# Deformations of I-surfaces with elliptic singularities

Robert Friedman  Columbia University, Department of Mathematics, New York, NY
10027<EMAIL_ADDRESS>and  Phillip Griffiths  Institute for Advanced
Study, Princeton, NJ 08540 and IMSA, The University of Miami, Coral Gables, FL
33146<EMAIL_ADDRESS>
###### Abstract.

An I-surface $S$ is an algebraic surface of general type with $K_{S}^{2}=1$
and $p_{g}(S)=2$. Recent research has centered on trying to give an explicit
description of the KSBA compactification of the moduli space of these
surfaces. The possible normal Gorenstein examples have been enumerated by work
of Franciosi-Pardini-Rollenske. The goal of this paper is to give a more
precise description of such surfaces in case their singularities are simple
elliptic and/or cusp singularities, and to work out their deformation theory.
In particular, under some mild general position assumptions, we show that
deformations of the surfaces in question are versal for deformations of the
singular points, with two exceptions where the discrepancy is analyzed in
detail.

## Introduction

A fundamental problem in the study of algebraic surfaces is to find a good
description of their compactified moduli spaces. For $K3$ surfaces, this
problem has a long history, dating back to early work of Shah [Sha80],
[Sha81], and continuing through the very recent work of Alexeev-Engel [AE23].
Aside from its intrinsic interest, this study has applications to the
asymptotic behavior of the period map and to various related issues such as
Torelli-type questions. A common theme throughout has been the interplay
between certain elliptic singularities, notably simple elliptic and cusp
singularities, and normal crossing models with trivial dualizing sheaf. In the
case of simple elliptic singularities, there is a good understanding of how
these two models are related (cf. for example Mérindol [Mér82] and Grojnowski
and Shepherd-Barron [GSB21]), but the relationship in the cusp case is still a
mystery. Controlling the deformation theory of the singular models is also
important, in order to describe the local structure of the compactified moduli
space, to understand how the various boundary strata are related, and to
analyze the limiting behavior of the period map. In general, however, the
deformation theory of the elliptic singularity model should contain more
information than that of the normal crossing model.

For surfaces of general type there is a very general compactification of the
moduli space, the KSBA compactification. However, at present it is not
practical to describe the myriad possibilities for boundary components in
general. Thus, attention has focused in recent years on trying to enumerate
all of the possible boundary strata for very simple examples of surfaces $S$
with “small” canonical class $K_{S}$; here small could refer either to the
size of $K_{S}^{2}$ or to that of $\dim H^{0}(S;K_{S})$ or both. An appealing
class for surfaces for which both invariants are small are the I-surfaces:

######  Definition 0.1.

An I-surface $S$ is a minimal surface of general type such that $K_{S}^{2}=1$
and $\dim H^{0}(S;K_{S})=2$. In this case, necessarily
$H^{1}(S;\mathcal{O}_{S})=0$.

I-surfaces were considered classically by Castelnuovo-Enriques. They provided
the first examples of surfaces of general type where the $4$-canonical
morphism $\varphi_{4K}$ fails to be birational, thus showing that Bombieri’s
result that $\varphi_{5K}$ is birational is sharp. The modern study of these
surfaces goes back at least to Kodaira (unpublished) and Horikawa [Hor76], who
showed in particular that $K_{S}$ has a single base point, necessarily simple,
and that $S$ is the minimal resolution of a weighted hypersurface of degree
$10$ with at worst rational double points contained in the smooth locus of the
weighted projective space $\mathbb{P}(1,1,2,5)$. The analysis was extended to
the Gorenstein case by Franciosi-Pardini-Rollenske, who showed in particular
[FPR17]:

######  Theorem 0.2.

Let $Y$ be a normal Gorenstein surface with at worst rational double point,
simple elliptic and cusp singularities such that $\omega_{Y}$ is ample,
$\omega_{Y}^{2}=1$, and $\dim H^{0}(Y;\omega_{Y})=2$. Then $Y$ is isomorphic
to a weighted hypersurface of degree $10$ contained in the smooth locus of the
weighted projective space $\mathbb{P}(1,1,2,5)$. ∎

