A Survey on the Minimal Number of Singular Fibers of a ${\mathbb{P}}^{\mathbf{1}}$-Semistable Curve

Abstract

Given a semistable and nonisotrivial fibered surface $f:X\to {\mathbb{P}}^1$, this paper presents the main results on the problem of determining the minimal number of singular fibers that f must admit. The known examples of fibrations with a low number of singular fibers are presented. Some related problems are briefly discussed.

1. Introduction

We work over the field $\mathbb{C}$. All varieties are considered projective, irreducible, and nonsingular unless explicitly stated. We assume the reader is acquainted with the fundamentals of algebraic geometry. For more specific topics on the theory of algebraic surfaces, references [1] or [2] can be consulted.

Definition 1.

A fibered surface is a triple $(X,B,f)$, where X is a surface, B is a curve, and $f:X\to B$ is a surjective morphism with $1—$connected fibers. The morphism f is called a fibration.

If f is a fibration, then a Zariski dense open subset $U\subseteq B$ exists such that for all $p\in U$, the fiber $F_p:=f^{-1}\left(p\right)$ is nonsingular.

Note that for each $p\in B$, the arithmetic genus $p_a\left(F_p\right)$ of $F_p$ is given by $2(p_a\left(F_p\right)-1)=F_p^2+F_p.K_X$, where $K_X$ denotes, as usual, the canonical divisor of X. Since all the fibers are algebraically equivalent, this number is constant for all $p\in B$. In particular, for $p\in U$, $p_a\left(F_p\right)$ coincides with the geometric genus of $F_p$. This genus g is called the genus of the fibration.

A fibration $f:X\to B$ is said to be semistable if all the fibers $F_p$ are semistable in the sense of Deligne–Mumford; that is, for all $p\in B$, $F_p$ is reduced at most nodal singularities and does not contain $(-1)$ curves as components (this last condition is also referred to as f being relatively minimal).

Thus, if f is semistable, a morphism

$$f_{*}:B\to \overline{{\mathcal{M}}_g},$$

exists, with $\overline{{\mathcal{M}}_g}$ denoting the Deligne–Mumford compactification of the moduli space of curves of genus g. The morphism $f_{*}$ sends a point $p\in B$ to the the class $\left[F_p\right]$ of $F_p$ in $\overline{{\mathcal{M}}_g}$. The fibration f is said to be nonisotrivial if $f_{*}$ is nonconstant.

This note deals with the following problem: given a nonisotrivial and semistable fibration $f:X\to {\mathbb{P}}^1$, we find a lower bound for the number singular fibers, s, that f must admit.

We do not present new results; we only intend to survey the main facts on the subject. We aim to give as self-contained an account as possible but avoid technicalities. Thus, the proofs are given only in the sketch. For each statement, a reference containing a complete proof is provided.

The proofs of Theorem 3 items (i)–(iii) are simplified versions of the originals. A complete proof of Theorem 3 (iv) is not given, but the sketch of an argument that could lead to an elementary one.

A trivial example of an isotrivial fibration is a trivial one:

$$C\times B\stackrel{p_2}{\to }B,$$

with C being a curve of genus g.

If B is neither rational nor elliptic (i.e., the genus of B is greater than 1), then examples exist of nonisotrivial fibrations $f:X\to B$ with no singular fibers (f is a smooth morphism). These are called Kodaira’s fibrations (for instance, see [3]).

The problem of determining the minimal number of singular fibers goes back at least to Shafarevich’s school. A. Parshin established the following inequality in [4]:

$$2\left(g\right(B)-1)+s>0.$$

Thus, if $g\left(B\right)=0$, then $s\ge 3$. In what follows, we shall see that this bound can be improved.

In this point, we want to develop an elementary and particular case to motivate the fact that if $g\ge 1$, then the number s must be at least 5 (Theorem 3 (ii)).

