Hitchin Fibrations with One Singular Fiber

Hitchin Fibrations on Two-Dimensional Moduli
Spaces of Irregular Higgs Bundles with One Singular
Fiber

Péter IVANICS †, András I. STIPSICZ † and Szilárd SZABÓ ‡

P. Ivanics, A.I. Stipsicz and S. Szabó

† Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda utca 13-15,
Hungary<EMAIL_ADDRESS><EMAIL_ADDRESS>‡ Budapest University of
Technology and Economics,
‡ 1111 Budapest, Egry József utca 1, H épület, Hungary<EMAIL_ADDRESS>
Received February 02, 2019, in final form October 25, 2019; Published online
November 04, 2019

We analyze and completely describe the four cases when the Hitchin fibration
on a $2$-dimensional moduli space of irregular Higgs bundles over
$\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber at
infinity is of type $I_{0}^{*}$ is further analyzed, and we give constructions
of all the possible configurations of singular curves in elliptic fibrations
having this type of singular fiber at infinity.

irregular Higgs bundles; Hitchin fibration; elliptic fibrations

14J27; 14H40; 14H60

##  1 Introduction

Let $D$ be an effective divisor in $\mathbb{C}P^{1}$ of total length $4$.
Consider the (complex) 2-dimensional moduli spaces
${\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}}$
of rank $2$ degree $d$ parabolically stable Higgs bundles over
$\mathbb{C}P^{1}$ with irregular singularities of prescribed local form at $D$
and of $0$ parabolic degree. When equipped with the Hitchin fibration, this
space has been shown to be biregular to the complement of a singular fiber in
an elliptic fibration on a rational elliptic surface (which complex surface is
diffeomorphic to the 9-fold blow-up
$\mathbb{C}P^{2}\\#_{9}{\overline{{\mathbb{C}}P}}^{2}$ of the complex
projective plane), see [14, Section 3.3] for the details of biregularity.

In the following we will describe all the cases when the Hitchin fibration on
the moduli space has a single singular fiber. In these cases the fibration on
the compactified space has exactly two singular fibers; fibrations with
exactly two singular fibers have been extensively studied [10, 26] – indeed,
the genus-1 case (which is of central relevance in our subsequent studies)
admits a simple classification scheme given in [26, Theorem 3.2]. In [26,
Theorem 3.2] four cases and an infinite family of genus-1 fibrations with
exactly two singular fibers have been encountered – since by [24, Section 4]
all fibrations we will study arise from pencils (hence admit a section), the
infinite family will not arise in our context.

We will distinguish five main cases, depending on the number of poles of the
Higgs field (constrained by our assumption having a complex 2-dimensional
moduli space):

  1. 1)

there is one pole (which has to be of multiplicity 4),

  2. 2)

there are two poles, each of multiplicity 2,

  3. 3)

there are two poles, one with multiplicity 3 and the other with multiplicity
1,

  4. 4)

there are three poles, one with multiplicity 2 and the further two with
multiplicity 1, and finally

  5. 5)

there are four poles, each with multiplicity 1.

At each pole we need to distinguish further two cases, depending on whether
the leading-order term of the polar part of the Higgs field at the pole is a
regular semi-simple endomorphism (untwisted case), or has non-vanishing
nilpotent part (twisted case). The third possible case, when the polar part is
non-regular semi-simple, does not provide elliptic fibration, as shown by a
simple calculation similar to [14, Lemma 5.6].

The polar parts will depend on some complex parameters, and our first aim is
to determine those parameter values for which the Hitchin fibration on the
moduli space has a unique singular fiber. These parameter values are complex
numbers once we fix certain trivializations – for details see Section 2. The
second aim of the paper is to provide explicit constructions for all the
possibilities of configurations of singular fibers in case (5) of the above
list (four logarithmic poles).

The spaces considered in this paper have a rich geometry. In addition to the
Dolbeault algebraic structure that we focus on, they admit a de Rham algebraic
structure parameterizing certain connections with irregular singularities
(related to the Dolbeault space via non-abelian Hodge theory), and a Betti
algebraic structure parameterizing representations of the fundamental group of
the punctured surface together with Stokes data (related to de Rham space via
the irregular Riemann–Hilbert correspondence). This picture was first studied
by Hitchin [11] for regular objects, then by Simpson [21] in the tame case,
subsequently extended to wild singularities by Biquard and Boalch [2]. In a
series of papers, Boalch studied the moduli spaces of connections with
irregular singularities on a curve and the corresponding wild character
varieties. In particular, Boalch [5] (in the untwisted case) and Boalch and
Yamakawa [6] (in the twisted case) defined the notion of an irregular curve,
and showed that for a fixed irregular curve, parabolic weights and orbits of
the residues in appropriate sense, one gets a Betti moduli functor (first
spelled out in [3] in the untwisted case) whose corresponding coarse moduli
space exists, and is given by a quasi-Hamiltonian quotient construction [1].
In the particular cases under study here, the above results imply that the
moduli spaces we are looking at admit the structure of a smooth hyperKähler
manifold of real dimension $4$. It is known [4, 7, 16] that an open subset of
each such space is in fact a quiver variety [17]. More precisely, it is a
fiber of the semi-universal deformation of a Kleinian surface singularity,
with an action of the corresponding Weyl group on the base of the deformation
family. Another equivalent way of labelling these spaces that appears in the
literature is by the type of the central fiber of this family.

The results of this paper, together with those of the previous papers [13, 14]
of the same authors, will be applied [25] by the third author to prove the
numerical version of the $P=W$ conjecture by de Cataldo, Hausel and Migliorini
[8] and Simpson’s geometric counterpart thereof [22] in the particular cases
covered by our papers, namely those corresponding to the Painlevé systems. It
turns out that these conjectures are easier to treat in the presence of a
${\mathbb{C}}^{\times}$-action. In the Painlevé cases (and more generally, if
the moduli space is complex $2$-dimensional), the existence of such an action
implies that the Hitchin fibration admits a unique singular fiber. Therefore,
one may hope that finding moduli spaces with a unique singular Hitchin fiber
may be of use in the generalization of the $P=W$ conjecture.

Throughout, we will be interested in moduli spaces
${\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2\big{)}$ of rank $2$
parabolic Higgs bundles over $\mathbb{C}P^{1}$ with higher order poles at some
divisor $D$, with prescribed local form near the points of the divisor. For
details, see Section 2.1. The moduli space admits a natural decomposition

$\displaystyle{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2\big{)}=\coprod_{d\in{\mathbb{Z}}}{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$

according to the degree of the underlying vector bundle ${\mathcal{E}}$. In
the cases where the data fixed to define
${\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$ contains a
regular non-semisimple (i.e., non-closed) adjoint orbit at some logarithmic
points $t\in D$, we will equally study the partial compactification

$\displaystyle\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$
(1.1)

parameterizing parabolic Higgs bundles with possibly non-regular residues at
the points $t$, having the same characteristic polynomial. This functor
differs from
${\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$ in that
instead of imposing the adjoint orbit of the residue of the Higgs field at
such points $t$, we merely impose that the residue lie in the _closure_ of the
given non-closed orbit. Then, (1.1) is a stratified algebraic space: the top-
dimensional stratum is
${\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$, and the
lower-dimensional strata are moduli spaces of parabolic logarithmic Higgs
bundles with some regular and some non-regular residues having the same
eigenvalues as fixed for the definition of
${\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$.

###  1.1 The case of one pole

This case was treated in [13], where we gave a complete classification of
singular fibers of the Hitchin fibration. In particular, in the untwisted case
the fibration depends on the complex parameters

$\displaystyle a_{\pm},b_{\pm},c_{\pm},\lambda_{\pm}\in{\mathbb{C}},\qquad
a_{+}\neq a_{-},$ (U)

and the fiber at infinity is of type $\widetilde{E}_{7}$. The $a_{+}=a_{-}$
case does not lead to an elliptic fibration, cf. [14, Lemma 5.6]. For the
geometric meaning of these parameters, see Section 2.1.1; the various singular
fibers are defined in Section 4. Let

$\displaystyle\Delta=\big{(}(b_{-}-b_{+}){}^{2}-4(a_{-}-a_{+})(c_{-}-c_{+})\big{)}{}^{3}-432(a_{-}-a_{+}){}^{4}\lambda_{+}^{2}.$

With these notations we have

######  Theorem 1.1 ([13, Theorem 1.1]).

In this case the Hitchin fibration on the moduli space
${\mathcal{M}}^{\vec{\alpha}-s}{\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$ admits a
single singular fiber if and only if $\lambda_{+}=\lambda_{-}=0$ and
$\Delta=0$. The single singular fiber is of type $III$.

In the twisted case we have complex parameters (described in detail Section
2.1.1)

$\displaystyle b_{-8},\ldots,b_{-3}\in{\mathbb{C}},\qquad b_{-7}\neq 0,$ (T)

the fiber at infinity is $\widetilde{E}_{8}$. The case $b_{-7}=0$ does not
give elliptic fibration [13, Section 2.3.2]. We introduce
${\Gamma}=\big{(}b_{-6}^{2}+4b_{-5}\big{)}{}^{2}-24b_{-7}(b_{-6}b_{-4}+2b_{-3})$
and we have

######  Theorem 1.2 ([13, Theorem 1.3]).

In this case the Hitchin fibration on the moduli space
${\mathcal{M}}^{\vec{\alpha}-s}{\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$ admits a
single singular fiber if and only if ${\Gamma}=0$. The single singular fiber
is of type $II$ $($i.e., a cusp fiber$)$.

###  1.2 The case of two poles

Let us consider first the case when both multiplicities are equal to 2. In
this case a simple argument (see also [14, Theorems 2.1, 2.2, 2.3]) shows that
the fiber at infinity is $I_{n}^{*}$ with $n\in\\{2,3,4\\}$ (depending on
whether the leading order term is regular semi-simple or not at the two
poles), hence in this case there is no Hitchin fibration on
${\mathcal{M}}^{\vec{\alpha}-s}{(\mathbb{C}P^{1},D,2,d)}$ with exactly one
singular fiber (cf. Corollary 4.2).

In the case of multiplicities 3 and 1 on the two poles, we have more examples;
indeed, if the pole with multiplicity 3 is untwisted and the other one is
twisted, then the fibration depends on the complex parameters $a_{+}$,
$a_{-}$, $b_{+}$, $b_{-}$, $\lambda_{+}$, $\lambda_{-}$, $b_{-1}$; their
geometric significance is given in Section 2.1.2. We assume $a_{+}\neq a_{-}$
because $a_{+}=a_{-}$ does not lead elliptic fibration see [14, Remark 6.7].
In addition, the fibration also depends on so-called parabolic weights
$\vec{\alpha}=\\{(\alpha_{j}^{+},\alpha_{j}^{-})\\}_{j=1,2}$. By taking

$A=a_{-}-a_{+},\qquad B=b_{-}-b_{+},\qquad L=\lambda_{-}-\lambda_{+},$

and defining ${\Theta}=4A^{2}\big{(}B^{2}-6AL\big{)}$, we get

######  Theorem 1.3 ([14, Theorem 2.5], [25, Lemma 2]).

Suppose that we have two poles, one with multiplicity $3$ and the other one
with multiplicity $1$. Assume furthermore that the multiplicity $3$ pole is
untwisted, while the other one is twisted. $($In this case the fiber at
infinity is of type $\widetilde{E}_{6}.)$ Then the Hitchin fibration on
$\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$
has a unique singular fiber if and only if ${\Theta}=0$ and $L=0$. For generic
parabolic weights, the single singular fiber is of type $IV$.

If the multiplicity 1 pole is untwisted, then according to [14, Theorems 2.4,
2.6] we do not have examples for unique singular fibers in the Hitchin
fibration on the moduli space. Finally, if both poles are twisted, then the
natural parameters (described in Section 2.1.2) take the form $b_{-6}$,
$b_{-5}$, $b_{-4}$, $b_{-3}$, $b_{-2}$, $b_{-1}$. Define $Q=8b_{-5}$ ($Q\neq
0$ due to [14, Lemma 6.8] and $R=b_{-4}^{2}+4b_{-3}$; with these notations in
place, we have

######  Theorem 1.4 ([14, Theorem 2.7], [25, Lemma 2]).

Suppose that the multiplicities of the poles are $3$ and $1$, and both are
twisted. In this case the fiber at infinity is of type $\widetilde{E}_{7}$,
and the Hitchin fibration on
$\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$
admits a single singular fiber if and only if $R=0$. For generic parabolic
weights, the single singular fiber is of type $III$.

###  1.3 The case of three poles

In this case one of the poles is of multiplicity 2 and the two others are both
of multiplicity 1. A simple argument shows that the fiber at infinity is
either $I_{1}^{*}$ (when the polar part is untwisted at the pole of
multiplicity 2) or $I_{2}^{*}$ (when the polar part is twisted at the pole of
multiplicity 2). Since by [23, Section 6] elliptic fibrations on the rational
elliptic surface with a singular fiber $I_{1}^{*}$ or $I_{2}^{*}$ have at
least three singular fibers (cf. Corollary 4.2), we conclude

######  Theorem 1.5.

In the case of three poles, the Hitchin fibration admits at least two singular
fibers on the moduli space ${\overline{{\mathcal{M}}^{\vec{\alpha}-s}}}$.