Let $Y$ be a surface as in the statement of Theorem 0.2 and let
$\pi\colon\widetilde{Y}\to Y$ be its minimal resolution. Let
$p_{1},\dots,p_{k}\in Y$ be the points where $Y$ has an elliptic singularity,
and let $D_{i}$ be the reduced divisor $\pi^{-1}(p_{i})$. For each $i$, let
$m_{i}=-D_{i}^{2}$. In [FPR15], [FPR17], Franciosi-Pardini-Rollenske show the
following:

######  Theorem 0.3.

In the above notation, the possibilities for $Y$ and the $m_{i}$ are as
follows:

  1. (i)

(Stratum $\mathfrak{N}_{\emptyset}$): $\widetilde{Y}$ is a minimal surface of
general type and $Y$ is its canonical model.

  2. (ii)

(Stratum $\mathfrak{N}_{1}$): $\widetilde{Y}$ is a minimal surface with
$\kappa(\widetilde{Y})=1$ and $k=m_{1}=1$. In this case, $\widetilde{Y}$ is an
elliptic surface over $\mathbb{P}^{1}$ with $p_{g}(\widetilde{Y})=1$ and a
single multiple fiber, necessarily of multiplicty $2$. The divisor $D=D_{1}$
consists of a bisection of the elliptic fibration together with some fiber
components if $D$ is reducible.

  3. (iii)

(Stratum $\mathfrak{N}_{2}$): $\widetilde{Y}$ is the blowup of a $K3$ surface
at one point, and $k=1$, $m_{1}=2$.

  4. (iv)

(Stratum $\mathfrak{N}_{1,1}^{E}$): $\widetilde{Y}$ is the blowup of an
$Enriques$ surface at one point, and $k=2$, $m_{1}=m_{2}=1$.

  5. (v)

(Strata $\mathfrak{N}_{2,2}$, $\mathfrak{N}_{2,1}$, $\mathfrak{N}_{1,1}^{R}$):
$\widetilde{Y}$ is a rational surface, $k=2$, and the possibilities for
$(m_{1},m_{2})$ are $(2,2)$, $(2,1)$, or $(1,1)$ as given in the subscripts.

  6. (vi)

(Strata $\mathfrak{N}_{2,1,1}$, $\mathfrak{N}_{1,1,1}$): $\widetilde{Y}$ is
the blowup of an elliptic ruled surface, $k=3$, and the possibilities for
$(m_{1},m_{2},m_{3})$ are $(2,1,1)$ or $(1,1,1)$ as given in the subscripts. ∎

There is also a description in [FPR17] of some of the incidence relations
among the various strata.

The goal of this paper is to study the corresponding deformation theory of
I-surfaces with the most basic elliptic singularities, namely simple elliptic
or cusp singularities. The ideal result would be one along the following lines
(a more precise discussion is given in §1):

Ideal Theorem: For $Y$ as above and $p_{1},\dots,p_{k}$ either simple elliptic
or cusp singularities, the deformations of $Y$ are versal for the local
deformations of the singular points except in Case (vi).

Here, versal means roughly that, given any collection of local deformations of
the germs of the singular points $p_{1},\dots,p_{k}$, there is a global
deformation of the projective surface $Y$ which realizes these deformations.
In particular, the singular points can be smoothed or deformed independently.
This also explains why Case (vi) is exceptional: in this case, only simple
elliptic singularities can arise and the exceptional divisors must all be
isogenous. Thus, any deformation of $Y$ which is equisingular at the points
$p_{1},p_{2},p_{3}$ has to preserve the condition that the exceptional
divisors are isogenous. In fact, this is the only constraint on the
deformation theory as we shall see.