Assume our fibration $f:X\to {\mathbb{P}}^1$ is obtained as the resolution of the base locus of a pencil $\Lambda$ of plane curves, where the general member of $\Lambda$ is nonsingular and intersects transversally any other member of the pencil. The singular fibers of f correspond to the pencil elements with singularities away from the base locus. Since the fibration is semistable, these singularities are necessarily nodes.

By Proposition 1 below, the number of such nodes must be

$$e_f=d^2+3+4(g-1),$$

(using $e\left({\mathbb{P}}^2\right)=3$). By the genus formula for nonsingular plane curves,

$$e_f=3d^3-6d+3.$$

If $F_0$ is a reduced plane curve of degree d, its maximal possible number of nodes is ${\displaystyle \frac{d(d-1)}{2}}$. This number is attained if and only if $F_0$ is the product of d lines. Thus, $s\le 4$ implies

$$3d^2-6d+3\le 4{\displaystyle \frac{d(d-1)}{2}}=2d^2-2d;$$

that is, $d\le 3$ or $g\le 1$.

In Section 2, we will explain some basic formulas concerning fibrations and the significant Miyaoka–Vojta–Tan inequality. In Section 3, we will formulate the main known inequalities on the number s. In Section 4, some techniques for constructing examples of fibrations with the minimal possible number of singular fibers are explained. Finally, in Section 5, some related problems are presented.

2. Preliminaries

Given a nonisotrivial fibration $f:X\to B$, the relative canonical sheaf (or divisor), $K_f$, is the only invertible sheaf that fits in the exact sequences:

$$0\to f^{*}{K}_B\to {\Omega }_X^1\to {\Omega }_{X/B}^1\to 0,$$

(1)

$$0\to {\Omega }_{X/B}^1\to K_f\to {\mathcal{O}}_Z\to 0,$$

(2)

where Z is the scheme defined by the critical locus of f. Equivalently, we have an exact sequence:

$$0\to f^{*}{K}_B\to {\Omega }_X^1\to K_f\otimes {\mathcal{I}}_Z\to 0,$$

(3)

with ${\mathcal{I}}_Z$ denoting the ideal sheaf of Z.

In this way, $K_f=K_X\otimes f^{*}{K}_B^{-}1$ (that is taken sometimes as the definition of $K_f$). Thus, if $B={\mathbb{P}}^1$, then $K_f=K_X(-2F)$, with F denoting a fiber. It follows that

$$K_f^2=K_X^2-8(g-1)(g_b-1).$$

A divisor D on an algebraic surface is called nef (numerically effective) if, for every effective divisor $\Delta$, $D.\Delta \ge 0$; it is called big if $D^2>0$. On the road to the “rigidity theorem”, Arakelov and Parshin [4,5] proved the following:

Theorem 1.

If a fibration $f:X\to B$ is nonisotrivial, then $K_f$ is big and nef and $deg{f}_{*}{K}_f>0$. If, moreover, the fibration is semistable, then the unique curves such that $K_f.E=0$ are the vertical $(-2)$ curves.

In what follows, we denote by $g_B$ the genus of B. Given a semistable fibration, we can count the total number of nodes in the fibers of f through the following formula:

Proposition 1.

Let $f:X\to B$ be a semistable fibration; denote by $e_f$ the number of nodes in the fibers of f, then

$$e_f=e\left(X\right)-4(g-1)(g_B-1),$$

where $e\left(X\right)$ denotes the topological Euler characteristic of X.

Proof. 

If f is semistable, then Z is a zero-dimensional, reduced scheme of length equal to $e_f$. The formula follows from computing the Chern classes in the exact sequence (3) and taking into account that $e\left(X\right)=c_2\left(X\right)$. For a detailed argument, refer to [6], and for a concise explanation of computing Chern classes, see Chapter 2 of [2]. □

There is a topological proof using a direct computation on Euler’s characteristic [3], Chapter III, Proposition 11.4. The formula can also be proved using cohomology with compact support [7].

Another useful formula is

Proposition 2.

