(\overline{X},B)\rightarrow S$ of $X$ over $S$ , our approach is more elementary, based on the notion of Kodaira dimension [7] adapted to the case of geometrically connected varieties defined over arbitrary base fields of characteristic zero. Indeed, the hypothesis means equivalently that the general fibers of $f$ have negative Kodaira dimension. This property is in turn inherited by the generic fiber of $f$ , which is a smooth affine surface defined over the function field of $S$ , thanks to a standard Lefschetz principle argument. Then we are left with checking that a smooth affine surface $X$ defined over an arbitrary base field $k$ of characteristic zero and with negative Kodaira dimension admits an $\mathbb{A}^{1}$ -fibration, possibly after a suitable finite base extension $\text{Spec}(k_{0})\rightarrow \text{Spec}(k)$ , a fact which ultimately follows from finite type hypotheses and the aforementioned characterization of Miyanishi and Sugie [14].

The article is organized as follows. The first section contains a review of the structure of smooth affine surfaces of negative Kodaira dimension over arbitrary base fields $k$ of characteristic zero. We show in particular that every such surface $X$ admits an $\mathbb{A}^{1}$ -fibration after a finite extension of the base field $k$ , and we give criteria for the existence of $\mathbb{A}^{1}$ -fibrations defined over $k$ . These results are then applied in the second section to the study of deformations $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$ -ruled affine surfaces: after giving the proof of the main result, Theorem 7, we consider in more detail the particular situation where the general fibers of $f:X\rightarrow S$ are irrational. In this case, after shrinking $S$ if necessary, we show that the morphism $f$ actually factors through an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over an $S$ -scheme $h:Y\rightarrow S$ which coincides, up to birational equivalence, with the maximally rationally connected quotient of a relative smooth projective model $\overline{f}:\overline{X}\rightarrow S$ of $X$ over $S$ . The last section is devoted to the case of affine threefolds equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$ -ruled surfaces over a smooth base curve $B$ : we explain in particular how to construct an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ factoring $f$ by means of a relative minimal model program applied to a smooth projective model $\overline{f}:\overline{X}\rightarrow B$ of $X$ over $B$ .

1 $\mathbb{A}^{1}$ -ruledness of affine surfaces over nonclosed field

In what follows, the term $k$ -variety refers to a geometrically integral scheme of finite type over a base field $k$ of characteristic zero. A $k$ -variety $X$ is said to be $k$ -rational if it is birationally isomorphic over $k$ to the projective space $\mathbb{P}_{k}^{n}$ , where $n=\dim _{k}X$ . When no particular base field is indicated, we use simply the term rational to refer to a geometrically rational variety. We call a variety irrational if it is not rational in the previous sense.

1.1 Logarithmic Kodaira dimension

1.1.1. Let $X$ be a smooth algebraic variety defined over a field $k$ of characteristic zero. By virtue of Nagata compactification [15] and Hironaka desingularization [5] theorems, there exists an open immersion $X{\hookrightarrow}(\overline{X},B)$ into a smooth complete algebraic variety $\overline{X}$ with reduced SNC boundary divisor $B=\overline{X}\setminus X$ . The (logarithmic) Kodaira dimension $\unicode[STIX]{x1D705}(X)$ of $X$ is then defined as the Iitaka dimension [6] of the pair $(\overline{X};\unicode[STIX]{x1D714}_{\overline{X}}(\log B))$ , where $\unicode[STIX]{x1D714}_{\overline{X}}(\log B)=(\det \unicode[STIX]{x1D6FA}_{\overline{X}/k}^{1})\otimes {\mathcal{O}}_{\overline{X}}(B)$ . So letting

$$\begin{eqnarray}{\mathcal{R}}(\overline{X},B)=\bigoplus _{m\geqslant 0}H^{0}(\overline{X},\unicode[STIX]{x1D714}_{\overline{X}}(\log B)^{\otimes m}),\end{eqnarray}$$

we have $\unicode[STIX]{x1D705}(X)=\text{tr}.\deg _{k}{\mathcal{R}}(\overline{X},B)-1$ if $H^{0}(\overline{X},\unicode[STIX]{x1D714}_{\overline{X}}(\log B)^{\otimes m})\neq 0$ for sufficiently large $m$ . Otherwise, if $H^{0}(\overline{X},\unicode[STIX]{x1D714}_{\overline{X}}(\log B)^{\otimes m})=0$ for every $m\geqslant 1$ , we set by convention $\unicode[STIX]{x1D705}(X)=-\infty$ and we say that $\unicode[STIX]{x1D705}(X)$ is negative. The so-defined element $\unicode[STIX]{x1D705}(X)\in \{-\infty \}\cup \left\{0,\ldots ,\text{dim}_{k}X\right\}$ is independent of the choice of a smooth complete model $(\overline{X},B)$  [7].

Furthermore, the Kodaira dimension of $X$ is invariant under arbitrary extensions of the base field $k$ . Indeed, given an extension $k\subset k^{\prime }$ , the pair $(\overline{X}_{k^{\prime }},B_{k^{\prime }})$ obtained by the base change $\text{Spec}(k^{\prime })\rightarrow \text{Spec}(k)$ is a smooth complete model of $X_{k^{\prime }}=X\times _{\text{Spec}(k)}\text{Spec}(k^{\prime })$ with reduced SNC boundary $B_{k^{\prime }}$ . Furthermore letting $\unicode[STIX]{x1D70B}:\overline{X}_{k^{\prime }}\rightarrow \overline{X}$ be the corresponding faithfully flat morphism, we have $\unicode[STIX]{x1D714}_{\overline{X}_{k^{\prime }}}(\log B_{k^{\prime }})\simeq \unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{X}(\log B)$ and so ${\mathcal{R}}(X_{k^{\prime }})\simeq {\mathcal{R}}(X)\otimes _{k}k^{\prime }$ by the flat base change theorem [4, Proposition III.9.3]. Thus $\unicode[STIX]{x1D705}(X)=\unicode[STIX]{x1D705}(X_{k^{\prime }})$ .

Example 1. The affine line $\mathbb{A}_{k}^{1}$ is the only smooth geometrically connected noncomplete curve $C$ with negative Kodaira dimension. Indeed, let $\overline{C}$ be a smooth projective model of $C$ and let $\overline{C}_{\overline{k}}$ be the curve obtained by the base change to an algebraic closure $\overline{k}$ of $k$ . Since $C$ is noncomplete, $B=\overline{C}_{\overline{k}}\setminus C_{\overline{k}}$ consists of a finite collection of closed points $p_{1},\ldots ,p_{s}$ , $s\geqslant 1$ , on which the Galois group $\text{Gal}(\overline{k}/k)$ acts by $k$ -automorphisms of $\overline{C}_{\overline{k}}$ . Clearly, $H^{0}(\overline{C}_{\overline{k}},\unicode[STIX]{x1D714}_{\bar{C}_{\overline{k}}}(\log B)^{\otimes m})\neq 0$ unless $\overline{C}_{\overline{k}}\simeq \mathbb{P}_{\overline{k}}^{1}$ and $s=1$ . Since $p_{1}$ is then necessarily $\text{Gal}(\overline{k}/k)$ -invariant,  $\overline{C}\setminus C$ consists of unique $k$ -rational point, showing that $\overline{C}\simeq \mathbb{P}_{k}^{1}$ and $C\simeq \mathbb{A}_{k}^{1}$ .

1.2 Smooth affine surfaces with negative Kodaira dimension

Recall that by virtue of [14], a smooth affine surface $X$ defined over an algebraically closed field of characteristic zero has negative Kodaira dimension if and only if it is $\mathbb{A}^{1}$ -ruled: there exists a Zariski dense open subset $U\subset X$ of the form $U\simeq Z\times \mathbb{A}^{1}$ for a suitable smooth curve $Z$ . In fact, the projection $\text{pr}_{Z}:U\simeq Z\times \mathbb{A}^{1}\rightarrow Z$ always extends to an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow C$ over an open subset $C$ of a smooth projective model $\overline{Z}$ of $Z$ . This characterization admits the following straightforward generalization to arbitrary base fields of characteristic zero:

Theorem 2. Let $X$ be a smooth geometrically connected affine surface defined over a field $k$ of characteristic zero. Then the following are equivalent:

1. (a) The Kodaira dimension $\unicode[STIX]{x1D705}(X)$ of $X$ is negative.

