# The cotangent bundle of K3 surfaces of degree two

Fabrizio Anella  and  Andreas Höring  Fabrizio Anella, Institut de
Mathématiques de Jussieu-Paris rive gauche, Paris, France<EMAIL_ADDRESS>Andreas Höring, Université Côte d’Azur, CNRS, LJAD, France, Institut
universitaire de France<EMAIL_ADDRESS>
(Date: July 18, 2022)

###### Abstract.

K3 surfaces have been studied from many points of view, but the positivity of
the cotangent bundle is not well understood. In this paper we explore the
surprisingly rich geometry of the projectivised cotangent bundle of a very
general polarised K3 surface $S$ of degree two. In particular we describe the
geometry of a surface $D_{S}\subset\mathbb{P}(\Omega_{S})$ that plays a
similar role to the surface of bitangents for a quartic in $\mathbb{P}^{3}$.

###### Key words and phrases:

K3 surface, cotangent bundle, pseudoeffective cone

###### 2010 Mathematics Subject Classification:

14J28, 14J27, 14J42

##  1\. Introduction

###  1.A. Motivation

The cotangent bundle of a K3 surface $S$ is well understood from the point of
view of stability theory: we know that $\Omega_{S}$ is stable for every
polarisation $L$, moreover there are effective bounds guaranteeing that the
restriction $\Omega_{S}\otimes{\mathcal{O}}_{C}$ is stable for every
irreducible curve $C\in|dL|$ [Hei06, Fey16]. Nevertheless these stability
results do not provide a complete description of the positivity properties of
$\Omega_{S}$, in fact the cotangent bundle of a K3 surface is never
pseudoeffective ([Nak04, BDPP13], cf. [HP19] for a more general result).

A way of measuring the negativity of the cotangent bundle is to describe the
pseudoeffective cone of the projectivised cotangent bundle
$\pi:\mathbb{P}(\Omega_{S})\rightarrow S$. Denote by
$\zeta_{S}\rightarrow\mathbb{P}(\Omega_{S})$ the tautological class, and let
$\alpha_{S}$ be a Kähler class on $S$. In our recent paper [AH21] we showed
that if $\alpha_{S}^{2}\geq 8$, then $\zeta_{S}+\pi^{*}\alpha_{S}$ is
pseudoeffective, the bound being optimal for a very general non-projective K3
surface [AH21, Thm.1.5]. The bound is also optimal for infinitely many, but
not all families of projective K3 surfaces with Picard number one [GO20,
Thm.B] which leads us to a more geometric question:

######  1.1.

Question. Let $S$ be a projective K3 surface, and let $L$ be an ample Cartier
divisor on $S$. Assume that $\zeta_{S}+\lambda\pi^{*}L$ is pseudoeffective for
some $\lambda<\sqrt{\frac{8}{L^{2}}}$. Can we relate the pseudoeffectivity of
$\zeta_{S}+\lambda\pi^{*}L$ with the projective geometry of $(S,L)$?

In view of the arguments used by Lazić and Peternell [LP20] we might ask even
more boldly for a relation between $\mbox{Pseff}(\mathbb{P}(\Omega_{S}))$ and
families of elliptic curves on $S$. Gounelas and Ottem [GO20] give a natural
framework for these questions: the embedding

$\mathbb{P}(\Omega_{S})\subset S^{[2]}$

in the Hilbert square allows to use results on the pseudoeffective cone of
$S^{[2]}$ by Bayer and Macrì[BM14]. For example this approach allows to
recover the following classical result:

######  1.2.

Theorem. [Tih80, Wel81] [GO20, Sect.4] Let $S\subset\mathbb{P}^{3}$ be a
smooth quartic surface with Picard number one, and denote by $L$ the
restriction of the hyperplane class. Then the surface of bitangents
$U\subset\mathbb{P}(\Omega_{S})$ has class $6\zeta_{S}+8\pi^{*}L$ and
generates an extremal ray of $\mbox{Pseff}(\mathbb{P}(\Omega_{S}))$.

The goal of this paper is to give an analogue of Theorem 1.2 in the case of a
very general polarised K3 surfaces $(S,L)$ of degree two, i.e. $S$ is a K3
surface with Picard number one obtained as a two-to-one cover

$f:S\rightarrow\mathbb{P}^{2}$

with ramification divisor a smooth curve of genus ten. Oguiso and Peternell
[OP96] observed that in this case the projectivised cotangent bundle should
have some exceptional properties that are not representative for general K3
surfaces. The approach of Gounelas and Ottem allows to determine the nef cone
of $\mathbb{P}(\Omega_{S})$ [GO20, Sect.4.1], but yields only that
$\zeta_{S}+\pi^{*}2L$ is pseudoeffective without determining the extremality
in $\mbox{Pseff}(\mathbb{P}(\Omega_{S})$. Note that $\alpha_{S}:=2L$ has
$\alpha_{S}^{2}=8$, so this situation corresponds exactly to the setup of
Question 1.1.

###  1.B. Main results

Let $S$ be a very general K3 surface of degree two, and let
$f:S\rightarrow\mathbb{P}^{2}$ be the double cover. Let
$d\subset\mathbb{P}^{2}$ be a line that is a simple tangent of the branch
divisor $B$, then its preimage $C:=f^{-1}(d)$ has a unique node so the
normalisation $n:\tilde{C}\rightarrow C$ is a smooth elliptic curve. The
natural surjection

$n^{*}\Omega_{S}\twoheadrightarrow\Omega_{\tilde{C}}$

determines a morphism $\tilde{C}\rightarrow\mathbb{P}(\Omega_{S})$ which we
call the canonical lifting of $C\in|L|$ (cf. Subsection 3.C).

######  1.3.

Theorem. Let $(S,L)$ be a very general polarised K3 surface of degree two, and
denote by $D_{S}\subset\mathbb{P}(\Omega_{S})$ the surface dominated by
canonical liftings of singular elliptic curves in $|L|$. Then the
normalisation of $D_{S}$ is a smooth (non-minimal) elliptic surface. Moreover
we have

$D_{S}\equiv 30\zeta_{S}+54\pi^{*}L\equiv 30(\zeta_{S}+1.8\pi^{*}L).$

The surface $D_{S}$ itself is very singular, the description of the
normalisation occupies the larger part of this paper. We will see that $D_{S}$
contains a lot of geometric information about the K3 surface, for example it
contains a curve isomorphic to the branch divisor $B$ in its non-normal locus.
While the surface $D_{S}$ plays a similar role than the surface of bitangents
for the quartic surfaces, it does not generate an extremal ray in the
pseudoeffective cone:

######  1.4.

Theorem. Let $(S,L)$ be a very general polarised K3 surface of degree two.
Then there exists a prime divisor $Z_{S}\subset\mathbb{P}(\Omega_{S})$ such
that

$Z_{S}\equiv a(\zeta_{S}+\lambda\pi^{*}L)$

with $\lambda\leq 1.7952024$111The exact statement is
$\lambda<\frac{15}{4}-27\frac{(1-i\sqrt{3})}{8((79+8i\sqrt{5})/3)^{(1/3)}}-\frac{1}{8}3^{(2/3)}(1+i\sqrt{3})(79+8i\sqrt{5})^{(1/3)}$
but we refrain from working with such futile precision..

Moreover let $Z_{S}\subset\mathbb{P}(\Omega_{S})$ be a prime divisor with this
property. Then the canonical liftings of
$C_{node,S}\subset\mathbb{P}(\Omega_{S})$ of the 324 rational curves in $|L|$
are contained in $Z_{S}$.

From a numerical point of view both parts of this statement come as a
surprise:

  * •

The non-nef locus of the divisor $\zeta_{S}+1.8\pi^{*}L$ consists of a unique
curve $R_{S}\subset\mathbb{P}(\Omega_{S})$. If we denote by
$\mu_{S}:Y\rightarrow\mathbb{P}(\Omega_{S})$ the blow-up of this curve (cf.
Subsection 3.A) the strict transform $D\subset Y$ of $D_{S}$ generates an
extremal ray in $\mbox{Pseff}(Y)$. Nevertheless $D_{S}$ itself is a big, non-
nef divisor with $D_{S}^{3}<0$.

  * •

The divisor $Z_{S}$ satisfies $Z_{S}\cdot C_{node,S}>0$, so we would not
expect these curves to be contained in the stable base locus. This property
will be a consequence of our investigation of the threefold
$Y\rightarrow\mathbb{P}(\Omega_{S})$.

While the class of the surface $D_{S}$ does not generate an extremal ray in
$\mbox{Pseff}(\mathbb{P}(\Omega_{S}))$, the detailed geometric study involved
in the proof of Theorem 1.3 allows to give a pretty sharp estimate for the
extremal ray:

######  1.5.

Theorem. Let $(S,L)$ be a very general polarised K3 surface of degree two.
Assume that there exists a prime divisor $Z_{S}\subset\mathbb{P}(\Omega_{S})$
such that $Z_{S}\equiv a(\zeta_{S}+\lambda\pi^{*}L)$.

Then we have $\lambda\geq\frac{39}{22}=1.7\overline{72}$.

The proofs of all our results are based on the analysis of the birational
morphisms

(1)
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{S}}$$\scriptstyle{\mu_{P}}$$\textstyle{\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}$$\textstyle{\mathbb{P}(\Omega_{S})}$

which allow us to transfer informations from the well-understood
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\subset
S\times(\mathbb{P}^{2})^{\vee}$ to the much more mysterious
$\mathbb{P}(\Omega_{S})$. In particular this will allow us to describe the
surface $D_{S}\subset\mathbb{P}(\Omega_{S})$ as the strict transform of the
universal family of singular elements in $|L|$ (cf. Lemma 3.12). The first
step is Theorem 3.13 where we show that $\bar{D}$, the normalisation of this
universal family, is a smooth minimal elliptic surface. The second step is to
show that the normalisation $\tilde{D}\rightarrow D_{S}$ is also a smooth
surface (Theorem 3.18), as a consequence the birational morphism
$\tilde{D}\rightarrow\bar{D}$ is simply a blow-up of 720 points. Determining
the rich geometry of the elliptic surface $\tilde{D}$ is the main technical
contribution of this paper and a somewhat delicate task. With all this
geometric information at hand the proofs of Theorem 1.4 and Theorem 1.5 follow
without too much effort.

###  1.C. Outlook: return to the Hilbert square

We will see in Subsection 3.A that
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$
identifies to the universal family $\mathcal{U}\rightarrow|L|$ of divisors in
the linear system $|L|$. This allows relating the morphisms in Diagram 1 to a
construction on the Hilbert scheme [Bak17, Ex.9]: the Hilbert square $S^{[2]}$
contains a Lagrangian plane $\mathbb{P}^{2}$ determined by mapping a point
$z\in\mathbb{P}^{2}$ to its preimage $[f^{-1}(z)]\in S^{[2]}$. Let

$S^{[2]}\dashrightarrow X$

be the Mukai flop of this Lagrangian plane. Then $X$ is a Hyperkähler manifold
that admits a Lagrangian fibration $X\rightarrow\mathbb{P}^{2}$, in fact $X$
is the degree two compactified Jacobian of the universal family
$\mathcal{U}\rightarrow|L|$. Moreover the graph of the Mukai flop is the
relative Hilbert scheme $\mbox{Hilb}^{2}(\mathcal{U}/|L|)$.

The restriction of the blow-up $\mbox{Hilb}^{2}(\mathcal{U}/|L|)\rightarrow
S^{[2]}$ to $\mathbb{P}(\Omega_{S})$ is our blow-up
$Y\rightarrow\mathbb{P}(\Omega_{S})$ (cf. [GO20, Sect.4.1]), so we have a
commutative diagram:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{P}}$$\scriptstyle{\mu_{S}}$$\textstyle{\mbox{Hilb}^{2}(\mathcal{U}/|L|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{U}=\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{[2]}}$$\textstyle{\mathbb{P}(\Omega_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|L|}$

It is tempting to believe that a description of the pseudoeffective cone of
the relative Hilbert scheme $\mbox{Hilb}^{2}(\mathcal{U}/|L|)$ allows to shed
further light on the geometry of $Y$ and ultimately determines the
pseudoeffective cone of $\mathbb{P}(\Omega_{S})$.

Acknowledgements. We thank Thomas Dedieu for pointing out the relation between
Theorem 3.13 and Teissier’s theorem, cf. Remark 3.14. The first-named author
is sponsored by ERC Synergy Grant HyperK, Grant agreement ID 854361, and he
warmly thanks all the people involved in this project, for providing support
and a stimulating environment. The second-named author thanks the Institut
Universitaire de France and the A.N.R. project Foliage (ANR-16-CE40-0008) for
providing excellent working conditions.

##  2\. Notation and general setup

We work over $\mathbb{C}$, for general definitions we refer to [Har77] and
[Voi02].

All the schemes appearing in this paper are projective, manifolds and normal
varieties will always be supposed to be irreducible. For notions of positivity
of divisors and vector bundles we refer to Lazarsfeld’s book [Laz04a, Laz04b].
Given two Cartier divisors $D_{1},D_{2}$ on a projective variety $X$ we denote
by $D_{1}\simeq D_{2}$ (resp. $D_{1}\equiv D_{2}$) the linear equivalence
(resp. numerical equivalence) of the Cartier divisor classes, while
$D_{1}=D_{2}$ is used for an equality of the cycles. We will frequently
identify an effective divisor to its cohomology class in $N^{1}(X)$ where
$N^{1}(X):=\mbox{NS}(X)\otimes\mathbb{R}$ is the Neron-Severi space of
$\mathbb{R}$-divisors.

In the whole paper we will work in the following

######  2.1.

Setup. Let $S$ be a very general polarised K3 surface of degree two, and let
$L$ be the primitive polarisation on $S$. We denote by

$f:S\rightarrow\mathbb{P}^{2}$

the double cover defined by the linear system $|L|$, so we have $L\simeq
f^{*}H_{1}$ where $H_{1}$ is the hyperplane class on $\mathbb{P}^{2}$. Since
$|L|=f^{*}|H_{1}|$ every curve $C\in|L|$ is a double cover of a line
$d\subset\mathbb{P}^{2}$.

We denote by $B\subset\mathbb{P}^{2}$ the branch locus of $f$ and by $R\subset
S$ the ramification divisor. By the ramification formula we know that $B$ is a
sextic curve, so

$g(B)=10,\qquad\deg\omega_{B}=18.$

Since $f^{*}B=2R$, we see that the ramification divisor $R$ is an element of
$|3L|$.

Denote by

$\pi:\mathbb{P}(\Omega_{S})\rightarrow S$

the projectivisation of the cotangent bundle, and by $\zeta_{S}$ the
tautological class. We denote by $l\subset\mathbb{P}(\Omega_{S})$ a fibre of
$\pi$.

This setup will become increasingly rich through a series of geometric
constructions which we will summarise in the diagrams (5) and (9).

Remark. Many of our arguments are valid for an arbitrary smooth polarised K3
surface of degree two. However the assumption that $S$ is very general is
necessary to assure that the Picard number of $S$ is one. Moreover this
implies that the branch curve $B\subset\mathbb{P}^{2}$ satisfies the
assumptions of Plücker’s theorem which is crucial in Subsection 3.B and the
proof of Theorem 3.18.

##  3\. Birational geometry of the projectivised cotangent bundle

###  3.A. Elementary transform of $\Omega_{S}$

Consider the projective plane $\mathbb{P}^{2}$ and its hyperplane class
$H_{1}$. We denote by
$p_{1}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow\mathbb{P}^{2}$ the
projectivisation and by $\zeta_{\mathbb{P}^{2}}$ its tautological class. Let
us recall some elementary facts: twisting the Euler sequence by
${\mathcal{O}}_{\mathbb{P}^{2}}(-1)$ and recalling that
$T_{\mathbb{P}^{2}}\simeq\Omega_{\mathbb{P}^{2}}\otimes
K_{\mathbb{P}^{2}}^{*}\simeq\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{\mathbb{P}^{2}}(3)$
we obtain an exact sequence

$0\rightarrow{\mathcal{O}}_{\mathbb{P}^{2}}(-1)\rightarrow{\mathcal{O}}_{\mathbb{P}^{2}}^{\oplus
3}\rightarrow\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{\mathbb{P}^{2}}(2)\rightarrow
0.$

Thus the vector bundle
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{\mathbb{P}^{2}}(2)$ is globally
generated and
$h^{0}(\mathbb{P}^{2},\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{\mathbb{P}^{2}}(2))=3$.