###  1.4 The case of four poles

This is the most interesting case. It is not hard to see, that the fiber at
infinity is an $I_{0}^{*}$ fiber, and (once again by [23, Section 6]) there is
one case when the complement of this fiber admits a single singular fiber:
when this other fiber is also of type $I_{0}^{*}$.

Before turning to the statement, we would like to point out a new feature in
this case. In all previous cases the holomorphic structure on the fiber at
infinity (which was either $\widetilde{E}_{8}$, $\widetilde{E}_{7}$ or
$\widetilde{E}_{6}$) was unique. The holomorphic structure on $I_{0}^{*}$,
however is not unique. All five $\mathbb{C}P^{1}$-components of $I_{0}^{*}$
can be equipped with a unique holomorphic structure, but the four points where
the curves corresponding to the leaves of the graph describing $I_{0}^{*}$
(see Fig. 1) intersect the central rational curve determine a cross ratio,
which is a holomorphic invariant. We can assume that the four points of
intersections are at $0$, $1$, $\infty$ and $t\in\mathbb{C}P^{1}$; hence $t$
is the complex parameter (distinct from $0$, $1$, $\infty$) determining the
complex structure on $I_{0}^{*}$. The pencil on the Hirzebruch surface in this
case is determined by four complex numbers $a$, $b$, $c$, $d$ (cf. Section
2.1.3 and equations (2.11) for their geometric role). The fibration again
depends on a choice of parabolic weights
$\vec{\alpha}=\\{(\alpha_{j}^{+},\alpha_{j}^{-})\\}_{j=1,\ldots,4}$, see
Section 3. We assume that the sum of all these weights is an integer, and
moreover that this integer is equal to the negative of the degree of the
underlying vector bundle of the Higgs bundle. With these notations and
assumptions, we have

######  Theorem 1.6.

The pencil given by the above parameters $t$, $a$, $b$, $c$, $d$ provides an
elliptic fibration with a single singular fiber $($besides the $I_{0}^{*}$
fiber at infinity$)$ if and only if

$\displaystyle c=a+(b-a-d)t+dt^{2}.$

For generic parabolic weights the single singular fiber of the partial
compactification

$\displaystyle\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}$

introduced in (1.1) is of type $I_{0}^{*}$.

The singular fibers of Hitchin fibrations in cases of one or two poles have
been completely determined in [13, 14] in terms of the parameters. In
principle a similar analysis can be carried out in the remaining cases – in
Section 6 we will restrict ourselves to provide examples of all possible
singular fiber configurations when we have four poles. For the list of these
possibilities, see Table 1 in Section 6. Here we do not determine the various
configurations in terms of the parameters. By providing various constructions
and examples, we prove:

######  Theorem 1.7.

Any configuration of singular fibers on the complement of an $I_{0}^{*}$ fiber
in a rational elliptic surface arises as the set of singular fibers of a
Hitchin fibration.

The paper is organized as follows. In Section 2 we collect some facts and
definitions of Higgs bundles and the Hitchin map, relevant to our present
investigation. We also include the necessary stability analysis here. Section
4 is devoted to list background material regarding elliptic fibrations, and in
Section 5 we describe the proofs of the statements about Hitchin fibrations
with unique singular fibers given in this introduction. Finally in Section 6
we describe a number of constructions for elliptic fibrations with one of the
singular fibers being $I_{0}^{*}$ (relevant in the case of four poles in the
above discussion), hence proving Theorem 1.7. In these constructions we pay
special attention to the further singular fibers of the resulting fibrations.

##  2 Preliminaries

In this paper we will deal with certain meromorphic Higgs bundles over
$\mathbb{C}P^{1}$ with structure group $\operatorname{GL}(2,{\mathbb{C}})$. In
the sequel we let $D$ be an effective divisor over $\mathbb{C}P^{1}$ and
denote by $K_{\mathbb{C}P^{1}}=\mathbf{\Omega}_{\mathbb{C}P^{1}}^{1}$ the
canonical bundle of $\mathbb{C}P^{1}$.

######  Definition 2.1.

A meromorphic Higgs bundle with polar divisor $D$ (or a $K(D)$-pair of rank
$2$) over $\mathbb{C}P^{1}$ consists of a pair $({\mathcal{E}},\theta)$ where
${\mathcal{E}}$ is a holomorphic rank $2$ vector bundle over $\mathbb{C}P^{1}$
and

$\displaystyle\theta\colon\ {\mathcal{E}}\to{\mathcal{E}}\otimes
K_{\mathbb{C}P^{1}}(D)$

is an ${\mathcal{O}}_{\mathbb{C}P^{1}}$-linear vector bundle morphism, called
the Higgs field.

From now on, we assume that the total length of $D$ is equal to $4$:

$\displaystyle D=m_{1}t_{1}+\cdots+m_{k}t_{k},\qquad m_{1}+\cdots+m_{k}=4,$
(2.1)

where $1\leq k\leq 4$, $1\leq m_{j}$ and $t_{j}\in\mathbb{C}P^{1}$ are
distinct points for $1\leq j\leq k$. We have an
${\mathcal{O}}_{\mathbb{C}P^{1}}$-linear isomorphism of line bundles

$\displaystyle K_{\mathbb{C}P^{1}}(D)\cong{\mathcal{O}}_{\mathbb{C}P^{1}}(2).$
(2.2)

We denote

  * •

by ${\mathbb{F}}_{2}$ the Hirzebruch surface obtained by the projectivization
of $K_{\mathbb{C}P^{1}}(D)$, and by
$p\colon{\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ the ruling coming from this
construction,

  * •

by ${\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(1)$ the relative ample
bundle of $p$ and by
${\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(n)={\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(1)^{\otimes
n}$ its tensor powers,

  * •

by $\zeta\in
H^{0}\big{(}{\mathbb{F}}_{2},p^{*}K_{\mathbb{C}P^{1}}(D)\otimes{\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(1)\big{)}$
and $\xi\in
H^{0}\big{(}{\mathbb{F}}_{2},{\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(1)\big{)}$
the canonical sections,

  * •

and by $\mbox{I}_{{\mathcal{E}}}$ the identity endomorphism of
${\mathcal{E}}$.

Let $({\mathcal{E}},\theta)$ be a $K(D)$-pair of rank $2$ over
$\mathbb{C}P^{1}$.

######  Definition 2.2.

The characteristic polynomial of $\theta$ is the section

$\displaystyle\chi_{\theta}(\zeta)=\det(\zeta\mbox{I}_{{\mathcal{E}}}-\xi
p^{*}\theta)=\zeta^{2}+\zeta\xi F_{\theta}+\xi^{2}G_{\theta}\in
H^{0}\big{(}{\mathbb{F}}_{2},p^{*}K_{\mathbb{C}P^{1}}(D)^{\otimes
2}\otimes{\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(2)\big{)}.\\!\\!\\!$
(2.3)

Here

$\displaystyle F_{\theta}\in
H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)\big{)},\qquad G_{\theta}\in
H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)^{\otimes 2}\big{)}$

are called the characteristic coefficients of $\theta$. Elements of
$H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)\big{)}$ will be called
meromorphic differentials and elements of
$H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)^{\otimes 2}\big{)}$
meromorphic quadratic differentials. The spectral curve of
$({\mathcal{E}},\theta)$ is the curve in ${\mathbb{F}}_{2}$ defined by the
equation $\chi_{\theta}(\zeta)=0$. The Hitchin base is the vector space

$\displaystyle\mathcal{B}=H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)\big{)}\oplus
H^{0}\big{(}\mathbb{C}P^{1},K_{\mathbb{C}P^{1}}(D)^{\otimes 2}\big{)}$

containing all possible characteristic coefficients of rank $2$ $K(D)$-pairs.

It follows from the definition and (2.2) that the characteristic coefficients
$F_{\theta}$, $G_{\theta}$ may be considered as global sections of
${\mathcal{O}}_{\mathbb{C}P^{1}}(2)$ and of
${\mathcal{O}}_{\mathbb{C}P^{1}}(4)$ respectively, so we have

$\displaystyle\dim_{{\mathbb{C}}}\mathcal{B}=8.$

######  Remark 2.3.

In Section 6 we will take a converse approach by setting $F_{\theta}=0$ and
considering sections

$\displaystyle\sigma\in
H^{0}\big{(}\mathbb{C}P^{1},{\mathcal{O}}_{\mathbb{C}P^{1}}(4)\big{)}$

(corresponding to the coefficient $G_{\theta}$); we then choose the points
$t_{j}$ (or equivalently, the fibers $F_{j}=p^{-1}(t_{j})$) appropriately in
order to obtain various singular fibers.

Let us denote by

$\displaystyle{\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2\big{)}$

the moduli stack of $K(D)$-pairs of rank $2$ over $\mathbb{C}P^{1}$. It
decomposes as

$\displaystyle{\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2\big{)}=\coprod_{d\in{\mathbb{Z}}}{\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$

where ${\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2,d\big{)}$ is the moduli stack
of $K(D)$-pairs of rank $2$ over $\mathbb{C}P^{1}$ satisfying
$\deg({\mathcal{E}})\allowbreak=d$.

######  Definition 2.4.

The Hitchin map is the morphism

$\displaystyle h\colon\ {\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2\big{)}$
$\displaystyle\to\mathcal{B},$ $\displaystyle({\mathcal{E}},\theta)$
$\displaystyle\mapsto(F_{\theta},G_{\theta})$

associating to a Higgs bundle its characteristic coefficients.

For generic $(F,G)\in\mathcal{B}$, the curve $\Sigma_{(F,G)}$ defined by the
equation

$\displaystyle\zeta^{2}+\zeta\xi F+\xi^{2}G$

is smooth, and for every $d\in{\mathbb{Z}}$ the fiber of

$\displaystyle h\colon\
{\mathcal{M}}\big{(}\mathbb{C}P^{1},D,2,d\big{)}\to\mathcal{B}$ (2.4)

is a torsor over the Jacobian $\operatorname{Jac}(\Sigma_{(F,G)})$ of
$\Sigma_{(F,G)}$. When the curve $\Sigma_{(F,G)}$ is not regular, then
$\operatorname{Jac}(\Sigma_{(F,G)})$ is non-compact, and for every $d$ the
fiber $h^{-1}(F,G)$ is (non-canonically) isomorphic to a certain
compactification of $\operatorname{Jac}(\Sigma_{(F,G)})$. In case
$\Sigma_{(F,G)}$ is integral, the suitable compactification of
$\operatorname{Jac}(\Sigma_{(F,G)})$ is the moduli space of torsion-free
sheaves of rank $1$ and given degree over $\Sigma_{(F,G)}$. On the other hand,
in case $\Sigma_{(F,G)}$ is reducible or non-reduced, the corresponding fiber
of (2.4) is not of finite type, and in order to get a moduli scheme of finite
type one needs to impose a stability condition on the objects. Stability
conditions naturally arise from parabolic structures on meromorphic Higgs
bundles; we will return to describing moduli spaces of stable objects and the
fibers of the Hitchin map on these moduli spaces once we have the assumptions
and notations concerning the local behaviour of a meromorphic Higgs field near
the points of $D$.

###  2.1 Parameters and their geometric significance

Our investigation is based on the local description of spectral curves on the
Hirzebruch surface ${\mathbb{F}}_{2}$ of degree 2. Recall from equation (2.1)
that $D=m_{1}t_{1}+\cdots+m_{k}t_{k},$ with $m_{1}+\cdots+m_{k}=4$.