2. (b) For some finite extension $k_{0}$ of $k$ , the surface $X_{k_{0}}$ contains an open subset $U\simeq Z\times \mathbb{A}_{k_{0}}^{1}$ for some smooth curve $Z$ defined over $k_{0}$ .

3. (c) There exist a finite extension $k_{0}$ of $k$ and an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{k_{0}}\rightarrow C_{0}$ over a smooth curve $C_{0}$ defined over $k_{0}$ .

Proof. Clearly (c) implies (b) and (b) implies (a). To show that (a) implies (c), we observe that letting $\overline{k}$ be an algebraic closure of $k$ , we have $\unicode[STIX]{x1D705}(X_{\overline{k}})=\unicode[STIX]{x1D705}(X)<0$ . It then follows from the aforementioned result of Miyanishi and Sugie [14] that $X_{\overline{k}}$ admits an $\mathbb{A}^{1}$ -fibration $q:X_{\overline{k}}\rightarrow C$ over a smooth curve $C$ , with smooth projective model $\overline{C}$ . Since $X_{\overline{k}}$ and $\overline{C}$ are of finite type over $\overline{k}$ , there exists a finite extension $k\subset k_{0}$ such that $q:X_{\overline{k}}\rightarrow \overline{C}$ is obtained from a morphism $\unicode[STIX]{x1D70C}:X_{k_{0}}\rightarrow \overline{C}_{0}$ to a smooth projective curve $\overline{C}_{0}$ defined over $k_{0}$ by the base extension $\text{Spec}(\overline{k})\rightarrow \text{Spec}(k_{0})$ . By virtue of Example 1, $\unicode[STIX]{x1D70C}:X_{k_{0}}\rightarrow \overline{C}_{0}$ is an $\mathbb{A}^{1}$ -fibration.◻

Examples of smooth affine surfaces $X$ of negative Kodaira dimension without any $\mathbb{A}^{1}$ -fibration defined over the base field but admitting $\mathbb{A}^{1}$ -fibrations of complete type after a finite base extension were already constructed in [1]. The following example illustrates the fact that a similar phenomenon occurs for $\mathbb{A}^{1}$ -fibrations of affine type, providing in particular a negative answer to [3, Problem 3.13].

Example 3. Let $B\subset \mathbb{P}_{k}^{2}=\text{Proj}(k[x,y,z])$ be a smooth conic without $k$ -rational point defined by a quadratic form $q=x^{2}+ay^{2}+bz^{2}$ , where $a,b\in k^{\ast }$ , and let $\overline{X}\subset \mathbb{P}_{k}^{3}=\text{Proj}(k[x,y,z,t])$ be the smooth quadric surface defined by the equation $q(x,y,z)-t^{2}=0$ . The complement $X\subset \overline{X}$ of the hyperplane section $\left\{t=0\right\}$ is a $k$ -rational smooth affine surface with $\unicode[STIX]{x1D705}(X)<0$ , which does not admit any $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow C$ over a smooth, affine or projective curve $C$ . Indeed, if such a fibration existed then a smooth projective model of $C$ would be isomorphic to $\mathbb{P}_{k}^{1}$ ; since the fiber of $\unicode[STIX]{x1D70C}$ over a general $k$ -rational point of $C$ is isomorphic to $\mathbb{A}_{k}^{1}$ , its closure in $\overline{X}$ would intersect the boundary $\overline{X}\setminus X\simeq B$ in a unique point, necessarily $k$ -rational, in contradiction with the choice of $B$ .

In contrast, for a suitable finite extension $k\subset k^{\prime }$ , the surface $X_{k^{\prime }}$ becomes isomorphic to the complement of the diagonal in $\overline{X}_{k^{\prime }}\simeq \mathbb{P}_{k^{\prime }}^{1}\times \mathbb{P}_{k^{\prime }}^{1}$ and hence, it admits at least two distinct $\mathbb{A}^{1}$ -fibrations over $\mathbb{P}_{k^{\prime }}^{1}$ , induced by the restriction of the first and second projections from $\overline{X}_{k^{\prime }}$ . Furthermore, since $X_{k^{\prime }}$ is isomorphic to the smooth affine quadric in $\mathbb{A}_{k^{\prime }}^{3}=\text{Spec}(k^{\prime }[u,v,w])$ with equation $uv-w^{2}=1$ , it also admits two distinct $\mathbb{A}^{1}$ -fibrations over $\mathbb{A}_{k^{\prime }}^{1}$ , induced by the restrictions of the projections $\text{pr}_{u}$ and $\text{pr}_{v}$ .

1.3 Existence of $\mathbb{A}^{1}$ -fibrations defined over the base field

1.3.1. The previous example illustrates the general fact that if $X$ is a smooth geometrically connected affine surface with $\unicode[STIX]{x1D705}(X)<0$ which does not admit any $\mathbb{A}^{1}$ -fibration, then there exists a finite extension $k^{\prime }$ of $k$ such that $X_{k^{\prime }}$ admits at least two $\mathbb{A}^{1}$ -fibrations of the same type, either affine or complete, with distinct general fibers. Indeed, by virtue of Theorem 2, there exists a finite extension $k_{0}$ of $k$ such that $X_{k_{0}}$ admits an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{k_{0}}\rightarrow C$ . Let $k^{\prime }$ be the Galois closure of $k_{0}$ in an algebraic closure of $k$ and let $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ be the $\mathbb{A}^{1}$ -fibration deduced from $\unicode[STIX]{x1D70C}$ . If $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ is globally invariant under the action of the Galois group $\text{Gal}(k^{\prime }/k)$ on $X_{k^{\prime }}$ , in the sense that for every $\unicode[STIX]{x1D6F7}\in \text{Gal}(k^{\prime }/k)$ considered as a Galois automorphism of $X_{k^{\prime }}$ there exists a commutative diagram

for a certain $k^{\prime }$ -automorphism $\unicode[STIX]{x1D719}$ of $C_{k^{\prime }}$ , then we would obtain a Galois action of $\text{Gal}(k^{\prime }/k)$ on $C_{k^{\prime }}$ for which $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ becomes an equivariant morphism. Since $C_{k^{\prime }}$ is quasi-projective and $\unicode[STIX]{x1D70C}_{k}^{\prime }$ is affine, it would follow from Galois descent that there exist a curve $\tilde{C}$ defined over $k$ and a morphism $q:X\rightarrow \tilde{C}$ defined over $k$ such that $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ is obtained from $q$ by the base change $\text{Spec}(k^{\prime })\rightarrow \text{Spec}(k)$ . Since by virtue of Example 1 the affine line does not have any nontrivial form, the generic fiber of $q$ would be isomorphic to the affine line over the field of rational functions of $\tilde{C}$ and so, $q:X\rightarrow \tilde{C}$ would be an $\mathbb{A}^{1}$ -fibration defined over $k$ , in contradiction with our hypothesis. So there exists at least an element $\unicode[STIX]{x1D6F7}\in \text{Gal}(k^{\prime }/k)$ considered as a $k$ -automorphism of $X_{k^{\prime }}$ such that the $\mathbb{A}^{1}$ -fibrations $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ and $\unicode[STIX]{x1D70C}_{k^{\prime }}\circ \unicode[STIX]{x1D719}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ have distinct general fibers.

Arguing backward, we obtain the following criterion:

Proposition 4. Let $X$ be a smooth geometrically connected affine surface with $\unicode[STIX]{x1D705}(X)<0$ . If there exists a finite Galois extension $k^{\prime }$ of $k$ such that $X_{k^{\prime }}$ admits a unique $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}^{\prime }:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ up to composition by automorphisms of $C_{k^{\prime }}$ , then $\unicode[STIX]{x1D70C}^{\prime }:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ is obtained by base extension from an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow C$ defined over $k$ .