The sequence above shows that there is an inclusion

$\mathbb{P}(\Omega_{\mathbb{P}^{2}})\subset\mathbb{P}^{2}\times(\mathbb{P}^{2})^{\vee},$

and if we denote by
$p_{2}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$
the fibration defined by the projection on the second factor, it allows to
identify $\mathbb{P}(\Omega_{\mathbb{P}^{2}})$ with the universal family of
lines on $\mathbb{P}^{2}$, i.e. for a point $y\in(\mathbb{P}^{2})^{\vee}$ the
curve $p_{1}(p_{2}^{-1}(y))\subset\mathbb{P}^{2}$ is the line corresponding to
the point $y$. The divisor class $\zeta_{\mathbb{P}^{2}}+2p_{1}^{*}H_{1}$
defines a base-point free linear system, in fact we have

$\zeta_{\mathbb{P}^{2}}+2p_{1}^{*}H_{1}\simeq p_{2}^{*}H_{2}$

where $H_{2}$ is the hyperplane class on $(\mathbb{P}^{2})^{\vee}$.

From now on we work in the Setup 2.1: let $f:S\rightarrow\mathbb{P}^{2}$ be
the double cover, and denote by

$\tilde{f}:\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow\mathbb{P}(\Omega_{\mathbb{P}^{2}})$

the induced cover and by
$p:\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow S$ the natural
fibration. We set

$\zeta_{P}:=\tilde{f}^{*}\zeta_{\mathbb{P}^{2}}$

for the tautological class. The linear system defined by
$\zeta_{P}+p^{*}2L\simeq\tilde{f}^{*}(\zeta_{\mathbb{P}^{2}}+2p_{1}^{*}H_{1})$
is base-point free and defines a fibration

$q:\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}.$

Since the pull-backs of lines $d\in|{\mathcal{O}}(1)|$ correspond exactly to
the elements of the linear system $|L|$, the fibration
$q:\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$
identifies the universal family of curves in the linear system $|L|$.

We will now study the relation between
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ and $\mathbb{P}(\Omega_{S})$. Since
$R$ is the ramification divisor of the double cover $f$, we have

(2)
$\Omega_{S}\otimes{\mathcal{O}}_{R}\simeq\omega_{R}\oplus{\mathcal{O}}_{R}(-R)$

and the relative cotangent sheaf $\Omega_{f}$ is isomorphic to
${\mathcal{O}}_{R}(-R)$. Consider now the exact sequence

(3) $0\rightarrow
f^{*}\Omega_{\mathbb{P}^{2}}\rightarrow\Omega_{S}\rightarrow\Omega_{f}\rightarrow
0$

Since $\Omega_{f}$ is a rank one bundle with support on the Cartier divisor
$R$, we can see $f^{*}\Omega_{\mathbb{P}^{2}}$ as the strict transform of
$\Omega_{S}$ along $\Omega_{f}$ (cf. [Mar72, Thm.1.3] for the terminology).
More precisely let

$R_{S}:=\mathbb{P}(\Omega_{f})\subset\mathbb{P}(\Omega_{S})$

be the curve defined by the surjection $\Omega_{S}\rightarrow\Omega_{f}$. Then
denote by $\mu_{S}:Y\rightarrow\mathbb{P}(\Omega_{S})$ the blow-up along this
curve, and by $E_{S}$ its exceptional divisor. Denote by $E_{P}$ the strict
transform of the divisor $\pi^{-1}(R)\subset\mathbb{P}(\Omega_{S})$. Note that
since $\mathbb{P}(\Omega_{f})\subset\pi^{-1}(R)$ is a Cartier divisor, we have

(4)
$E_{P}\simeq\pi^{-1}(R)\simeq\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})\simeq\mathbb{P}(\omega_{R}\oplus{\mathcal{O}}_{R}(-R))$

where in the last step we used (2).

By the relative contraction theorem applied to the morphism $\pi\circ\mu_{S}$
there exists a birational morphism contracting the divisor $E_{P}$ onto a
curve. By [Mar72, Thm.1.3] this contraction can be described more explicitly:
the image of
$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\Omega_{S}\otimes{\mathcal{O}}_{R}$
is isomorphic to $\omega_{R}$, so we have a surjection
$f^{*}\Omega_{\mathbb{P}^{2}}\rightarrow\omega_{R}$. Thus

$R_{P}:=\mathbb{P}(\omega_{R})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$

is a curve, and the blow-up along $R_{P}$ gives a morphism
$\mu_{P}:Y\rightarrow\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ such that the
following diagram commutes:

(5)
$\textstyle{E_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{S}}$$\scriptstyle{\mu_{P}}$$\textstyle{E_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{\mathbb{P}(\Omega_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{R_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{2})^{\vee}=|L|}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{f}$$\textstyle{S}$$\textstyle{\mathbb{P}^{2}}$

Arguing as above we see that

(6) $E_{S}\simeq
p^{-1}(R)\simeq\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})\simeq\mathbb{P}(\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{B}),$

where in the last step we used that $f|_{R}$ is an isomorphism onto $B$. We
deduce from [Mar72, Thm.1.1] that

(7)
$\mu_{P}^{*}\zeta_{P}=\mu_{S}^{*}\zeta_{S}-E_{S},\qquad\mu_{S}^{*}\zeta_{S}=\mu_{P}^{*}\zeta_{P}-E_{P}+3(p\circ\mu_{P})^{*}L.$

######  3.1.

Lemma. In the situation summarised by Diagram 5, we have the following
intersection numbers:

$E_{S}^{3}=18,\qquad E_{P}^{3}=-72.$

Set $\tilde{R}:=E_{P}\cap E_{S}$. Then we have

$E_{S}\cdot\tilde{R}=-36,\qquad E_{P}\cdot\tilde{R}=54$

and

$(\tilde{R})^{2}_{E_{S}}=54,\qquad(\tilde{R})^{2}_{E_{P}}=-36.$

Remark. We denote by $(\tilde{R})^{2}_{E_{S}}$ (resp.
$(\tilde{R})^{2}_{E_{P}}$) the self-intersection of the curve $\tilde{R}$,
seen as a divisor in $E_{S}$ (resp. $E_{P}$).

###### Proof.

We describe $E_{S}$ and $E_{P}$ in terms of the blowup: by construction one
has $E_{S}\simeq\mathbb{P}(N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})})$ and
${\mathcal{O}}_{E_{S}}(E_{S})\simeq{\mathcal{O}}_{\mathbb{P}(N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})})}(-1)$.
We have an exact sequence

$0\rightarrow
N^{*}_{\pi^{-1}(R)/\mathbb{P}(\Omega_{S})}\otimes{\mathcal{O}}_{R_{S}}\rightarrow
N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}\rightarrow
N^{*}_{R_{S}/\pi^{-1}(R)}\rightarrow 0.$

Since $R\in|3L|$, we have
$N^{*}_{\pi^{-1}(R)/\mathbb{P}(\Omega_{S})}\otimes{\mathcal{O}}_{R_{S}}\simeq{\mathcal{O}}_{R}(-3L)$,
where we identified $R_{S}\simeq R$. Since $R_{S}$ corresponds to the quotient
$\Omega_{S}\otimes{\mathcal{O}}_{R}\rightarrow{\mathcal{O}}_{R}(-3L)$, an
adjunction computation shows
$N^{*}_{R_{S}/\pi^{-1}(R)}\simeq{\mathcal{O}}_{R}(6L)$. Thus we have an
extension

$0\rightarrow{\mathcal{O}}_{R}(-3L)\rightarrow
N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}\rightarrow{\mathcal{O}}_{R}(6L)\rightarrow
0,$

in particular $\det
N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}\simeq{\mathcal{O}}_{R}(3L)$ and

$E_{S}^{3}=c_{1}({\mathcal{O}}_{\mathbb{P}(N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})})}(-1))^{2}=\deg
N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}=\deg{\mathcal{O}}_{R}(3L)=18.$

Now recall that by (6) we also have
$E_{S}\simeq\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})$,
so there exists a line bundle $M\rightarrow R$ such that
$N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}\simeq
f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\otimes M$. Since $\det
f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\simeq{\mathcal{O}}_{R}(-3L)$
we obtain

$N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})}\simeq
f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}(3L).$

Recall now that $\mu_{P}$ is the blowup along the curve $R_{P}$ corresponding
to the quotient
$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\omega_{R}\simeq{\mathcal{O}}_{R}(3L)$.
Then the curve $\tilde{R}$ identifies to $R_{P}$ under the isomorphism
$\mu_{P}|_{E_{S}}:E_{S}\rightarrow\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})$
and we have

$E_{S}\cdot\tilde{R}=c_{1}({\mathcal{O}}_{\mathbb{P}(N^{*}_{R_{S}/\mathbb{P}(\Omega_{S})})}(-1))\cdot\tilde{R}=c_{1}({\mathcal{O}}_{\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\otimes{\mathcal{O}}_{R}(3L))}(-1))\cdot\tilde{R}$
$=-\left(c_{1}({\mathcal{O}}_{\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}}(1))\cdot
R_{P}+\deg{\mathcal{O}}_{R}(3L)\right)=-\deg{\mathcal{O}}_{R}(6L)=-36.$

We repeat the argument for $E_{P}$: by construction one has
$E_{P}\simeq\mathbb{P}(N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})})$
and
${\mathcal{O}}_{E_{P}}(E_{P})\simeq{\mathcal{O}}_{N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}}(-1)$.
We have an exact sequence

$0\rightarrow
N^{*}_{p^{-1}(R)/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\otimes{\mathcal{O}}_{R_{P}}\rightarrow
N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\rightarrow
N^{*}_{R_{P}/p^{-1}(R)}\rightarrow 0.$

Since $R\in|3L|$, we have
$N^{*}_{p^{-1}(R)/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\otimes{\mathcal{O}}_{R_{P}}\simeq{\mathcal{O}}_{R}(-3L)$,
where we identified $R_{P}$ with $R$. Since $R_{P}$ corresponds to the
quotient
$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\omega_{R}\simeq{\mathcal{O}}_{R}(3L)$,
an adjunction computation shows
$N^{*}_{R_{P}/p^{-1}(R)}\simeq{\mathcal{O}}_{R}(-9L)$. Thus we have an
extension

$0\rightarrow{\mathcal{O}}_{R}(-3L)\rightarrow
N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\rightarrow{\mathcal{O}}_{R}(-9L)\rightarrow
0,$

in particular $\det
N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\simeq{\mathcal{O}}_{R}(-12L)$
and

$E_{P}^{3}=c_{1}({\mathcal{O}}_{\mathbb{P}(N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})})}(-1))^{2}=\deg
N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}=\deg{\mathcal{O}}_{R}(-12L)=-72.$

Now recall that by (4) we also have
$E_{P}\simeq\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})$, so there exists a
line bundle $M\rightarrow R$ such that
$N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\simeq\Omega_{S}\otimes{\mathcal{O}}_{R}\otimes
M$. Since $\det\Omega_{S}\otimes{\mathcal{O}}_{R}\simeq{\mathcal{O}}_{R}$ we
finally obtain

$N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})}\simeq\Omega_{S}\otimes{\mathcal{O}}_{R}\otimes{\mathcal{O}}_{R}(-6L).$

Recall now that $\mu_{S}$ is the blowup along the curve $R_{S}$ corresponding
to the quotient
$\Omega_{S}\otimes{\mathcal{O}}_{R}\rightarrow{\mathcal{O}}_{R}(-3L)$. Then
the curve $\tilde{R}$ identifies to $R_{S}$ under the isomorphism
$\mu_{S}|_{E_{P}}:E_{P}\rightarrow\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})$
and we have

$E_{P}\cdot\tilde{R}=c_{1}({\mathcal{O}}_{\mathbb{P}(N^{*}_{R_{P}/\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})})}(-1))\cdot\tilde{R}=c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R}\otimes{\mathcal{O}}_{R}(-6L))}(-1))\cdot\tilde{R}$
$=-\left(c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R}}(1))\cdot
R_{S}+\deg{\mathcal{O}}_{R}(-6L)\right)=-\deg{\mathcal{O}}_{R}(-9L)=54.$

In order to determine the self-intersection of $\tilde{R}$, recall that the
self-intersection of a curve is invariant under isomorphism. The isomorphism
$\mu_{P}|_{E_{S}}:E_{S}\rightarrow\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})$
sends $\tilde{R}$ to $R_{P}$, thus we obtain

$(\tilde{R})^{2}_{E_{S}}=(R_{P})^{2}_{p^{-1}(R)}=\deg
N_{R_{P}/p^{-1}(R)}=\deg{\mathcal{O}}_{R}(9L)=54.$

Since the isomorphism
$\mu_{S}|_{E_{P}}:E_{P}\rightarrow\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})$
sends $\tilde{R}$ to $R_{S}$ one has

$(\tilde{R})^{2}_{E_{P}}=(R_{S})^{2}_{\pi^{-1}(R)}=\deg
N_{R_{S}/\pi^{-1}(R)}=\deg{\mathcal{O}}_{R}(-6L)=-36.$

∎

######  3.2.

Corollary. In the situation summarised by Diagram 5, denote by $l_{S}\subset
E_{S}$ a fibre of the projection

$E_{S}\simeq\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})\rightarrow
R_{S}.$

Then the pseudoeffective cone of $E_{S}$ coincides with the nef cone and its
extremal rays are generated by the classes $l_{S}$ and $\tilde{R}-27l_{S}$.

In particular if $\tau:E_{S}\rightarrow W$ is a generically finite morphism
onto a surface $W$, it is finite.

###### Proof.

By (6) we have
$E_{S}\simeq\mathbb{P}(\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{B})$.
Since the K3 surface $S$ is general, the sextic curve $B$ is general in its
linear system, so by [Fle84, Thm.1.2] the restricted vector bundle
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{B}$ is semistable. Thus by
[Laz04a, Sect.1.5.A] the pseudoeffective cone of $E_{S}$ coincides with the
nef cone. Since a nef divisor with positive self-intersection is big, this
also shows that the nef cone coincides with the positive cone of $E_{S}$.

The curve $\tilde{R}\subset E_{S}$ is a section and by Lemma 3.1 we have
$(\tilde{R})^{2}_{E_{S}}=54$. Thus $\tilde{R}-27l_{S}$ is a generator of the
second extremal ray of the positive cone.

The last statement follows by observing that a curve contracted by a
generically finite morphism of surfaces has negative self-intersection. ∎

######  3.3.

Remark. We have seen at the beginning of this subsection that the vector
bundle $f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{S}(2L)$ is globally
generated and
$h^{0}(S,f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{S}(2L))\geq 3$.
Using the projection formula and
$f_{*}{\mathcal{O}}_{S}\simeq{\mathcal{O}}_{\mathbb{P}^{2}}\oplus{\mathcal{O}}_{\mathbb{P}^{2}}(-3)$
we see that equality holds.

Twisting the tangent sequence (3) by ${\mathcal{O}}_{S}(2L)$ and using that
$\omega_{f}\simeq{\mathcal{O}}_{R}(-R)\simeq{\mathcal{O}}_{R}(-3L)$ we obtain
that

$h^{0}(\mathbb{P}(\Omega_{S}),\zeta_{S}+\pi^{*}2L)=h^{0}(S,\Omega_{S}\otimes{\mathcal{O}}_{S}(2L))=3$

and the linear system $|\zeta_{S}+\pi^{*}2L|$ is globally generated in the
complement of $R_{S}=\mathbb{P}(\omega_{f})$. In fact, since the tangent map
has rank one along $R$ we see that the scheme-theoretic base locus of
$|\zeta_{S}+\pi^{*}2L|$ is the reduced curve $R_{S}$. Thus the blow-up $R_{S}$
resolves the base-locus and

$\mu_{S}^{*}(\zeta_{S}+\pi^{*}2L)=\mu_{P}^{*}(\zeta_{P}+p^{*}2L)+E_{S}=(q\circ\mu_{P})^{*}H_{2}+E_{S},$

where $H_{2}$ is the hyperplane class on $(\mathbb{P}^{2})^{\vee}$.