If we are in the case of one pole with multiplicity $4$ at $t_{1}$, then we
introduce two local charts on $\mathbb{C}P^{1}$: $U$ with $u\in{\mathbb{C}}$
(where $\\{u=0\\}=[0:1]=t_{1}$) and $V$ with $v\in{\mathbb{C}}$ (where
$\\{v=0\\}=[1:0]$). The bundle $K_{\mathbb{C}P^{1}}(4t_{1})$ admits the
trivializing sections $\kappa_{U}$ over $U$ and $\kappa_{V}$ over $V$:

$\displaystyle\kappa_{U}=\frac{{\rm d}u}{u^{4}},\qquad\kappa_{V}={\rm d}v.$

If we are in the case of at least two poles, then we introduce two local
charts on $\mathbb{C}P^{1}$: $U$ with $u\in{\mathbb{C}}$ (where
$\\{u=0\\}=[0:1]=t_{1}$) and $V$ with $v\in{\mathbb{C}}$ (where
$\\{v=0\\}=[1:0]=t_{k}$). We also denote by $t_{i}$ the representation of a
pole in the local chart $U$, we refer $t_{1}$ as $0$, $t_{k}$ as $\infty$ and
$t_{i}^{-1}$ as the image of $t_{i}\in\mathbb{C}P^{1}$ in chart $V$. The
bundle $K_{\mathbb{C}P^{1}}(D)$ admits the following trivializing sections:

$\displaystyle\kappa_{U}=\frac{{\rm
d}u}{\prod\limits_{i=2}^{k-1}(u-t_{i})^{m_{i}}},\qquad\kappa_{V}=\frac{{\rm
d}v}{\prod\limits_{i=2}^{k-1}\big{(}v-t_{i}^{-1}\big{)}^{m_{i}}}.$

The conversion from $\kappa_{U}$ to $\kappa_{V}$ is the following (if the
total multiplicity is $4$):

$\displaystyle{\kappa_{U}=-\frac{v^{2}}{\prod\limits_{i=2}^{k-1}(-t_{i})^{m_{i}}}\kappa_{V}.}$

The trivialization $\kappa_{j}$ induces a trivialization $\kappa_{j}^{2}$ on
$K_{\mathbb{C}P^{1}}(D)^{\otimes 2}$, $j=U,V$. Moreover, (recalling that
$\zeta$ denotes the canonical section of
$p^{*}K_{\mathbb{C}P^{1}}(D)\otimes{\mathcal{O}}_{{\mathbb{F}}_{2}|\mathbb{C}P^{1}}(1)$)
we can introduce coordinates $w$ in $p^{-1}(U)$ and $z$ in $p^{-1}(V)$ by

$\displaystyle\zeta=w\otimes\kappa_{U}=z\otimes\kappa_{V}.$ (2.5)

Choose a suitable trivialization of ${\mathcal{E}}$. Consider an irregular
Higgs bundle $({\mathcal{E}},\theta)$ in the $\kappa_{j}$ trivializations
($j=U,V$) and the chosen trivialization of ${\mathcal{E}}$. The local forms of
$\theta$ near $t_{i}$’s are the following:

  * •

the case of $D=4t_{1}$:

$\displaystyle\theta=\sum_{n\geq-4}A_{n}u^{n}\otimes{\rm d}u,$ (2.6)

  * •

the case of $D=2t_{1}+2t_{2}$ is not of interest, because we concentrate on
the cases with two singular fibers, but according to [14, Theorems 2.1, 2.2,
2.3], $D=2t_{1}+2t_{2}$ to leads at least three singular fibers,

  * •

the case of $D=3t_{1}+1t_{2}$:

$\displaystyle\theta=\sum_{n\geq-3}A_{n}u^{n}\otimes{\rm
d}u\qquad\text{and}\qquad\theta=\sum_{n\geq-1}B_{n}v^{n}\otimes{\rm d}v,$
(2.7)

  * •

the case of $D=2t_{1}+1t_{2}+1t_{3}$ is not relevant: a simple geometric
argument similar to Example 4.3 shows that in this case the elliptic fibration
contains a fiber of type $I_{1}^{*}$ or $I_{2}^{*}$, and according to
Corollary 4.2 this case can not lead to exactly two singular fibers,

  * •

the case of $D=1t_{1}+1t_{2}+1t_{3}+1t_{4}$. Near one of the $t_{i}$
($i=1,2,3$) the local form is

$\displaystyle\theta=\sum_{n\geq-1}A_{n,i}(u-t_{i})^{n}\otimes{\rm d}u$ (2.8a)
and near $t_{4}=\infty$:
$\displaystyle\theta=\sum_{n\geq-1}A_{n,4}v^{n}\otimes{\rm d}v,$ (2.8b)

where $A_{n},B_{n},A_{n,i}\in\mathfrak{gl}(2,{\mathbb{C}})$.

The coefficient $A_{-1,i}$ in above expression is called the residue of
$\theta$ at $t_{i}$, with respect to the chosen trivialization of
${\mathcal{E}}$. The residue of $\theta$ at $t_{i}$ is denoted by
$\operatorname{res}_{t_{i}}(\theta)$. It is a well-defined endomorphism of
${\mathcal{E}}|_{t_{i}}$ ($i=1,\dots,4$).

The spectral curve ($\chi_{\theta}(\zeta)=0$) has an expansion near $t_{i}$ in
which these parameters have a geometric meaning. The lowest index matrices
$A_{n}$, $B_{n}$ and $A_{n,i}$ in equations (2.6), (2.7) and (2.8) encode the
base locus of a pencil.

Let us consider the local forms separated according to the number of poles and
their multiplicities.

####  2.1.1 $\boldsymbol{D=4t_{1}}$

The matrices $A_{-3}$, $A_{-2}$ and $A_{-1}$ in equation (2.6) encode the
tangent, the second and third derivative of the curve in the pencil. If the
leading order term is a regular semi-simple endomorphism (untwisted case) then
the local form in the trivialization $\kappa_{U}$ of $K_{\mathbb{C}P^{1}}$ and
the chosen trivialization of ${\mathcal{E}}$ is

$\displaystyle\theta=\left[\begin{pmatrix}a_{+}&0\\\
0&a_{-}\end{pmatrix}u^{-4}+\begin{pmatrix}b_{+}&0\\\
0&b_{-}\end{pmatrix}u^{-3}+\begin{pmatrix}c_{+}&0\\\
0&c_{-}\end{pmatrix}u^{-2}+\begin{pmatrix}\lambda_{+}&0\\\
0&\lambda_{-}\end{pmatrix}u^{-1}+O(1)\right]\otimes{\rm d}u.$

These matrices provide the parameters (U) in Theorem 1.1.

If the leading order term has non-vanishing nilpotent part (twisted case) then
the local form in the trivialization $\kappa_{U}$ is

$\displaystyle\theta=\left[\begin{pmatrix}b_{-8}\\!&1\\\
0&\\!b_{-8}\end{pmatrix}u^{-4}\\!+\begin{pmatrix}0&0\\\
b_{-7}\\!&\\!b_{-6}\end{pmatrix}u^{-3}\\!+\begin{pmatrix}0&0\\\
b_{-5}\\!&\\!b_{-4}\end{pmatrix}u^{-2}\\!+\begin{pmatrix}0&0\\\
b_{-3}\\!&\\!b_{-2}\end{pmatrix}u^{-1}+O(1)\right]\otimes{\rm d}u,$

with $b_{-7}\neq 0$. These matrices provide the parameters (T) in Theorem 1.2.

####  2.1.2 $\boldsymbol{D=3t_{1}+1t_{2}}$

The matrices $A_{-2}$ and $A_{-1}$ in equation (2.7) encode the slope of the
tangent line of a pencil and the second derivative of the curve in the pencil.

If the pole with multiplicity $3$ (i.e., $t_{1}$) is untwisted then the local
form in the trivialization $\kappa_{U}$ (near $t_{1}$) and the chosen
trivialization of ${\mathcal{E}}$:

$\displaystyle\theta=\left[\begin{pmatrix}a_{+}&0\\\
0&a_{-}\end{pmatrix}u^{-3}+\begin{pmatrix}b_{+}&0\\\
0&b_{-}\end{pmatrix}u^{-2}+\begin{pmatrix}\lambda_{+}&0\\\
0&\lambda_{-}\end{pmatrix}u^{-1}+O(1)\right]\otimes{\rm d}u.$

If the multiplicity $3$ pole is twisted, then

$\displaystyle\theta=\left[\begin{pmatrix}b_{-6}&1\\\
0&b_{-6}\end{pmatrix}u^{-3}+\begin{pmatrix}0&0\\\
b_{-5}&b_{-4}\end{pmatrix}u^{-2}+\begin{pmatrix}0&0\\\
b_{-3}&b_{-2}\end{pmatrix}u^{-1}+O(1)\right]\otimes{\rm d}u,$

with $b_{-5}\neq 0$.

The matrix $B_{-1}$ contains the base locus of the pencil in the
trivialization $\kappa_{V}$ and the chosen trivialization of ${\mathcal{E}}$.
If the pole $t_{2}$ is untwisted, then

$\displaystyle\theta=\left[\begin{pmatrix}\mu_{+}&0\\\
0&\mu_{-}\end{pmatrix}v^{-1}+O(1)\right]\otimes{\rm d}v.$

If the pole $t_{2}$ is twisted, then up to holomorphic gauge transformation

$\displaystyle\theta=\left[\begin{pmatrix}b_{-1}&\varepsilon\\\
0&b_{-1}\end{pmatrix}v^{-1}+O(1)\right]\otimes{\rm d}v$

for $\varepsilon\in{\mathbb{C}}$. These parameters appear in Theorems 1.3 and
1.4.

####  2.1.3 $\boldsymbol{D=1t_{1}+1t_{2}+1t_{3}+1t_{4}}$

The matrices in equations (2.8) only encode the base locus of the pencil in
the four distinguished fibers at $t_{1}=0$, $t_{2}=1$, $t_{3}=t$ and
$t_{4}=\infty\in\mathbb{C}P^{1}$.

If one of the poles $t_{i}$ ($i=1,\dots,4$) is untwisted then the matrix from
the local form in the trivialization $\kappa_{j}$ ($j=U,V$) and the chosen
trivialization of ${\mathcal{E}}$ is the following:

$\displaystyle A_{-1,i}=\begin{pmatrix}a_{+,i}&0\\\ 0&a_{-,i}\end{pmatrix},$
(2.9)

where we assume that $a_{+,i}\neq a_{-,i}$, otherwise a similar computation as
in [14, Lemma 5.6] shows that the fibration is not elliptic.

Similarly, if one of the poles $t_{i}$ ($i=1,\dots,4$) is twisted then the
matrix has non-trivial nilpotent part. In the trivialization $\kappa_{j}$
($j=U,V$) and the chosen trivialization of ${\mathcal{E}}$:

$\displaystyle A_{-1,i}=\begin{pmatrix}a_{i}&\varepsilon\\\
0&a_{i}\end{pmatrix}$ (2.10)

for $\varepsilon\in{\mathbb{C}}$.

For simplicity we introduce new notations for the parameters appearing above:

$\displaystyle a_{\pm}:=a_{\pm,1},$ (2.11a) $\displaystyle
b_{\pm}:=a_{\pm,2},$ (2.11b) $\displaystyle c_{\pm}:=a_{\pm,3},$ (2.11c)
$\displaystyle d_{\pm}:=a_{\pm,4},$ (2.11d) $\displaystyle a:=a_{1},$ (2.11e)
$\displaystyle b:=a_{2},$ (2.11f) $\displaystyle c:=a_{3},$ (2.11g)
$\displaystyle d:=a_{4}.$ (2.11h)

The parameters $a$, $b$, $c$ and $d$ which appear in the Theorem 1.6 encode
the base locus in the distinguished fibers.

##  3 Higgs bundles with non-reduced spectral curve

Let $({\mathcal{E}},\theta)$ be a meromorphic Higgs bundle of rank $2$ over
$\mathbb{C}P^{1}$ with divisor $D$. Parabolic stability for Higgs fields with
higher-order poles has been investigated in [13, 14]. In this subsection we
assume that the multiplicity $m_{j}$ is equal to $1$ for all $j$, i.e., that

$\displaystyle D=t_{1}+\cdots+t_{4}$

for some distinct points $t_{j}$ ($j=1,2,3,4$).

######  Definition 3.1.

A compatible quasi-parabolic structure is the choice of a $1$-dimensional
eigenspace $\ell_{j}\subset{\mathcal{E}}|_{t_{j}}$ of
$\operatorname{res}_{t_{j}}(\theta)$ for all $1\leq j\leq 4$. A compatible
parabolic structure consists of a compatible quasi-parabolic structure and a
choice of a pair of rational numbers $(\alpha_{j}^{+},\alpha_{j}^{-})$ (called
parabolic weights) satisfying

$\displaystyle 0\leq\alpha_{j}^{-}<\alpha_{j}^{+}<1$

for each $j$. The parabolic degree of ${\mathcal{E}}$ is defined by the
formula

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{E}})=\deg({\mathcal{E}})+\sum_{j=1}^{4}(\alpha_{j}^{+}+\alpha_{j}^{-}).$

A rank $1$ meromorphic Higgs subbundle of $({\mathcal{E}},\theta)$ is a pair
$({\mathcal{F}},\theta|_{{\mathcal{F}}})$ where ${\mathcal{F}}$ is a rank $1$
subbundle of ${\mathcal{E}}$ such that the restriction
$\theta|_{{\mathcal{F}}}$ maps ${\mathcal{F}}$ into ${\mathcal{F}}\otimes
K_{\mathbb{C}P^{1}}(D)$.