Corollary 5. A smooth geometrically connected irrational affine surface $X$ has negative Kodaira dimension if and only if it admits an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow C$ over a smooth irrational curve $C$ defined over the base field $k$ . Furthermore for every extension $k^{\prime }$ of $k$ , $\unicode[STIX]{x1D70C}_{k^{\prime }}:X_{k^{\prime }}\rightarrow C_{k^{\prime }}$ is the unique $\mathbb{A}^{1}$ -fibration on $X_{k^{\prime }}$ up to composition by automorphisms of $C_{k^{\prime }}$ .

Proof. Uniqueness is clear since otherwise $C_{k^{\prime }}$ would be dominated by a general fiber of another $\mathbb{A}^{1}$ -fibration on $X_{k^{\prime }}$ , and hence would be rational, implying in turn the rationality of $X$ . By virtue of Theorem 2, there exist a finite Galois extension $k^{\prime }$ of $k$ and an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}^{\prime }:X_{k^{\prime }}\rightarrow C^{\prime }$ over a smooth curve $C^{\prime }$ . The latter is irrational as $X$ is irrational, which implies that $\unicode[STIX]{x1D70C}^{\prime }:X_{k^{\prime }}\rightarrow C^{\prime }$ is the unique $\mathbb{A}^{1}$ -fibration on $X_{k^{\prime }}$ . So $\unicode[STIX]{x1D70C}^{\prime }$ descend to an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow C$ over a smooth irrational curve $C$ defined over $k$ .◻

The following example shows that the irrationality hypothesis cannot be weakened to the property that $X$ is geometrically rational but not $k$ -rational.

Example 6. Let $a\in \mathbb{Q}$ be a rational number which is not a cube and let $S=S_{a}\subset \mathbb{P}_{\mathbb{Q}}^{3}=\text{Proj}_{\mathbb{Q}}(\mathbb{Q}[x,y,z,t])$ be the smooth cubic surface defined by the equation $x^{3}+y^{3}+z^{3}+at^{3}=0$ . All lines on $S$ are defined over the splitting field $K$ of the polynomial $u^{3}+a\in \mathbb{Q}[u]$ , and one checks by direct computation that no orbit of the action of the Galois group $\text{Gal}(K/\mathbb{Q})\simeq \mathfrak{S}_{3}$ on $S_{K}$ consists of a disjoint union of such lines. It follows that the Picard number $\unicode[STIX]{x1D70C}(S)$ of $S$ is equal to $1$ , hence by Segree–Manin Theorem that $S$ is rational but not $\mathbb{Q}$ -rational (see e.g., [12, Exercise 2.18 and Theorem 2.1]). Let $H=\left\{x+y=0\right\}\subset \mathbb{P}_{\mathbb{Q}}^{3}$ be the tangent hyperplane to $S$ at the point $p=\left[1:-1:0:0\right]$ and let $X=S\setminus (H\cap S)$ . So $X$ is a smooth affine surface defined over $\mathbb{Q}$ , and since the intersection of $H_{\mathbb{C}}$ with $S_{\mathbb{C}}$ consists of three lines meeting at the Eckardt point $p$ , one checks easily that $\unicode[STIX]{x1D705}(X)=\unicode[STIX]{x1D705}(X_{\mathbb{C}})=-\infty$ . Thus $X_{\mathbb{C}}$ admits an $\mathbb{A}^{1}$ -fibration by virtue of [14], but we claim that $X$ does not admit any such fibration defined over $\mathbb{Q}$ . Indeed, suppose on the contrary that $\unicode[STIX]{x1D70B}:X\rightarrow C$ is an $\mathbb{A}^{1}$ -fibration over a smooth curve defined over $\mathbb{Q}$ . Since $C$ is geometrically rational and contains a $\mathbb{Q}$ -rational point, for instance the image by $\unicode[STIX]{x1D70B}$ of the point $[0:-1:1:0]\in X(\mathbb{Q})$ , it is $\mathbb{Q}$ -rational. But then $X$ whence $S$ would be $\mathbb{Q}$ -rational, a contradiction.

2 Families of $\mathbb{A}^{1}$ -ruled affine surfaces

2.1 Existence of étale $\mathbb{A}^{1}$ -cylinders

This subsection is devoted to the proof of the following:

Theorem 7. Let $X$ and $S$ be normal algebraic varieties defined over a field $k$ of infinite transcendence degree over $\mathbb{Q}$ , and let $f:X\rightarrow S$ be a dominant affine morphism with the property that for a general closed point $s\in S$ , the fiber $X_{s}$ is a smooth geometrically connected affine surface with negative Kodaira dimension. Then there exist an open subset $S_{\ast }\subset S$ , a finite étale morphism $T\rightarrow S_{\ast }$ and a normal $T$ -scheme $h:Y\rightarrow T$ such that $f_{T}=\text{p}r_{T}:X_{T}=X\times _{S_{\ast }}T\rightarrow T$ factors as

$$\begin{eqnarray}f_{T}=h\circ \unicode[STIX]{x1D70C}:X_{T}\stackrel{\unicode[STIX]{x1D70C}}{\longrightarrow }Y\stackrel{h}{\longrightarrow }T\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}:X_{T}\rightarrow Y$ is an $\mathbb{A}^{1}$ -fibration.

Proof. Shrinking $S$ if necessary, we may assume that $S$ is affine, that $f:X\rightarrow S$ is smooth and that $\unicode[STIX]{x1D705}(X_{s})<0$ for every closed point $s\in S$ . It is enough to show that the fiber $X_{\unicode[STIX]{x1D702}}$ of $f$ over the generic point $\unicode[STIX]{x1D702}$ of $S$ is geometrically connected, with negative Kodaira dimension. Indeed, if so, then by Theorem 2 above, there exist a finite extension $L$ of $K=\text{Frac}(\unicode[STIX]{x1D6E4}(S,{\mathcal{O}}_{S}))$ and an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{\unicode[STIX]{x1D702}}\times _{\text{Spec}(K)}\text{Spec}(L)\rightarrow C$ onto a smooth curve $C$ defined over $L$ . Letting $T$ be the normalization of $S$ in $L$ and shrinking $T$ again if necessary, we obtain a finite étale morphism $T\rightarrow S$ such that the generic fiber of $\text{pr}_{T}:X_{T}\rightarrow T$ is isomorphic to the $\mathbb{A}^{1}$ -fibered surface $\unicode[STIX]{x1D70C}:X_{\unicode[STIX]{x1D702}}\times _{\text{Spec}(K)}\text{Spec}(L)\rightarrow C$ and then the assertion follows from Lemma 8 below.

The properties of being geometrically connectedness and having negative Kodaira dimension are invariant under finite algebraic extensions of the base field. So letting $\overline{k}$ be an algebraic closure of $k$ , it is enough to show that the generic fiber of the induced morphism $f_{\overline{k}}:X_{\overline{k}}\rightarrow S_{\overline{k}}$ is geometrically connected, of negative Kodaira dimension. We may thus assume from now on that $k$ is algebraically closed. Since $X$ and $S$ are affine and of finite type over $k$ , there exist a subfield $k_{0}$ of $k$ of finite transcendence degree over $\mathbb{Q}$ , and a smooth morphism $f_{0}:X_{0}\rightarrow S_{0}$ of $k_{0}$ -varieties such that $f:X\rightarrow S$ is obtained from $f_{0}:X_{0}\rightarrow S_{0}$ by the base extension $\text{Spec}\left(k\right)\rightarrow \text{Spec}(k_{0})$ . The field $K_{0}=\text{Frac}(\unicode[STIX]{x1D6E4}(S_{0},{\mathcal{O}}_{S_{0}}))$ has finite transcendence degree over $\mathbb{Q}$ and hence, it admits a $k_{0}$ -embedding $\unicode[STIX]{x1D709}:K_{0}{\hookrightarrow}k$ . Letting $(X_{0})_{\unicode[STIX]{x1D702}_{0}}$ be the fiber of $f_{0}$ over the generic point $\unicode[STIX]{x1D702}_{0}:\text{Spec}(K_{0})\rightarrow S_{0}$ of $S_{0}$ , the composition $\unicode[STIX]{x1D6E4}(S_{0},{\mathcal{O}}_{S_{0}}){\hookrightarrow}K_{0}{\hookrightarrow}k$ induces a $k$ -homomorphism $\unicode[STIX]{x1D6E4}(S_{0},{\mathcal{O}}_{S_{0}})\otimes _{k_{0}}k\rightarrow k$ defining a closed point $s:\text{Spec}(k)\rightarrow \text{Spec}(\unicode[STIX]{x1D6E4}(S_{0},{\mathcal{O}}_{S_{0}})\otimes _{k_{0}}k)=S$ of $S$ for which we obtain the following commutative diagram