Let $f:X\to B$ be a fibration (not necessarily semistable). Then,

$$deg{f}_{*}{K}_f=\chi \left({\mathcal{O}}_X\right)-(g-1)(g_B-1).$$

Proof. 

$f_{*}{K}_f$ and $R^1{f}_{*}{K}_f$ is are locally free sheaves of rank g and 1, respectively, over B. By Riemann Roch:

$$\chi \left(f_{*}{K}_f\right)=deg{f}_{*}{K}_f-g(g_B-1).$$

(4)

Using the Leray spectral sequence associated with f, we obtain

$$\chi \left(f_{*}{K}_f\right)=\chi \left(K_f\right)+\chi \left(R^1{f}_{*}{K}_f\right),$$

(5)

but by relative duality,

$${\left(R^1{f}_{*}{K}_f\right)}^{\vee }=f_{*}{\mathcal{O}}_X\simeq {\mathcal{O}}_B,$$

and $\chi \left({\mathcal{O}}_X\right)=1-g_B$. Substituting (5) in (4) and computing $\chi \left(K_f\right)$ using Riemman–Roch on X (see [2], Chapter 2, Theorem 2), we obtain the statement. □

The ideas of the proofs of Propositions 1 and 2 presented here are taken from [6].

Now, we recall the important Miyaoka–Vojta–Tan (MVT) inequalities. The semistable fibration $f:X\to B$ could contain $(-2)$ curves, or even chains of $(-2)$ curves on its fibers. These chains can be contracted to double rational singularities in a new, normal surface $X^{\#}$. We denote by $q_1,...,q_r\in X^{\#}$ the rational singularities obtained after contracting the vertical $(-2)$ divisors by and ${\mu }_q$ the length of the corresponding chain of $(-2)$ curves. If q is a singular point of some fiber and is not contained in any $(-2)$ divisor, we write ${\mu }_q=0$. Set the following:

$$r_f=\sum {\displaystyle \frac{1}{1+{\mu }_q}}.$$

Then, if $g\ge 1$, then for any integer $e\ge 2$:

$$e^2(K_X^2-2(g-1)(6(g_B-1)+s-s/e))\le 3r_f\le 3e_f,$$

(MVT)

(see [8,9]). Taking the limit when e goes to infinity, we obtain the canonical class inequality:

Theorem 2.

Let $f:X\to B$ be a semistable fibration of genus $g>0$, and denote by s the number of singular fibers. Then,

$$K_f^2<(2g_B-2+s)(2g-2).$$

As will be seen in the next section, this inequality can be used to obtain several lower bounds for the number s. It is essential to remember that not only the canonical inequality is true but all the sequences of inequalities (MVT).

Finally, we note that if $F_i$ is a singular fiber of f and $F_i=F_{i1}+\cdots +F_{i{l}_i}$ is its decomposition into irreducible components, then, by the adjunction formula (see [2], Chapter 1, Formula (1.4)), and the fact that $-F_{ij}^2=F_{ij}.\left({\sum }_{k\ne j}{F}_{ik}\right)$, we have

$$(g-1)=\sum _j(g_{ij}-1)+n_i,$$

where $g_{ij}$ denotes the geometric genus of $F_{ij}$, and $n_i$ the number of nodes on $F_i$. Considering all the singular fibers of f and summing the previous equalities, we obtain

$$s(g-1)=\sum _{ij}(g_{ij}-1)+e_f.$$

(6)

3. The Results

The main results of the theory can be summarized in the following:

Theorem 3.

Let $f:X\to {\mathbb{P}}^1$ be a semistable and non-isotrivial fibration of genus g and denote by s the number of singular fibers of f. Then,

(i) 

$s\ge 4$. If $s=4$, then $g=1$, all the irreducible components of the singular fibers of f are of geometric genus 0, and the surface X is rational.

(ii) 

If $g\ge 2$, then $s\ge 5$.

(iii) 

If the Kodaira dimension of X is non-negative, then $s\ge 6$.