Let $\big{(}{\mathcal{E}},\theta,\\{\ell_{j}\\}_{j=1}^{4}\big{)}$ be a
meromorphic Higgs bundle of rank $2$ over $\mathbb{C}P^{1}$ with a compatible
quasi-parabolic structure, and with given parabolic weights
$\\{(\alpha_{j}^{+},\alpha_{j}^{-})\\}_{j=1}^{4}$. For any rank $1$
meromorphic Higgs subbundle $({\mathcal{F}},\theta|_{{\mathcal{F}}})$ of
$({\mathcal{E}},\theta)$, the fiber ${\mathcal{F}}|_{t_{j}}$ is preserved by
$\operatorname{res}_{t_{j}}(\theta)$. If ${\mathcal{F}}|_{t_{j}}=\ell_{j}$,
then we set

$\displaystyle\alpha_{j}({\mathcal{F}})=\alpha_{j}^{+},$

otherwise we set

$\displaystyle\alpha_{j}({\mathcal{F}})=\alpha_{j}^{-}.$

Finally, we define

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{F}})=\deg({\mathcal{F}})+\sum_{j=1}^{4}\alpha_{j}({\mathcal{F}}).$

######  Definition 3.2.

$\big{(}{\mathcal{E}},\theta,\\{\ell_{j}\\}_{j=1}^{4}\big{)}$ is
$\vec{\alpha}$-stable if for any rank $1$ meromorphic Higgs subbundle
$({\mathcal{F}},\theta|_{{\mathcal{F}}})$ we have

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{F}})<\frac{\deg_{\vec{\alpha}}({\mathcal{E}})}{2}.$

Moreover, $\big{(}{\mathcal{E}},\theta,\\{\ell_{j}\\}_{j=1}^{4}\big{)}$ is
$\vec{\alpha}$-semistable if for any rank $1$ meromorphic Higgs subbundle
$({\mathcal{F}},\theta|_{{\mathcal{F}}})$ we have

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{F}})\leq\frac{\deg_{\vec{\alpha}}({\mathcal{E}})}{2}.$

Let $M^{\vec{\alpha}-s}$ and $M^{\vec{\alpha}-ss}$ be the functors

$\displaystyle\mbox{Schemes}\to\mbox{Sets}$

associating to a scheme $S$ the set of $S$-equivalence classes of holomorphic
vector-bundles

$\displaystyle{\mathcal{E}}\to\mathbb{C}P^{1}\times S$

endowed with a morphism

$\displaystyle\theta\colon\ {\mathcal{E}}\to{\mathcal{E}}\otimes
p_{1}^{*}K_{\mathbb{C}P^{1}}$

(where $p_{1}\colon\mathbb{C}P^{1}\times S\to\mathbb{C}P^{1}$ stands for the
first projection) and with filtrations

$\displaystyle\ell_{j}\subset{\mathcal{E}}|_{\\{t_{j}\\}\times S},$

where $\ell_{j}$ is a line-subbundle preserved by
$\operatorname{res}_{\\{t_{j}\\}\times S}(\theta)$, and such that for any
geometric point $s\in S$ the triple
$\big{(}{\mathcal{E}},\theta,\\{\ell_{j}\\}_{j=1}^{4}\big{)}$ defines an
$\vec{\alpha}$-stable (respectively, $\vec{\alpha}$-semistable) meromorphic
Higgs bundle of rank $2$ over $\mathbb{C}P^{1}$ with a compatible parabolic
structure and local form of the Higgs field specified in Section 2.1; for
simplicity of the notation, we do not include the parameters $a,b,\ldots$ in
the notation of the functors, but we tacitly fix them.

######  Theorem 3.3 ([19]).

There exist quasi-projective coarse moduli schemes

$\displaystyle{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2\big{)}\subseteq{\mathcal{M}}^{\vec{\alpha}-ss}\big{(}\mathbb{C}P^{1},D,2\big{)}$

for the functors $M^{\vec{\alpha}-s}$ and $M^{\vec{\alpha}-ss}$, respectively.

The Hitchin map introduced in Definition 2.4 induces morphisms

$\displaystyle h\colon\
{\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}}\to\mathcal{B},$

and for any $(F,G)\in\mathcal{B}$ we denote by

$\displaystyle{\overline{{\mathcal{M}}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}}_{(F,G)}$

its fiber over $(F,G)$.

In [13, 14] we studied stability of Higgs bundles with spectral curves such
that the curve in the associated elliptic fibration is of type $I_{n}$ $(1\leq
n\leq 3)$, $II$, $III$ or $IV$. The only new case that appears in the cases
treated here is that of a spectral curve $\Sigma$ such that the curve in the
associated elliptic fibration is of type $I_{0}^{*}=\tilde{D}_{4}$. As we will
see in Lemma 4.4, this case only appears if the spectral curve $\Sigma$ is a
section of $p\colon{\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ taken with multiplicity
two. From now on, we assume that $\Sigma$ is of this type, and in addition
that

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{E}})=0.$

Up to tensoring with a rank $1$ meromorphic Higgs bundle, the curve
$\Sigma=\Sigma_{(0,0)}$ is the image of the $0$-section $F=0$, $G=0$.

######  Assumption 3.4.

From now on we only consider Higgs bundles satisfying

$\displaystyle\deg_{\vec{\alpha}}({\mathcal{E}})=0.$

This assumption implies that

$\displaystyle\sum_{j=1}^{4}(\alpha_{j}^{+}+\alpha_{j}^{-})\in{\mathbb{Z}},$

so the space of possible parabolic weights belong to a finite union of affine
hyperplanes in $[0,1)^{8}$. We will say that the parabolic weights are generic
if for any choice of $\varepsilon_{j}\in\\{+,-\\}$ we have

$\displaystyle\sum_{j=1}^{4}\alpha_{j}^{\varepsilon_{j}}\notin{\mathbb{Z}}.$

Generic parabolic weights belong to the complement of finitely many proper
affine subspaces of the affine hyperplanes of all possible parabolic weights.
For generic parabolic weights, $\vec{\alpha}$-semi-stability is equivalent to
$\vec{\alpha}$-stability, because the parabolic degree of an invariant
subobject ${\mathcal{F}}$ is never an integer.

######  Proposition 3.5.

For generic parabolic weights the Hitchin fiber

$\displaystyle\overline{{\mathcal{M}}_{t}^{\vec{\alpha}-s}\big{(}\mathbb{C}P^{1},D,2,d\big{)}}_{(0,0)}$

over $(0,0)\in\mathcal{B}$ is of type $I_{0}^{*}$.

###### Proof.

It follows from [2, Theorem 0.2] that the moduli space in the assertion is a
smooth complete hyperKähler manifold. Moreover, according to [25, Lemma 1],
the Hitchin map defined on the moduli space is proper. It is well-known that
the generic Hitchin fibers are elliptic curves; indeed, they are (up to a
degree twist) the Jacobians of the elliptic curves appearing in the pencil of
spectral curves to be studied in Example 4.3. Now, the elliptic fibration of
Example 4.3 admits a central fiber of type $I_{0}^{*}$, therefore the
monodromy matrix of the pencil of spectral curves is conjugate to

$\displaystyle\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}.$

It follows from this that the monodromy matrix of its dual fibration is also
conjugate to the same matrix. As the monodromy matrix uniquely determines the
Kodaira type of a singular fiber in an elliptic surface, we deduce that the
only singular fiber of the Hitchin map is of type $I_{0}^{*}$ too.

Alternatively, it follows from [7] that an open subset of the moduli space is
a smooth quiver variety associated to the quiver $D_{4}^{(1)}$ and dimension
vector equal to the positive minimal integer generator of the radical of the
associated quadratic form. This quiver variety is isomorphic to a minimal
resolution of the surface singularity of type $D_{4}$. The exceptional
divisors of this resolution lie in the singular Hitchin fiber, so the singular
Hitchin fiber contains a curve of type $D_{4}$; the only Kodaira fiber
containing a curve of this type is $I_{0}^{*}$. ∎

##  4 Singular fibers in elliptic fibrations

An elliptic fibration is a map $\pi\colon X\to C$ where $X$ is a compact
complex surface, $C$ is a compact complex curve and the generic fiber is an
elliptic curve, i.e., smoothly diffeomorphic to the 2-torus. In the following
we will consider only those elliptic surfaces which admit a section, i.e.,
_basic elliptic surfaces_ in the terminology of [9, Section 3.1.4]. This
restriction is justified by the existence of the Hitchin section [12] in
moduli spaces of Higgs bundles, whose construction can easily be generalized
to moduli spaces of (logarithmic or irregular) singular Higgs bundles.
Alternatively, it follows from [24, Section 4] that the elliptic fibrations we
will encounter all originate from a pencil by blow-ups, hence all admit
sections. The singular fibers in an elliptic fibration have been classified by
Kodaira in [15]. From Kodaira’s list of singular fibers we will only need the
fibers $\widetilde{E}_{8}$, $\widetilde{E}_{7}$, $\widetilde{E}_{6}$, $II$,
$III$, $IV$ and $I_{n}^{*}$ $(n\geq 0)$, see Figs. 1 and 2 for their
description. (The fiber of type $II$ is not given pictorially – it is the cusp
fiber, which is topologically a sphere with a singular point, where the
singularity can be modeled by a cone on the trefoil knot. In $\mathbb{C}P^{2}$
such a curve can be given in homogeneous coordinates $[x:y:z]$ by the equation
$zy^{2}=x^{3}$.)

Figure 1: Plumbings of singular fibers of types (a) $I_{n}^{*}$, (b)
${\tilde{E}}_{8}$, (c) ${\tilde{E}}_{7}$, and (d) ${\tilde{E}}_{6}$. Integers
next to vertices indicate the multiplicities of the corresponding homology
classes in the fiber. All dots correspond to rational curves with self-
intersection $-2$. In $I_{n}^{*}$ we have a total of $n+5$ vertices; in
particular, $I_{0}^{*}$ admits a vertex of valency four. Figure 2: Singular
fibers of types $III$ and $IV$ in elliptic fibrations. In (a) the two curves
are tangent with multiplicity two, and in (b) the three curves pass through
one point and intersect each other there transversely. All curves have
multiplicity one and self-intersection $-2$.

Various possibilities of singular fibers in an elliptic fibration on a
rational elliptic surface were analyzed in detail in [18, 20, 23]. In
particular, we have

######  Theorem 4.1 ([23, Section 6]).

Suppose that $\pi\colon X\to\mathbb{C}P^{1}$ is an elliptic fibration on a
rational elliptic surface $X$. Assume furthermore that $\pi$ admits a singular
fiber of type $I_{n}^{*}$. Then $n\leq 4$. Furthermore, if $\pi$ has exactly
two singular fibers, and one of them is of type $I_{n}^{*}$, then $n=0$ and
both fibers are of type $I_{0}^{*}$.

######  Corollary 4.2.

There is no elliptic fibration on the rational elliptic surface with exactly
two singular fibers, one of which is $I_{n}^{*}$ with $n>0$.

We analyze the fibration with two $I_{0}^{*}$ fibers a little further. We can
construct such a fibration by considering the following pencils.

######  Example 4.3.

Consider the Hirzebruch surface ${\mathbb{F}}_{2}$ with infinity section of
square $(-2)$ and section of square $2$. Let $C_{0}$ be the curve given by a
section with multiplicity 2. Consider $C_{\infty}$ as the union of the section
at infinity with multiplicity 2 together with 4 distinct fibers of the ruling
on ${\mathbb{F}}_{2}$. It is easy to see that the two curves are homologous,
and indeed, can be given as zero-sets of two sections of the same holomorphic
line bundle, hence there is a pencil of curves containing both. Blowing up the
four basepoints (which are the intersections of the fibers with the section)
twice, we get an elliptic fibration on the 8-fold blow-up of
${\mathbb{F}}_{2}$ (which is a rational elliptic surface), with two singular
fibers, each of type $I_{0}^{*}$. For the blow-up process, see Fig. 3.

Figure 3: On the left the Hirzebruch surface (with the ruling), together with
the curves $C_{\infty}=\sigma_{\infty}\cup F_{1}\cup F_{2}\cup F_{3}\cup
F_{4}$ and $C_{0}=\sigma_{0}$ is depicted. The concentric circles indicate
double blow-ups – after one blow-up at each of the four points we get the
middle configuration, while four further blow-ups provide the right hand
diagram, with two $I_{0}^{*}$ fibers.

In some sense the converse of the above example also holds. Indeed, if we get
a fibration by blowing up a pencil on ${\mathbb{F}}_{2}$ containing
$C_{\infty}$, then the pencil has at least 4 basepoints, each on different
fibers of the ruling on ${\mathbb{F}}_{2}$, hence the fibration has at least
four disjoint $(-1)$-sections. In addition, these sections intersect different
curves in the fiber (of type $I_{0}^{*}$) coming from the curve $C_{\infty}$
at infinity.

######  Lemma 4.4.

Suppose that $\pi\colon X\to\mathbb{C}P^{1}$ is an elliptic fibration with two
singular fibers $F_{0}$, $F_{1}$, both of type $I_{0}^{*}$, and the fibration
admits four disjoint sections $($of homological square $(-1))$ which intersect
different leaves of $F_{0}$. Then this fibration comes from the construction
of Example 4.3.

###### Proof.

Let $G_{i}^{1}$, $G_{i}^{2}$, $G_{i}^{3}$ and $G_{i}^{4}$ denote the
$(-2)$-curves in the fiber $F_{i}$ (with $i=0,1$) corresponding to the leaves
of the plumbing presentation of these fibers of type $I_{0}^{*}$. Suppose
furthermore that the four sections are denoted by $E_{1}$, $E_{2}$, $E_{3}$
and $E_{4}$, with the understanding that $E_{j}$ intersects $G_{0}^{j}$ (and
no other $G_{0}^{k}$), see Fig. 4.

Figure 4: The two $I_{0}^{*}$ fibers and the sections $E_{i}$ are shown in the
diagram. $E_{1}$ intersects $G_{1}^{1}$; the further intersections of the
$E_{i}$’s with $i>1$ provide various cases, analysed in the proof.

We can also assume (after possibly renumbering) that $G_{1}^{1}$ is the curve
among the $G_{1}^{j}$’s intersected by the most $E_{i}$’s. Note that each
section intersects a fiber of type $I_{0}^{*}$ only in curves corresponding to
leaves (since the central curve has multiplicity 2, while a section intersects
the fiber exactly once (with multiplicity)).