The bottom square of the cube being Cartesian by construction, we deduce that

$$\begin{eqnarray}(X_{0})_{\unicode[STIX]{x1D702}_{0}}\times _{\text{Spec}(K_{0})}\text{Spec}(k)\simeq X_{0}\times _{S_{0}}\text{Spec}(k)\simeq X\times _{S}\text{Spec}(k)=X_{s}.\end{eqnarray}$$

Since by assumption, $X_{s}$ is geometrically connected with $\unicode[STIX]{x1D705}(X_{s})<0$ , we conclude that $(X_{0})_{\unicode[STIX]{x1D702}_{0}}$ is geometrically connected and has negative Kodaira dimension. This implies in turn that $X_{\unicode[STIX]{x1D702}}$ is geometrically connected and that $\unicode[STIX]{x1D705}(X_{\unicode[STIX]{x1D702}})<0$ as desired.◻

In the proof of the above theorem, we used the following lemma:

Lemma 8. Let $f:X\rightarrow S$ be a dominant affine morphism between normal varieties defined over a field $k$ of characteristic zero. Then the following are equivalent:

1. (a) The generic fiber $X_{\unicode[STIX]{x1D702}}$ of $f$ admits an $\mathbb{A}^{1}$ -fibration $q:X_{\unicode[STIX]{x1D702}}\rightarrow C$ over a smooth curve $C$ defined over the fraction field $K$ of $S$ .

2. (b) There exist an open subset $S_{\ast }$ of $S$ and a normal $S_{\ast }$ -scheme $h:Y\rightarrow S_{\ast }$ of relative dimension $1$ such that the restriction of $f$ to $V=f^{-1}(S_{\ast })$ factors as $f\mid _{V}=h\circ \unicode[STIX]{x1D70C}:V\rightarrow Y\rightarrow S_{\ast }$ where $\unicode[STIX]{x1D70C}:V\rightarrow Y$ is an $\mathbb{A}^{1}$ -fibration.

Proof. If (b) holds then letting $L$ be the fraction field of $Y$ , we have a commutative diagram

in which each square is Cartesian. It follows that $h_{\unicode[STIX]{x1D702}}:C\rightarrow \text{Spec}(K)$ is a normal whence smooth curve defined over $K$ and that $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D702}}:X_{\unicode[STIX]{x1D702}}\rightarrow C$ is an $\mathbb{A}^{1}$ -fibration. Conversely, suppose that $X_{\unicode[STIX]{x1D702}}$ admits an $\mathbb{A}^{1}$ -fibration $q:X_{\unicode[STIX]{x1D702}}\rightarrow C$ and let $\overline{C}$ be a smooth projective model of $C$ over $K$ . Then there exist an open subset $S_{0}$ of $S$ and a projective $S_{0}$ -scheme $h:Y\rightarrow S_{0}$ whose generic fiber is isomorphic to $\overline{C}$ . After shrinking $S_{0}$ if necessary, the rational map $\unicode[STIX]{x1D70C}:V{\dashrightarrow}Y$ of $S_{0}$ -schemes induced by $q$ becomes a morphism and we obtain a factorization $f\mid _{V}=h\circ \unicode[STIX]{x1D70C}$ . By construction, the generic fiber $V_{\unicode[STIX]{x1D709}}$ of $\unicode[STIX]{x1D70C}:V\rightarrow Y$ is isomorphic to

$$\begin{eqnarray}V\times _{Y}\text{Spec}(L)\simeq (V\times _{Y}C)\times _{C}\text{Spec}(L)\simeq X_{\unicode[STIX]{x1D702}}\times _{C}\text{Spec}(L)\simeq \mathbb{A}_{L}^{1}\end{eqnarray}$$

since $V\times _{Y}C\simeq V_{\unicode[STIX]{x1D702}}\simeq X_{\unicode[STIX]{x1D702}}$ and $\unicode[STIX]{x1D70C}:X_{\unicode[STIX]{x1D702}}\rightarrow C{\hookrightarrow}\overline{C}$ is an $\mathbb{A}^{1}$ -fibration. So $\unicode[STIX]{x1D70C}:V\rightarrow Y$ is an $\mathbb{A}^{1}$ -fibration.◻

Example 9. Let $R=\mathbb{C}[s^{\pm 1},t^{\pm 1}]$ , $S=\text{Spec}(R)$ and let $D$ be the relatively ample divisor in $\mathbb{P}_{S}^{2}=\text{Proj}_{R}(R[x,y,z])$ defined by the equation $x^{2}+sy^{2}+tz^{2}=0$ . The restriction $h:X=\mathbb{P}_{S}^{2}\setminus D\rightarrow S$ of the structure morphism defines a family of smooth affine surfaces with the property that for every closed point $s\in S$ , $X_{s}$ is isomorphic to the complement in $\mathbb{P}_{\mathbb{C}}^{2}$ of the smooth conic $D_{s}$ . In particular, $X_{s}$ admits a continuum of pairwise distinct $\mathbb{A}^{1}$ -fibrations $X_{s}\rightarrow \mathbb{A}_{\mathbb{C}}^{1}$ , induced by the restrictions to $X_{s}$ of the rational pencils on $\mathbb{P}_{\mathbb{C}}^{2}$ generated by $D_{s}$ and twice its tangent line at an arbitrary closed point $p_{s}\in D_{s}$ . On the other hand, the fiber of $D$ over the generic point $\unicode[STIX]{x1D702}$ of $S$ is a conic without $\mathbb{C}(s,t)$ -rational point in $\mathbb{P}_{\mathbb{C}(s,t)}^{2}$ and hence, we conclude by a similar argument as in Example 3 that $X_{\unicode[STIX]{x1D702}}$ does not admit any $\mathbb{A}^{1}$ -fibration defined over $\mathbb{C}(s,t)$ . Therefore there is no open subset $S_{\ast }$ of $S$ over which $h$ can be factored through an $\mathbb{A}^{1}$ -fibration.

2.2 Deformations of irrational $\mathbb{A}^{1}$ -ruled affine surfaces

In this subsection, we consider the particular situation of a flat family $f:X\rightarrow S$ over a normal variety $S$ whose general fibers are irrational $\mathbb{A}^{1}$ -ruled affine surfaces. A combination of Corollary 5 and Theorem 7 above implies that if $f:X\rightarrow S$ is smooth and defined over a field of infinite transcendence degree over $\mathbb{Q}$ , then the generic fiber $X_{\unicode[STIX]{x1D702}}$ of $f$ is $\mathbb{A}^{1}$ -ruled. Equivalently, there exist an open subset $S_{\ast }\subset S$ and a normal $S_{\ast }$ -scheme $h:Y\rightarrow S_{\ast }$ such that the restriction of $f$ to $X_{\ast }=X\times _{S}S_{\ast }$ factors through an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{\ast }\rightarrow Y$ (see Lemma 8). The restriction of $\unicode[STIX]{x1D70C}$ to the fiber of $f$ over a general closed point $s\in S_{0}$ is an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow Y_{s}$ over the normal, whence smooth, curve $Y_{s}$ . Since $X_{s}$ is irrational, $Y_{s}$ is irrational, and so $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow Y_{s}$ is the unique $\mathbb{A}^{1}$ -fibration on $X_{s}$ up to composition by automorphisms of $Y_{s}$ . So in this case, we can identify $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow Y_{s}$ with the Maximally Rationally Connected fibration (MRC-fibration) $\unicode[STIX]{x1D719}:\overline{X}_{s}{\dashrightarrow}Y_{s}$ of a smooth projective model $\overline{X}_{s}$ of $X_{s}$ in the sense of [11, IV.5]: recall that $\unicode[STIX]{x1D719}$ is unique, characterized by the property that its general fibers are rationally connected and that for a very general point $y\in Y_{s}$ any rational curve in $\overline{X}_{s}$ which meets $\overline{X}_{y}$ is actually contained in $\overline{X}_{y}$ . The $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X_{\ast }\rightarrow Y$ can therefore be re-interpreted as being the MRC-fibration of a relative smooth projective model $\overline{X}$ of $X$ over $S$ .