(iv) 

If X is of general type (i.e., of Kodaira dimension two), then $s\ge 7$.

We want to propose a variation of the original proofs, or rather, a different organization of the usual arguments. We start by establishing the following:

Lemma 1.

Let $f:X\to {\mathbb{P}}^1$ be a nonisotrivial fibration of genus $g\ge 2$; then, $K_f(-F)$ is nef.

Proof. 

Let us denote $K_f(-F)$ by B. From $deg{f}_{*}{K}_f>0$ and the projection formula, B is an effective divisor. Note that $\left|2B\right|=|2K_f-2F|=|K_f+K_X|$ is the adjoint linear system of $K_f$, thus proving that the linear system $\left|2B\right|$ is base point free is sufficient.

We can apply Reider’s method as $K_f$ is big and nef [2]. First, $B.F\ge 2g-2\ge 2$; therefore, B admits at least one horizontal component, and by Theorem 1, we have $B.K_f\ge 1$. Now, as $K_f.B=K_f^2-2(g-1)>0$, $g\ge 3$ implies $K_f^2\ge 5$. According to Reider’s theorem, if x is a base point of $\left|2B\right|$, then x must be contained in either of the following: (a) an effective divisor E such that $K_f.E=0$ and $E^2=-1$; (b) an effective divisor E such that $K_f.E=1$ and $E^2=0$. Since $2p_a\left(E\right)-2=E^2+E.K_X=E^2+E.K_f-2E.F$, it follows that both cases are impossible.

The case $g=2$ can be similarly proved using the linear system $\left|3B\right|$ (see [10]). □

Next, we outline the proof of Theorem 3:

Proof. 

For the proof of (ii), since $K_f(-F)$ is nef, we must have ${\left(K_f(-F)\right)}^2\ge 0$. Combining this with Theorem 2, we obtain

$$4(g-1)\le K_f^2<(s-2)(2g-2);$$

that is, $4<s$.

For (iii), we note that $n{K}_X$ is effective if $n>>0$. Since $K_f(-F)$ is nef, $K_X.K_f(-F)=K_f(-2F).K_f(-F)\ge 0$. Combining, once again, with Theorem 2, we obtain

$$6(g-1)\le K_f^2<(s-2)(2g-2).$$

Thus, $5<s$.

Once (ii) is established, we see that $s\le 4$ implies $g=1$. Assume $g=1$ and f is nonisotrivial and semistable. By (6) and Proposition 1, we have

$$0=\sum _{ij}(g_{ij}-1)+e\left(X\right).$$

The ingredients in Beauville’s original proof of (i) (see [1]) are

• If f is nonisotrivial, then $q\left(X\right)<g$ (Appendix to [11]).
• With the notation preceding (6) and denoting by $\rho \left(X\right)$ the Picard number of X, we have

  $$\rho \left(X\right)\ge \sum _i(l_i-1)+2$$
  (see [1], LEMME 2).

All these combined leads to

$$\begin{array}{cc}\hfill 0& =\sum _{ij}(g_{ij}-1)+e\left(X\right)\hfill \\ & =\sum _{ij}(g_{ij}-1)+2+b_2\left(X\right)\hfill \\ & \ge \sum _{ij}(g_{ij}-1)+2+\sum _i^s(l_i-1)+2\hfill \\ & =\sum _{ij}{g}_{ij}-\sum _i^s{l}_i+4+\sum _i^s(l_i-1).\hfill \end{array}$$

From this, it follows that $s\ge 4$ and that if $s=4$, then $g_{ij}=0$ for all $i,j$. Moreover, $b_2\left(X\right)=\rho \left(X\right)$, and from this and Hodge theory, we must have $p_g\left(X\right)=0$. Thus, X is a surface with $q=p_g=0$, but X cannot be of general type because it contains an algebraic family of elliptic curves. We conclude that X is rational.