The deformations of a Gorenstein I-surface $Y$ are unobstructed. More
precisely, it follows easily from the description of $Y$ as a degree $10$
surface in $\mathbb{P}(1,1,2,5)$ that the obstruction space
$\mathbb{T}^{2}_{Y}$ to the deformation space of $Y$ is $0$ (Theorem 1.22).
Given this, the “Ideal Theorem” would reduce the study of the discriminant
locus in the moduli space of an I-surface with simple elliptic or cusp
singularities to the corresponding discriminant locus for the local
deformation space of the singular points. For example, in case $Y$ has just
one singular point which is simple elliptic, there is a neighborhood of $Y$ in
the moduli space where the locus where the surface continues to have a simple
elliptic singularity is a smooth submanifold whose codimension can be read off
from the multiplicity of the simple elliptic singularity. Blowing up this
submanifold, and taking a cyclic branched cover along the exceptional divisor,
leads to a normal crossing model for the degenerations of the I-surface. Given
a theory of semistable models for cusp singularities, a similar picture should
hold in the cusp case.

As stated, the “Ideal Theorem” is not quite the most ideal version that one
could hope for: it would be natural to show that the deformations of $Y$ are
also versal for the rational double points, or at least that these can all be
smoothed in a family which is equisingular for the elliptic singularities.
However, in order to keep this paper at a reasonable length, we shall ignore
this question and in fact make the running assumption that $Y$ has elliptic
singularities, but not rational double points (Assumption 2.7 below). Even so,
we are only able to prove (most of) the “Ideal Theorem” in a case-by-case
fashion and under some mild general position assumptions. More precise
statements are contained in the following: For the case
$\kappa(\widetilde{Y})=1$, Corollary 3.8 and Corollary 3.10. For the case
where $\widetilde{Y}$ is the blowup of a $K3$ surface, Theorem 4.5 and Theorem
4.6. For the case where $\widetilde{Y}$ is the blowup of an Enriques surface,
Theorem 4.15, Theorem 4.16, and Theorem 4.17. For the case where
$\widetilde{Y}$ is a rational surface, Proposition 5.5 and Corollary 5.7 if
$(m_{1},m_{2})=(2,2)$, Proposition 5.11 and Proposition 5.12 if
$(m_{1},m_{2})=(2,1)$, and Proposition 5.19 if $(m_{1},m_{2})=(1,1)$ and
$D_{1}$, $D_{2}$ are irreducible. (We do not treat the case where
$\widetilde{Y}$ is rational, $(m_{1},m_{2})=(1,1)$ and either $D_{1}$ or
$D_{2}$ is reducible.) For the case where $\widetilde{Y}$ is the blowup of an
elliptic ruled surface, Theorem 7.3. These results are then used to refine and
clarify the discussion of the adjacencies between the various strata given in
[FPR17, Figure 4], i.e. when the closure of one stratum has nonempty
intersection with another. For example, using the exotic adjacencies between
cusp singularities of degree $2$ and cusp or simple elliptic singularities of
degree one due to Brieskorn-Wahl, we show that the closure of
$\mathfrak{N}_{1}$ meets $\mathfrak{N}_{2}$ (Theorem 4.6 and Corollary 4.7),
the closure of $\mathfrak{N}_{2,1}$ meets $\mathfrak{N}_{2,2}$ (Remark 5.8),
and the closure of $\mathfrak{N}_{1,1}^{R}$ meets $\mathfrak{N}_{2,1}$
(Proposition 5.13). In addition, $\mathfrak{N}_{1,1,1}$ is contained in the
closure both of $\mathfrak{N}_{1,1}^{R}$ and $\mathfrak{N}_{1,1}^{E}$ (Theorem
7.3(v)).

It seems likely that special cases of the “Ideal Theorem,” especially when $Y$
has just one singular point which is not overly complicated, should also
follow from Theorem 0.2 by explicit calculations with the equations defining a
weighted hypersurface of degree $10$ in $\mathbb{P}(1,1,2,5)$.

The contents of this paper are as follows. In Section 1, we collect various
facts about simple elliptic and cusp singularities and their deformations. In
Section 2, we give a quick review of the arguments of [FPR15] used to prove
Theorem 0.3. Then we give a more detailed discussion in case
$\kappa(\widetilde{Y})=-\infty$, i.e. the cases (v), (vi) of Theorem 0.3. The
main point is the following, which generalizes the picture for
$\kappa(\widetilde{Y})\geq 0$: The multiplicities $m_{i}=-D_{i}^{2}$ must be
either $1$ or $2$. If $m_{i}=2$, then $\widetilde{Y}$ is the one point blowup
of a surface $X$ such that $-K_{X}$ is effective. If $m_{i}=1$, then there
exists a not necessarily relatively minimal elliptic fibration
$\widetilde{Y}\to\mathbb{P}^{1}$, with at least one multiple fiber, such that
all multiple fibers have multiplicity $2$ and there is a bisection of the
fibration.