Reversing the argument, general existence and uniqueness results for MRC-fibrations allow actually to get rid of the smoothness hypothesis of a general fiber of $f:X\rightarrow S$ and to extend the conclusion of Theorem 7 to arbitrary base fields of characteristic zero. Namely, we obtain the following characterization:

Theorem 10. Let $X$ and $S$ be normal varieties defined over a field $k$ of characteristic zero and let $f:X\rightarrow S$ be a dominant affine morphism with the property that for a general closed point $s\in S$ , the fiber $X_{s}$ is an irrational $\mathbb{A}^{1}$ -ruled surface. Then there exist a dense open subset $S_{\ast }$ of $S$ and a normal $S_{\ast }$ -scheme $h:Y\rightarrow S_{\ast }$ such that the restriction of $f$ to $X_{\ast }=X\times _{S}S_{\ast }$ factors as

$$\begin{eqnarray}f\mid _{X_{\ast }}=h\circ \unicode[STIX]{x1D70C}:X_{\ast }\stackrel{\unicode[STIX]{x1D70C}}{\longrightarrow }Y\stackrel{h}{\longrightarrow }S_{\ast }\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}:X_{\ast }\rightarrow Y$ is an $\mathbb{A}^{1}$ -fibration.

Proof. Shrinking $S$ if necessary, we may assume that it is smooth and that for every closed point $s\in S$ , $X_{s}$ is irrational and $\mathbb{A}^{1}$ -ruled, hence carrying a unique $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{s}:X_{s}\rightarrow C_{s}$ over an irrational normal curve $C_{s}$ . Since $f:X\rightarrow S$ is affine, there exist a normal projective $S$ -scheme $\overline{X}\rightarrow S$ and an open embedding $X{\hookrightarrow}\overline{X}$ of schemes over $S$ . Letting $W\rightarrow \overline{X}$ be a resolution of the singularities of $\overline{X}$ , hence in particular of those of $X$ , we may assume up to shrinking $S$ again if necessary that $W\rightarrow S$ is a smooth morphism. We let $j:X{\dashrightarrow}W$ be the birational map of $S$ -schemes induced by the embedding $X{\hookrightarrow}\overline{X}$ . By virtue of [11, Theorem 5.9], there exist an open subset $W^{\prime }$ of $W$ , an $S$ -scheme $h:Z\rightarrow S$ and a proper morphism $\overline{q}:W^{\prime }\rightarrow Z$ such that for every $s\in S$ , the induced rational map $\overline{q}_{s}:W_{s}{\dashrightarrow}Z_{s}$ is the MRC-fibration for $W_{s}$ . On the other hand, since $W_{s}$ is a smooth projective model of $X_{s}$ , the induced rational map $\unicode[STIX]{x1D70B}_{s}:\overline{X}_{s}{\dashrightarrow}C_{s}$ is the MRC-fibration for $W_{s}$ . Consequently, for a general closed point $z\in Z$ with $h(z)=s$ , the fiber $W_{z}$ of $\overline{q}_{s}$ , which is an irreducible proper rational curve contained in $W_{s}$ , must coincide with the closure of the image by $j$ of a general closed fiber of $\unicode[STIX]{x1D70B}_{s}$ . The latter being isomorphic to the affine line $\mathbb{A}_{\unicode[STIX]{x1D705}}^{1}$ over the residue field $\unicode[STIX]{x1D705}$ of the corresponding point of $C_{s}$ , we conclude that there exists an affine open subset $U$ of $X$ on which the composition $\overline{q}\circ j:U\rightarrow Z$ is a well-defined morphism with general closed fibers isomorphic to affine lines over the corresponding residue fields. So $\overline{q}\circ j:U\rightarrow Z$ is an $\mathbb{A}^{1}$ -fibration by virtue of [9]. The generic fiber of $f:X\rightarrow S$ is thus $\mathbb{A}^{1}$ -ruled and the assertion follows from Lemma 8 above.◻

Example 11. Let $h:Y\rightarrow S$ be a smooth family of complex projective curves of genus $g\geqslant 2$ over a normal affine base $S$ and let ${\mathcal{T}}_{Y/S}$ be the relative tangent sheaf of $h$ . Since by Riemman–Roch $H^{0}(Y_{s},{\mathcal{T}}_{Y/S,s})=0$ and $\dim H^{1}(Y_{s},{\mathcal{T}}_{Y/S,s})=3g-3$ for every point $s\in S$ , $h_{\ast }{\mathcal{T}}_{Y/S,s}=0$ , $R^{1}h_{\ast }{\mathcal{T}}_{Y/S}$ is locally free of rank $3g-3$  [4, Corollary III.12.9] and so, $H^{1}(Y,{\mathcal{T}}_{Y/S})\simeq H^{0}(S,R^{1}h_{\ast }{\mathcal{T}}_{Y/S})$ by the Leray spectral sequence. Replacing $S$ by an open subset, we may assume that $R^{1}h_{\ast }{\mathcal{T}}_{Y/S}$ admits a nowhere vanishing global section $\unicode[STIX]{x1D70E}$ . Via the isomorphism $H^{1}(Y,{\mathcal{T}}_{Y/S})\simeq \text{Ext}_{Y}^{1}({\mathcal{O}}_{Y},{\mathcal{T}}_{Y/S})$ , we may interpret this section as the class of a nontrivial extension $0\rightarrow {\mathcal{T}}_{Y/S}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{O}}_{Y}\rightarrow 0$ of locally free sheaves over $Y$ . The inclusion ${\mathcal{T}}_{Y/S}\rightarrow {\mathcal{E}}$ defines a section $D$ of the locally trivial $\mathbb{P}^{1}$ -bundle $\overline{\unicode[STIX]{x1D70C}}:\overline{X}=\text{Proj}(\text{Sym}_{{\mathcal{O}}_{Y}}{\mathcal{E}}^{\vee })\rightarrow Y$ and the nonvanishing of $\unicode[STIX]{x1D70E}$ guarantees that $D$ is the support of an $S$ -ample divisor. Indeed the $S$ -ampleness of $D$ is equivalent to the property that for every $s\in S$ the induced section $D_{s}$ of the $\mathbb{P}^{1}$ -bundle $\overline{\unicode[STIX]{x1D70C}}_{s}:\overline{X}_{s}\rightarrow Y_{s}$ over the smooth projective curve $Y_{s}$ is ample. Since by construction, $\overline{\unicode[STIX]{x1D70C}}_{s}\mid _{\overline{X}_{s}\setminus D_{s}}:\overline{X}_{s}\setminus D_{s}\rightarrow Y_{s}$ is a nontrivial torsor under the line bundle $\text{Spec}(\text{Sym}{\mathcal{T}}_{Y_{s}}^{\vee })\rightarrow Y_{s}$ , it follows that $D_{s}$ intersects positively every section $D$ of $\overline{\unicode[STIX]{x1D70C}}_{s}$ except maybe $D_{s}$ itself. On the other hand, we have $(D_{s}^{2})=-\text{deg}\,{\mathcal{T}}_{Y_{s}}=2g(Y_{s})-2>0$ , and so the ampleness of $D_{s}$ follows from the Nakai–Moishezon criterion and the description of the cone effective cycles on an irrational projective ruled surface given in [4, Propositions 2.20–2.21].