The original proof of (iv) uses properties of the Hodge bundle on the moduli of genus g curves to deduce that if $s=6$ and X is of general type, then $p_g\left(X\right)=0$. After this, the proof is not difficult. Instead of sketching this proof, we give an argument showing that elementary methods can reduce the problem of studying bounded families of surfaces and fibrations.

Given the fibration $f:X\to {\mathbb{P}}^1$, with X of general type, let S be the minimal model of X and $\Lambda$ the pencil in S induced by f. Denote by C the general element of $\Lambda$. We assume for simplicity that C is nonsingular and intersects any other element of $\Lambda$ transversely. Thus, $2(g-1)=C.K_S+C^2$.

Let us prove that if $7\le K_S^2$, then $7\ge s$.

Assume $s=6$, inequality (MVT) evaluated in $e=3$ combined with Hodge’s index theorem (see [2] Chapter 1, Theorem 11) is as follows:

$$C^2{K}_S^2\le {(C.K_S)}^2={(2(g-1)-C^2)}^2.$$

Considering this expression as a quadratic inequality in $C^2$, we obtain

$$K_S^2+(K_S^2{(K_S^2+8(g-1))}^{1/2}\le 6\chi \left({\mathcal{O}}_S\right),$$

(7)

and using $2(g-1)=C^2+C.K_S$,

$$C^2\le {\displaystyle \frac{3}{4}}{K}_S^2+{\displaystyle \frac{91}{K_S^2}}+18-C.K_S.$$

(8)

On the other hand, the evaluation of (MVT) in $e=4$ gives

$$K_S^2+18C.K_S\le C^2+108.$$

Combining this last inequality with (8), we obtain

$${\displaystyle \frac{K_S^2}{4}}+19C.K_S\le {\displaystyle \frac{81}{K_S^2}}+126.$$

Thus, if $8\le K_S^2$, then $C.K_S\le 7$. By Hodge’s index theorem,

$$C^2{K}_S^2\le 49.$$

Now using (MVT) evaluated in $e=5$ and (8), we obtain

$${\displaystyle \frac{5}{2}}+6C.K_S\le {\displaystyle \frac{49}{K_S^2}}+27.$$

By Hodge’s index theorem $3\le C.K_S$, we obtain a contradiction with $8\le K_S^2$.

Applying similar arguments allows us to prove that if $s=6$, then $K_S^2\le 2$, and for each possible value $K_S^2=1$ or 2, the possible values of g are bounded. □

The history of this theorem is as follows: Part (i) was proved by A. Beauville in a seminar paper [1]. Part (ii) was proved by S.L. Tan in [8] using Beauville’s result and by K. Liu in [12] using the differential geometry of $\overline{{\mathcal{M}}_g}$; the Lemma 1 was proved in [10] (see also [13]), and from this part (iii) was deduced. Part (iii) was also proved in [14]; thus, the proof we present here does not follow the chronological order. After Lemma 1, parts (ii) and (iii) are straightforward consequences of Theorem 2 and are independent of (i). Moreover, once (ii) is established, the proof of (i) becomes simplified.

Part (iv) of Theorem 3 was proved in [15]. The argument presented here is a variation, developed by Castañeda-Salazar and the author, based on an original argument by A. Huitrado-Mora and R. Pignatelli.

The next natural step in studying the problem of the minimal number of singular fibers is to classify fibrations with the minimal number of singular fibers. Beauville achieved this for the case $s=4$ in [16]. There are exactly 6 semistable fibrations of genus $g=1$ with only 4 singular fibers. They come from a quotient of $\mathbb{H}$ under the action of certain modular groups.

In [17], it was proved that if $s=5$ and the surface X is bi-rationally ruled but nonrational, then ${(K+F)}^2=0$. Concerning the rational case, a list of possibilities was recently provided in [18].

Theorem 4.

Let X be a rational nonsingular complex projective surface and $f:X\to {\mathbb{P}}^1$ be a semistable nonisotrivial fibration of genus$g\ge 4$. Assume$s=5$ and F is non-hyperelliptic. Then, one of the following occurs:

(i) 

$K_X^2=2-3g,g\le 11$, and the general fiber F is trigonal.