######  3.4.

Lemma. In the situation summarised by Diagram 5, we set

$\psi:=q\circ\mu_{P}:Y\rightarrow(\mathbb{P}^{2})^{\vee}$

and

$M:=\psi^{*}H_{2}=\mu_{S}^{*}(\zeta_{S}+\pi^{*}2L)-E_{S}$

where $H_{2}$ is the hyperplane class on $(\mathbb{P}^{2})^{\vee}$. Then we
have the following intersection numbers:

$E_{S}^{3}=18\ \ \ E_{S}^{2}\cdot E_{P}=-36\ \ \ E_{S}\cdot E_{P}^{2}=54\ \ \
E_{P}^{3}=-72$ $M^{3}=0\ \ \ M^{2}\cdot E_{S}=6\ \ \ M\cdot E_{S}^{2}=-12$
$M^{2}\cdot E_{P}=0\ \ \ M\cdot E_{P}^{2}=-30\ \ \ M\cdot E_{S}\cdot E_{P}=30$
$(E_{P}+E_{S})\cdot E_{S}^{2}=-18\ \ \ (E_{P}+E_{S})\cdot E_{P}\cdot E_{S}=18\
\ \ (E_{P}+E_{S})\cdot E_{P}^{2}=-18$ $M^{2}\cdot(p\circ\mu_{P})^{*}L=2\ \ \
M\cdot(p\circ\mu_{P})^{*}L^{2}=2$ $(p\circ\mu_{P})^{*}L\cdot E_{P}^{2}=-6\ \ \
(p\circ\mu_{P})^{*}L^{2}\cdot E_{P}=0$

###### Proof.

The result follows from Lemma 3.1 and standard computations involving the
geometric construction. In particular observe that

$E_{P}+E_{S}=(\pi\circ\mu_{S})^{*}R=(p\circ\mu_{P})^{*}R,$

so $E_{P}+E_{S}\equiv 3(p\circ\mu_{P})^{*}L$.

As an example, let us show that $M^{2}\cdot E_{S}=6$: note that $\mu_{P}$ maps
$E_{S}$ isomorphically onto
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})$. A fibre of
$p$ maps onto a line in $(\mathbb{P}^{2})^{\vee}$, so $M$ has degree one on
the fibres of $E_{S}\rightarrow S$. The curve $\tilde{R}\simeq R_{P}$ maps
birationally onto $R^{\vee}\subset(\mathbb{P}^{2})^{\vee}$. Since $R^{\vee}$
has degree 30 by Plücker’s formula (see Subsection 3.B below), we obtain
$M\cdot\tilde{R}=30$. Since $(\tilde{R})^{2}_{E_{S}}=54$ by Lemma 3.1 this
shows that $M|_{E_{S}}\equiv\tilde{R}-24l_{S}$, and thus $M^{2}\cdot
E_{S}=(M|_{E_{S}})^{2}=6$. ∎

Remark. For the convenience of the reader, let us summarise that by the
argument above and Lemma 3.1 we have

$M|_{E_{S}}\equiv\tilde{R}-24l_{S},\qquad E_{P}|_{E_{S}}=\tilde{R},\qquad
E_{S}|_{E_{S}}\equiv-\tilde{R}+18l_{S}.$

###  3.B. The surface $D$

Let $B^{\vee}\subset(\mathbb{P}^{2})^{\vee}$ be the dual curve of the sextic
curve $B\subset\mathbb{P}^{2}$. Since $B$ is a general sextic we can apply
Plücker’s formulas to see that

$\deg B^{\vee}=30$

and $B^{\vee}$ has exactly 324 nodes (resp. 72 simple cusps) corresponding to
bitangent lines (resp. inflection lines) and no other singularities. By
definition the dual curve parametrises lines $d\subset\mathbb{P}^{2}$ that are
tangent to $B$ in at least one point. Since the preimage $f^{-1}(d)$ of a line
is singular if and only if it is tangent to $B$ in at least one point, we see
that $B^{\vee}$ naturally parametrises the singular elements of $|L|$. We
denote by

$R^{\vee}\subset|L|$

the image of $B^{\vee}$ under the isomorphism
$(\mathbb{P}^{2})^{\vee}\simeq|L|$. Since the elements of $|L|$ have
arithmetic genus two it is not difficult to see that

  * •

a smooth point $t\in R^{\vee}$ parametrises a curve $C_{t}$ with exactly one
node, so the normalisation is an elliptic curve;

  * •

a node $t\in R^{\vee}$ parametrises a curve $C_{t}$ with exactly two nodes, so
the normalisation is a rational curve;

  * •

a cusp $t\in R^{\vee}$ parametrises a curve $C_{t}$ with one simple cusp, so
the normalisation is an elliptic curve.

The surface $q^{-1}(R^{\vee})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$
identifies with the universal family of singular elements in $|L|$, in
particular, this surface is not normal.

######  3.5.

Lemma. We have $R_{P}\subset(q^{-1}(R^{\vee}))_{\operatorname*{sing}}$ and
$R_{P}$ is the unique irreducible component of
$(q^{-1}(R^{\vee}))_{\operatorname*{sing}}$ mapping onto $R^{\vee}$. Moreover
$q^{-1}(R^{\vee})$ has a nodal singularity in a point of $R_{P}$ that maps
onto a smooth point on $R^{\vee}$.

###### Proof.

For a smooth point $t\in R^{\vee}$ the curve $C_{t}\subset|L|$ has a unique
singular point $x$, so it is clear that
$(q^{-1}(R^{\vee}))_{\operatorname*{sing}}$ has exactly one irreducible
component mapping onto $R^{\vee}$. We claim that the point $x\in
C_{t}\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ is on the curve $R_{P}$.

Proof of the claim. Let $d\subset\mathbb{P}^{2}$ be a line and let
$\tilde{d}\subset\mathbb{P}(\Omega_{\mathbb{P}^{2}})$ be the lifting defined
by the canonical quotient
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{d}\rightarrow\omega_{d}$. Then
$\tilde{d}$ is the fibre of the fibration
$p_{2}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$
over the point $[d]\in(\mathbb{P}^{2})^{\vee}$, hence
$C_{t}=\tilde{f}^{-1}(\tilde{d})$.

Let $B_{P}\subset\mathbb{P}(\Omega_{\mathbb{P}^{2}})$ be the curve defined by
the canonical quotient
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{B}\rightarrow\omega_{B}$. Since
the curve $R_{P}$ corresponds to the quotient

$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{B}\simeq
f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\omega_{R},$

we have a set-theoretical equality $R_{P}=\tilde{f}^{-1}(B_{P})$.

Now recall that the point $x\in C_{t}=f^{-1}(d)$ is singular if and only

$T_{d,f(x)}=T_{B,f(x)}\subset T_{\mathbb{P}^{2},x}.$

Thus the curves $\tilde{d}$ and $B_{P}$ intersect over $f(x)$. This proves the
claim.

In order to see that $q^{-1}(R^{\vee})$ has a nodal singularity over the
smooth points of $R^{\vee}$, we just observe that
$q^{-1}(R^{\vee})=\tilde{f}^{*}p_{2}^{*}B^{\vee}$. Since $p_{2}^{*}B^{\vee}$
is a $\mathbb{P}^{1}$-bundle over $B^{\vee}$, the statement follows by
considering the intersection of a general $\mathbb{P}^{1}$ with the branch
divisor of the double cover $\tilde{f}$ (see the proof of Theorem 3.18 for a
more refined description of the singularities). ∎

We now introduce the main object of our study:

######  3.6.

Proposition. Let $D\subset Y$ be the strict transform of the surface
$q^{-1}(R^{\vee})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$. Using the
notation of Lemma 3.4 we have

(8) $D=30M-2E_{P}.$

###### Proof.

Since $R^{\vee}\subset|L|$ has degree $30$ we have
$(q\circ\mu_{P})^{*}R^{\vee}=30M$. By Lemma 3.5 the surface $q^{-1}(R^{\vee})$
has multiplicity two along the curve $R_{P}$, thus its strict transform has
class $30M-2E_{P}$. ∎

Since $D=30M-2E_{P}$ the following statement follows from Lemma 3.4 by
elementary computations.

######  3.7.

Lemma. In the situation summarised by Diagram 5, we have the following
intersection numbers:

$D^{3}=-10224\ \ \ D^{2}\cdot E_{S}=2016\ \ \ D\cdot E_{S}^{2}=-288$
$D^{2}\cdot E_{P}=3312\ \ \ D\cdot E_{P}^{2}=-756\ \ \ D\cdot E_{P}\cdot
E_{S}=792$ $(E_{P}+E_{S})\cdot D^{2}=5328\ \ \ (E_{P}+E_{S})\cdot D\cdot
E_{P}=36\ \ \ (E_{P}+E_{S})\cdot D\cdot E_{S}=504$

The prime divisors $E_{P},E_{S},D$ form a basis of $N^{1}(Y)$ that we will use
for our computations in the later sections, the class
$M=\psi^{*}c_{1}({\mathcal{O}}_{(\mathbb{P}^{2})^{\vee}}(1))$ will also be
useful due to its simple geometric interpretation. For the convenience of the
reader we summarise their intersections with a dual base:

######  3.8.

Lemma. In the situation summarised by Diagram 5, denote by $l_{P}$ (resp.
$l_{S}$) an exceptional curve of the blowup $\mu_{P}$ (resp. $\mu_{S}$), and
by $l_{D}$ a general fibre of the fibration

$(q\circ\mu_{P})_{D}:D\rightarrow R^{\vee}.$

Then we have the following intersection numbers:

$E_{S}\cdot l_{P}=1,\qquad E_{P}\cdot l_{P}=-1,\qquad M\cdot l_{P}=0,\qquad
D\cdot l_{P}=2.$ $E_{S}\cdot l_{S}=-1,\qquad E_{P}\cdot l_{S}=1,\qquad M\cdot
l_{S}=1,\qquad D\cdot l_{S}=28.$ $E_{S}\cdot l_{D}=4,\qquad E_{P}\cdot
l_{D}=2,\qquad M\cdot l_{D}=0,\qquad D\cdot l_{D}=-4.$

###### Proof.

Note first that the intersection numbers involving $D$ follow from the other
numbers and (8). Note also that the intersection numbers for the curves
$l_{S}$ and $l_{P}$ with $E_{S},E_{P},M$ are straightforward from the
construction of the elementary transform, summarised in the diagram (5).

Thus we are left to compute the intersection numbers of $l_{D}$. Recall that
by definition $D\subset Y$ is the strict transform of
$q^{-1}(R^{\vee})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$. Thus
$l_{D}$ is the strict transform of a general fibre $C_{t}$ of
$q^{-1}(R^{\vee})\subset R^{\vee}$, so it is clear that $M\cdot l_{D}=0$. The
fibre $C_{t}$ has a nodal singularity in its intersection point with the curve
$R_{P}$, the natural map $\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow
S$ maps $R_{P}$ (resp. $C_{t}$) isomorphically onto $R$ (resp. the
corresponding curve in $|L|$). Thus each branch of $C_{t}$ meets $R_{P}$
transversally, since this holds for their images in $S$. This shows that
$E_{P}\cdot l_{D}=2$. Since $E_{P}+E_{S}=(p\circ\mu_{P})^{*}3L$ and
$p(l_{D})\in|L|$ we deduce that $E_{S}\cdot l_{D}=4$. ∎

###  3.C. Canonical liftings

In this short subsection we relate the surface $D$ to the surface $D_{S}$
appearing in the introduction.

######  3.9.

Definition. Let $S$ be a smooth projective surface, and let $C\subset S$ be an
irreducible curve. Denote by $n:\tilde{C}\rightarrow C$ the normalisation, and
by

$n^{*}\Omega_{S}\rightarrow Q_{C}$

the image of the cotangent map $n^{*}\Omega_{S}\rightarrow\omega_{\tilde{C}}.$

The canonical lifting of $C$ to $\mathbb{P}(\Omega_{S})$ is the image of the
morphism $\tilde{n}:\tilde{C}\rightarrow\mathbb{P}(\Omega_{S})$ corresponding
to the line bundle $Q_{C}$. We denote this curve by
$C_{S}\subset\mathbb{P}(\Omega_{S})$.

######  3.10.

Remarks.

  1. a)

If the curve $C$ is singular the birational map $C_{S}\rightarrow C$ is not
necessarily an isomorphism.

  2. b)

We have $Q_{C}\simeq\omega_{\tilde{C}}$ if and only if $C$ is immersed.

  3. c)

If $C\subset S$ is a nodal curve, then $C_{S}\subset\mathbb{P}(\Omega_{S})$ is
a smooth curve: since $\pi\circ\tilde{n}=n$ the morphism is immersive, so we
only have to show that it is one-to-one. Recalling that
$\mathbb{P}(\Omega_{S})=\mathbf{P}(T_{S})$ (the projective bundle of lines in
$T_{S}$), we see from the definition of the canonical lifting that $\tilde{n}$
maps a point $t\in\tilde{C}$ to the point $[Tn(T_{C,t})]\in P(T_{S,n(t)})$,
where $Tn$ is the tangent map. Yet if $t_{1},t_{2}$ are two points in
$\tilde{C}$ such that $n(t_{1})=n(t_{2})$ then
$[Tn(T_{C,t_{1}})]\neq[Tn(T_{C,t_{2}})]$ since the curve is nodal. Thus
$\tilde{n}(t_{1})\neq\tilde{n}(t_{2})$. This also shows that $C_{S}$ meets the
fibre $\pi^{-1}(n(t_{i}))\simeq\mathbf{P}(T_{S,n(t_{i})})$ transversally in
two points.

  4. d)

If $C\subset S$ is a curve such that the unique singular point is a simple
cusp $p$, the canonical lifting $C_{S}\subset\mathbb{P}(\Omega_{S})$ is a
smooth curve that has tangency order two with the fibre $\pi^{-1}(p)$: the
claim is local, so we can consider that $C$ is given by a parametrisation

$n:\Delta\rightarrow\mathbb{C}^{2},\ t\ \mapsto\ (t^{2},t^{3}).$

Denote by $x,y$ the coordinates on $\mathbb{C}^{2}$, then we have induced
coordinates $(x,y),[u_{x},u_{y}]$ on

$\mathbb{P}(\Omega_{S})\simeq\mathbf{P}(T_{S})\simeq\mathbb{C}^{2}\times\mathbf{P}(\mathbb{C}\frac{\partial}{\partial
x}\oplus\mathbb{C}\frac{\partial}{\partial y}).$

The tangent map $Tn$ is given by

$t\mapsto 2t\frac{\partial}{\partial x}+3t^{2}\frac{\partial}{\partial y},$

so we see that $Q_{C}\simeq{\mathcal{I}}_{0}\otimes\omega_{\Delta}$ and
$\tilde{n}$ is given by

$t\mapsto\left((t^{2},t^{3}),[2\frac{\partial}{\partial
x}+3t\frac{\partial}{\partial y}]\right).$

This map is well-defined in the origin and maps it onto the point
$\left((0,0),[2:0]\right)$. In the affine chart $u_{x}\neq 0$ the map
$\tilde{n}$ is given by

$t\mapsto(t^{2},t^{3},3t).$

In the coordinates $(x,y,z)$ of the affine chart the image of this map is the
smooth curve cut out by $9x=z^{2},27y=z^{3}$. The intersection of this curve
with the fibre $x=0,y=0$ is a finite non-reduced scheme of length two.

######  3.11.

Notation. We will denote by $C_{node}$ (resp. $C_{cusp}$) an element of $|L|$
that has exactly two nodes (resp. a cuspidal point)222This notation is
justified by the fact that these curves correspond to the nodes (resp. cusps)
of the curve $R^{\vee}$, see Subsection 3.B..

We will denote by $C_{node,S}$ (resp. $C_{cusp,S}$) the canonical lifting of
the curve $C_{node}$ (resp. $C_{cusp}$) to $\mathbb{P}(\Omega_{S})$.

######  3.12.

Lemma. In the situation summarised by Diagram 5, let
$C\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ be a fibre of the universal
family $\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow|L|$, and let
$C_{S}\subset\mathbb{P}(\Omega_{S})$ be the canonical lifting of the curve
$[C]\in|L|$. Then $C_{S}$ is the strict transform of $C$ under the birational
map $\mu_{S}\circ\mu_{P}^{-1}$.

In particular let

$D_{S}:=\bigcup_{[C]\in R^{\vee}}C_{S}\subset\mathbb{P}(\Omega_{S})$

be the irreducible surface obtained by canonical liftings of singular elements
of $|L|$. Then we have $D_{S}=\mu_{S}(D)$.

###### Proof.

Since $D$ is the strict transform of $q^{-1}(R^{\vee})$ which is the universal
family of singular elements of $|L|$, the second statement follows from the
first.