If $G_{1}^{1}$ is intersected by a single $E_{i}$, then by our choice each
$G_{1}^{j}$ is intersected by a single section, hence blowing down first the
sections and then the images of the curves $G_{1}^{j}$’s, we get the
configuration of curves described in Example 4.3.

Suppose now that $G_{1}^{1}$ is intersected by two $E_{i}$’s, say by $E_{1}$
and $E_{2}$. Blow these curves and $G_{0}^{1}$, $G_{0}^{2}$ down. In the
result the curve $G_{1}^{1}$ will be a $(+2)$-curve, hence the complement of
this curve should be negative definite. On the other hand, $E_{3}$
(intersecting $G_{0}^{3}$) also intersects the fiber $F_{1}$, in the curve
$G_{1}^{2}$ (after possible renumbering). Now blowing down $E_{3}$ and
$G_{1}^{2}$, the curve $G_{0}^{3}$ will be a complex curve of self-
intersection 0. Since a complex curve does not vanish in homology, it defines
a nontrivial homology class in the complement of the $(+2)$-curve with zero
self-intersection, providing a contradiction.

Assume now that $G_{1}^{1}$ is intersected by three sections, say $E_{1}$,
$E_{2}$ and $E_{3}$. Blowing these and the corresponding curves from $F_{0}$
(i.e., $G_{0}^{1}$, $G_{0}^{2}$, $G_{0}^{3}$) down, the curve defined by
$G_{1}^{1}$ will have self-intersection $(+4)$ in
$\mathbb{C}P^{2}\\#3{\overline{{\mathbb{C}}P}}^{2}$ with four curves
$\big{(}G_{0}^{4},G_{1}^{2},G_{1}^{3},G_{1}^{4}\big{)}$ in its complement,
linearly independent in homology. This is a contradiction again.

Assume finally that all four sections intersect $G_{1}^{1}$. Blowing the
sections and then the curves $G_{0}^{i}$ down, we get the image of $G_{1}^{1}$
in $\mathbb{C}P^{2}\\#{\overline{{\mathbb{C}}P}}^{2}$ of self-intersection
$(+6)$, with three curves in the complement, defining linearly independent
homology classes, providing the last desired contradiction. ∎

######  Example 4.5.

In a similar manner, we can specify a pencil of curves in the complex
projective plane which (after 9 blow-ups) provides an elliptic fibration with
two singular fibers, both of type $I_{0}^{*}$. Indeed, let
$S_{\infty}=\cup_{i=1}^{3}\ell_{i}$ be the union of three lines $\ell_{1}$,
$\ell_{2}$, $\ell_{3}$ passing through a fixed point $P\in\mathbb{C}P^{2}$,
each line with multiplicity one. Define $S_{0}$ to be the union $L_{1}\cup
L_{2}$, where $L_{1}$ passes through $P$ while $L_{2}$ does not; we consider
$L_{1}$ with multiplicity 1 and $L_{2}$ with multiplicity 2, see Fig. 5.

Figure 5: Curve configuration in ${\mathbb{C}}P^{2}$ used in Example 4.5.

In this way both $S_{\infty}$ and $S_{0}$ are (singular) curves of degree 3,
and the resulting pencil (with four basepoints, the three intersection points
$\ell_{i}\cap L_{2}$ and $P$) will define an elliptic fibration with exactly
two $I_{0}^{*}$ fibers and no further singular fibers.

######  Example 4.6.

A more symmetric presentation of the same fibration can be given as follows.
Consider $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ (which is the Hirzebruch
surface ${\mathbb{F}}_{0}$), together with the ruling
$r\colon\mathbb{C}P^{1}\times\mathbb{C}P^{1}\allowbreak\to\mathbb{C}P^{1}$.
Choose two sections $S_{0}$ and $S_{\infty}$ of the ruling, and four distinct
fibers $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$. Define the two curves $C_{0}$,
$C_{\infty}$ by

$C_{0}=S_{0}\cup F_{1}\cup F_{2},\qquad C_{\infty}=S_{\infty}\cup F_{3}\cup
F_{4},$

where each $F_{i}$ appears with multiplicity 1, while both $S_{0}$ and
$S_{\infty}$ appear with multiplicity 2, see Fig. 6.

Figure 6: The symmetric curve configuration in
${\mathbb{F}}_{0}={\mathbb{C}}P^{1}\times{\mathbb{C}}P^{1}$ providing a
elliptic fibration with two $I_{0}^{*}$ fibers.

The two curves obviously generate a pencil on
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ with four basepoints. Blowing up each
basepoints twice, we get the desired elliptic fibration with two singular
fibers, each of the type of $I_{0}^{*}$, originated from the curves $C_{0}$
and $C_{\infty}$.

##  5 The proofs

In this section we give the proofs of Theorems 1.1–1.6. For sake of
completeness and in order to have a uniform treatment, we include and provide
references to our previously obtained results whenever applicable.

###  5.1 The case of one pole

###### Proof of Theorems 1.1 and 1.2.

These theorems are special cases of [13, Theorem 1.1] and [13, Theorem 1.3],
respectively; hence we do not give any further details here. ∎

###  5.2 The case of two poles

This case has been extensively studied in [14]. Recall that in this case we
have two subcases depending on the multiplicities of the poles: these
multiplicities can be (a) 2 and 2 or (b) 3 and 1.

If the two multiplicities are 2 and 2, then the fiber at infinity is either of
type $I_{2}^{*}$, $I_{3}^{*}$ or $I_{4}^{*}$. According to Corollary 4.2,
elliptic fibrations with those singular fibers on a rational elliptic surface
have at least two further singular fibers, hence they do not occur in the
cases under the current investigation. (For the complete picture in this case
see [14, Theorems 2.1, 2.2, 2.3], depending on whether the polar parts are
twisted or untwisted. See also [14, Table 1].)

###### Proof of Theorems 1.3 and 1.4.

Suppose now that the two multiplicities are 3 and 1. By examining [14,
Theorems 2.4, 2.5, 2.6, 2.7], we see that there are two cases with single
singular fibers in the Hitchin fibration, as given in the two theorems. ∎

###  5.3 The case of three poles

Suppose now that there are three poles. In this case the fiber at infinity in
the elliptic fibration is either of type $I_{1}^{*}$ (when the polar part at
the pole with multiplicity 2 is untwisted (i.e., regular semi-simple)) or of
type $I_{2}^{*}$ (when this polar part is twisted, i.e., has non-vanishing
nilpotent part).

###### Proof of Theorem 1.5.

According to Corollary 4.2, next to these singular fibers we cannot have a
single singular fiber in any elliptic fibration on a rational elliptic
surface, verifying the statement of the theorem. ∎

###  5.4 The case of four poles

###### Proof of Theorem 1.6.

Consider the curve $C_{\infty}\subset{\mathbb{F}}_{2}$ which is the union of
the section at infinity (with self-intersection $(-2)$) with multiplicity 2
and four fibers of the ruling on ${\mathbb{F}}_{2}$. We can choose coordinates
on $\mathbb{C}P^{1}$ in such a way that the four fibers are over $0$, $1$,
$\infty$ and $t$, where $t\in{\mathbb{C}}$ described the holomorphic type of
the fiber at infinity.

By Lemma 4.4 the pencil containing the above curve provides an elliptic
fibration with two $I_{0}^{*}$ fibers if and only if the pencil contains a
section of the ruling $p\colon{\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ (which is of
multiplicity 2). By virtue of Proposition 3.5, in this case the Hitchin
fibration on the extended moduli space $\overline{{\mathcal{M}}}$ of (1.1)
also has a singular fiber of type $I_{0}^{*}$ for generic parabolic weights.
In particular, the pencil has four basepoints, one on each of the four
distinguished fibers of the ruling of ${\mathbb{F}}_{2}$. Suppose that the
four basepoints are at $a$, $b$, $c$, $d$ in the fibers over the points
$0,1,t,\infty\in\mathbb{C}P^{1}$.

The ruling $p\colon{\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ admits a complex
3-dimensional family of sections (all disjoint from the section at infinity),
which can be given by homogeneous degree-2 polynomials on $\mathbb{C}P^{1}$.
In particular, if the homogeneous coordinates on $\mathbb{C}P^{1}$ are denoted
by $[u:v]$, then a section is of the form $\alpha v^{2}+\beta uv+\gamma
u^{2}$. The section intersects the fiber over $0$ (given by the homogeneous
coordinates $[0:1]$) at $\alpha$, hence we need $\alpha=a$. Similarly, over
$\infty=[1:0]$ the intersection point is $\gamma$, so we have $\gamma=d$.
Finally, over the point $1=[1:1]\in\mathbb{C}P^{1}$ we get that
$\alpha+\beta+\gamma=b$, implying $\beta=b-a-d$. In conclusion, the unique
section intersecting the fibers over $0$, $1$, $\infty$ in the prescribed
points $a$, $b$, $d$ can be given by the equation

$av^{2}+(b-a-d)uv+du^{2}.$

This section takes the value $a+(b-a-d)t+dt^{2}$ at $t=[t:1]$, providing the
condition

$c=a+(b-a-d)t+dt^{2}$

for the existence of the desired section (and therefore for the other fiber to
be of type $I_{0}^{*}$), and concluding the proof of the statement. ∎

This argument concludes the proof of Theorems 1.1–1.6. We are left with
Theorem 1.7, the proof of which will be provided in the next section.

##  6 Pencils and fibrations with fiber $\boldsymbol{I_{0}^{*}}$

We devote this section to the proof of Theorem 1.7. We will describe pencils
in the Hirzebruch surface ${\mathbb{F}}_{2}$, and sometimes (for the sake of
convenience) we also provide a pencil in the complex projective plane giving
rise to a fibration with the same desired properties. There is a rather simple
procedure for converting pencils in ${\mathbb{F}}_{2}$ to pencils in
$\mathbb{C}P^{2}$ (by blowing up and down); in most cases we will omit the
actual steps of this procedure.

The possible combinations of singular fibers in elliptic fibrations on a
rational elliptic surface have been determined in [18, 20], see also [23].
Recall the types of elliptic singular fibers listed in Section 4; there are a
few further types we need to encounter in the following. $I_{1}$ denotes the
nodal fiber (also called a fishtail fiber), which is a nodal elliptic curve,
and topologically an immersed 2-sphere with one positive transverse double
point. The fiber $I_{n}$ consists of $n$ rational curves
$E_{0},\ldots,E_{n-1}$ such that $E_{i}$ transversely intersects $E_{i-1}$ and
$E_{i+1}$ in two distinct points (one each), and the indices are viewed mod
$n$. In particular, $I_{2}$ consists of two rational curves $E_{0}$ and
$E_{1}$ intersecting each other in two distinct points – as the two points
converge to each other, the limit of such a sequence of singular fibers is a
type $III$ fiber.

There are 19 distinct configurations of singular fibers on
$\mathbb{C}P^{2}\\#9{\overline{\mathbb{C}P^{2}}}$ containing $I_{0}^{*}$ as
one of the singular fiber, which we list now (separating the singular fibers
in a fibration with an addition sign), grouped in four groups. Below we list
the fibers next to the (always existing) $I_{0}^{*}$ fiber:

$I_{0}^{*}$, |  $I_{4}+2I_{1}$, |  $IV+II$, |  $IV+2I_{1}$, |  $I_{3}+II+I_{1}$, |  $I_{3}+3I_{1}$;
---|---|---|---|---|---
$2III$, |  $III+I_{2}+I_{1}$, |  $III+II+I_{1}$, |  $III+3I_{1}$; |  |
$3I_{2}$, |  $2I_{2}+2I_{1}$, |  $I_{2}+2II$, |  $I_{2}+II+2I_{1}$, |  $I_{2}+4I_{1}$; |
$3II$, |  $2II+2I_{1}$, |  $II+4I_{1}$, |  $6I_{1}$. |  |
Table 1: Possible singular fiber combinations next to $I_{0}^{*}$.

In the following we give examples of pencils of curves in ${\mathbb{F}}_{2}$
(and sometimes in $\mathbb{C}P^{2}$) providing elliptic fibrations on rational
elliptic surfaces containing the configurations listed above. The curves are
chosen so that we do have control on the further singular fibers of the
fibration. In the following we will describe pencils in ${\mathbb{F}}_{2}$ and
in $\mathbb{C}P^{2}$ by specifying two curves (in each complex surface) and
consider the pencil generated by them – and the singular fibers will then be
in the fibrations we get by blowing up the base points of these pencils.

The curve $C_{\infty}\subset{\mathbb{F}}_{2}$ will be part of all the pencils;
$C_{\infty}$ consists of the union of four fibers $F_{1}$, $F_{2}$, $F_{3}$,
$F_{4}$ of the ruling ${\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ (all with
multiplicity 1) and the section at infinity of the ruling (this curve with
multiplicity 2) – as it was already described in Example 4.3. Similarly, one
curve in each pencil in $\mathbb{C}P^{2}$ will be the same:
$S_{\infty}=\ell_{1}\cup\ell_{2}\cup\ell_{3}$ is the cubic curve we get by
taking the union of three projective lines $\ell_{i}$ ($i=1,2,3$) (all with
multiplicity 1) passing through a fixed point $P\in\mathbb{C}P^{2}$. The curve
$S_{\infty}$ is shown by the thick red lines on the right of the figures
below. (This curve already appeared in Example 4.5.)