(ii) 

$K_X^2=-16,g=6$, and f is obtained by blowing up the base locus of a pencil with a general nonsingular member of plane degree 5 curves;

(iii) 

$K_X^2=-10,g=4$, and f is obtained by blowing up the base locus of a pencil with a general nonsingular member of cubic hypersurface sections of the quadric cone in ${\mathbb{P}}^3$;

(iv) 

$K_X^2=3-3g,g\le 10$, and f is obtained by blowing up the base locus of plane degree 6 curves, whose general element admits only $10-g$ singularities of order 2;

(v) 

$K_X^2=-24,g=9$, and f is obtained by blowing up the base locus of a pencil with a general nonsingular member of the quartic hypersurface sections of a nonsingular quadric in ${\mathbb{P}}^3$.

4. Examples

4.1. Beauville’s Construction

Beauville in [1] introduced the following construction.

Consider a covering

$$\phi :C\to {\mathbb{P}}^1$$

of degree n. Assume that all the ramification points of $\phi$ are simple. Let $u\in PGL\left(2\right)$ be an automorphism of ${\mathbb{P}}^1$. Denote by R the branch locus of $\phi$ and assume that $uR\subset R$, but R does not intersect the fixed points of u.

Let ${\Gamma }_{\phi }\subset C\times {\mathbb{P}}^1$ be the graph of $\phi$ and ${\Gamma }_{u\circ \phi }$ the graph of $u\circ \phi$. Then, ${\Gamma }_{\phi }$ is linearly equivalent to ${\Gamma }_{u\circ \phi }$, and a covering $2:1$ map,

$$\pi :\tilde{X}\to C\times {\mathbb{P}}^1,$$

ramified over ${\Gamma }_{\phi }+{\Gamma }_{u\circ \phi }$ exists.

The surface $\tilde{X}$ is singular because if x and y are local coordinates of ${\Gamma }_{\phi }$, ${\Gamma }_{u\circ \phi }$ around a point $(0,0)\in {\Gamma }_{\phi }\cap {\Gamma }_{u\circ \phi }$, then the local equation of $\tilde{X}$ is

$$w^2=xy.$$

Note that these intersections occur only on the fixed points of u.

A blow-up resolves this singularity; the exceptional divisor is a $(-2)-$ divisor. Let X be the desingularization of $\tilde{X}$, and f, the fibration be obtained by the following composition:

$$X\to \tilde{X}\stackrel{\pi }{\to }C\times {\mathbb{P}}^1\stackrel{p_2}{\to }{\mathbb{P}}^1,$$

with $p_2$ the projection onto the second factor. The fibers of f are $2:1$ coverings of C. They have singularities on the branch points of $\phi$ because locally around such a point, the equation of ${\Gamma }_{\phi }$ is $t-x^2$, where x is a local parameter of C around a ramification point. Thus, the local equation in X of the fiber on $t=0$ is $w^2=x^2$, and analogously for ${\Gamma }_{u\circ \phi }$. Therefore, f has $s=\left|R\right|+2$ singular fibers and the genus of f is $2g\left(C\right)+n-1$.

The importance of the condition $uR\subset R$ is that the number of singular fibers does not increase by considering the branch points on ${\Gamma }_{\phi }$ and ${\Gamma }_{u\circ \phi }$, and the importance of avoiding fixed points of u in R lies in obtaining simple singularities in the fibers.

Thus, the problem of finding fibrations with a low number of singular fibers reduces to the construction of appropriate coverings $\phi :C\to {\mathbb{P}}^1$. Using this method, Beauville constructed examples of the following:

(1)

Semistable fibrations of every genus with $s=6$. The fibers are hyperelliptic if g is even and double covers of elliptic curves if g is odd. Thus, in particular, the gonality of the fibers is less than or equal to 4.

(2)

An example of a genus 3 fibration with $s=5$ and hyperelliptic fibers.