For the proof of the first statement recall (cf. the proof of Lemma 3.5) that
the fibres of
$p_{2}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$
are given by liftings corresponding to canonical quotients
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{d}\rightarrow\omega_{d}$ of
lines $d\subset\mathbb{P}^{2}$. Since $q$ is obtained by base change from
$p_{2}$ we see that the fibres of $q$ are given by liftings corresponding to
quotients

$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{C}\rightarrow(f|_{C})^{*}\omega_{d}.$

In the complement of the ramification divisor the vector bundle
$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{C}$ (resp.
$(f|_{C})^{*}\omega_{d}$) identifies to $\Omega_{S}\otimes{\mathcal{O}}_{C}$
(resp. $\omega_{C}$). Thus the two liftings coincide in the complement of $R$,
hence the curves are strict transforms of each other. ∎

###  3.D. Geometry of the surface $\bar{D}$

This subsection contains the technical core of our study. We will use the
notation introduced in the Subsections 3.A and 3.B, in particular Diagram (5).
We still denote by

$p_{1}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow\mathbb{P}^{2},\qquad
p_{2}:\mathbb{P}(\Omega_{\mathbb{P}^{2}})\rightarrow(\mathbb{P}^{2})^{\vee}$

the projections and by $H_{1}$ (resp. $H_{2}$) the hyperplane classes on
$\mathbb{P}^{2}$ (resp. $(\mathbb{P}^{2})^{\vee}$). In order to simplify the
notation we will identify

$(\mathbb{P}^{2})^{\vee}=|L|,\qquad R=B,\qquad R^{\vee}=B^{\vee}.$

The goal of this subsection is to show

######  3.13.

Theorem. Let $\bar{\nu}:\bar{D}\rightarrow q^{-1}(R^{\vee})$ be the
normalisation. Then $\bar{D}$ is a smooth minimal projective surface and the
elliptic fibration $\bar{q}:\bar{D}\rightarrow R$ induced by
$q|_{q^{-1}(R^{\vee})}$ has exactly 648 singular fibres which are all of
Kodaira type $I_{1}$ (i.e. nodal cubics).

######  3.14.

Remark. Let $R\rightarrow R^{\vee}$ be the normalisation. The fibre product

$(q^{-1}(R^{\vee})\times_{R^{\vee}}R)\rightarrow R$

is a family of curves that satisfies the assumptions of Teissier’s
simultaneous normalisation theorem [DPT80, I,1.3.2,Thm.1]. Thus the fibration
$\bar{D}\rightarrow R$ is smooth near the bisection determined by the preimage
of the rational section $R_{P}\subset q^{-1}(R^{\vee})$. Note that Teissier’s
theorem does not imply the smoothness of $\bar{D}$ in the 648 remaining nodes.

The following commutative diagram will guide the reader through the
construction. The varieties and morphisms in the columns 3 to 5 have been
introduced in the Subsections 3.A and 3.B. The second column is obtained from
the third column by normalisation. The curves in the first column will be
successively introduced in this subsection.

(9)
$\textstyle{N_{\tilde{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{\mu}_{P}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{P}|_{D}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{P}}$$\textstyle{N_{\bar{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}_{\bar{D}}}$$\scriptstyle{\bar{\nu}}$$\textstyle{q^{-1}(R^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\scriptstyle{p}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{R_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{T}}$$\scriptstyle{\nu_{T}}$$\textstyle{p_{2}^{-1}(R^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}(\Omega_{\mathbb{P}^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{\mathbb{P}^{2}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n_{R}}$$\textstyle{R^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{2})^{\vee}}$

Let

$R_{\mathbb{P}^{2}}\subset\mathbb{P}(\Omega_{\mathbb{P}^{2}})$

be the curve defined by the canonical quotient
$\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\omega_{R}$ (here
we use the identification $B=R$). The curve $R_{\mathbb{P}^{2}}$ maps
birationally onto $R$ (resp. $R^{\vee}$), so its class in
$\mathbb{P}(\Omega_{\mathbb{P}^{2}})$ is

(10) $R_{\mathbb{P}^{2}}=30l_{1}+6l_{2}$

where $l_{i}$ is a fibre of the fibration $p_{i}$. Let $n_{R}:R\rightarrow
R^{\vee}$ be the normalisation, and let

$\nu_{T}:T\rightarrow
p_{2}^{-1}(R^{\vee})\subset\mathbb{P}(\Omega_{\mathbb{P}^{2}})$

be the normalisation of $p_{2}^{-1}(R^{\vee})$. Since
$p_{2}^{-1}(R^{\vee})\rightarrow R^{\vee}$ is locally trivial with fibre
$\mathbb{P}^{1}$, we obtain a ruled surface

$q_{T}:T\rightarrow R.$

In fact we have
$T\simeq\mathbb{P}(n_{R}^{*}\Omega_{(\mathbb{P}^{2})^{\vee}})$, and we set

$\zeta_{T}:=\nu_{T}^{*}c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})}(1))$

for the tautological class. The curve $R_{\mathbb{P}^{2}}\subset
p_{2}^{-1}(R^{\vee})$ is not contained in the singular locus of
$p_{2}^{-1}(R^{\vee})$ and its strict transform is a section $R_{T}\subset T$.

######  3.15.

Lemma. In the situation summarised by Diagram 9 we have

$R_{T}^{2}=-18<0.$

Thus $R_{T}\subset T$ is the unique curve with negative self-intersection and
we have

$\overline{\mbox{NE}}(T)=\langle R_{T},l_{T}\rangle,\qquad\mbox{\rm
Nef}(T)=\langle R_{T}+18l_{T},l_{T}\rangle.$

where $l_{T}$ is a fibre of $q_{T}$. Moreover we have

$K_{T}\equiv-2\zeta_{T}-72l_{T}.$

###### Proof.

Observe that
$\mathbb{P}(\Omega_{\mathbb{P}^{2}})\simeq\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})$
and
$c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})}(1))=p_{1}^{*}H_{1}-2p_{2}^{*}H_{2}.$
Using (10) we obtain

$c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})}(1))\cdot
R_{\mathbb{P}^{2}}=-54.$

Since $R^{\vee}\equiv 30H_{2}$ we have

$c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})}(1))^{2}\cdot
p_{2}^{-1}(R^{\vee})=-90.$

Since
$\zeta_{T}\equiv\nu_{T}^{*}c_{1}({\mathcal{O}}_{\mathbb{P}(\Omega_{(\mathbb{P}^{2})^{\vee}})}(1))$
and $R_{T}\rightarrow R_{\mathbb{P}^{2}}$ is birational this implies that

(11) $\zeta_{T}^{2}=-90,\qquad\zeta_{T}\cdot R_{T}=-54.$

Since $R_{T}$ is a $q_{T}$-section we conclude that

(12) $R_{T}\equiv\zeta_{T}+36l_{T}$

and hence $R_{T}^{2}=-18$. The description of the Mori cone and the nef cone
is now standard, cf. [Har77, V, Sect.2]. The canonical class of $T$ is given
by

$K_{T}\simeq
q_{T}^{*}(K_{R}+c_{1}(n_{R}^{*}\Omega_{(\mathbb{P}^{2})^{\vee}}))-2\zeta_{T}.$

Since $\deg K_{R}=18$ and

$\deg(n_{R}^{*}\Omega_{(\mathbb{P}^{2})^{\vee}})=\deg(\Omega_{(\mathbb{P}^{2})^{\vee}}\otimes{\mathcal{O}}_{R^{\vee}})=-3H_{2}\cdot
30H_{2}=-90$

we get $K_{T}\equiv-2\zeta_{T}-72l_{T}$. ∎

######  3.16.

Lemma. In the situation summarised by Diagram 9, let
$\bar{\nu}:\bar{D}\rightarrow q^{-1}(R^{\vee})$ be the normalisation. By the
universal property of the normalisation we have an induced two-to-one cover

$\tilde{f}_{\bar{D}}:\bar{D}\rightarrow T$

and we denote the branch locus by $B_{T}\subset T$. Then $B_{T}$ does not
contain any fibre of the ruling and its class is

$B_{T}\equiv 4\zeta_{T}+288l_{T}.$

Moreover one has

$K_{\bar{D}}\equiv 72\tilde{f}_{\bar{D}}^{*}l_{T}.$

Note that at this point we do not know that $\bar{D}$ is smooth. However,
being a cyclic double cover of a smooth surface, it is Gorenstein.

###### Proof.

The double cover $q^{-1}(R^{\vee})\rightarrow p_{2}^{-1}(R^{\vee})$ is
branched over the scheme $p_{1}^{*}R\cap p_{2}^{-1}(R^{\vee})$ which does not
contain any fibre of $p_{2}^{-1}(R^{\vee})\rightarrow R^{\vee}$. Thus, since
$\tilde{f}_{\bar{D}}$ is obtained from $q^{-1}(R^{\vee})\rightarrow
p_{2}^{-1}(R^{\vee})$ by normalisation, its branch locus does not contain a
fibre of the ruling.

Since $R\equiv 6H_{1}$ we have by (10)

$\nu_{T}^{*}p_{1}^{*}R\cdot R_{T}=p_{1}^{*}R\cdot R_{\mathbb{P}^{2}}=36.$

Note also that $\nu_{T}^{*}p_{1}^{*}R$ has degree six on the fibres of $q_{T}$
since a fibre maps onto a line in $\mathbb{P}^{2}$. Thus using (11) we deduce
that

$\nu_{T}^{*}p_{1}^{*}R\equiv 6\zeta_{T}+360l_{T}.$

If $l_{T}$ is a general $q_{T}$-fibre we can identify it to the corresponding
$p_{2}$-fibre. Hence we know that this fibre intersects $p_{1}^{*}R$
transversally in four points and with multiplicity two in the point which will
give the node. This last point is on the curve $R_{\mathbb{P}^{2}}$, so we see
that the effective divisor $\nu_{T}^{*}p_{1}^{*}R$ contains its strict
transform $R_{T}$ with multiplicity $2$. Thus we can write

(13) $\nu_{T}^{*}p_{1}^{*}R=2R_{T}+B_{T},$

hence by (12) we have $B_{T}\equiv 4\zeta_{T}+288l_{T}$. Since
$q^{-1}(R^{\vee})\rightarrow p_{2}^{-1}(R^{\vee})$ ramifies exactly over
$p_{1}^{*}R\cap p_{2}^{-1}(R^{\vee})$ and the surface $q^{-1}(R^{\vee})$ has a
nodal singularity in the generic point of $R_{P}\simeq
R_{\mathbb{P}^{2}}\simeq R_{T}$ (cf. Lemma 3.5), the branch locus of
$\tilde{f}_{\bar{D}}$ is exactly $B_{T}$.

Now we can compute the canonical class of $\bar{D}$: since
$\tilde{f}_{\bar{D}}$ is a double cover ramified along $B_{T}$ we have
$K_{\bar{D}}\equiv\tilde{f}_{\bar{D}}^{*}K_{T}+\frac{1}{2}\tilde{f}_{\bar{D}}^{*}B_{T}.$
Thus we obtain

$K_{\bar{D}}\equiv\tilde{f}_{\bar{D}}^{*}(-2\zeta_{T}-72l_{T}+\frac{1}{2}(4\zeta_{T}+288l_{T}))=72\tilde{f}_{\bar{D}}^{*}l_{T}.$

∎

We will now analyse the singularities of $\bar{D}$ via the double cover
$\tilde{f}_{\bar{D}}$: the morphism $\tilde{f}_{\bar{D}}$ is the degree two
cyclic cover associated to a line bundle isomorphic to $\frac{1}{2}B_{T}\equiv
2\zeta_{T}+144l_{T}$, so by [Laz04a, Sect.4.1.B], given a point $p\in T$ and a
local equation $g(x,y)$ for $B_{T}$ near $p$, a local equation of $\bar{D}$ is

$z^{2}=g(x,y).$

In particular $\bar{D}$ is singular if and only $B_{T}$ is singular in $p$.

###### Proof of Theorem 3.13.

By Lemma 3.16 we know that the canonical class is nef, so the surface is
minimal. We will show that the branch locus $B_{T}$ is smooth, hence $\bar{D}$
is smooth.

Let $n_{R}:R\rightarrow R^{\vee}$ be the normalisation. Fix a point $r\in R$
and denote by $T_{r}$ the $q_{T}$-fibre over the point $r$. The double cover
$\tilde{f}_{d}^{-1}(T_{r})\rightarrow T_{r}$ is determined by the double cover
$C\stackrel{{\scriptstyle f|_{C}}}{{\rightarrow}}d$ of the line
$d\subset\mathbb{P}^{2}$ corresponding to the point $n_{R}(r)$. We make a case
distinction:

  * •

If $n_{R}(r)$ is smooth, the corresponding cover $C\rightarrow d$ ramifies in
the four points where $d$ is not tangent to the branch divisor $B$. Since by
Lemma 3.16 the curve $B_{T}$ does not contain fibre of the ruling and has
degree four on every fibre, we see that $B_{T}\cap T_{r}$ is smooth. Thus
$B_{T}$ is smooth near $T_{r}$.

  * •

If $n_{R}(r)$ is a cusp, the corresponding cover $C\rightarrow d$ ramifies in
the three points where $d$ is not tangent to $B$. Thus the finite scheme
$B_{T}\cap T_{r}$ has at least three smooth points. Since the scheme has
length four, it is smooth. Thus $B_{T}$ is smooth near $T_{r}$.

  * •

If $n_{R}(r)$ is a node, the corresponding cover $C\rightarrow d$ ramifies in
the two points where $d$ is not tangent to $B$. Since the normalisation
$\tilde{C}\rightarrow C$ is a rational curve, the double cover
$\tilde{C}\rightarrow l$ has exactly two ramification points. This shows that
$B_{T}\cap T_{r}$ has two smooth points and a point $t\in T_{r}$ with
multiplicity two supported on one of the two points where $d$ is tangent to
$B$ (the other point is on $R_{T}$, hence the asymmetry). Thus $B_{T}\cap
T_{r}$ is not smooth, nevertheless we claim that $B_{T}$ is smooth near
$T_{r}$: the fibration $p_{2}$ is the universal family of lines on
$\mathbb{P}^{2}$ so by construction the line $p_{1}(p_{2}^{-1}(n_{R}(r)))$ is
tangent to $R$. Thus $p_{2}^{-1}(n_{R}(r))$ is tangent to $p_{1}^{*}R$. Since
$B_{T}$ is the unique component of $\nu_{T}^{*}p_{1}^{*}R$ passing through
$t$, we obtain that $B_{T}$ is tangent to the fibre $T_{r}$. Since the local
intersection number in the point $t$ is two, we deduce (e.g. by [Per08,
p.255]) that $B_{T}$ is smooth near this point.

The analysis of the branch locus allows also to determine the singular fibres
of $\bar{q}:\bar{D}\rightarrow R$: since $q_{T}$ is a smooth fibration, a
fibre of $\bar{q}=q_{T}\circ\tilde{f}_{\bar{D}}$ is singular if and only if
the restriction of the branch locus $B_{T}$ to the fibre $T_{r}$ is singular.
We have seen that this happens if and only if $n_{R}(r)$ is a node. Since
$R^{\vee}$ has 324 nodes, we see that there are 648 singular fibres. Since the
intersection $B_{T}\cap T_{r}$ has two reduced points and one point with
multiplicity two, the double cover is a nodal cubic. ∎

For later use, we note

######  3.17.

Corollary. The curve $B_{T}\subset T$ is smooth and irreducible.

###### Proof.

The smoothness of $B_{T}$ was shown in the proof of Theorem 3.13, so we are
left to show that $B_{T}$ is connected. Since $B_{T}$ does not contain the
curve $R_{T}$ we know by Lemma 3.15 that the irreducible components of $B_{T}$
are nef divisors. Since $B_{T}=4\zeta_{T}+288l_{T}$ by Lemma 3.16 we can use
(11) to compute that $B_{T}^{2}=864>0$. Thus $B_{T}$ is a nef and big divisor,
hence connected. ∎

###  3.E. Geometry of the surface $\tilde{D}$

The first goal of this section is to describe the normalisation of the surface
$D$:

######  3.18.

Theorem. Let $\nu:\tilde{D}\rightarrow D$ be the normalisation. Then
$\tilde{D}$ is a smooth projective surface that is the blowup of $\bar{D}$ in
720 points: the 648 nodal points of the singular fibres and 72 points on the
elliptic fibres which correspond to cuspidal curves in $|L|$.