We turn then to the various cases. Section 3 deals with the case
$\kappa(\widetilde{Y})=1$. The main technical tool is a description via double
covers of relatively minimal elliptic surfaces with a bisection and a multiple
fiber of multiplicity 2 which essentially goes back to Horikawa [Hor78].
Section 4 deals with the case $\kappa(\widetilde{Y})=0$, and Section 5 with
the case where $\widetilde{Y}$ is a rational surface. After a preliminary
Section 6 on elliptic ruled sufaces which are also elliptic surfaces with two
or three multiple fibers of multiplicity $2$, Section 7 turns to the case
where $\widetilde{Y}$ is the blowup of an elliptic ruled surface. In all
cases, the main point is typically to prove some kind of vanishing theorem,
through a variety of somewhat _ad hoc_ techniques. This is in contrast to the
case where $\omega_{Y}$ is trivial or negative, where the corresponding
results follow easily from standard methods. It is natural to ask if there is
a more unified approach in the I-surface case.

### Acknowledgements

The second author would like to thank Mark Green for the many discussions and
correspondences about I-surfaces, especially regarding their Hodge theory.

### Notation and conventions

We will generally identify line bundles with the corresponding divisor
classes, in particular for the canonical bundle of a surface. Thus we usually
use additive notation for the tensor product of two line bundles where we mean
the sum of the corresponding divisor classes. For two divisors $A$, $B$ on a
surface $S$, the notation $A\equiv B$ means: $A$ is numerically equivalent to
$B$.

##  1\. Deformations of simple elliptic and cusp singularities

###  1.1. Preliminary results on simple elliptic and cusp singularities

Throughout this section, $U$ is the germ of an isolated Gorenstein surface
singularity $p$ and $\pi\colon\widetilde{U}\to U$ is its minimal resolution.
We will also use $U$ to denote a contractible Stein representative of the
germ. Let $D=\bigcup_{i}E_{i}$ be the exceptional set $\pi^{-1}(p)$. The germ
$U$ has a minimally elliptic singularity if
$R^{1}\pi_{*}\mathcal{O}_{\widetilde{U}}$ has length one. Some special
minimally elliptic singularities are given as follows:

######  Definition 1.1.

The germ $(U,p)$ is a simple elliptic singularity if $D$ is a smooth elliptic
curve. The germ $(U,p)$ is a cusp singularity if $D=\bigcup_{i=1}^{r}E_{i}$ is
either an irreducible nodal curve or a cycle of smooth rational curves, i.e.
$E_{i}\cdot E_{i+1}=1$ and $E_{r}\cdot E_{1}=1$, with necessarily
$E_{i}^{2}\leq-2$ for all $i$ and $E_{i}^{2}\leq-3$ for at least one $i$. In
this case the integer $r$ is the length of the singularity (or of $D$), where
$r=1$ if $D$ is irreducible. The germ $(U,p)$ is a Dolgachev or triangle
singularity if $D=\bigcup_{i=1}^{r}E_{i}$ with $r\leq 3$, where $D$ is a
cuspidal rational curve if $r=1$, two smooth rational curves with a tacnode
singularity if $r=2$, and three smooth rational curves meeting at a point if
$r=3$. We will also inaccurately refer to $D$ as a simple elliptic, cusp or
triangle singularity. In all cases the integer $m=-D^{2}$ is the multiplicity
of the singularity (or of $D$). Here, in the cusp case, with
$D=\bigcup_{i=1}^{r}E_{i}$ and $E_{i}^{2}=-e_{i}$, $m=\sum_{i=1}^{r}(e_{i}-2)$
if $r\geq 2$. A similar result holds in the triangle case. (Note however that
$m$ is the multiplicity in the usual sense only when $m\geq 2$.) We will refer
to the sequence $(-e_{1},\dots,-e_{r})$ as the type of the cusp or triangle
singularity; it is well-defined up to cyclic permutation and reflection.