(3)

An example of a genus 1 fibration with $s=4$.

4.2. Tan’s Variation on Beauville’s Construction

In [8], S. L. Tan introduced a variation on Beauville’s construction. Instead of $\phi$ and u, he considers two coverings $\alpha ,\beta :{\mathbb{P}}^1\to {\mathbb{P}}^1$ with $deg\alpha +deg\beta =2g+2$, and imposes similar conditions on the ramification points. Using this construction, he was able to construct an example of a genus 2 fibration (thus, hyperelliptic) with $s=5$.

Another possibility is to consider the fibered product of two convenient coverings of ${\mathbb{P}}^1\to {\mathbb{P}}^1$ and take its desingularization. Using this idea, examples of fibrations with $s=6$ and $s=7$ can be constructed. The former, being defined on a $K3$ surface, is of genus 4, and the latter is of genus 3 and is defined on a general type surface (see [10]).

Further developments of this idea enabled Y. Tu to construct examples of fibrations defined on all types of surfaces of Kodaira dimension 0 with $s=6$ and genus arbitrary large (see [19]).

4.3. Wiman-Edge’s Pencil

Consider Wiman’s curve:

$$W:2\sum _{i,j,l}(x_i^4{x}_j{x}_l+x_i^3{x}_l^3)-2\sum _{i\ne j}\left(x_i^4{x}_j^2\right)+\sum _{i,j,l}{x}_i^3{x}_j^2{x}_l-6x_0^2{x}_1^2{x}_2^2=0.$$

The curve W has nodes in the standard frame of reference $B:={\left\{e_i\right\}}_{i=1}^4$ and admits an action of the symmetric group ${\mathcal{S}}_5$. Geometrically, in this action, the alternating group ${\mathcal{A}}_5$ is generated by the linear automorphisms of ${\mathbb{P}}^2$, fixing $e_i\in B$ and cyclically permuting ${\left\{e_j\right\}}_{j\ne i}$ and the quadratic transformations, with fundamental triangles supported in B, and fixing the point not in the given triangle.

Another plane curve invariant by this action of ${\mathcal{A}}_5$ is defined by the product of lines $L:\prod l_{ij}=0$, with $l_{ij}$ passing through $e_i$ and $e_j$. The so-called Wiman–Edge pencil is the pencil $\mathcal{WE}$ generated by W and L. All the members of $\mathcal{WE}$ are invariant under the above-described action of ${\mathcal{A}}_5$.

L is a member of the pencil having singular points outside the base locus. We consider the fibration $f:X\to {\mathbb{P}}^1$ obtained after resolving the base locus of $\mathcal{WE}$. A careful analysis of the action of ${\mathcal{A}}_5$ on the elements of $\mathcal{WE}$ leads to the conclusion that f has $s=5$. The genus of the fibration is 6, and the gonality of the general fiber is 4. To our knowledge, this, until now, has been the only example of a fibration with $s=5$ and $g\ge 6$. Also, this provides an example of fibration satisfying the hypothesis in item (iv) of Theorem 4.

For more on the Wiman–Edge pencil, see, for instance, [20,21,22].

5. Final Remarks

Hitherto, we have concentrated on the problem of the minimal number of singular fibers for semistable and nonsisotrivial fibrations with a rational base. This problem has several variations.

For instance, one can drop the hypothesis of semistability (but not the one on nonisotriviality). In this case, the primary result is that $3\le s$ (see [1]), but there are various refinements on the kind of fibers that can occur if $s=3$ (see [23]).

Also, we can ask for the minimal number of fibers of a non-compact type (see [24]); or, in the case of a positive characteristic, $s\ge 3$ and $s\ge 4$ if X is of a general type (see [6]). The bibliography quoted here is not exhaustive but is indicative of some possible variations of the problem.

In [25], the first example of a semistable fibration over an elliptic base with $s=1$ was constructed. See also [26].

The results and examples presented here suggest that imposing generic conditions on the fibers and the surface X could improve the known bounds on s.