We say that a pencil in ${\mathbb{F}}_{2}$ _has a section_ if one of the
curves in the pencil other than $C_{\infty}$ has more than one components – in
this case each component of this curve is indeed a section of the ruling of
the Hirzebruch surface ${\mathbb{F}}_{2}$.

In some cases we need to see more than two singular fibers in the fibration.
To this end, we will apply a principle (formulated in Lemma 6.1 below) and a
method of generating more curves in a pencil (described after Lemma 6.1).

Suppose that the pencil is given by the two curves $C_{0}$ and $C_{\infty}$ in
${\mathbb{F}}_{2}$ (or $S_{0}$ and $S_{\infty}$ in $\mathbb{C}P^{2}$),
intersecting each other in points $P_{1},\ldots,P_{k}$ with multiplicities
$n_{1},\ldots,n_{k}$, satisfying $\sum_{i=1}^{k}n_{i}=8$ in ${\mathbb{F}}_{2}$
and $\sum_{i=1}^{k}n_{i}=9$ in $\mathbb{C}P^{2}$. (In case $n_{i}>1$, not only
$P_{i}$ is fixed, but also the higher order tangencies are part of the given
data, for example by specifying the points in the exceptional divisor of the
blow-up the strict transforms of the curves pass through.) We mentioned that
the elements of matrices in equations (2.9) and (2.10) encode the base locus
of the pencil, hence $P_{1},\ldots,P_{k}$ are determined by these data. Next,
we will construct some examples where we choose these parameters
$P_{1},\ldots,P_{k}$ appropriately on four distinguished fibers of the ruling,
so that all combinations of further singular fibers appear. It is easy to see
that these chosen parameters can be transformed into
$a_{\pm},\dots,d_{\pm},a,\dots,d$. (For convenience, from now on we will not
insist on having the distinguished fibers over the points $0$, $1$, $\infty$ –
the application of a simple Möbius transformation would provide this extra
feature.)

Suppose now that $C$ (or $S$) is a complex curve in ${\mathbb{F}}_{2}$ (or in
$\mathbb{C}P^{2}$, respectively) which is homologous to $C_{0}$ and
$C_{\infty}$ (or $S_{0}$ and $S_{\infty}$) and passes through
$P_{1},\ldots,P_{k}$, intersecting $C_{0}$ (or $S_{0}$) in those points with
multiplicities $n_{1},\ldots,n_{k}$ (and with the required higher order
tangencies in case $n_{i}>1$).

######  Lemma 6.1.

Under the above circumstances $C$ $($or $S)$ is in the pencil in
${\mathbb{F}}_{2}$ generated by $C_{0}$ and $C_{\infty}$ $($or $S_{0}$ and
$S_{\infty}$ in case we work in $\mathbb{C}P^{2})$.

###### Proof.

Take any point $P\in C\setminus\\{P_{1},\ldots,P_{k}\\}$. There exists a
unique $t\in\mathbb{C}P^{1}$ such that $C_{t}$ passes through $P$. We then see
that $C$ intersects $C_{t}$ with multiplicity $>\sum_{i=1}^{k}n_{i}$, which is
by assumption the self-intersection number of $C$. It then follows that
$C=C_{t}$. ∎

We can construct curves in ${\mathbb{F}}_{2}$ disjoint from the section
$\sigma_{\infty}$ at infinity and intersecting the fiber of the ruling twice
(possibly once, with multiplicity two) in the following way. Since the curve
is disjoint from $\sigma_{\infty}$, it is a double section of the bundle
${\mathcal{O}}(2)\to\mathbb{C}P^{1}$. Such double sections can be given by
sections of ${\mathcal{O}}(4)\to\mathbb{C}P^{1}$: for a given section $\sigma$
of ${\mathcal{O}}(4)$, use the identification
${\mathcal{O}}(2)\otimes{\mathcal{O}}(2)\cong{\mathcal{O}}(4)$ and get a
double section of ${\mathcal{O}}(2)$: over $P\in\mathbb{C}P^{1}$ take those
points $\zeta\in{\mathcal{O}}(2)$ which satisfy
$\zeta\otimes\zeta=\sigma_{P}$, see (2.3) with the notational change of
denoting $G_{\theta}$ by $\sigma$. There are two such points ($\zeta$ and
$-\zeta$) if $\sigma_{P}\neq 0$ and there is a single one if $\sigma_{P}=0$.
(Indeed, this is the classical construction of double branched covers.) In
turn, sections of ${\mathcal{O}}(4)\to\mathbb{C}P^{1}$ can be given by
homogeneous degree-4 polynomials in the homogeneous coordinates $[u:v]$ on
$\mathbb{C}P^{1}$.

Here are some instructive examples of this phenomenon:

  * •

The polynomial $u^{4}$ (as a section of ${\mathcal{O}}(4)$) provides two
sections in ${\mathcal{O}}(2)$ which are tangent to each other over the point
$[0:1]\in\mathbb{C}P^{1}$.

  * •

The polynomial $u^{2}v^{2}$ gives rise to two sections of ${\mathcal{O}}(2)$,
intersecting each other transversally in two points over $[0:1]$ and
$[1:0]\in\mathbb{C}P^{1}$.

  * •

The polynomial $u^{2}v(u+v)$ defines a double section in ${\mathcal{O}}(2)$
which has a node over $[0:1]\in\mathbb{C}P^{1}$, and is tangent to the fibers
of the ruling at $[1:0]$ and $[-1:1]\in\mathbb{C}P^{1}$.

  * •

The polynomial $u^{3}v$ will define a cuspidal curve which is a double section
in ${\mathbb{F}}_{2}$ – the cusp point is over $[0:1]$ and the fiber over
$[1:0]$ is tangent to the cuspidal curve.

We will start our examples with configurations admitting sections.

######  Example 6.2 (fibration with another $I_{0}^{*}$).

If $C_{0}$ is a section of the ruling ${\mathbb{F}}_{2}\to\mathbb{C}P^{1}$
with square 2 and we take it with multiplicity 2, the pencil generated by
$(C_{0},C_{\infty})$ provides a fibration with two singular fibers, both of
type $I_{0}^{*}$, cf. Fig. 7.

Figure 7: The diagram on the left shows two curves in ${\mathbb{F}}_{2}$, one
of them is the curve $C_{\infty}$, while the other one is a section of the
ruling with multiplicity two (denoted by $\sigma$). Thick (red) lines provide
one of the curves, while the thin (blue) segment defines the other curve. The
fibration defined by the pencil contains two $I_{0}^{*}$ fibers. The two
degree-3 curves in $\mathbb{C}P^{2}$ on the right generate a pencil which
gives rise to a fibration admitting two $I_{0}^{*}$ fibers.

In a similar fashion, by taking two lines $L_{1},L_{2}\subset\mathbb{C}P^{2}$
such that $L_{1}$ passes through the fixed point $P$ while $L_{2}$ avoids it,
and defining $S_{0}$ as the union of $L_{1}$ and $L_{2}$ (with $L_{1}$ of
multiplicity 1 and $L_{2}$ of multiplicity 2) we get a pencil on
$\mathbb{C}P^{2}$ induced by $(S_{0},S_{\infty})$ so that the resulting
elliptic fibration has two $I_{0}^{*}$ fibers (cf. Example 4.5 and Fig. 7).

######  Example 6.3 (fibration with an $I_{4}$ fiber).

A fibration with an $I_{4}$ fiber (next to the $I_{0}^{*}$ coming from
$C_{\infty}$) can be given in ${\mathbb{F}}_{2}$ by considering two sections
$\sigma_{1}$, $\sigma_{2}$ of the ruling (each with multiplicity 1)
intersecting each other once on $F_{1}$ and once on $F_{4}$, as shown by the
left diagram of Fig. 8. A similar pencil in $\mathbb{C}P^{2}$ can be given by
taking $S_{0}=L_{1}\cup L_{2}\cup L_{3}$, where $L_{i}$ are three distinct
lines, $L_{1}$ passes through $P$, while $L_{2}$, $L_{3}$ do not, and moreover
the intersection point $L_{2}\cap L_{3}$ is on one of the $\ell_{i}$; the
configuration is shown by the right diagram of Fig. 8. Note that by the
classification of possible singular fibers, it follows that next to the
$I_{0}^{*}$ and $I_{4}$ fibers there is a single possibility for other
singular fibers: these must be two fishtail ($I_{1}$) fibers.

Figure 8: The diagram on the left shows two sections in ${\mathbb{F}}_{2}$
intersecting each other in two points of $C_{\infty}$. (The sections are
denoted by $\sigma_{1}$ and $\sigma_{2}$.) Once again, thick (red) lines
provide one of the curves, while the thin (blue) segments define the other
curve. The pencil generated by these curves provides a fibration with an
$I_{0}^{*}$ fiber and an $I_{4}$ fiber. The two degree-3 curves in
$\mathbb{C}P^{2}$ on the right diagram generate a pencil which gives rise to a
fibration admitting an $I_{0}^{*}$ fiber and an $I_{4}$ fiber.

######  Example 6.4 (fibration with a type $IV$ fiber).

To get a fibration with a fiber of type $IV$ next to $I_{0}^{*}$, take two
sections in ${\mathbb{F}}_{2}$, which are now tangent to each other at a point
of $F_{1}$, see the left diagram of Fig. 9. In $\mathbb{C}P^{2}$ consider
$S_{0}$ to be three distinct lines $L_{1}$, $L_{2}$, $L_{3}$ again, so that
$L_{1}$ passes through $P$, while all three lines pass through
$P^{\prime}\in\mathbb{C}P^{2}$ which is distinct from $P$, see the right
diagram of Fig. 9.

Figure 9: The diagram depicts two pencils (one in ${\mathbb{F}}_{2}$ and the
other one in $\mathbb{C}P^{2}$) giving fibrations with an $I_{0}^{*}$ fiber
and a type $IV$ fiber. The conventions are the same as earlier.

In order to get a pencil generating a fibration with an $I_{0}^{*}$, a type
$IV$ and a type $II$ fiber, consider the sections $u^{4}$ and $uv^{3}$ in
${\mathcal{O}}(4)$, giving rise to two double sections in ${\mathbb{F}}_{2}$.
Let $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$ be chosen as follows: $F_{1}$ is over
$[0:1]$ (where the two sections are tangent to each other in one of the double
section), and the other fibers are over $[u_{i}:1]$, where $u_{i}^{3}=1$. The
corresponding fibers pass through the intersections of the sections of
${\mathcal{O}}(4)$, hence each such fiber passes through two basepoints in the
pencil in ${\mathbb{F}}_{2}$, therefore both curves are in the pencil.

In a similar manner, when starting with the sections $u^{4}$ and
$uv^{2}(u+v)$, we get a pencil giving a fibration with an $I_{0}^{*}$ fiber, a
type $IV$ fiber and an $I_{1}$ fiber, hence (again by the classification)
there is a single further singular fiber which is an $I_{1}$. (The fibers are
over the points $[u_{i}:1]$ where $u_{i}$ solves $u^{3}=u+1$.) These two
examples exhaust the cases having an $I_{0}^{*}$ fiber and a type $IV$ fiber
next to it.

######  Example 6.5 (fibration with an $I_{3}$ fiber).

Consider now two sections $\sigma_{1}$, $\sigma_{2}$ of the ruling on
${\mathbb{F}}_{2}$ intersecting each other in $F_{1}$, and in a further point
which is not on the chosen fibers $F_{i}$ ($i=2,3,4$). Taking this curve as
$C_{0}$, the resulting pencil will have an $I_{3}$ fiber next to the
$I_{0}^{*}$ fiber. In $\mathbb{C}P^{2}$ we consider $S_{0}$ which is again a
union of three lines $L_{1}$, $L_{2}$, $L_{3}$, with $L_{1}$ passing through
$P$ and $L_{2}$, $L_{3}$ intersecting each other in a generic point
$P^{\prime}$ (which is not on $L_{1}$ and on any $\ell_{i}$); see Fig. 10.

Figure 10: The diagram depicts two pencils (one in ${\mathbb{F}}_{2}$ and the
other one in $\mathbb{C}P^{2}$) giving fibrations with an $I_{0}^{*}$ fiber
and an $I_{3}$ fiber. The conventions are the same as earlier.