A simple elliptic singularity is determined up to analytic isomorphism by the
multiplicity $m$ and the $j$-invariant (i.e. the isomorphism class) of the
elliptic curve $D$. A cusp singularity is determined up to analytic
isomorphism by its type, i.e. by the sequence $(-e_{1},\dots,-e_{r})$. This is
usually phrased as saying that cusp singularities are taut. A similar result
is almost true for triangle singularities: there are two analytic types,
depending on whether the singularity does or does not have a
$\mathbb{C}^{*}$-action.

######  Remark 1.2.

The above singularities are exactly the minimally elliptic singularities with
reduced fundamental cycle, which is equivalent in this case to:
$K_{\widetilde{U}}\cong\mathcal{O}_{\widetilde{U}}(-D)$. By the Grauert-
Riemenschneider theorem, $R^{i}\pi_{*}\mathcal{O}_{\widetilde{U}}(-D)=0$ for
$i>0$. For a simple elliptic or cusp singularity, this says that $(U,p)$ is a
Du Bois singularity [Ste83, (3.8)]. However, the analogous statement does not
hold for a triangle singularity, because the minimal resolution is not a log
resolution, i.e. the exceptional set $D$ in the minimal resolution has worse
than nodal singularities.

While we will only consider the case of simple elliptic and cusp singularities
in this paper, many of the classification results in §2 carry over to the case
where some of the singularities are allowed to be triangle singularities.

We shall be concerned here with singularities of multiplicity $1$ or $2$. An
immediate calculation gives:

######  Lemma 1.3.

In the above notation, if $m=1$ or $2$ and $D$ is not irreducible, then the
possibilities for the sequence $(-e_{1},\dots,-e_{r})$, up to cyclic
permutation, are as follows:

  1. (i)

$m=1$: $(-e_{1},\dots,-e_{r})=(-3,-2,\dots,-2)$.

  2. (ii)

$m=2$: Either $(-e_{1},\dots,-e_{r})=(-4,-2,\dots,-2)$ or

$(-e_{1},\dots,-e_{r})=(-3,\underbrace{-2,\cdots,-2}_{a},-3,\underbrace{-2,\cdots,-2}_{b}).\qed$

###  1.2. Adjacencies of some elliptic singularities

######  Definition 1.4.

An adjacency between two (germs of) isolated singularities $U_{1}$ and $U_{2}$
is a deformation (a flat family) of germs
$f\colon(\mathcal{U},0)\to(\Delta,0)$, where $\Delta$ is the unit disk,
$f^{-1}(0)\cong(U_{1},0)$ and $f^{-1}(t)\cong(U_{2},0)$ for all $t\neq 0$. In
particular, the restriction of $f$ to the critical locus is proper. Typically,
however, $U_{2}$ can be a disconnected germ. In case $U_{1}$ is a local
complete intersection, adjacencies are transitive: if $U_{1}$ is adjacent to
$U_{2}$ and $U_{2}$ is adjacent to $U_{3}$, then $U_{1}$ is adjacent to
$U_{3}$. In this case, we say that the adjacency from $U_{1}$ to $U_{3}$ is a
consequence of the adjacency from $U_{1}$ to $U_{2}$ and will not always note
it explicitly.

Adjacencies for all simple elliptic and cusp singularities of multiplicity at
most $2$ or more generally which are local complete intersections have been
classified by the work of Karras [Kar77], Brieskorn [Bri79], Wahl [Wah80] and
Looijenga [Loo81]:

######  Theorem 1.5.

  1. (i)

Suppose that $U$ is a simple elliptic singularity of multiplicity $m=1,2$.
Then the possible adjacent singularities $U$ are either simple elliptic
singularities of multiplicity $m$, rational double points, or smooth germs.

  2. (ii)

The cusp singularity of type $(-3,\underbrace{-2,\dots,-2}_{r-1})$ is adjacent
to the cusp singularity of type $(-3,\underbrace{-2,\dots,-2}_{s-1})$ with
$s\leq r$, to a simple elliptic singularity or length one cusp singularity of
multiplicity $1$, or to a rational double point or smooth germ.