Letting $X=\overline{X}\setminus D$ , we obtain a smooth family

$$\begin{eqnarray}f=g\circ \overline{\unicode[STIX]{x1D70C}}\mid _{X}:X\stackrel{\overline{\unicode[STIX]{x1D70C}}\mid _{X}}{\longrightarrow }Y\stackrel{h}{\rightarrow }S\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D70C}}\mid _{X}:X\rightarrow Y$ is a nontrivial, locally trivial, $\mathbb{A}^{1}$ -bundle such that for every $s\in S$ , $X_{s}$ is an affine surface with an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow Y_{s}$ of complete type.

In contrast with the previous example, the following proposition shows in particular that if the total space of a family of irrational $\mathbb{A}^{1}$ -ruled affine surfaces $f:X\rightarrow S$ has finite divisor class group, then the induced $\mathbb{A}^{1}$ -fibration on a general fiber of $f:X\rightarrow S$ is necessarily of affine type.

Proposition 12. Let $X$ be a geometrically integral normal affine variety with finite divisor class group $\text{Cl}(X)$ and let $f:X\rightarrow S$ be a dominant affine morphism to a normal variety $S$ with the property that for a general closed point $s\in S$ , the fiber $X_{s}$ is an irrational $\mathbb{A}^{1}$ -ruled surface, say with unique $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{s}:X_{s}\rightarrow C_{s}$ . Then there exists an effective action of the additive group scheme $\mathbb{G}_{a,S}$ on $X$ such that for a general closed point $s\in S$ , the $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{s}:X_{s}\rightarrow C_{s}$ factors through the algebraic quotient $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow X_{s}/\!/\mathbb{G}_{a,s}=\text{Spec}(\unicode[STIX]{x1D6E4}(X_{s},{\mathcal{O}}_{X_{s}})^{\mathbb{G}_{a,s}})$ .

Proof. Let $f\mid _{X_{\ast }}=h\circ \unicode[STIX]{x1D70C}:X_{\ast }\stackrel{\unicode[STIX]{x1D70C}}{\longrightarrow }Y\stackrel{h}{\longrightarrow }S_{\ast }$ be as in Theorem 10. Since $\unicode[STIX]{x1D70C}$ is an $\mathbb{A}^{1}$ -fibration, there exists an affine open subset $U\subset Y$ such that $\unicode[STIX]{x1D70C}^{-1}(U)\simeq U\times \mathbb{A}^{1}$ as schemes over $U$ . Since $\unicode[STIX]{x1D70C}^{-1}(U)$ is affine, its complement in $X$ is of pure codimension $1$ , and the finiteness of $\text{Cl}(X)$ implies that it is actually the support of an effective principal divisor $\text{div}_{X}(a)$ for some $a\in \unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})$ . Let $\unicode[STIX]{x2202}_{0}$ be the locally nilpotent derivation of $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70C}^{-1}(U),{\mathcal{O}}_{X})\simeq \unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})_{a}$ corresponding to the $\mathbb{G}_{a,U}$ -action by translations on the second factor. Since $a$ is invertible in $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70C}^{-1}(U),{\mathcal{O}}_{X})$ , it belongs to the kernel of $\unicode[STIX]{x2202}_{0}$ , and the finite generation of $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})$ guarantees that for a suitably chosen $n\geqslant 0$ , $a^{n}\unicode[STIX]{x2202}_{0}$ is a locally nilpotent derivation $\unicode[STIX]{x2202}$ of $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})$ . By construction, the restriction of $f$ to the dense open subset $\unicode[STIX]{x1D70C}^{-1}(U)$ of $X$ is invariant under the corresponding $\mathbb{G}_{a}$ -action, and so $f:X\rightarrow S$ is $\mathbb{G}_{a}$ -invariant. For a general closed point $s\in S$ , the induced $\mathbb{G}_{a}$ -action on $X_{s}$ is nontrivial, and its algebraic quotient $\unicode[STIX]{x1D70C}_{s}:X_{s}\rightarrow X_{s}/\!/\mathbb{G}_{a}=\text{Spec}(\unicode[STIX]{x1D6E4}(X_{s},{\mathcal{O}}_{X_{s}})^{\mathbb{G}_{a}})$ is a surjective $\mathbb{A}^{1}$ -fibration onto a normal affine curve $X_{s}/\!/\mathbb{G}_{a}$ . Since $C_{s}$ is irrational, the general fibers of $\unicode[STIX]{x1D70C}_{s}$ and $\unicode[STIX]{x1D70B}_{s}$ must coincide. It follows that $\unicode[STIX]{x1D70B}_{s}$ is $\mathbb{G}_{a}$ -invariant, whence factors through $\unicode[STIX]{x1D70C}_{s}$ .◻

3 Affine threefolds fibered in irrational $\mathbb{A}^{1}$ -ruled surfaces

In this section, we consider in more detail the case of normal complex affine threefolds $X$ admitting a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$ -ruled surfaces, over a smooth curve $B$ . We explain how to derive the variety $h:Y\rightarrow B$ for which $f$ factors through an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ from a relative minimal model program applied to a suitable projective model of $X$ over $B$ . In the case where the divisor class group of $X$ is finite, we provide a complete classification of such fibrations in terms of additive group actions on $X$ .

3.1 $\mathbb{A}^{1}$ -cylinders via relative minimal model program

Let $X$ be a normal complex affine threefold and let $f:X\rightarrow B$ be a flat morphism onto a smooth curve $B$ with the property that a general closed fiber $X_{b}$ of $f$ is an irreducible irrational $\mathbb{A}^{1}$ -ruled surface. We let $\overline{f}:W\rightarrow B$ be a smooth projective model of $X$ over $B$ obtained from an arbitrary normal relative projective completion $X{\hookrightarrow}\overline{X}$ of $X$ over $B$ by resolving the singularities. We let $j:X{\dashrightarrow}W$ be the birational map induced by the open immersion $X{\hookrightarrow}\overline{X}$ .

By applying a minimal model program for $W$ over $B$ , we obtain a sequence of birational $B$ -maps

$$\begin{eqnarray}W=W_{0}\stackrel{\unicode[STIX]{x1D719}_{1}}{{\dashrightarrow}}W_{1}\stackrel{\unicode[STIX]{x1D719}_{2}}{{\dashrightarrow}}W_{2}{\dashrightarrow}\cdots {\dashrightarrow}W_{\ell -1}\stackrel{\unicode[STIX]{x1D719}_{\ell }}{{\dashrightarrow}}W_{\ell }=W^{\prime },\end{eqnarray}$$

between $B$ -schemes $\overline{f}_{i}:W_{i}\rightarrow B$ , where $\unicode[STIX]{x1D719}_{i}:W_{i}{\dashrightarrow}W_{i+1}$ is either a divisorial contraction or a flip, and the rightmost variety $W^{\prime }$ is the output of a minimal model program over $B$ . The hypotheses imply that $W^{\prime }$ has the structure of a Mori conic bundle $\overline{\unicode[STIX]{x1D70C}}:W^{\prime }\rightarrow Y$ over a projective $B$ -scheme $h:Y\rightarrow B$ corresponding to the contraction of an extremal ray of $\overline{\text{NE}}(W^{\prime }/B)$ . Indeed, a general fiber of $\overline{f}$ being a birationally ruled projective surface, the output $W^{\prime }$ is not a minimal model of $W$ over $B$ . So $W^{\prime }$ is either a Mori conic bundle over a $B$ -scheme $Y$ of dimension $2$ or a del Pezzo fibration over $B$ , the second case being excluded by the fact that the general fibers of $\overline{f}$ are irrational.

Proposition 13. The induced map $\unicode[STIX]{x1D70C}=\overline{\unicode[STIX]{x1D70C}}\mid _{X}:X{\dashrightarrow}Y$ is a rational $\mathbb{A}^{1}$ -fibration.