###### Proof.

For simplicity’s sake we denote by

$q:q^{-1}(R^{\vee})\rightarrow R^{\vee},\qquad\psi:D\rightarrow R^{\vee}$

the restriction of the fibration $q$ over the curve $R^{\vee}$ (resp. the
restriction of $\psi$ to $D$). Given a point $r\in R^{\vee}$ we will describe
the map $D\rightarrow q^{-1}(R^{\vee})$ in a neighbourhood of the fibre
$q^{-1}(r)$. This local description will then allow us to show that
$\tilde{D}$ is smooth.

1st case. $r\in R^{\vee}$ is a smooth point. We know by Lemma 3.5 that in
$q^{-1}(R^{\vee})$ has nodal singularities near $q^{-1}(r)$, so the blowup
along $R_{P}$ coincides with the normalisation and $D$ is smooth near
$\psi^{-1}(r)$.

2nd case. $r\in R^{\vee}$ is a node. Denote by $\Delta\subset R^{\vee}$ and
$\Delta^{\prime}\subset R^{\vee}$ the two local branches through $r$. Then
$\Delta$ is smooth, and we can suppose that it is a small disc that contains
no other singular points of $R^{\vee}$. The preimage

$q^{-1}(\Delta)\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$

is an analytic hypersurface, so Gorenstein, and generically reduced, hence
reduced. We have seen in Subsection 3.B that the fibres of

$q^{-1}(\Delta)\rightarrow\Delta$

for $t\neq 0$ are curves with exactly one node and the central fibre has
exactly two nodes. Denote by $x_{1},x_{2}$ the nodes of the central fibre, and
by $x_{t}$ the unique node of the fibre over $t\neq 0$. Then $x_{1}$ is, up to
renumbering, in the closure of $\cup_{t\neq 0}x_{t}$, so these points form a
curve $R_{\Delta}$ that is a section of

$q^{-1}(\Delta)\rightarrow\Delta$

and the curve $R_{\Delta}$ is in the non-normal locus of $q^{-1}(\Delta)$.
Since $R_{\Delta}$ is a section over a smooth curve, the point $x_{2}$ is not
in $R_{\Delta}$. Since $\Delta$ is smooth and
$q^{-1}(\Delta)\rightarrow\Delta$ has smooth fibres in the complement of
$R_{\Delta}\cup x_{2}$, we see that the singular locus of $q^{-1}(\Delta)$ is
contained in $R_{\Delta}\cup x_{2}$. Since $x_{2}$ is an isolated singularity
and $q^{-1}(\Delta)$ is Gorenstein, we see that $x_{2}$ is in the normal locus
of $q^{-1}(\Delta)$. Let $\widetilde{q^{-1}(\Delta)}\rightarrow
q^{-1}(\Delta)$ be the normalisation. By the universal property of the
normalisation we have an embedding
$\widetilde{q^{-1}(\Delta)}\hookrightarrow\bar{D}$. Since $\bar{D}$ is smooth
by Theorem 3.13 we see that $\widetilde{q^{-1}(\Delta)}$ is smooth, in
particular $x_{2}$ is a smooth point $q^{-1}(\Delta)$.

We can now describe the blowup of $q^{-1}(\Delta)$ along the scheme
$q^{-1}(\Delta)\cap R_{P}$. By construction $R_{P}$ contains all the nodes
over $R^{\vee}$ so we see that set-theoretically

$R_{P}\cap q^{-1}(\Delta)=R_{\Delta}\cup x_{2}.$

Since $q^{-1}(\Delta)$ has nodal singularities along $R_{\Delta}\subset
q^{-1}(\Delta)\cap R_{P}$, the blowup coincides with the normalisation. We
also observe that the curve $R_{P}$ meets $q^{-1}(\Delta)$ transversally in
$x_{2}$: if the intersection is not transversal, the intersection of the
images $q(q^{-1}(\Delta))$ and $q(R_{\Delta^{\prime}})$ is not transversal.
But $q(q^{-1}(\Delta))=\Delta$ and $q(R_{\Delta^{\prime}})=\Delta^{\prime}$,
so they intersect transversally.

Thus near the point $x_{2}$ the blowup of $q^{-1}(\Delta)$ along the scheme
$R_{P}\cap q^{-1}(\Delta)$ coincides with the blowup of the reduced point
$x_{2}\in q^{-1}(\Delta)$, hence the blow-up is smooth.

We can now conclude as follows: near the fibre $q^{-1}(r)$ the surface
$q^{-1}(R^{\vee})$ is reducible with irreducible components $q^{-1}(\Delta)$
and $q^{-1}(\Delta^{\prime})$. The strict transform of $q^{-1}(R^{\vee})$ in
$Y$ coincides with the union of the strict transforms of $q^{-1}(\Delta)$ and
$q^{-1}(\Delta^{\prime})$ in $Y$. We have seen that the strict transforms of
$q^{-1}(\Delta)$ and $q^{-1}(\Delta^{\prime})$ are smooth. Thus in a
neighbourhood of $\psi^{-1}(r)$, the normalisation $\tilde{D}\rightarrow D$
consists just of separating the two irreducible components.

This finishes the discussion of this case. Note that the argument above also
shows that near one of the 648 nodal fibres of the elliptic fibration
$\bar{q}:\bar{D}\rightarrow R$ (cf. Theorem 3.13), the birational map
$\tilde{D}\rightarrow D$ is given by the blowup of the node of the fibre.

3rd case. $r\in R^{\vee}$ is a cusp. Let $x\in q^{-1}(r)$ be the unique
singular point. The statement is clear in the complement of $x$, so a local
computation near $x$ will allow us to conclude: the universal family of lines
$\mathbb{P}(T_{\mathbb{P}^{2}})\subset\mathbb{P}^{2}\times(\mathbb{P}^{2})^{\vee}$
can be given by the equation

$\sum_{i=0}^{2}x_{i}y_{i}=0$

where $x_{i}$ are coordinations on $\mathbb{P}^{2}$ and $y_{i}$ on
$(\mathbb{P}^{2})^{\vee}$. Identifying $x\in q^{-1}(r)\subset S$ to its image
in $S$, we can assume that $f(x)=[1:0:0]$ and work in the affine chart
$x_{0}=1$ on $\mathbb{P}^{2}$. Since the branch curve $B$ has a simple
inflection line in $(0,0)$ we can suppose that it is given in the local chart
by $x_{2}=x_{1}^{3}$. The tangent line to $B$ in the origin is given by
$x_{2}=0$, so it corresponds to the point $[0:0:1]\in(\mathbb{P}^{2})^{\vee}$.
Therefore we choose the affine chart $y_{2}=1$ for $(\mathbb{P}^{2})^{\vee}$,
so the universal family of lines is given by

$y_{0}+x_{1}y_{1}+x_{2}=0.$

We choose local coordinates $(u_{1},u_{2})$ on $S$ near the point $x$ such
that the double cover $f$ is given by

$(u_{1},u_{2})\mapsto(u_{2},u_{1}^{2}+u_{2}^{3}).$

The coordinates $(u_{1},u_{2},y_{0},y_{1})$ are thus local coordinates on
$S\times(\mathbb{P}^{2})^{\vee}$, in these coordinates the divisor
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ is given by the equation

$y_{0}+u_{2}y_{1}+u_{1}^{2}+u_{2}^{3}=0.$

It is not difficult to check that the local equation of $R^{\vee}$ is

$4y_{1}^{3}+27y_{0}^{2}=0,$

so the surface
$q^{-1}(R^{\vee})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ is given by
the local equations

$y_{0}+u_{2}y_{1}+u_{1}^{2}+u_{2}^{3}=0,\qquad 4y_{1}^{3}+27y_{0}^{2}=0.$

The curve $R_{P}\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ can be
parametrised by

$t\mapsto(0,t,2t^{3},-3t^{2}),$

so the map

$\mathbb{C}^{4}\rightarrow\mathbb{C}^{4},\quad(u,t,v,w)\mapsto(u,t,2t^{3}+uv,-3t^{2}+uw)$

is a local chart333The computation for the other charts is analogous, but
simpler. for the blow-up of $S\times(\mathbb{P}^{2})^{\vee}$ along the curve
$R_{P}$. In this local chart the equation of the threefold
$Y\subset\mbox{Bl}_{R_{P}}S\times(\mathbb{P}^{2})^{\vee}$ is

$v+tw+u=0,$

while the strict transform of the hypersurface $4y_{1}^{3}+27y_{0}^{2}=0$ is
given by

$108t^{4}w-36t^{2}uw^{2}+4u^{2}w^{3}+108t^{3}v+27uv^{2}=0.$

Substituting $v=-tw-u$ in the last equation we obtain local equations of $D$
in $\mbox{Bl}_{R_{P}}S\times(\mathbb{P}^{2})^{\vee}$:

(14) $v+tw+u=0,\qquad-9t^{2}w^{2}+4uw^{3}-108t^{3}+54twu+27u^{2}=0.$

Let $C_{cusp,Y}$ be the strict transform of the cuspidal curve $q^{-1}(r)$.
Computing the Jacobian matrix for the system of equations (14) we see that $D$
is singular along $C_{cusp,Y}$ but its unique singular point on the
exceptional divisor $u=0$ is the origin. Since the origin is contained in
$C_{cusp,Y}$, it is a non-normal point of $D$. Thus we are done if we show
that the normalisation is smooth near the preimage of $C_{cusp,Y}$: note that
$C_{cusp,Y}$ can locally be parametrised by

$w\mapsto(\frac{w^{3}}{27},-\frac{w^{2}}{9},\frac{2w^{3}}{27},w),$

so

$\mathbb{C}^{4}\rightarrow\mathbb{C}^{4},\quad(\alpha,\beta,\gamma,\delta)\mapsto(\alpha\beta+\frac{\delta^{3}}{27},\beta-\frac{\delta^{2}}{9},\beta\gamma+\frac{2\delta^{3}}{27},\delta)$

gives a local chart for the blow-up of $\mathbb{C}^{4}$ along this curve (the
other charts are less interesting for the proof and left to the reader). The
strict transform of the hypersurface $v+tw+u=0$ under this blowup is given by

$\alpha+\delta+\gamma=0,$

and the strict transform of $-9t^{2}w^{2}+4uw^{3}-108t^{3}+54twu+27u^{2}=0$ is

$-108\beta+27\delta^{2}+54\alpha\delta+27\alpha^{2}=0.$

Hence the strict transform $D^{\prime}$ of the surface $D$ is given by

$\alpha+\delta+\gamma=0,\qquad-108\beta+27\delta^{2}+54\alpha\delta+27\alpha^{2}=0.$

It is straightforward to check that $D^{\prime}$ is smooth and the morphism
$D^{\prime}\rightarrow D$ is finite. By the universal property of the
normalisation $\tilde{D}\rightarrow D$ we obtain an embedding
$D^{\prime}\hookrightarrow\tilde{D}$, so $\tilde{D}$ is smooth. ∎

We are finally ready to prove of our first main result:

###### Proof of Theorem 1.3.

We have shown in Lemma 3.12 that $D_{S}=\mu_{S}(D)$. We claim that the
birational morphism $D\rightarrow D_{S}$ is finite. In particular the
normalisation of $D_{S}$ is given by the surface $\tilde{D}$, so the statement
follows from Theorem 3.18.

For the proof of the claim recall that $D$ is the strict transform of
$q^{-1}(R^{\vee})\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ and the
curves $l_{S}$ contracted by $\mu_{S}$ are the strict transforms of the fibres
of
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R})\rightarrow
R_{S}$ (cf. Corollary 3.2). Thus it is sufficient that $q^{-1}(R^{\vee})$ does
not contain any $p$-fibre, yet this is clear: a $p$-fibre is mapped by $q$
onto a line in $(\mathbb{P}^{2})^{\vee}$, so it is not contained in the
irreducible curve $R^{\vee}$.

In order to determine the class of $D_{S}$, it is sufficient to compute the
push-forward of the class $D\equiv 30M-2E_{P}$ (cf. Proposition 3.6).

Since $\mu_{S}^{*}(\zeta_{S}+\pi^{*}2L)\equiv M+E_{S}$ (cf. Remark 3.3), we
have $(\mu_{S})_{*}M\equiv\zeta_{S}+\pi^{*}2L$. We have seen in Subsection 3.A
that $\mu_{S}(E_{P})=\pi^{-1}(R)$. Since $R\simeq 3L$ we obtain

$(\mu_{S})_{*}D\equiv(\mu_{S})_{*}(30M-2E_{P})\equiv 30\zeta_{S}+\pi^{*}54L.$

∎

Theorem 3.18 gives a complete description of the elliptic fibration
$\tilde{D}\rightarrow R$. In order to determine an intersection table for
$\tilde{D}$, we are left to describe a curve that is horizontal with respect
to the fibration:

######  3.19.

Lemma. Let $\tilde{f}_{\bar{D}}:\bar{D}\rightarrow T$ be the double cover
introduced in Lemma 3.16, and let $\bar{q}:\bar{D}\rightarrow R$ be the
elliptic fibration (cf. Theorem 3.13). Set

$N_{\bar{D}}:=\tilde{f}_{\bar{D}}^{*}R_{T}.$

Then $N_{\bar{D}}$ is a smooth irreducible curve that is a bisection of
$\bar{q}$ and

(15) $N_{\bar{D}}^{2}=-36.$

Moreover if

  * •

$t\in R$ is a point mapping onto a node in $R^{\vee}$, the intersection of
$N_{\bar{D}}$ with the fibre $\bar{q}^{-1}(t)$ consists of two smooth points.
In particular $N_{\bar{D}}$ is disjoint from the node in $\bar{q}^{-1}(t)$.

  * •

$t\in R$ is a point mapping onto a cusp in $R^{\vee}$, the intersection of
$N_{\bar{D}}$ with the fibre $\bar{q}^{-1}(t)$ is non-reduced, its support
being the unique point of $\bar{q}^{-1}(t)$ mapping onto the cusp
$q^{-1}(\nu(t))$.

###### Proof.

We know by (13) that the pull-back of $p_{1}^{*}R$ to $R_{T}$ decomposes as
$2R_{T}+B_{T}$ where $B_{T}$ is the branch divisor of $\tilde{f}_{\bar{D}}$.
In particular $\tilde{f}_{\bar{D}}$ is étale in the generic point of $R_{T}$,
so the pull-back $\tilde{f}_{\bar{D}}^{*}R_{T}$ is a reduced bisection and by
Lemma 3.15 one has

$N_{\bar{D}}^{2}=(\tilde{f}_{\bar{D}}^{*}R_{T})^{2}=2\cdot(-18)=-36.$

In order to see that $N_{\bar{D}}$ is smooth we observe, e.g. by using the
description in the proof of Theorem 3.13 that $R_{T}$ is disjoint from the
branch divisor $B_{T}$ in the complement of the fibres corresponding to cusps.
Thus its preimage $N_{\bar{D}}$ is smooth in the complement of the fibres
corresponding to cusps. For a fibre over a cusp the section $R_{T}$ passes
exactly through the inflection point. Thus, analogously to the proof of
Theorem 3.18, a local computation shows that $N_{\bar{D}}$ is also smooth near
these fibres.

Finally observe that for a point $t\in R$ mapping onto a node in $R^{\vee}$,
the curve $R_{T}$ is disjoint from the branch curve $B_{T}$, so there are two
distinct points $N_{\bar{D}}$ mapping onto $R_{T}\cap q_{T}^{-1}(t)$. For a
point $t\in R$ mapping onto a cusp in $R^{\vee}$, the intersection $R_{T}\cap
q_{T}^{-1}(t)$ is contained in $B_{T}$, so the set-theoretic preimage is a
single point. ∎

######  3.20.

Proposition. In the situation of Theorem 3.18, let $N_{\tilde{D}}$ be the
strict transform of $N_{\bar{D}}$ (cf. Lemma 3.19 above) under the birational
map $\tilde{\mu}_{P}:\tilde{D}\rightarrow\bar{D}$. Moreover if

  * •

$t\in R$ is a point mapping onto a node in $R^{\vee}$, denote by
$C_{node,t}\subset\tilde{D}$ the strict transform of the nodal cubic
$q^{-1}(t)$, and by $l_{node,t}\subset\tilde{D}$ the exceptional curve mapping
onto the node.