  3. (iii)

The adjacencies of the cusp singularity of type
$(-4,\underbrace{-2,\dots,-2}_{r-1})$ are a consequence of the following:

    1. (a)

To the cusp singularity of type $(-4,\underbrace{-2,\dots,-2}_{s-1})$ with
$s\leq r$.

    2. (b)

If $r\geq 4$, to the cusp singularity of type
$(-3,\underbrace{-2,\dots,-2}_{r-3})$. For $r=3$, to a simple elliptic
singularity or length one cusp singularity of multiplicity $1$, i.e. of type
$(-1)$. For $r=2$, to a simple elliptic singularity of multiplicity $1$.

  4. (iv)

The adjacencies of the cusp singularity of type
$(-3,\underbrace{-2,\cdots,-2}_{a},-3,\underbrace{-2,\cdots,-2}_{b})$ are a
consequence of the following:

    1. (a)

For $a>0$, to the cusp singularity of type
$(-3,\underbrace{-2,\cdots,-2}_{a-1},-3,\underbrace{-2,\cdots,-2}_{b})$.

    2. (b)

For $a=0$, the cusp singularity of type
$(-3,-3,\underbrace{-2,\cdots,-2}_{b})$ is adjacent to the one of type
$(-4,\underbrace{-2,\cdots,-2}_{b})$ for $b\geq 1$ and the one of type
$(-3,-3)$ is adjacent to the one of type $(-2)$ and to a simple elliptic
singularity of multiplicity $2$. All further adjacencies are consequences of
those given previously.

######  Remark 1.6.

The very complicated question as to which (unions of) rational double point
singularities are adjacent to a cusp singularity of multiplicity $1$ or $2$
has been answered by Looijenga [Loo81]. The much simpler question as to which
(unions of) rational double point singularities are adjacent to a simple
elliptic singularity of multiplicity $1$ or $2$ reduces to a question about
the rational double point configurations in generalized del Pezzo surfaces of
degree $1$ or $2$.

###  1.3. Deformations of elliptic singularities: the local case

Let $\pi\colon\widetilde{U}\to U$ be the minimal resolution of a simple
elliptic or cusp singularity. In particular, the divisor $D$ has (local)
normal crossings. As usual, we denote by $\Omega^{1}_{\widetilde{U}}(\log D)$
the sheaf of differential $1$-forms with logarithmic poles along $D$. Let
$T_{\widetilde{U}}(-\log D)=(\Omega^{1}_{\widetilde{U}}(\log D))^{\vee}$ be
the dual sheaf. Then $H^{1}(\widetilde{U};T_{\widetilde{U}})$ (roughly)
classifies first order deformations of the noncompact manifold $\widetilde{U}$
and $H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))$ (roughly) classifies
first order deformations of $\widetilde{U}$ keeping the effective divisors
$E_{i}$. A key technical result is the following:

######  Lemma 1.7.

  1. (i)

$\dim H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))=1$ if $D$ is a simple
elliptic singularity.

  2. (ii)

$H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))=0$ if $D$ is a cusp
singularity.

###### Proof.

There is a perfect pairing

$\Omega^{1}_{\widetilde{U}}(\log D)\otimes\Omega^{1}_{\widetilde{U}}(\log
D)\to\Omega^{2}_{\widetilde{U}}(\log
D)=K_{\widetilde{U}}\otimes\mathcal{O}_{\widetilde{U}}(D)\cong\mathcal{O}_{\widetilde{U}}.$

Thus the dual $T_{\widetilde{U}}(-\log D)$ of $\Omega^{1}_{\widetilde{U}}(\log
D)$ is identified with $\Omega^{1}_{\widetilde{U}}(\log D)$. From the Poincaré
residue sequence

$0\to\Omega^{1}_{\widetilde{U}}\to\Omega^{1}_{\widetilde{U}}(\log
D)\to\bigoplus_{i}\mathcal{O}_{E_{i}}\to 0$

and the fact that $H^{2}(\widetilde{U};\Omega^{1}_{\widetilde{U}})=0$ for
dimension reasons, there is a long exact sequence