Proof. Since a general closed fiber $X_{b}$ is a normal affine surface with an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{b}:X_{b}\rightarrow C_{b}$ over a certain irrational smooth curve $C_{b}$ , it follows that there exists a unique maximal affine open subset $U_{b}$ of $C_{b}$ such that $\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b})\simeq U_{b}\times \mathbb{A}^{1}$ and such that the rational map $j_{b}:\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b}){\dashrightarrow}W_{b}$ induced by $j$ is regular, inducing an isomorphism between $\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b})$ and its image. Each step $\unicode[STIX]{x1D719}_{i}:W_{i}{\dashrightarrow}W_{i+1}$ consists of either a flip whose flipping and flipped curves are contained in fibers of $\overline{f}_{i}:W_{i}\rightarrow B$ and $\overline{f}_{i+1}:W_{i+1}\rightarrow B$ respectively, or a divisorial contraction whose exceptional divisor is contained in a fiber of $\overline{f}_{i}:W_{i}\rightarrow B$ , or a divisorial contraction whose exceptional divisor intersects a general fiber of $\overline{f}_{i}:W_{i}\rightarrow B$ . Clearly, a general closed fiber of $\overline{f}_{i}:W_{i}\rightarrow B$ is not affected by the first two types of birational maps. On the other hand, if $\unicode[STIX]{x1D719}_{i}:W_{i}\rightarrow W_{i+1}$ is the contraction of a divisor $E_{i}\subset W_{i}$ which dominates $B$ , then a general fiber of $\unicode[STIX]{x1D719}_{i}\mid _{E_{i}}$ is a smooth proper rational curve. The intersection of $E_{i}$ with a general closed fiber $W_{i,b}$ of $\overline{f}_{i}$ thus consists of proper rational curves, and its intersection with the image of the maximal affine cylinder like open subset $\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b})$ of $X_{b}$ is either empty or composed of affine rational curves. Since $U_{b}$ is an irrational curve, it follows that each irreducible component of $E_{i}\cap (\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b}))$ is contained in a fiber of $\unicode[STIX]{x1D70B}_{b}$ . This implies that there exists an open subset $U_{b,0}$ of $U_{b}$ with the property that for every $i=1,\ldots ,\ell$ , the restriction of $\unicode[STIX]{x1D719}_{i}\circ \cdots \circ \unicode[STIX]{x1D719}_{1}\circ j$ to $\unicode[STIX]{x1D70B}_{b}^{-1}(U_{b,0})\subset X_{b}$ is an isomorphism onto its image in $W_{i,b}$ . A general fiber of $\overline{\unicode[STIX]{x1D70C}}:W^{\prime }\rightarrow Y$ over a closed point $y\in Y$ being a smooth proper rational curve, its intersection with $\unicode[STIX]{x1D70B}_{h(y)}^{-1}(U_{h(y),0})$ viewed as an open subset of $W_{h(y)}^{\prime }$ , is thus either empty or equal to a fiber of $\unicode[STIX]{x1D70B}_{h(y)}$ . So by virtue of [9], there exists an open subset $V$ of $X$ on which $\overline{\unicode[STIX]{x1D70C}}$ restricts to an $\mathbb{A}^{1}$ -fibration $\overline{\unicode[STIX]{x1D70C}}\mid _{V}:V\rightarrow Y$ .◻

Corollary 14. Let $X$ be a normal complex affine threefold $X$ equipped with a morphism $f:X\rightarrow B$ onto a smooth curve $B$ whose general closed fibers are irrational $\mathbb{A}^{1}$ -ruled surfaces. Then $X$ is birationally equivalent to the product of $\mathbb{P}^{1}$ with a family $h_{0}:{\mathcal{C}}_{0}\rightarrow B_{0}$ of smooth projective curves of genus $g\geqslant 1$ over an open subset $B_{0}\subset B$ .

Proof. By the previous Proposition, $X$ has the structure of a rational $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X{\dashrightarrow}Y$ over a $2$ -dimensional normal proper $B$ -scheme $h:Y\rightarrow B$ . In particular, $X$ is birational to $Y\times \mathbb{P}^{1}$ . On the other hand, for a general closed point $b\in B$ , the curve $Y_{b}$ is birational to the base $C_{b}$ of the unique $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{b}:X_{b}\rightarrow C_{b}$ on the irrational affine surface $X_{b}$ . Letting $\unicode[STIX]{x1D70E}:{\tilde{Y}}\rightarrow Y$ be a desingularization of $Y$ , there exists an open subset $B_{0}$ of $B$ over which the composition $h\circ \unicode[STIX]{x1D70E}:{\tilde{Y}}\rightarrow Y$ restricts to a smooth family $h_{0}:{\mathcal{C}}_{0}\rightarrow B_{0}$ of projective curves of a certain genus $g\geqslant 1$ . By construction, $X$ is birational to ${\mathcal{C}}_{0}\times \mathbb{P}^{1}$ .◻

Remark 15. Example 11 above shows conversely that for every smooth family $h:{\mathcal{C}}\rightarrow B$ of projective curves of genus $g\geqslant 2$ , there exists a smooth $\mathbb{A}^{1}$ -ruled affine threefold $X$ birationally equivalent to ${\mathcal{C}}\times \mathbb{P}^{1}$ . Actually, in the setting of the previous Corollary 14, if we assume further that a general fiber of $f:X\rightarrow B$ carries an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{b}:X_{b}\rightarrow C_{b}$ over a smooth curve $C_{b}$ whose smooth projective model has genus $g\geqslant 2$ , then there exists a uniquely determined family $h:{\mathcal{C}}\rightarrow B$ of proper stable curves of genus $g$ such that $X$ is birationally equivalent to ${\mathcal{C}}\times \mathbb{P}^{1}$ : indeed, the moduli stack $\overline{{\mathcal{M}}}_{g}$ of stable curves of genus $g\geqslant 2$ being proper and separated, the smooth family $h_{0}:{\mathcal{C}}_{0}\rightarrow B_{0}$ extends in a unique way to a family $h:{\mathcal{C}}\rightarrow B$ of stable curves of genus $g$ .

3.2 Factorial affine threefolds

Proposition 16. Let $X$ be a normal affine threefold with finite divisor class group $\text{Cl}(X)$ and let $f:X\rightarrow B$ be a morphism onto a smooth curve $B$ whose general closed fibers are irrational $\mathbb{A}^{1}$ -ruled surfaces. Then there exists a factorization $f=h\circ \unicode[STIX]{x1D70C}:X\rightarrow Y\rightarrow B$ where $\unicode[STIX]{x1D70C}:X\rightarrow Y$ is the algebraic quotient morphism of an effective $\mathbb{G}_{a,B}$ -action on $X$ . In particular, a general fiber of $f$ admits an $\mathbb{A}^{1}$ -fibration of affine type.

Proof. By virtue of Proposition 12, there exists an effective $\mathbb{G}_{a,B}$ -action on $X$ such that for a general closed point $b\in B$ , the $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{b}:X_{b}\rightarrow C_{b}$ on $X_{b}$ factors through the algebraic quotient

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{b}:X_{b}\rightarrow X_{b}/\!/\mathbb{G}_{a,b}=\text{Spec}(\unicode[STIX]{x1D6E4}(X_{b},{\mathcal{O}}_{X_{b}})^{\mathbb{G}_{a,b}}).\end{eqnarray}$$

Since $X$ is a threefold, the ring of invariants $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{\mathbb{G}_{a,B}}$ is finitely generated [16]. The quotient morphism $\unicode[STIX]{x1D70C}:X\rightarrow Y=\text{Spec}(\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{\mathbb{G}_{a,B}})$ is an $\mathbb{A}^{1}$ -fibration, and since $Y$ is a categorical quotient in the category of algebraic varieties, the invariant morphism $f:X\rightarrow B$ factors through $\unicode[STIX]{x1D70C}$ .◻

Corollary 17. Let $f:\mathbb{A}^{3}\rightarrow B$ be a morphism onto a smooth curve $B$ with irrational $\mathbb{A}^{1}$ -ruled general fibers. Then $B$ is isomorphic to either $\mathbb{P}^{1}$ or $\mathbb{A}^{1}$ and there exists a factorization $f=h\circ \unicode[STIX]{x1D70C}:\mathbb{A}^{3}\rightarrow \mathbb{A}^{2}\rightarrow B$ , where $\unicode[STIX]{x1D70C}:\mathbb{A}^{3}\rightarrow \mathbb{A}^{2}$ is the quotient morphism of an effective $\mathbb{G}_{a,B}$ -action on $\mathbb{A}^{3}$ .