Fibrations admitting an $I_{0}^{*}$ and an $I_{3}$ fiber on the rational
elliptic surface have further singular fibers; the possibilities are: either a
cusp and a fishtail, or three fishtails. The second case is the generic one.
Let us see how to construct such a fibration explicitly. For this purpose, we
take

$\displaystyle\sigma_{1}=(u+v)^{2}v(u-v),\qquad\sigma_{2}=u^{2}v^{2}.$ (6.1)

Then $\sigma_{1}$ is not a complete square, but has two equal zeroes, so the
equation $\zeta\otimes\zeta=\sigma_{1}$ defines a double section with a nodal
point over $[u:v]=[-1:1]$, giving rise to a fiber of type $I_{1}$. On the
other hand, $\sigma_{2}$ is a complete square with zeroes at $[u:v]=[0:1]$ and
$[u:v]=[1:0]$. The basepoints are determined by the equation

$\displaystyle(u+v)^{2}v(u-v)=u^{2}v^{2},$

which has roots $[u:v]=[1:0]$ and $[u:v]=[u_{i}:1]$ where $u_{1}$, $u_{2}$,
$u_{3}$ are the roots of

$\displaystyle(u+1)^{2}(u-1)-u^{2}=0.$

As one of the roots $[1:0]$ of $\sigma_{2}$ is at the same time a basepoint
while its other root $[0:1]$ is not, $\sigma_{2}$ gives rise to a fiber of
type $I_{3}$. Now, let the parameter of the pencil be denoted
$\lambda=[\lambda_{1}:\lambda_{2}]\in\mathbb{C}P^{1}$. Then, the curve
corresponding to $\lambda$ in the pencil has equation

$\displaystyle(\lambda_{1}+\lambda_{2})\zeta\otimes\zeta=\sigma_{\lambda},$

where

$\displaystyle\sigma_{\lambda}=\lambda_{1}\sigma_{1}+\lambda_{2}\sigma_{2}.$

The parameter value $\lambda=[-1:1]$ corresponds to a degenerate curve
consisting of the fibers containing the basepoints (and the section at
infinity in ${\mathbb{F}}_{2}$ with multiplicity $2$), and gives rise to the
fiber of type $I_{0}^{*}$. From now on we assume that $\lambda\neq[-1:1]$, and
the equation of the corresponding curve can be written as

$\displaystyle\zeta\otimes\zeta=\frac{\sigma_{\lambda}}{\lambda_{1}+\lambda_{2}}.$

Expressing the right hand side of this equation using equation (6.1), and
taking $v=1$ we get

$\lambda_{1}u^{3}+(\lambda_{1}+\lambda_{2})u^{2}-\lambda_{1}u-\lambda_{1}.$

The discriminant of this polynomial is given by

$\displaystyle\lambda_{1}\lambda_{2}\frac{32\lambda_{1}^{2}+13\lambda_{1}\lambda_{2}+4\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{4}}.$

The parameter values $\lambda_{1}=0$ and $\lambda_{2}=0$ define $\sigma_{2}$
and $\sigma_{1}$ respectively, hence we have already treated them. (Remember
that $\lambda_{1}$, $\lambda_{2}$ may not simultaneously vanish.) Consider the
roots
$\lambda^{+}=[\lambda_{1}^{+}:\lambda_{2}^{+}],\lambda^{-}=[\lambda_{1}^{-}:\lambda_{2}^{-}]\in\mathbb{C}P^{1}$
of

$\displaystyle 32\lambda_{1}^{2}+13\lambda_{1}\lambda_{2}+4\lambda_{2}^{2};$

a straightforward check shows that $\lambda^{+}\neq\lambda^{-}$ and that

$\displaystyle\lambda^{+},\lambda^{-}\in{\mathbb{C}}\setminus\\{[0:1],[-1:1]\\}\subset\mathbb{C}P^{1}.$

Then, $\sigma_{\lambda^{\pm}}$ both have a double zero, therefore each of the
corresponding curves has at least one singular point. Moreover, these singular
curves are different from the ones we found earlier. By the classification,
the corresponding curves in the fibration are both of type $I_{1}$, and we are
done.

Now, let us turn our attention to finding an example of the non-generic case,
i.e., a cuspidal curve beside $I_{0}^{*}$ and $I_{3}$. Using our earlier
method of generating double sections from homogeneous degree-4 polynomials, we
proceed as follows. Take the polynomials $u^{2}v^{2}$ and $u(u+v)^{3}$, and
consider the fibers $F_{1},\ldots,F_{4}$ over the points
$[u_{i}:1]\in\mathbb{C}P^{1}$ for which $u^{2}=u(u+1)^{3}$ holds. These are
the fibers passing through the intersection points of the sections in
${\mathcal{O}}(4)$, hence of pairs of basepoints in ${\mathbb{F}}_{2}$. The
second curve is a cuspidal curve, hence the resulting fibration will have an
$I_{0}^{*}$ fiber, together with an $I_{3}$ fiber and a cusp (and a further
$I_{1}$ fiber, as dictated by the classification).

######  Example 6.6 (fibration with $III$).

The curve $C_{0}$ in ${\mathbb{F}}_{2}$ is a union of two sections, which are
tangent to each other in a point not on the chosen fibers $F_{i}$
($i=1,2,3,4$). (Alternatively, if we get a fibration with a double-section
which has a cusp point on the fiber $F_{4}$ – as shown in Fig. 11 – we get a
fibration with an $I_{0}^{*}$ fiber and a type $III$ fiber.) The curve $S_{0}$
in the pencil in $\mathbb{C}P^{2}$ now consists of two curves, a line $L_{1}$
passing through $P$ and a quadric $Q$ which does not pass through $P$ and it
is tangent to $L_{1}$, as shown by the right diagram of Fig. 11.

Figure 11: Continuing with the same conventions as above, the diagrams show
pencils defining fibrations with an $I_{0}^{*}$ fiber and a type $III$ fiber.
In this case we describe two pencils in ${\mathbb{F}}_{2}$, one having eight,
the other one seven basepoints.

There are four possibilities of the further fibers (listed in the second row
of Table 1): it can be (a) another type $III$ fiber, (b) $I_{2}$ and $I_{1}$,
(c) $II$ and $I_{1}$ and finally (d) three $I_{1}$’s.

By taking the degree-4 polynomials $u^{3}v$ and $uv^{3}$ and the fibers over
$[0:1],[\pm 1:1],[1:0]\allowbreak\in\mathbb{C}P^{1}$, the pencil will give
rise to a fibration with an $I_{0}^{*}$ fiber and two type $III$ fibers.
Similarly, by taking $u^{4}$ and $(u+v)v^{3}$ and the fibers over $[u_{i}:1]$
with $u_{i}$ solving $u^{4}=(u+1)$, we get a pencil giving a fibration with an
$I_{0}^{*}$, a type $III$ and a cusp (type $II$) fibers, and hence necessarily
a further $I_{1}$ fiber. In a similar manner, starting with $u^{3}v$ and
$uv^{2}(u+v)$, and taking the fibers over $[0:1],[1:0]\in\mathbb{C}P^{1}$ and
$[u_{i}:1]\in\mathbb{C}P^{1}$ with $u_{i}$ solving $u^{2}=u+1$, we get a
pencil giving a fibration with an $I_{0}^{*}$ fiber, a type $III$ and an
$I_{2}$ fibers (and therefore a further $I_{1}$). In the generic situation we
have three $I_{1}$ fibers next to $I_{0}^{*}$ and $III$. This case can be
shown to exist by the same method as the generic case in Example 6.5 has been
handled. Specifically, setting $\sigma_{1}=u^{4}$ (giving rise to a curve of
type $III$ with singularity over $[0:1]$) and $\sigma_{2}=(u-v)^{2}v(u+v)$
(giving rise to a curve of type $I_{1}$ with singularity over $[1:1]$), the
generic curve in the pencil again reads as

$\displaystyle\sigma_{\lambda}=\lambda_{1}\sigma_{1}+\lambda_{2}\sigma_{2}.$

A computation shows that the discriminant of $\sigma_{\lambda}$ is given by

$\displaystyle\Delta=\lambda_{1}\lambda_{2}^{3}\big{(}256\lambda_{1}^{2}-107\lambda_{1}\lambda_{2}-32\lambda_{2}^{2}\big{)}.$

The quadratic term in the parentheses admits two distinct roots in
${\mathbb{C}}^{\times}$, so there exist two values
$\lambda^{\pm}=\big{[}\lambda_{1}^{\pm}:\lambda_{2}^{\pm}\big{]}$ such that
$\sigma_{\lambda^{\pm}}$ is singular. By the classification, in the fibration
these singular curves must be of type $I_{1}$. Hence, together with
$[\lambda_{1}:\lambda_{2}]=[0:1]$ and $[\lambda_{1}:\lambda_{2}]=[1:0]$ we
have a curve of type $III$ and three curves of type $I_{1}$.

######  Example 6.7 (fibration with an $I_{2}$ fiber).

To get a fibration with an $I_{2}$ fiber, we consider two sections
$\sigma_{1}$, $\sigma_{2}$ intersecting each other in two distinct points,
none of them on the chosen fibers $F_{i}$ ($i=1,2,3,4$). Alternatively, we can
choose a nodal curve with the node on the fiber $F_{4}$, shown on Fig. 12.

Figure 12: The pencils give rise to fibrations with an $I_{0}^{*}$ fiber and
an $I_{2}$ fiber.

To get such a fibration from a pencil in $\mathbb{C}P^{2}$, take $L_{1}$ as
earlier, together with a quadric, which does not pass through $P$ and it is
transverse to $L_{1}$, see Fig. 12.

There are five cases for the further singular fibers, as shown in the third
row of Table 1: next to $I_{0}^{*}$ and $I_{2}$ we can have: (a) two further
$I_{2}$, (b) an $I_{2}$ and two $I_{1}$, (c) two type $II$, (d) a type $II$
and two $I_{1}$ and finally (e) four $I_{1}$. We will discuss these cases
below.

######  Example 6.8 (three $I_{2}$ fibers).

Consider an elliptic pencil on ${\mathbb{F}}_{2}$ whose eight basepoints are
all distinct, pairwise lying on the fibers over $0$, $1$, $t$, $\infty$ of the
ruling. Assume in addition that there exists a section $\sigma_{1}$ of
${\mathbb{F}}_{2}$ passing through four of these basepoints and a section
$\sigma_{2}$ passing through the other four basepoints (see the upper left
diagram of Fig. 12).

If there exist two sections $\sigma_{1}$, $\sigma_{2}$ satisfying the above
conditions, then in the corresponding fibration they give rise to a curve of
type $I_{2}$. In order to find a fibration with three $I_{2}$ fibers, it is
thus sufficient to find values $a_{\pm}$, $b_{\pm}$, $c_{\pm}$, $d_{\pm}$ such
that there exist simultaneously

  * •

a pair of sections $\sigma_{1}$, $\sigma_{2}$ such that $\sigma_{1}$ passes
through $a_{+}$, $b_{+}$, $c_{+}$, $d_{+}$ and $\sigma_{2}$ passes through
$a_{-}$, $b_{-}$, $c_{-}$, $d_{-}$;

  * •

a pair of sections $\sigma_{3}$, $\sigma_{4}$ such that $\sigma_{3}$ passes
through $a_{-}$, $b_{+}$, $c_{-}$, $d_{+}$ and $\sigma_{4}$ passes through
$a_{+}$, $b_{-}$, $c_{+}$, $d_{-}$;

  * •

a pair of sections $\sigma_{5}$, $\sigma_{6}$ such that $\sigma_{5}$ passes
through $a_{+}$, $b_{+}$, $c_{-}$, $d_{-}$ and $\sigma_{6}$ passes through
$a_{-}$, $b_{-}$, $c_{+}$, $d_{+}$.

Each of these pairs of sections then give rise to different curves of type
$I_{2}$ in the fibration coming from the pencil defined by the corresponding
set of basepoints, and by an Euler characteristic computation this fibration
has no other singular fibers.

A necessary and sufficient condition for the existence of such a pair of
sections can be found as follows. In the trivialization of
$K_{\mathbb{C}P^{1}}(D)$ over $U$ discussed in (2.5) for $1\leq j\leq 2$ the
section $\sigma_{j}$ of ${\mathbb{F}}_{2}$ is of the form

$\displaystyle\sigma_{j}(u)=\alpha_{j}u^{2}+\beta_{j}u+\gamma_{j}$

for some $\alpha_{j},\beta_{j},\gamma_{j}\in\mathbb{C}$. For simplicity assume
that $t=-1$. Then, $\sigma_{1}$ passes through the basepoints if and only if
the following equations hold

$\displaystyle\gamma_{1}=a_{+},$
$\displaystyle\alpha_{1}+\beta_{1}+\gamma_{1}=b_{+},$
$\displaystyle\alpha_{1}-\beta_{1}+\gamma_{1}=c_{+},$
$\displaystyle\alpha_{1}=d_{+},$

where $a_{\pm}$, $b_{\pm}$, $c_{\pm}$ are the $w$-coordinates of the
basepoints over $0$, $1$, $t=-1$ respectively and $d_{\pm}$ are the $z$
coordinates over $\infty$. This linear system in three variables $\alpha_{1}$,
$\beta_{1}$, $\gamma_{1}$ has a solution if and only if

$\displaystyle 2(a_{+}+d_{+})=b_{+}+c_{+}.$

Similarly, the existence of $\sigma_{2}$ is equivalent to

$\displaystyle 2(a_{-}+d_{-})=b_{-}+c_{-}.$

We may repeat the above argument to find conditions for the existence of
$\sigma_{3}$, $\sigma_{4}$, $\sigma_{5}$, $\sigma_{6}$. These conditions,
together with the ones found above for $\sigma_{1}$, $\sigma_{2}$, form a
system of $6$ homogeneous linear equations in the $8$ variables $a_{\pm}$,
$b_{\pm}$, $c_{\pm}$, $d_{\pm}$. Completing this system with the degree
formula