  * •

$t\in R$ is a point mapping onto a cusp in $R^{\vee}$, denote by
$C_{cusp,t}\subset\tilde{D}$ the strict transform of the smooth elliptic curve
$q^{-1}(t)$ and by $l_{cusp,t}$ the exceptional curve mapping onto a point in
$q^{-1}(t)$.

Then we have the following intersection numbers:

$C_{node,t}\cdot l_{node,t}=2,\ C_{node,t}^{2}=-4,\ l_{node,t}^{2}=-1,$
$C_{cusp,t}\cdot l_{cusp,t}=1,\ C_{cusp,t}^{2}=-1,\ l_{cusp,t}^{2}=-1,$
$N_{\tilde{D}}\cdot l_{node,t}=0,\ N_{\tilde{D}}\cdot C_{node,t}=2,\
N_{\tilde{D}}\cdot l_{cusp,t}=1,\ N_{\tilde{D}}\cdot C_{cusp,t}=1,$
$N_{\tilde{D}}^{2}=-108.$

###### Proof.

The intersection numbers that do not involve $N_{\tilde{D}}$ are immediate
consequences of Theorem 3.18. The intersection numbers involving
$N_{\tilde{D}}$ follow from Lemma 3.19. ∎

The final step of our computations will be to use the intersection table in
Proposition 3.20 to further investigate the divisor classes on $Y$. As a
preparation, we consider the image of the curve $N_{\tilde{D}}$ in $Y$: by
construction this image is a curve $N\subset E_{P}$ which is a bisection of
the ruling $E_{P}\rightarrow R$. We can make this description more geometric:
recall that

$E_{P}\simeq\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})\simeq\mathbf{P}(T_{S}\otimes{\mathcal{O}}_{R})$

and by Lemma 3.12 the fibres of the universal family map birationally onto the
canonically lifted curves. Recall also that for a point $x\in R$ there exists
a unique element of $|L|$ that is singular in $x$ (the preimage of the tangent
line of $B\simeq R$ in $x$), denote this element by $C_{x}$. Using the notion
of tangent line of plane singular curves as explained in [Per08, p.97]. we
obtain that

(16) $N=\\{v\in\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{R})\simeq
P(T_{S}\otimes{\mathcal{O}}_{R})\ |\ v\mbox{ is a tangent line of $C_{x}$
where $x=\pi(v)$}\\}.$

######  3.21.

Lemma. The class of $N\subset E_{P}$ is $2\tilde{R}+72l_{p}$ where $l_{p}$ is
a fibre of the ruling $E_{P}\rightarrow R_{P}$.

Moreover we have the following intersection numbers:

$E_{S}\cdot N=0,\qquad E_{P}\cdot N=36,\qquad M\cdot N=60,\qquad D\cdot
N=1728.$

###### Proof.

We know that $N\subset E_{P}$ is a bisection, so we have $N\equiv
2\tilde{R}+\lambda l_{p}$. We claim that $N$ is disjoint from the negative
section $\tilde{R}\subset E_{P}$. By Lemma 3.1 we have
$(\tilde{R})_{E_{P}}^{2}=-36$, so the claim determines the class of $N\subset
E_{P}$. The intersection numbers then follow from Lemma 3.1 and the fact that
the map $(q\circ\mu_{P})|_{N}:N\rightarrow R^{\vee}$ has degree two and
$R^{\vee}=B^{\vee}$ has degree 30 (cf. Subsection 3.B).

Proof of the claim. Recall that the tangent map
$f^{*}\Omega_{\mathbb{P}^{2}}\rightarrow\Omega_{S}$ has rank one along $R$ and
its image defines a canonical inclusion
$\Omega_{R}\hookrightarrow\Omega_{S}\otimes{\mathcal{O}}_{R}$ (cf. Subsection
3.A. The negative section $\tilde{R}$ is the section corresponding to the
induced quotient bundle
$\Omega_{S}\otimes{\mathcal{O}}_{R}\twoheadrightarrow(\Omega_{S}\otimes{\mathcal{O}}_{R}/\Omega_{R})$.
If we write $f$ in local coordinates as $(x,y)\mapsto(x,y^{2})$ (so the
ramification divisor is $y=0$), the image of
$f^{*}\Omega_{\mathbb{P}^{2}}\otimes{\mathcal{O}}_{R}\rightarrow\Omega_{S}\otimes{\mathcal{O}}_{R}$
is generated by $dx$, so the quotient is given by $dy$. Dually the image of
the inclusion
$(\Omega_{S}\otimes{\mathcal{O}}_{R}/\Omega_{R})^{*}\hookrightarrow
T_{S}|_{R}$ is generated by $\frac{\partial}{\partial y}$. Thus it is
sufficient to verify that the tangent lines of the curves $C_{x}$ are not
collinear to $\frac{\partial}{\partial y}$.

If $x\in R\simeq B$ is an inflection point, the curve $C_{x}$ is cuspidal and
has a unique tangent line which coincides with the tangent line of the
ramification divisor $R$. Thus it is generated by $\frac{\partial}{\partial
x}$.

If $x\in R\simeq B$ is not an inflection point, the curve $C_{x}$ is nodal
with local equation $y^{2}-x^{2}=(y-x)(y+x)$, so none of the tangent lines is
generated by $\frac{\partial}{\partial y}$. ∎

######  3.22.

Remark. We conclude this subsection with an observation indispensable for for
the proof of Theorem 1.5: we have seen in Subsection 3.D that the intersection
of the surfaces $p_{2}^{-1}(B^{\vee})=p_{2}^{-1}(R^{\vee})$ and
$p_{1}^{-1}(B)$ has two components, which in the ruled surface $T$ correspond
to the curves $R_{T}$ and $B_{T}$. By Corollary 3.17 the curve $B_{T}$ is
irreducible.

Since the double cover $\tilde{f}$ is bijective along its ramification divisor
$p^{-1}(R)$, this shows that the intersection of $q^{-1}(R^{\vee})$ and
$p^{-1}(R)$ has two irreducible components, the curves $R_{P}$ and
$\tilde{f}^{-1}(B_{T})$. In the generic point of $R_{P}$ each branch of
$q^{-1}(R^{\vee})$ meets $p^{-1}(R)$ transversally, so their strict transforms
in $Y$, i.e. the surfaces $D$ and $E_{S}$, intersect only along the strict
transform of the irreducible curve $\tilde{f}^{-1}(B_{T})$. The curve
$\tilde{f}^{-1}(B_{T})$ is not contained in the singular locus of $D$, so the
pull-back

$\nu^{*}E_{S}=\nu^{*}(\tilde{f}^{-1}(B_{T}))_{\operatorname*{red}}$

is an irreducible curve in the normalisation $\tilde{D}$.

###  3.F. Numerical restrictions on pseudoeffective classes

Our goal in this subsection is to use the description of the surface
$\tilde{D}$ to obtain additional information on the effective divisors in $Y$.
Somewhat abusively we denote by

$\nu:\tilde{D}\rightarrow Y$

the composition of the normalisation $\nu:\tilde{D}\rightarrow D$ with the
inclusion $D\hookrightarrow Y$.

######  3.23.

Lemma. The image of the pull-back map $\nu^{*}:N^{1}(Y)\rightarrow
N^{1}(\tilde{D})$ is contained in the subspace generated by the classes
$C_{node,t},l_{node,t},C_{cusp,t},l_{cusp,t}$ and $N_{\tilde{D}}$ (cf.
Proposition 3.20 for the notation).

Remark. We do not claim that these curves generate $N^{1}(\tilde{D})$, indeed
the rank of Mordell–Weil group of the elliptic fibration might not be zero.

###### Proof.

Denote by $i:S\rightarrow S$ the involution induced by the double cover $f$.
Then $i$ acts via push-forward on the linear system
$|L|=(\mathbb{P}^{2})^{\vee}$. Since all the elements of $|L|$ are pull-backs
from $\mathbb{P}^{2}$, this action is trivial. Thus we have a natural
involution $(i,i_{*}):S\times|L|\rightarrow S\times|L|$ which preserves the
universal family of $|L|$, i.e. the subvariety
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\subset S\times|L|$. We denote by
$i_{P}:\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$
the induced involution and note that by construction the fibrations $p$ and
$q$ are equivariant with respect to the action of $i_{P}$. Since
$R_{P}\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ is the locus where $q$
is not smooth, it is is preserved by $i_{P}$. Thus $i_{P}$ lifts to an
involution $i_{Y}:Y\rightarrow Y$. Since the action of $i_{*}$ preserves the
curve $R^{\vee}$, the involution $i_{Y}$ leaves the surface $D$ invariant, so
it lifts to an involution $\tilde{i}:\tilde{D}\rightarrow\tilde{D}$. Taking
the quotient by these involutions we obtain a commutative diagram

$\textstyle{\tilde{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\tau_{D}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\tilde{D}/\langle\tilde{i}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{\prime}}$$\textstyle{Y/\langle
i_{Y}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{T}}$$\textstyle{\mathbb{P}(\Omega_{\mathbb{P}^{2}})}$

The threefold $Y/\langle i_{Y}\rangle$ is a (weighted) blowup of
$\mathbb{P}(\Omega_{\mathbb{P}^{2}})$, so it has Picard number three. Since
$Y$ has also Picard number three, this shows that the pull-back
$\tau^{*}:N^{1}(Y/\langle i_{Y}\rangle)\rightarrow N^{1}(Y)$ is an
isomorphism. Thus it is sufficient to show that the image of the pull-back

$(\nu^{\prime})^{*}:N^{1}(Y/\langle i_{Y}\rangle)\rightarrow
N^{1}(\tilde{D}/\langle\tilde{i}\rangle)$

is generated by the classes
$\tau_{D}(C_{node,t}),\tau_{D}(l_{node,t}),\tau_{D}(C_{cusp,t}),\tau_{D}(l_{cusp,t})$
and $\tau_{D}(N_{\tilde{D}})$. But the surface
$\tilde{D}/\langle\tilde{i}\rangle$ is a (weighted) blow-up of the ruled
surface $T$ with exceptional locus the curves $\tau_{D}(l_{node,t})$ and
$\tau_{D}(l_{cusp,t})$, so its Neron-Severi space is generated by these
classes. ∎

######  3.24.

Lemma. In the situation of Theorem 3.18 and Proposition 3.20, we set

$\operatorname*{L_{node}}:=\sum_{t\in R,\nu(t)\mbox{ is a
node}}l_{node,t},\qquad\operatorname*{L_{cusp}}:=\sum_{t\in R,\nu(t)\mbox{ is
a cusp}}l_{cusp,t},$

Denote by $l_{D}$ a general fibre of the elliptic fibration
$\psi_{\tilde{D}}:\tilde{D}\rightarrow R$. Then we have

(17) $\displaystyle\nu^{*}E_{S}$ $\displaystyle=$ $\displaystyle
2\operatorname*{N_{\tilde{D}}}+72l_{D}-\operatorname*{L_{node}}+\operatorname*{L_{cusp}}$
(18) $\displaystyle\nu^{*}E_{P}$ $\displaystyle=$
$\displaystyle\operatorname*{N_{\tilde{D}}}+\operatorname*{L_{node}}+2\operatorname*{L_{cusp}}$
(19) $\displaystyle\nu^{*}M$ $\displaystyle=$ $\displaystyle 30l_{D}$

Before we give the proof of the lemma, let us recall the geometry of the
situation: the curves $l_{node,t}$ and $l_{cusp,t}$ are the exceptional curves
of the blow-up $\tilde{\mu}_{P}:\tilde{D}\rightarrow\bar{D}$ (cf. Theorem
3.18). This blow-up is induced by the blow-up
$\mu_{P}:Y\rightarrow\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ and we have

$\nu_{*}l_{node,t}=l_{P}=\nu_{*}l_{cusp,t}$

where $l_{P}$ is an exceptional curve of $\mu_{P}$. The general fibre of
$\psi_{\tilde{D}}:\tilde{D}\rightarrow R$ maps isomorphically onto the general
fibre of $q\circ\mu_{P}:D\rightarrow R^{\vee}$. This justifies that we denote
them by the same letter $l_{D}$ (cf. Lemma 3.8).

###### Proof of Lemma 3.24.

By Lemma 3.23 we know that the pull-backs are can be written as a linear
combination of the classes $C_{node,t},l_{node,t},C_{cusp,t},l_{cusp,t}$ and
$N_{\tilde{D}}$. These classes are not linearly independent, but by Theorem
3.18 one has

$l_{D}\equiv C_{node,t}+2l_{node,t}\equiv C_{cusp,t}+l_{cusp,t}.$

Thus it is straightforward to see that the classes
$\operatorname*{N_{\tilde{D}}},\ l_{D},\ l_{node,t},\ l_{cusp,t}$ form a basis
of the vector space. In particular we can verify the formulas by computing the
intersection numbers with the elements of the basis.

By the discussion before this proof we see that the intersection numbers with
the left hand sides are determined by Lemma 3.8, Lemma 3.21 and the projection
formula. The intersections with the right hand sides are determined by
Proposition 3.20. ∎

######  3.25.

Remark. The Mori cone of the surface $\tilde{D}$ is quite complex: the
irreducible curves
$\operatorname*{N_{\tilde{D}}},C_{node,t},l_{node,t},C_{cusp,t},l_{cusp,t}$
have negative self-intersection, so they generate extremal rays in the Mori
cone [KM98, Lemma 1.22]. By Remark 3.22 the effective divisor $\nu^{*}E_{S}$
is irreducible, and by Formula 17 the class of this divisor is not in the cone
generated by the classes
$\operatorname*{N_{\tilde{D}}},C_{node,t},l_{node,t},C_{cusp,t},l_{cusp,t}$.
Since $(\nu^{*}E_{S})^{2}=E_{S}^{2}\cdot D=-288<0$ by Lemma 3.7 the
irreducible curve $\nu^{*}E_{S}$ thus generates another extremal ray in Mori
cone.

With the description of the geometry of $E_{S}$ and $D$ in mind, we obtain
strong obstructions to the effectivity of a class in $\mathbb{P}(\Omega_{S})$:

###### Proof of Theorem 1.5.

Let $Z_{S}\subset\mathbb{P}(\Omega_{S})$ be a prime divisor and let $Z\subset
Y$ be its strict transform. The statement is clear for $Z=D$ and $Z=E_{P}$, so
we assume that $Z$ is distinct from these two surfaces. Note that we can
assume that the class of $Z$ is not a linear combination $Z=xD+yE_{P}$:
computing the intersections with $l_{D}$ and $l_{P}$ (see Lemma 3.8) we obtain
that $x\geq 0$ and $y\geq 0$. Thus we have $\lambda\geq 1.8$. Thus, up to a
multiple, we can denote by $\eta:=E_{S}+xD+yE_{P}$ the cohomology class of $Z$
in $Y$. Since $E_{S}$ is $\mu_{S}$-exceptional, the prime divisor $Z\neq
E_{S}$ and its restriction to $E_{S}$ is effective. By Corollary 3.2 this
tells that $Z|_{E_{S}}$ is also nef, and in particular $\eta^{2}\cdot
E_{S}\geq 0$. Using Lemma 3.4 and Lemma 3.7 one can easily see that this
implies that

(20) $\frac{1-4x}{3}\leq y.$

We claim that $\alpha=\nu^{*}E_{S}+0.5\operatorname*{C_{node}}$ is nef.

Proof of the claim. Using Equation (17) and Proposition 3.20 one sees that
$\alpha\cdot\operatorname*{C^{i}_{node}}=0$ and $\alpha\cdot\nu^{*}E_{S}=360$.
Since by Remark 3.22 the curve $\nu^{*}E_{S}$ is irreducible, the class
$\alpha$ is non-negative along the irreducible components of its support. This
implies the claim.

Since $\alpha$ is nef, using again Equation (17) and Proposition 3.20, we
obtain

$0\leq\alpha\cdot\nu^{*}\eta=72(5-8x+29y).$

Combining this condition with the inequality (20) gives
$\frac{y}{x}\geq-\frac{3}{11}$.