Proof. Since $B$ is dominated by a general line in $\mathbb{A}^{3}$ , it is necessarily isomorphic to $\mathbb{P}^{1}$ or $\mathbb{A}^{1}$ . The second assertion follows from Proposition 16 and the fact that the algebraic quotient of every nontrivial $\mathbb{G}_{a}$ -action on $\mathbb{A}^{3}$ is isomorphic to $\mathbb{A}^{2}$  [13].◻

Example 18. In Corollary 17 above, the base curve $B$ need not be affine. For instance, the morphism

$$\begin{eqnarray}f:\mathbb{A}^{3}=\text{Spec}(\mathbb{C}[x,y,z])\longrightarrow \mathbb{P}^{1},(x,y,z)\mapsto [(xz-y^{2})x^{2}+1:(xz-y^{2})^{3}]\end{eqnarray}$$

defines a family whose general member is isomorphic to the product $C_{\unicode[STIX]{x1D706}}\times \mathbb{A}^{1}$ where $C_{\unicode[STIX]{x1D706}}\subset \mathbb{A}^{2}=\text{Spec}(\mathbb{C}[xz-y^{2},x])$ is the affine elliptic curve with equation $(xz-y^{2})^{3}+\unicode[STIX]{x1D706}((xz-y^{2})x^{2}+1)=0$ . The subring $\mathbb{C}[xz-y^{2},x]$ of $\mathbb{C}[x,y,z]$ coincides with the ring of invariants of the $\mathbb{G}_{a}$ -action associated with the locally nilpotent $\mathbb{C}[x]$ -derivation $x\unicode[STIX]{x2202}_{y}+2y\unicode[STIX]{x2202}_{z}$ and $f$ is the composition of the quotient morphism $\unicode[STIX]{x1D70C}:\mathbb{A}^{3}\rightarrow \mathbb{A}^{2}=\mathbb{A}^{3}/\!/\mathbb{G}_{a}=\text{Spec}(\mathbb{C}[u,v])$ defined by $(x,y,z)\mapsto (xz-y^{2},x)$ and of the morphism $h:\mathbb{A}^{2}=\text{Spec}(\mathbb{C}[u,v])\rightarrow \mathbb{P}^{1}$ defined by $\left(u,v\right)\mapsto [uv^{2}+1:u^{3}]$ .

Corollary 17 above implies in particular that if a general fiber of a regular function $f:\mathbb{A}^{3}\rightarrow \mathbb{A}^{1}$ is irrational and admits an $\mathbb{A}^{1}$ -fibration, then the latter is necessarily of affine type. In contrast, regular functions $f:\mathbb{A}^{3}\rightarrow \mathbb{A}^{1}$ whose general fibers are rational and equipped with $\mathbb{A}^{1}$ -fibrations of complete type only do exist, as illustrated by the following example.

Example 19. Let $f=x^{3}-y^{3}+z(z+1)\in \mathbb{C}[x,y,z]$ and let $f:\mathbb{A}^{3}=\text{Spec}(\mathbb{C}[x,y,z])\rightarrow \mathbb{A}^{1}=\text{Spec}(\mathbb{C}[\unicode[STIX]{x1D706}])$ be the corresponding morphism. The closure $\overline{S}_{\unicode[STIX]{x1D706}}$ in $\mathbb{P}^{3}=\text{Proj}(\mathbb{C}[x,y,z,t])$ of a general fiber $S_{\unicode[STIX]{x1D706}}=f^{\ast }(\unicode[STIX]{x1D706})$ of $f$ is a smooth cubic surface which intersects the hyperplane $H_{\infty }=\left\{t=0\right\}$ along the union $B_{\unicode[STIX]{x1D706}}$ of three lines meeting at the Eckardt point $p=\left[0:0:1:0\right]$ . Thus $S_{\unicode[STIX]{x1D706}}$ is rational and a direct computation reveals that $\unicode[STIX]{x1D705}(S_{\unicode[STIX]{x1D706}})=-\infty$ . So by virtue of [14], $S_{\unicode[STIX]{x1D706}}$ admits an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D706}}:S_{\unicode[STIX]{x1D706}}\rightarrow C_{\unicode[STIX]{x1D706}}$ over a smooth rational curve $C_{\unicode[STIX]{x1D706}}$ . If $C_{\unicode[STIX]{x1D706}}$ was affine, then there would exist a nontrivial $\mathbb{G}_{a}$ -action on $S_{\unicode[STIX]{x1D706}}$ having the general fibers of $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D706}}$ as general orbits. But it is straightforward to check that every automorphism of $S_{\unicode[STIX]{x1D706}}$ considered as a birational self-map of $\overline{S}_{\unicode[STIX]{x1D706}}$ is in fact a biregular automorphism of $\overline{S}_{\unicode[STIX]{x1D706}}$ preserving the boundary $B_{\unicode[STIX]{x1D706}}$ . So the automorphism group of $S_{\unicode[STIX]{x1D706}}$ injects into the group $\text{Aut}(\overline{S}_{\unicode[STIX]{x1D706}},B_{\unicode[STIX]{x1D706}})$ of automorphisms of the pair $(\overline{S}_{\unicode[STIX]{x1D706}},B_{\unicode[STIX]{x1D706}})$ . The latter being a finite group, we conclude that no such $\mathbb{G}_{a}$ -action exists, and hence that $S_{\unicode[STIX]{x1D706}}$ only admits $\mathbb{A}^{1}$ -fibrations of complete type. An $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D706}}:S_{\unicode[STIX]{x1D706}}\rightarrow \mathbb{P}^{1}$ can be obtained as follows: letting $B_{\unicode[STIX]{x1D706}}=L_{1}\cup L_{2}\cup L_{3}$ , $L_{1}$ is a member of a $6$ -tuple of pairwise disjoint lines whose simultaneous contraction realizes $\overline{S}_{\unicode[STIX]{x1D706}}$ as a blow-up $\unicode[STIX]{x1D70E}:\overline{S}_{\unicode[STIX]{x1D706}}\rightarrow \mathbb{P}^{2}$ of $\mathbb{P}^{2}$ in such a way that $\unicode[STIX]{x1D70E}(L_{2})$ and $\unicode[STIX]{x1D70E}(L_{3})$ are respectively a smooth conic and its tangent line at the point $p=\unicode[STIX]{x1D70E}(L_{1})$ . The birational transform $\overline{\unicode[STIX]{x1D70B}}_{\unicode[STIX]{x1D706}}:\overline{S}_{\unicode[STIX]{x1D706}}{\dashrightarrow}\mathbb{P}^{1}$ on $\overline{S}_{\unicode[STIX]{x1D706}}$ of the pencil generated by $\unicode[STIX]{x1D70E}(L_{2})$ and $2\unicode[STIX]{x1D70E}(L_{3})$ restricts to an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D706}}:S_{\unicode[STIX]{x1D706}}\rightarrow \mathbb{P}^{1}$ with two degenerate fibers: an irreducible one, of multiplicity two, consisting of the intersection with $S_{\unicode[STIX]{x1D706}}$ of the unique exceptional divisor of $\unicode[STIX]{x1D70E}$ whose center is supported on $\unicode[STIX]{x1D70E}(L_{3})\setminus \{p\}$ , and a smooth one consisting of the intersection with $S_{\unicode[STIX]{x1D706}}$ of the four exceptional divisors of $\unicode[STIX]{x1D70E}$ with centers supported on $\unicode[STIX]{x1D70E}(L_{2})\setminus \left\{p\right\}$ .