$\bigoplus_{i}H^{0}(\mathcal{O}_{E_{i}})\to
H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}})\to
H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log
D))\to\bigoplus_{i}H^{1}(\mathcal{O}_{E_{i}})\to 0.$

Note that $\bigoplus_{i}H^{1}(\mathcal{O}_{E_{i}})=0$ if $D$ is a cusp
singularity whereas
$\bigoplus_{i}H^{1}(\mathcal{O}_{E_{i}})=H^{1}(D;\mathcal{O}_{D})$ has
dimension one if $D$ is simple elliptic. There is an exact sequence (cf.
[FL22, Lemma 2.1(iii)])

$0\to\Omega^{1}_{\widetilde{U}}(\log
D)(-D)\to\Omega^{1}_{\widetilde{U}}\to\Omega^{1}_{D}/\tau^{1}_{D}\to 0,$

where $\Omega^{1}_{D}/\tau^{1}_{D}$ is the Kähler differentials mod the
torsion subcomplex. Since cusp singularities are log canonical, and hence Du
Bois, it follows from a result of Steenbrink [Ste97, p. 1369] that

$H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log D)(-D))=0$

(this is a very special case of the “extra vanishing lemma” [FL22, Lemma
2.5]). Moreover, $H^{2}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log
D)(-D))=0$ for dimension reasons. Thus

$H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}})\cong
H^{1}(D;\Omega^{1}_{D}/\tau^{1}_{D})\cong\bigoplus_{i}H^{1}(E_{i};\Omega^{1}_{E_{i}})\cong
H^{2}(D;\mathbb{C})\cong H^{2}(\widetilde{U};\mathbb{C}).$

The map $\bigoplus_{i}H^{0}(E_{i};\mathcal{O}_{E_{i}})\to
H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}})$ from the Poincaré residue
sequence is the same as the fundamental class map
$\bigoplus_{i}H^{0}(E_{i})\to\bigoplus_{i}H^{2}(D_{i})$ and hence is an
isomorphism (by negative definiteness). Hence

$H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))\cong
H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log
D))\cong\bigoplus_{i}H^{1}(E_{i};\mathcal{O}_{E_{i}})$

and thus has dimension $1$ if $D$ is simple elliptic and is $0$ if $D$ is a
cusp. ∎

We can see a more precise version of Lemma 1.7(i) from a slightly different
point of view. There is a commutative diagram

$\setcounter{MaxMatrixCols}{13}\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
T_{\widetilde{U}}(-D)@=T_{\widetilde{U}}(-D)\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>T_{\widetilde{U}}(-\log
D)@>{}>{}>T_{\widetilde{U}}@>{}>{}>\bigoplus_{i}N_{E_{i}/\widetilde{U}}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>T^{0}_{D}@>{}>{}>T_{\widetilde{U}}|D@>{}>{}>N_{D/\widetilde{U}}@>{}>{}>T^{1}_{D}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ 00\end{CD}$

Here $T^{0}_{D}$ is the sheaf of derivations of $\mathcal{O}_{D}$ and
$H^{1}(D;T^{0}_{D})$ is the tangent space to locally trivial deformations of
$D$. Thus $H^{1}(D;T^{0}_{D})=0$ in the cusp case and
$H^{1}(D;T^{0}_{D})=H^{1}(D;T_{D})$ has dimension $1$ in the simple elliptic
case.

######  Lemma 1.8.

In the simple elliptic case, the map
$H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))\to H^{1}(D;T_{D})$ is an
isomorphism.