Since $(\mu_{S})_{*}E_{S}=0$, $(\mu_{S})_{*}E_{P}\equiv\pi^{*}3L$ and
$(\mu_{S})_{*}M\equiv\zeta_{S}+\pi^{*}2L$ (cf. Remark 3.3), a straightforward
computation (cf. Lemma 4.1) gives

$(\mu_{S})_{*}\eta\equiv 30x(\zeta_{S}+(1.8+\frac{y}{10x})\pi^{*}L).$

Since the ratio $\frac{y}{10x}$ is bounded from below by $-\frac{3}{110}$ the
cohomology class the prime divisor $Z_{S}$ in $\mathbb{P}(\Omega_{S})$ can
thus be written as $[Z_{S}]\equiv a(\zeta_{S}+\lambda\pi^{*}L)$ with
$\lambda\geq\frac{9}{5}-\frac{3}{110}=\frac{39}{22}$, as desired. ∎

##  4\. Positivity results

In this whole section we work in the same setup as in Subsection 3.A (cf.
summary in the Diagram (5)). We denote by

$\varphi:Y\rightarrow S$

the composition $\pi\circ\mu_{S}=p\circ\mu_{P}$. The pseudoeffective cone of
the threefold $\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ is generated by the
classes $p^{*}L$ and $q^{*}H_{2}$ where $H_{2}$ is the hyperplane class on
$(\mathbb{P}^{2})^{\vee}$. The picture becomes significantly less trivial when
we consider the pseudoeffective cone of $\mathbb{P}(\Omega_{S})$ or $Y$. Given
a divisor class $Z$ on $Y$ we write

(21) $Z\equiv aM+b\varphi^{*}L-mE_{P}.$

If $Z\subset Y$ is an effective $\mathbb{R}$-divisor such that the support
does not contain the exceptional divisor $E_{P}$, its image
$Z_{P}\subset\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})$ has class $Z_{P}\equiv
aq^{*}H_{2}+bp^{*}L$ with $a\geq 0,b\geq 0$. Moreover the effective divisor
$Z_{P}$ has multiplicity $m$ along the curve $R_{P}$.

###  4.A. Necessary conditions

In this subsection we collect some basic necessary conditions, as an
application we prove the second part of Theorem 1.4.

######  4.1.

Lemma. Let $Z\subset Y$ be an effective $\mathbb{R}$-divisor such that the
support does not contain the exceptional divisors $E_{P}$ and $E_{S}$. Let
$Z_{S}\subset\mathbb{P}(\Omega_{S})$ be the image of $Z$. Then, using the
notation (21), one has

$Z_{S}\equiv a\zeta_{S}+(2a+b-3m)\pi^{*}L$

###### Proof.

Recall that $E_{P}+E_{S}\equiv 3\varphi^{*}L$, so

$Z\equiv aM+\frac{b}{3}E_{S}+(\frac{b}{3}-m)E_{P}.$

Since $(\mu_{S})_{*}E_{S}\equiv 0,\ (\mu_{S})_{*}E_{P}\equiv\pi^{*}3L$ and
$(\mu_{S})_{*}M=\zeta_{S}+\pi^{*}2L$ we obtain

$[Z_{S}]\equiv a\zeta_{S}+(2a+b-3m)\pi^{*}L.$

∎

Remark. The divisor $D\subset Y$ introduced in Proposition 3.6 is the strict
transform of $q^{-1}(R^{\vee})$, for which we have $a=30,b=0,m=2$. Thus Lemma
4.1 is a technical extension of the formula in Theorem 1.3.

######  4.2.

Corollary. Let $Z_{S}\subset\mathbb{P}(\Omega_{S})$ be an effective
$\mathbb{R}$-divisor such that the class of $Z_{S}$ is not contained in the
cone generated by $D_{S}$ and $\pi^{*}L$. Then, using the Notation (21), we
have

(22) $\frac{2a+b-3m}{a}<1.8\quad\Leftrightarrow\quad\frac{1}{5}a+b<3m$

###### Proof.

Recall that by Theorem 1.3 we have $[D_{S}]\equiv 30(\zeta_{S}+1.8\pi^{*}L)$.
Thus we simply apply Lemma 4.1 to the strict transform of $Z_{S}$. ∎

######  4.3.

Lemma. Let $Z\subset Y$ be an effective $\mathbb{R}$-divisor such that the
support does not contain the exceptional divisors $E_{P}$ and $E_{S}$. Then,
using the notation (21) we have

(23) $a\geq m,\qquad\mbox{and}\qquad\frac{1}{2}a+b\geq\frac{9}{2}m.$

Assume furthermore that there exists a canonical lifting $C_{node,S}$ of a
rational nodal curve $C\in|L|$ (cf. Notation 3.11) that is not contained in
$Z_{S}$. Then one has

(24) $b\geq 2m.$

As a preparation let us note that the class of the curve $R_{P}\subset
p^{-1}(R)$ is given by

$R_{P}=(q^{*}H_{2}+4p^{*}L)|_{p^{-1}(R)}.$

Indeed since $p(R_{P})=R$ (resp. $q(R_{P})=R^{\vee}$) we know that
$p^{*}L\cdot R_{P}=6$ and $q^{*}H_{2}\cdot R_{P}=30$. By Lemma 3.4 we have
$(q^{*}H_{2})^{2}\cdot p^{-1}(R)=M^{2}\cdot E_{S}=6$ which implies the result.

###### Proof of Lemma 4.3.

The support of $Z$ does not contain $E_{S}$, so the support of its image
$Z_{P}$ does not contain $\mu_{P}(E_{S})=p^{-1}(R)$. Since $Z_{P}$ has
multiplicity $m$ along $R_{P}$, the intersection $Z_{P}\cap p^{-1}(R)$
contains the curve $R_{P}$ with multiplicity at least $m$. Thus if $f$ is a
general fibre of the ruling $p^{-1}(R)\rightarrow R$, then $(Z_{P}\cap
p^{-1}(R))\cdot f\geq m$. Since $Z_{P}\equiv aq^{*}H_{2}+bp^{*}L$ we obtain
that $a\geq m$.

By what precedes the class

$[Z_{P}\cap p^{-1}(R)]-mR_{P}\equiv[(a-m)q^{*}H+(b-4m)p^{*}L]|_{p^{-1}(R)}$

is effective. The isomorphism $E_{S}\rightarrow p^{-1}(R)$ maps the curve
$\tilde{R}$ onto $R_{P}$ so the class $\tilde{R}-27l_{S}$ maps onto the class
$(q^{*}H_{2}-\frac{1}{2}p^{*}L)|_{p^{-1}(R)}$. By Corollary 3.2 the class
$\tilde{R}-27l_{S}$ is nef, so

$(q^{*}H_{2}-\frac{1}{2}p^{*}L)\cdot[(a-m)q^{*}H_{2}+(b-4m)p^{*}L]\cdot
p^{-1}(R)\geq 0.$

Using the intersection numbers from Lemma 3.4 we obtain the second inequality
in (23).

For the proof of the last statement we recall that by Lemma 3.12 the curve
$C_{node,S}$ is the strict transform of the corresponding fibre $C_{node}$ of
the universal family
$\mathbb{P}(f^{*}\Omega_{\mathbb{P}^{2}})\rightarrow|L|\simeq(\mathbb{P}^{2})^{\vee}$.
Since the generic point of $C_{node,S}$ is contained in the locus where
$\mu_{P}\circ\mu_{S}^{-1}$ is an isomorphism, its strict transform $C_{node}$
is not contained in $Z_{P}$. The curve $C_{node}$ has two singular points,
which are both on the curve $R_{P}$ (cf. the proof of Lemma 3.5). Since
$Z_{P}$ has multiplicity $m$ along $R_{P}$ we obtain that $Z_{P}\cdot
C_{node}\geq 4m$. Since $Z_{P}\equiv aq^{*}H_{2}+bp^{*}L$ and $C_{node}$ is
contracted by $q$ this yields the inequality (24). ∎

######  4.4.

Corollary. Let $Z$ be a pseudoeffective $\mathbb{R}$-divisor class on $Y$ that
is modified nef [Bou04, Defn.2.2], write its divisor class as $Z\equiv
aM+b\varphi^{*}L-mE_{P}$. Then we have

$a\geq m,\qquad\mbox{and}\qquad\frac{1}{2}a+b\geq\frac{9}{2}m.$

Remark. The property that a divisor class is modified nef is equivalent to
being in the closure of the cone of mobile divisor classes [Bou04, Sect.5.1].

###### Proof.

Since $Z$ is modified nef it is in the closure of the mobile cone. Thus we can
find a sequence of effective mobile $\mathbb{Q}$-divisors $Z_{t}$ such that
their class converges to $Z$. Since the decomposition of the divisor class in
the basis $M,\varphi^{*}L,E_{P}$ is unique, this implies that the coefficients
$a_{t},b_{t},m_{t}$ converge to $a,b,m$. The divisor $Z_{t}$ being mobile its
support does not contain the exceptional divisors $E_{P}$ and $E_{S}$. Thus
Lemma 4.3 yields $a_{t}\geq m_{t}$ and
$\frac{1}{2}a_{t}+b_{t}\geq\frac{9}{2}m_{t}$. We conclude by passing to the
limit. ∎

######  4.5.

Corollary. Let $Z_{S}\subset\mathbb{P}(\Omega_{S})$ be an effective
$\mathbb{R}$-divisor such that $Z_{S}=a(\zeta+\lambda\pi^{*}L)$ with
$\lambda<1.8$. Let $C_{S}\subset\mathbb{P}(\Omega_{S})$ be a canonical lifting
of a rational nodal curve $C\in|L|$. Then $C_{S}\subset Z_{S}$.

###### Proof.

We argue by contradiction and assume that $C_{S}\not\subset Z_{S}$. By (24)
this implies $b\geq 2m$. Moreover by (23) we always have $a\geq 9m-2b$. Thus
we have

$\frac{1}{5}a+b\geq\frac{9}{5}m+\frac{3}{5}b\geq 3m,$

a contradiction to (22). ∎

###  4.B. Sufficient conditions

In this subsection we will use Boucksom’s divisorial Zariski decomposition
[Bou04, Thm.4.8]: given a pseudoeffective $\mathbb{R}$-divisor class $Z$ on
$Y$, there exists a decomposition $Z_{+}+Z_{-}$ such that $Z_{+}$ is modified
nef [Bou04, Defn.2.2] and $Z_{-}$ is an effective $\mathbb{R}$-divisor.

######  4.6.

Lemma. Let $Z$ be a pseudoeffective $\mathbb{R}$-divisor class on $Y$, and let
$Z_{+}+Z_{-}$ be its divisorial Zariski decomposition. Assume that the support
of $Z_{-}$ does not contain the exceptional divisors $E_{P}$ and $E_{S}$. Then
one has

$\varphi^{*}L\cdot Z^{2}>0$

unless $Z\equiv b\varphi^{*}L$.

######  4.7.

Remark. We will need the following elementary remark: let $Z^{\prime}\subset
Y$ be a prime divisor that surjects onto the surface $S$. For $d\gg 0$ let
$A\in|dL|$ be a general element. Then the intersection
$Z^{\prime}\cap\varphi^{-1}(A)$ is an irreducible curve. Proof: the linear
system $\varphi|_{Z^{\prime}}^{*}|dL|$ is globally generated, so a general
element does not have an irreducible component that is in the non-normal locus
of $Z^{\prime}$. Thus we can assume without loss of generality that
$Z^{\prime}$ is normal. Yet then a general element is normal [BS95,
Thm.1.7.1], hence smooth. Since it is connected by Kawamata–Viehweg vanishing,
a general element is irreducible.

###### Proof.

We first prove the statement in the case where $Z$ is a prime divisor that is
distinct from $E_{S}$ and $E_{P}$. We use the notation (21), i.e. we write

$Z\equiv aM+b\varphi^{*}L-mE_{P}.$

Note that if $a=0$ then $Z\cdot l_{s}=-m$. Since $Z$ is distinct from $E_{S}$
we obtain that $m=0$, so $Z\equiv b\varphi^{*}L$. Thus we can assume that
$a>0$.

Using the formulas in Lemma 3.7 we obtain

$\varphi^{*}L\cdot N^{2}=2(a^{2}+2ab-3m^{2}).$

Since $a>0$ we have $a^{2}+ab>a\cdot(\frac{1}{2}a+b)$. Applying twice (23) we
obtain

$a\cdot(\frac{1}{2}a+b)\geq\frac{9}{2}m^{2}.$

Thus we have

$a^{2}+ab-3m^{2}>1,5m^{2}\geq 0.$

The same argument holds if $Z$ is a pseudoeffective divisor class that is
modified nef. Indeed the argument above only uses (23). By Corollary 4.4 this
inequality also holds for pseudoeffective divisors classes that are modified
nef.

Now we consider the general case: we write $Z=Z_{+}+\sum\lambda_{i}Z_{i}$
where $\lambda_{i}>0$ and $Z_{i}$ are the irreducible components of the
negative part. Fix $d\gg 0$ such that for a very general element $A\in|dL|$
the intersection $C_{i}:=Z_{i}\cap\varphi^{-1}(A)$ is irreducible (this is
possible by Remark 4.7) and $Z_{+}|_{\varphi^{-1}(A)}$ is nef (this is
possible since the non-nef locus of $Z_{+}$ consists of at most countably many
curves [Bou04, Defn.3.3]).

Since the divisors $Z_{i}$ are distinct, the irreducible curves
$C_{i}=Z_{i}\cap\varphi^{-1}(A)$ are distinct. Thus we have

$(Z|_{\varphi^{-1}(A)})^{2}=(Z_{+}|_{\varphi^{-1}(A)}+\sum\lambda_{i}C_{i})^{2}\geq(Z_{+}|_{\varphi^{-1}(A)})^{2}+\sum\lambda_{i}^{2}C_{i}^{2}.$

By what precedes all the terms are non-negative and the sum is positive unless
all the classes are a pull-back from $S$. ∎

We can now give a bigness criterion on $Y$. For the proof of Theorem 1.4 we
will need to work with $\mathbb{R}$-divisor classes, this leads to some
additional technical effort.

######  4.8.

Lemma. Let $Z\subset Y$ be a pseudoeffective $\mathbb{R}$-divisor class, and
let $Z_{+}+Z_{-}$ be its divisorial Zariski decomposition. Assume that the
support of $Z_{-}$ does not contain the exceptional divisors $E_{P}$ and
$E_{S}$. Assume that there exists a rational number $\varepsilon>0$ such that
$(Z+\varepsilon E_{S})\cdot l_{S}>0$ and $(Z+\varepsilon E_{S})^{3}>0$.

Then $Z+\varepsilon E_{S}$ is big.

###### Proof.

Step 1. Positivity properties of $Z+\varepsilon E_{S}$. The relative Picard
number of the conic bundle $\varphi:Y\rightarrow S$ is two and the relative
Mori cone is generated by the curve classes $l_{P}$ and $l_{S}$. By assumption
we have $(Z+\varepsilon E_{S})\cdot l_{S}>0$. Since $E_{P}$ is not contained
in the support of $Z_{-}$ and the deformations of $l_{P}$ cover a divisor we
have

$Z\cdot l_{P}\geq Z_{-}\cdot l_{P}\geq 0.$

Thus we also have $(Z+\varepsilon E_{S})\cdot l_{P}>0$ and the divisor class
that $Z+\varepsilon E_{S}$ is $\varphi$-ample.

Let $A\in|dL|$ be a very general element for some $d\gg 0$. Set
$Y_{A}:=\varphi^{-1}(A)$, and denote by $\varphi_{A}:Y_{A}\rightarrow A$ the
restriction of $\varphi$. We denote the restriction of a divisor class $T$ to
$Y_{A}$ by $T_{A}$. We claim that $(Z+\varepsilon E_{S})_{A}$ is an ample
$\mathbb{R}$-divisor class.

Proof of the claim. Note that we do not have $Z\equiv b\varphi^{*}L$, since
otherwise $(Z+\varepsilon E_{S})\cdot l_{S}=-\varepsilon<0$. Thus we can apply
Lemma 4.6 and obtain $\varphi^{*}L\cdot Z^{2}>0$. Since the class of $A$ is a
positive multiple of $L$, this implies $Z_{A}^{2}>0$.