$\displaystyle a_{+}+b_{+}+c_{+}+d_{+}+a_{-}+b_{-}+c_{-}+d_{-}=0$ (6.2)

results in a system of $7$ homogeneous linear equations in $8$ variables. In
view of the assumption that the basepoints are all distinct, in order to get a
fibration with three $I_{2}$ fibers we merely need to show that the above
linear system admits generic solutions, namely solutions such that

$\displaystyle a_{+}\neq a_{-},\qquad b_{+}\neq b_{-},\qquad c_{+}\neq
c_{-},\qquad d_{+}\neq d_{-}.$ (6.3)

Now, the coefficient matrix with respect to the variables
$a_{+},a_{-},\ldots,d_{+},d_{-}$ of the system of equations governing the
existence of $\sigma_{1},\ldots,\sigma_{6}$ complemented by the degree formula
(6.2) reads as

$\displaystyle\begin{pmatrix}2&0&-1&0&-1&0&2&0\\\ 0&2&0&-1&0&-1&0&2\\\
0&2&-1&0&0&-1&2&0\\\ 2&0&0&-1&-1&0&0&2\\\ 2&0&-1&0&0&-1&0&2\\\
0&2&0&-1&-1&0&2&0\\\ 1&1&1&1&1&1&1&1\end{pmatrix}.$

One numerically checks that this matrix is of rank $5$, hence it admits a
three-dimensional family of solutions. As a matter of fact, the last row of
the coefficient matrix is a linear combination of the first six rows, and the
corresponding last entry in the extended matrix of the linear system of
equations is $0$, i.e., the degree formula automatically holds once there
exists three fibers of type $I_{2}$ in the given configuration; consequently,
the degree formula does not effect the solvability of the system. In addition,
one sees that $c_{-}$, $d_{+}$, $d_{-}$ may be chosen as parameters and the
other variables may be expressed in terms of these as

$\displaystyle a_{+}=-d_{-},\qquad a_{-}=-d_{+},\qquad b_{+}=-c_{-},\qquad
b_{-}=2d_{-}-2d_{+}-c_{-},$ $\displaystyle c_{+}=-2d_{-}+2d_{+}+c_{-}.$

It is therefore sufficient to choose any $d_{-}\neq d_{+}$ to get solutions
satisfying (6.3). Finally, we note that a similar attempt to find four $I_{2}$
fibers fails because the corresponding linear system has two free parameters
$d_{+}$, $d_{-}$ and the other variables can be expressed as

$\displaystyle a_{+}=-d_{-},\quad a_{-}=-d_{-},\quad\ldots.$

In particular, there exist no solution satisfying (6.3).

######  Example 6.9 (three $I_{2}$ fibers).

Another, less explicit construction of the same type of fibration can be given
as follows: consider a nodal double section of ${\mathbb{F}}_{2}$ with the
property that the node is on $F_{3}$ while $F_{1}$ and $F_{4}$ are tangent to
the nodal double section, see Fig. 13. (It is easy to see from the Hurwitz-
formula that a nodal double section can have at most two tangent fibers.)

Figure 13: A pencil which gives rise to fibrations with an $I_{0}^{*}$ fiber
and three $I_{2}$ fibers.

In this case an $I_{2}$ fiber is originated from the blow-up of the node of
the chosen double section. On the other hand, when we blow up the basepoints
where our chosen double section is tangent to $F_{1}$ and $F_{4}$, we see that
disjoint rational $(-2)$-curves appear in fibers, hence by the classification
the further fibers should be two $I_{2}$ fibers. (Indeed, the pencil contains
two further nodal curves, their nodes are on $F_{1}$ and $F_{4}$ and are
tangent to two chosen fibers.)

If only $F_{1}$ is tangent to our chosen curve (see Fig. 14),

Figure 14: A pencil which gives rise to fibrations with an $I_{0}^{*}$ fiber,
an $I_{2}$ fiber and further $I_{2}$ or type $III$ fibers.

then the fibration (next to $I_{0}^{*}$ and $I_{2}$) has either a further
$I_{2}$ (and also two $I_{1}$) or a type $III$ (and a further $I_{1}$). The
outcome depends on the fact whether the curve with singularity on $F_{1}$ is
nodal (giving a further $I_{2}$) or cuspidal (providing a type $III$ fiber).
Clearly the first is the generic case among these examples. In the next
examples we will give constructions of these cases.

######  Example 6.10 (two $I_{2}$ fibers).

By choosing appropriate homogeneous polynomials, we can arrange that we have
two $I_{2}$ fibers (and two further $I_{1}$ fibers). Indeed, by taking
$u^{2}v(u+v)$ and $uv^{2}(u+2v)$, for the first curve we have a node and a
tangency (for the second a tangency and a node) at the fibers over $[0:1]$ and
$[1:0]$, and by choosing the fibers over $[u_{i}:1]$ with $u_{i}$ solving the
equation $u(u+1)=u+2$, we get the desired pencil and hence fibration. Indeed,
next to $I_{0}^{*}$ and the two $I_{2}$ fibers, no further fiber can contain a
$(-2)$-curve (since the pencil cannot contain sections, and no further
singular point can be on the fibers of the ruling).

A more specific construction may be given along the lines of Example 6.5,
choosing $\sigma_{1}=(u-v)^{2}(u+2v)^{2}$, $\sigma_{2}=u^{2}v^{2}$, with the
corresponding discriminant

$\displaystyle\Delta=64\lambda_{1}^{2}\lambda_{2}^{2}\big{(}81\lambda_{1}^{2}-14\lambda_{1}\lambda_{2}+\lambda_{2}^{2}\big{)}$

admitting two distinct roots (other than at $[\lambda_{1}:\lambda_{2}]=[0:1]$
and $[1:0]$).

######  Example 6.11 (one $I_{2}$ fiber).

A fibration with an $I_{2}$ and a cusp fiber $II$ (next to the $I_{0}^{*}$
fiber) and by classification having either a further $II$ or two further
$I_{1}$ fibers) can be given by taking the homogeneous degree-4 polynomials
$u^{2}v^{2}$ and $(u+v)(u+xv)^{3}$ with parameter $x\neq 0,1$ (the first
giving an $I_{2}$ fiber, the second a cusp curve with cusp over $[-x:1]$),
where the fibers of the ruling ${\mathbb{F}}_{2}\to\mathbb{C}P^{1}$ are chosen
over $[0:1]$ and over $[u_{i}:1]$ with $u_{i}$ solving $u^{2}=(u+1)(u+x)^{3}$.
The two cases for the additional fibers can both occur. In the pencil
generated by the above two singular curves, the discriminant is

$\displaystyle-\lambda_{1}^{2}\lambda_{2}^{2}x^{3}\big{(}{-}16\lambda_{1}^{2}+4\lambda_{1}\lambda_{2}+27\lambda_{2}^{2}x^{5}-108\lambda_{2}^{2}x^{4}+162\lambda_{2}^{2}x^{3}+4\lambda_{1}\lambda_{2}x^{3}-108\lambda_{2}^{2}x^{2}$
$\displaystyle\qquad{}-132\lambda_{1}\lambda_{2}x^{2}+27\lambda_{2}^{2}x-132\lambda_{1}\lambda_{2}x\big{)}.$

Setting $\lambda_{2}=1$, the term in the parentheses has discriminant in
$\lambda_{1}$ equal to

$\displaystyle 16\big{(}x^{2}+14x+1\big{)}^{3}.$

The roots of this polynomial are distinct from $0$, $1$. If $x$ is a root of
this polynomial, then the pencil has only one further singular fiber which is
then necessarily a cusp. Otherwise the pencil has two further singular fibers
which are fishtails.

Finally, the generic case is when next to $I_{0}^{*}$ and the $I_{2}$ fiber
there are four $I_{1}$ fibers. To get an example with these singular fibers,
pick $\sigma_{1}=(u-v)^{2}(u+v)^{2}$, $\sigma_{2}=u^{2}v(u+2v)$, giving rise
to fibers of type $I_{2}$, $I_{1}$ respectively. Then the generic curve in the
pencil has discriminant

$\displaystyle\Delta=\lambda_{1}\lambda_{2}^{2}\big{(}768\lambda_{1}^{3}-832\lambda_{1}^{2}\lambda_{2}+325\lambda_{1}\lambda_{2}^{2}-32\lambda_{2}^{3}\big{)}.$

The cubic polynomial in parentheses admits three distinct roots, each
providing a value of $[\lambda_{1}:\lambda_{2}]$ for which the corresponding
curve in the fibration is singular. By the classification, these singular
curves are of type $I_{1}$.

It is easy to see that if the fibration contains (next to $I_{0}^{*}$) a fiber
which is $I_{4}$, $I_{3}$, $I_{2}$ or type $IV$ or $III$ (their common
property being that they contain rational $(-2)$-curves), then the pencil in
${\mathbb{F}}_{2}$ either contains curves with more than one components (i.e.,
unions of two sections of the ruling), or nodal/cuspidal curves with their
singular points on a chosen fiber $F_{i}$. These cases comprise the first
three rows of the list summarized in Table 1, and were all considered above.

If no such curves are present in the pencil, then the singular curves in the
pencil are either cusps or nodal curves, with their singular points away from
the chosen fibers. There are four possibilities, listed in the fourth row of
Table 1: next to the $I_{0}^{*}$ fiber we can have (a) three cusps, (b) two
cusps and two fishtails, (c) one cusp and four fishtails and finally (d) six
fishtails. Clearly, the last case is the generic. In the following we will
discuss some cases left open above. We will only discuss pencils in the
Hirzebruch surface ${\mathbb{F}}_{2}$.

######  Example 6.12 (three or two cusp fibers).

It is relatively easy to find a pencil giving rise to a fibration with at
least two cusps (next to $I_{0}^{*}$): consider the double sections coming
from the homogeneous degree-4 polynomials $u^{3}v$ and $(u+xv)^{3}(u+v)$ with
parameter $x\neq 0,1$. If we take the fibers over the points
$[u_{i}:1]\in\mathbb{C}P^{1}$ where $u_{i}$ solves $(u+x)^{3}(u+1)=u^{3}$, we
get at least two cusps.

We are in cases (a) or (b) above. The actual case depends on the value of $x$.
Indeed the general polynomial of the pencil has discriminant in $u$ (setting
$v=1$) equal to

$\displaystyle-\lambda_{1}^{2}\lambda_{2}^{2}x^{6}\big{(}27\lambda_{1}^{2}+27\lambda_{2}^{2}+54\lambda_{1}\lambda_{2}+27\lambda_{2}^{2}x^{4}-108\lambda_{2}^{2}x^{3}+4\lambda_{1}\lambda_{2}x^{3}$
$\displaystyle\qquad{}+162\lambda_{2}^{2}x^{2}-18\lambda_{1}\lambda_{2}x^{2}-108\lambda_{2}^{2}x+216\lambda_{1}\lambda_{2}x\big{)}.$

Setting $\lambda_{2}=1$, the term in the parentheses has discriminant in
$\lambda_{1}$ equal to

$\displaystyle 16x(x-9)^{2}(x+3)^{3}.$

The roots of this polynomial are $-3$ and $9$. If $x$ is equal to one of these
values, then the pencil has only one further singular fiber which is then
necessarily a cusp. Otherwise the pencil has two further singular fibers which
are fishtails.

######  Example 6.13 (one cusp fiber).

In a similar manner, taking $u^{3}v$ and $(u-v)^{2}(u+v)(u+2v)$ we get a
fibration with a cusp and a fishtail fiber next to the $I_{0}^{*}$ fiber; the
discriminant of the generic curve in the pencil is given by

$\displaystyle\Delta=-\lambda_{1}\lambda_{2}^{2}\big{(}108\lambda_{1}^{3}+320\lambda_{1}^{2}\lambda_{2}+1827\lambda_{1}\lambda_{2}^{2}+864\lambda_{2}^{3}\big{)}.$

As the cubic in parentheses admits three distinct roots, the remaining fibers
are all $I_{1}$ fibers, giving the example of a cusp together with four
fishtails.

######  Example 6.14 (six fishtails).

The generic case in this situation is simply six $I_{1}$ fibers next to
$I_{0}^{*}$. To construct an example, set $\sigma_{1}=(u-v)^{2}(u+v)(u+3v)$,
$\sigma_{2}=u^{2}v(u+2v)$, both giving rise to a type $I_{1}$ fiber. The
discriminant of the generic curve in the pencil reads as

$\displaystyle\Delta=-\lambda_{1}\lambda_{2}\big{(}24576\lambda_{1}^{4}-8272\lambda_{1}^{3}\lambda_{2}+3768\lambda_{1}^{2}\lambda_{2}^{2}-517\lambda_{1}\lambda_{2}^{3}+96\lambda_{2}^{4}\big{)}.$

It can be shown that the quartic polynomial in parentheses has non-vanishing
discriminant, therefore $\Delta$ has four distinct roots
$[\lambda_{1}:\lambda_{2}]$, giving rise to four further fibers of type
$I_{1}$.

###### Proof of Theorem 1.7.

The combination of the examples given in this section provide the proof of the
theorem. ∎

### Acknowledgments

The authors acknowledge support of NKFIH through the _Élvonal (Frontier)_
program, with grant KKP126683. The third author was also supported by the
grant NKFIH 120697. The authors would like to thank the anonymous referees for
many helpful comments, suggestions and corrections.

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