By construction $Y_{A}$ is the blowup of
$\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{A})$ in the finite set
$R_{S}\cap\pi^{-1}(A)$. Since $E_{S}$ is the exceptional divisor of $\mu_{S}$
the exceptional locus of

$\mu_{S,A}:Y_{A}\rightarrow\mathbb{P}(\Omega_{S}\otimes{\mathcal{O}}_{A})$

is equal to the support of $E_{S}\cap Y_{A}$ which is a disjoint union of
curves numerically equivalent to $l_{S}$. Since by assumption $(Z+\varepsilon
E_{S})\cdot l_{S}>0$ we obtain that

$(Z+\varepsilon E_{S})_{A}\cdot(E_{S})_{A}>0.$

Thus we have

$(Z+\varepsilon E_{S})_{A}^{2}>Z_{A}\cdot(Z+\varepsilon E_{S})_{A}\geq
Z_{A}^{2}$

where in the last inequality we used again that $E_{S}$ is not in the negative
part of $Z$. Since $Z_{A}^{2}>0$ this finally shows that

(25) $(Z+\varepsilon E_{S})_{A}^{2}>0.$

Let now $Z=Z_{+}+\sum\lambda_{i}Z_{i}$ be the divisorial Zariski
decomposition. By assumption none the prime divisors $Z_{i}$ coincides with
$E_{S}$ or $E_{P}$. Thus the intersection $Z_{i}\cap\varphi^{-1}(A)$ is either
a union of general $\varphi_{A}$-fibres or an irreducible curve $C_{i}$ (cf.
Remark 4.7). Moreover by Lemma 4.6 we have

$C_{i}^{2}=d\varphi^{*}LZ_{i}^{2}\geq 0$

and equality holds if and only if $Z_{i}\equiv b_{i}\varphi^{*}L$. In both
cases the curve $C_{i}$ is a nef divisor on $Y_{A}$. Since $Z_{+}$ is nef in
codimension one and $A$ is very general, the restriction $(Z_{+})_{A}$ is nef
. In conclusion we obtain that

$Z_{A}\equiv(Z_{+})_{A}+\sum\lambda_{i}C_{i}$

is a nef divisor.

In view of (25) and the Nakai–Moishezon criterion [Laz04a, Thm.2.3.18] the
claim follows if we show that $(Z+\varepsilon E_{S})_{A}\cdot G>0$ for every
irreducible curve $G\subset Y_{A}$. Arguing by contradiction we assume that
there exists an irreducible curve $G$ such that $(Z+\varepsilon
E_{S})_{A}\cdot G\leq 0$.

Since $(Z+\varepsilon E_{S})\cdot l_{S}>0$ we have $G\not\subset E_{S}$. Since
$Z_{A}$ is nef and $(Z+\varepsilon E_{S})_{A}\cdot G\leq 0$, we obtain that
$E_{S}\cdot G=0$. Hence $G$ is disjoint from the exceptional locus of
$\mu_{S,A}$. Thus we have

$0\geq(Z+\varepsilon E_{S})_{A}\cdot G=(\mu_{S,A})_{*}(Z+\varepsilon
E_{S})_{A}\cdot(\mu_{S,A})_{*}G.=(\mu_{S,A})_{*}Z_{A}\cdot(\mu_{S,A})_{*}G.$

We claim that $(\mu_{S,A})_{*}Z_{A}$ is an ample class, this yields the
desired contradiction.

For the proof of the claim, note that the push-forward $(\mu_{S})_{*}Z$ is
pseudo-effective and not a pull-back from $S$, so it is collinear to
$\zeta_{S}+\lambda\pi^{*}L$ for some $\lambda\in\mathbb{Q}$. Since $\zeta_{S}$
is not pseudoeffective we have $\lambda>0$.

Yet the cotangent bundle $\Omega_{S}$ is stable, so if $A$ is a sufficiently
positive hyperplane section, the restricted vector bundle
$\Omega_{S}\otimes{\mathcal{O}}_{A}$ is stable with trivial determinant. Thus
the restriction of the tautological class $\zeta_{S}$ to $\pi^{-1}(A)$ is nef.
Since $\lambda>0$ this shows that the restriction of
$\zeta_{S}+\lambda\pi^{*}L$ to $\pi^{-1}(A)$ is ample. This proves the claim.

Step 2. Approximation. Since ampleness is an open property we can find by Step
1 real numbers $0\leq\delta_{i}\ll 1$ such that

$Z^{\delta}:=Z-\delta_{1}\varphi^{*}L-\delta_{2}E_{P}-\delta_{3}E_{S}$

is a $\mathbb{Q}$-divisor class with the following properties:

  * •

$Z^{\delta}+\varepsilon E_{S}$ is $\varphi$-ample;

  * •

$(Z^{\delta}+\varepsilon E_{S})_{A}$ is ample;

  * •

$(Z^{\delta}+\varepsilon E_{S})^{3}>0$.

We claim that such a $\mathbb{Q}$-divisor class is big. Since
$\delta_{1}\varphi^{*}L+\delta_{2}E_{P}+\delta_{3}E_{S}$ is effective this
will finish the proof of the lemma.

Proof of the claim. Our goal is to show that for $m$ sufficiently high and
divisible we have

$H^{j}(Y,{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S})))=0$

for $j=2,3$. Then we have

$h^{0}(Y,{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S})))\geq\chi(Y,{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S}))),$

so the bigness of $Z^{\delta}+\varepsilon E_{S}$ follows from asymptotic
Riemann-Roch and the property $(Z^{\delta}+\varepsilon E_{S})^{3}>0$.

Since $Z^{\delta}+\varepsilon E_{S}$ is $\varphi$-ample we know by relative
Serre vanishing that
$R^{1}\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S}))=0$ for
sufficiently high and divisible $m$. By cohomology and base change [Har77,
III,Thm.12.11] the direct image sheaf
$\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S}))$ is locally
free and commutes with base change.

Since the higher direct images vanish, we have

$H^{j}(Y,{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S})))\simeq
H^{j}(S,\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S}))),$

in particular $H^{3}(Y,{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon E_{S})))=0$.
Moreover, since $\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S}))$ is locally free, Serre duality applies:

$H^{2}(S,\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S})))=H^{0}(S,(\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S})))^{*}).$

Hence the vanishing follows if we show that

$H^{0}(A,(\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S})))^{*}\otimes{\mathcal{O}}_{A})=0.$

Since the direct image sheaf has the base change property, we have

$\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S}))\otimes{\mathcal{O}}_{A}\simeq(\varphi_{A})_{*}{\mathcal{O}}_{Y_{A}}(m(Z^{\delta}+\varepsilon
E_{S})).$

Yet $(Z^{\delta}+\varepsilon E_{S})_{A}$ is ample, so by [Anc82, Cor.2.11,
Thm.3.1.] the direct image sheaf
$(\varphi_{A})_{*}{\mathcal{O}}_{Y_{A}}(m(Z^{\delta}+\varepsilon E_{S}))$ is
ample for $m\gg 0$. By [Laz04b, Ex.6.1.4] this implies that its dual
$(\varphi_{*}{\mathcal{O}}_{Y}(m(Z^{\delta}+\varepsilon
E_{S})))^{*}\otimes{\mathcal{O}}_{A}$ does not have global sections. ∎

The next statement is an immediate application of Lemma 4.8 and a first step
towards Theorem 1.4:

######  4.9.

Corollary. The divisor class $D+4E_{S}$ is big. In particular its push-forward
$(\mu_{S})_{*}(D+4E_{S})\equiv 30(\zeta_{S}+1.8\pi^{*}L)$ is a big divisor
class.

###### Proof.

We are done if we verify the conditions of Lemma 4.8: using the formulas in
Lemma 3.7 we obtain that $(D+4E_{S})^{3}=10242>0$. The divisor $D$ is
effective and prime, so the exceptional divisors $E_{P}$ and $E_{S}$ are not
in the support of the negative part of its divisorial Zariski decomposition.
Finally by Lemma 3.8 one has $(D+4E_{S})\cdot l_{S}=24>0$.∎

###### Proof of Theorem 1.4.

For the simplicity of notation we will work with the rounded value
$1,7952024$.

We argue by contradiction and assume that $\lambda\geq 1.7952024$ for all
prime divisors in $\mathbb{P}(\Omega_{S})$. Let $Z$ be a pseudoeffective
divisor class on $Y$ such that $(\mu_{S})_{*}Z:=Z_{S}$ generates the second
extremal ray of the pseudoeffective cone of $\mathbb{P}(\Omega_{S})$. Since
$(\mu_{S})_{*}Z$ is not big, the divisor class $Z+\varepsilon E_{S}$ is not
big for every $\varepsilon\geq 0$.

Assume first that the negative part the divisorial Zariski decomposition of
$Z$ contains the exceptional divisor $E_{P}$ or $E_{S}$: then we can write
$Z=Z^{\prime}+\alpha E_{P}+\beta E_{S}$ with $Z^{\prime}$ pseudoeffective and
$\alpha\geq 0$ and $\beta\geq 0$. Since

$(\mu_{S})_{*}Z\equiv(\mu_{S})_{*}Z^{\prime}+\alpha(\mu_{S})_{*}E_{P}\equiv(\mu_{S})_{*}Z^{\prime}+3\alpha\pi^{*}L$

generates an extremal ray we see that $\alpha=0$. Since
$(\mu_{S})_{*}Z=(\mu_{S})_{*}Z^{\prime}$ we can replace $Z$ by $Z^{\prime}$
and assume without loss of generality that the negative part of $Z$ does not
contain $E_{P}$ or $E_{S}$.

Since $\mu_{S}$ contracts the extremal ray generated by $l_{S}$ we have
$Z\cdot l_{S}=0$ if and only if $Z=\mu_{S}^{*}Z_{S}$. Since
$Z_{S}=(\mu_{S})_{*}Z$ generates the second extremal ray of the
pseudoeffective cone, we have $Z_{S}\equiv a(\zeta_{S}+\lambda\pi^{*}L)$ with
$\lambda<3$. In particular $Z_{S}\cdot R_{S}<0$, so
$Z|_{E_{S}}=(\mu_{S}|_{E_{S}})^{*}Z_{S}|_{R_{S}}$ is not pseudoeffective, a
contradiction.

Thus we have $Z\cdot l_{S}>0$. We claim that there exists an $\varepsilon>0$
such that $(Z+\varepsilon l_{S})>0$ and $(Z+\varepsilon E_{S})^{3}>0$. Then we
know by Lemma 4.8 that $Z+\varepsilon E_{S}$ is big, a contradiction.

Proof of the claim. A positive multiple of $Z+\varepsilon E_{S}$ can be
written as

$E_{S}+xD+\eta xE_{P}.$

Since $(\mu_{S})_{*}(E_{S}+xD+\eta xE_{P})\equiv
30x(\zeta_{S}+(1.8+\frac{\eta}{10x})\pi^{*}L)$ our assumption and Corollary
4.9 imply that

$0>\eta\geq-0.047976$

By Lemma 3.8 the condition $(Z+\varepsilon E_{S})\cdot l_{S}>0$ is equivalent
to $x>\frac{1}{28+\eta}$. Since $\eta\geq-0.047976$ this holds if $x\geq
0.03577$. Applying again Lemma 3.7 we obtain that $(E_{S}+xD+\eta xE_{P})^{3}$
is equal to

(26)
$x^{3}(-10224+9936\eta-2268\eta^{2}-72\eta^{3})+x^{2}(6048+4752\eta+162\eta^{2})+x(-864-108\eta)+18.$

Setting $y=\eta x$ the Figure 1 shows the planar cubic defined by this
polynomial. Hence for $\eta\geq-0.047976$ we can find $x\geq 0$ that satisfies
both conditions444More formally we could use the classical formulas for the
roots of cubic polynomials to show that for fixed $\eta\geq-0.047976$ the
cubic polynomial has a root satisfying the inequality $x\geq 0.03577$. We
refrain from giving the tedious details of such a proof..

Figure 1. Vanishing set of (26) in red, note that since $E_{S}^{3}>0$ the
origin lies in a connected component of the plane where (26) is positive. Line
$x=0.03577$ in blue and line $y=-0.047976x$ in green.

∎

######  4.10.

Remark. Theorem 1.4 may be a rather small improvement of Corollary 4.9, yet in
view of Theorem 1.5 it is also clear that there is not much space left. In
fact more should be true: by Corollary 4.9 the divisor $D+4E_{S}$ (which in
Figure 1 corresponds to the point $x=0.25,y=0$) is big. If it is also nef, the
estimate in Theorem 1.5 can be improved to $\lambda\geq\frac{25}{14}\approx
1.7857$.

## References

  * [AH21] Fabrizio Anella and Andreas Höring.  Twisted cotangent bundle of hyperkähler manifolds.  J. Éc. polytech. Math., 8:1429—1457, 2021.  With an appendix by Simone Diverio.
  * [Anc82] Vincenzo Ancona.  Faisceaux amples sur les espaces analytiques.  Trans. Amer. Math. Soc., 274(1):89–100, 1982.
  * [Bak17] Benjamin Bakker.  A classification of Lagrangian planes in holomorphic symplectic varieties.  J. Inst. Math. Jussieu, 16(4):859–877, 2017.
  * [BDPP13] Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell.  The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension.  Journal of Algebraic Geometry, 22:201–248, 2013.
  * [BM14] Arend Bayer and Emanuele Macrì.  MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations.  Invent. Math., 198(3):505–590, 2014.
  * [Bou04] Sébastien Boucksom.  Divisorial Zariski decompositions on compact complex manifolds.  Ann. Sci. École Norm. Sup. (4), 37(1):45–76, 2004.
  * [BS95] Mauro C. Beltrametti and Andrew J. Sommese.  The adjunction theory of complex projective varieties, volume 16 of de Gruyter Expositions in Mathematics.  Walter de Gruyter & Co., Berlin, 1995.
  * [DPT80] Michel Demazure, Henry Charles Pinkham, and Bernard Teissier, editors.  Séminaire sur les Singularités des Surfaces, volume 777 of Lecture Notes in Mathematics.  Springer, Berlin, 1980.  Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977.
  * [Fey16] Soheyla Feyzbakhsh.  An effective restriction theorem via wall-crossing and Mercat’s conjecture, 2016.
  * [Fle84] Hubert Flenner.  Restrictions of semistable bundles on projective varieties.  Comment. Math. Helv., 59(4):635–650, 1984.
  * [GO20] Frank Gounelas and John Christian Ottem.  Remarks on the positivity of the cotangent bundle of a K3 surface.  Épijournal Géom. Algébrique, 4:Art. 8, 16, 2020.
  * [Har77] Robin Hartshorne.  Algebraic geometry.  Springer-Verlag, New York, 1977.  Graduate Texts in Mathematics, No. 52.
  * [Hei06] Georg Hein.  Restriction of stable rank two vector bundles in arbitrary characteristic.  Comm. Algebra, 34(7):2319–2335, 2006.
  * [HP19] Andreas Höring and Thomas Peternell.  Algebraic integrability of foliations with numerically trivial canonical bundle.  Invent. Math., 216(2):395–419, 2019.
  * [KM98] János Kollár and Shigefumi Mori.  Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1998.  With the collaboration of C. H. Clemens and A. Corti.
  * [Laz04a] Robert Lazarsfeld.  Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 2004.  Classical setting: line bundles and linear series.
  * [Laz04b] Robert Lazarsfeld.  Positivity in algebraic geometry. II, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 2004.  Positivity for vector bundles, and multiplier ideals.
  * [LP20] Vladimir Lazić and Thomas Peternell.  Maps from $K$-trivial varieties and connectedness problems.  Ann. H. Lebesgue, 3:473–500, 2020.
  * [Mar72] Masaki Maruyama.  On a family of algebraic vector bundles.  Dissertation Kyoto University, https://doi.org/10.14989/doctor.r2072, 1972.
  * [Nak04] Noboru Nakayama.  Zariski-decomposition and abundance, volume 14 of MSJ Memoirs.  Mathematical Society of Japan, Tokyo, 2004.
  * [OP96] Keiji Oguiso and Thomas Peternell.  Semi-positivity and cotangent bundles.  In Complex analysis and geometry (Trento, 1993), volume 173 of Lecture Notes in Pure and Appl. Math., pages 349–368. Dekker, New York, 1996.
  * [Per08] Daniel Perrin.  Algebraic geometry.  Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2008.  An introduction, Translated from the 1995 French original by Catriona Maclean.
  * [Tih80] A. S. Tihomirov.  Geometry of the Fano surface of a double ${\bf P}^{3}$ branched in a quartic.  Izv. Akad. Nauk SSSR Ser. Mat., 44(2):415–442, 479, 1980.
  * [Voi02] Claire Voisin.  Théorie de Hodge et géométrie algébrique complexe, volume 10 of Cours Spécialisés.  Société Mathématique de France, Paris, 2002.
  * [Wel81] Gerald. E. Welters.  Abel-Jacobi isogenies for certain types of Fano threefolds, volume 141 of Mathematical Centre Tracts.  Mathematisch Centrum, Amsterdam, 1981.