###### Proof.

By Lemma 1.7, both $H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))$ and
$H^{1}(D;T_{D})$ have dimension $1$. The map
$H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))\to H^{1}(D;T_{D})$ is
surjective since $H^{2}(\widetilde{U};T_{\widetilde{U}}(-D))=0$. Thus it is an
isomorphism. ∎

Using the middle row of the above commutative diagram then gives:

######  Corollary 1.9.

  1. (i)

In the simple elliptic case, there is an exact sequence

$0\to H^{1}(D;T_{D})\to
H^{1}(\widetilde{U};T_{\widetilde{U}})=H^{0}(U;R^{1}\pi_{*}T_{\widetilde{U}})\to
H^{1}(D;N_{D/\widetilde{U}})\to 0.$

  2. (ii)

In the cusp case,

$H^{1}(\widetilde{U};T_{\widetilde{U}})=H^{0}(U;R^{1}\pi_{*}T_{\widetilde{U}})\cong\bigoplus_{i}H^{1}(E_{i};N_{E_{i}/\widetilde{U}}).\qed$

######  Remark 1.10.

For a simple elliptic singularity, we can consider the equisingular
deformations, i.e. those for which the singularity remains simple elliptic but
the $j$-invariant of the resolution, i.e. the isomorphism type, is allowed to
vary. (For a cusp singularity, an equisingular deformation is the same thing
as the trivial deformation.) We saw in the course of the proof of Lemma 1.7
that $H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log D)(-D))=0$. By local
duality and the first line of the proof of Lemma 1.7,

$0=(H^{1}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log
D)(-D)))^{\vee}=H^{1}_{D}(\widetilde{U};\Omega^{1}_{\widetilde{U}}(\log
D))=H^{1}_{D}(\widetilde{U};T_{\widetilde{U}}(-\log D)).$

By Wahl’s theory [Wah76, Proposition 2.5, Proposition 2.7], since
$H^{1}_{D}(\widetilde{U};T_{\widetilde{U}}(-\log D))=0$, there is an inclusion
$H^{1}(\widetilde{U};T_{\widetilde{U}}(-\log D))\to H^{0}(U;T^{1}_{U})$ whose
image is identified with the tangent space to the equisingular deformations of
$U$ inside the tangent space to all deformations.

###  1.4. Deformations of elliptic singularities: the global case

We now consider the following global issue: Let $Y$ be a compact analytic
surface or a projective surface with a collection of simple elliptic and cusp
singularities of multiplicity at most $2$, or more generally which are local
complete intersections. Then $T^{2}_{Y}=0$ and $H^{1}(Y;T^{1}_{Y})=0$ since
the singularities are isolated. The local to global spectral sequence for
$\mathbb{T}^{i}_{Y}=\operatorname{Ext}^{i}(\Omega^{1}_{Y},\mathcal{O}_{Y})$
then reduces to the exact sequence

$0\to H^{1}(Y;T^{0}_{Y})\to\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})\to
H^{2}(Y;T^{0}_{Y})\to\mathbb{T}^{2}_{Y}\to 0.$

Thus we have:

######  Lemma 1.11.

  1. (i)

If the map $H^{0}(Y;T^{1}_{Y})\to H^{2}(Y;T^{0}_{Y})$ is surjective, then
$\mathbb{T}^{2}_{Y}=0$.

  2. (ii)

If $H^{2}(Y;T^{0}_{Y})=0$, then $\mathbb{T}^{2}_{Y}=0$ and the map
$\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is surjective. ∎

This is related to the Leray spectral sequence as follows. Since
$\pi\colon\widetilde{Y}\to Y$ is the minimal resolution, it is equivariant and
hence $T^{0}_{Y}\cong R^{0}\pi_{*}T_{\widetilde{Y}}$. Then the Leray spectral
sequence $E^{p,q}_{2}=H^{p}(Y;R^{q}\pi_{*}T_{\widetilde{Y}})\implies
H^{p+q}(\widetilde{Y};T_{\widetilde{Y}})$ gives an exact sequence

$0\to H^{1}(Y;T^{0}_{Y})\to H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})\to H^{2}(Y;T^{0}_{Y})\to
H^{2}(\widetilde{Y};T_{\widetilde{Y}})\to 0.$

######  Lemma 1.12.

  1. (i)

If $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$, then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}})=0$.

  2. (ii)

If $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$, then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D))=H^{2}(\widetilde{Y};T_{\widetilde{Y}})=0$ and the map

$H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))\to H^{1}(D;T^{0}_{D})$

is surjective.

###### Proof.

We have the two exact sequences

$\displaystyle 0\to T_{\widetilde{Y}}(-D)\to T_{\widetilde{Y}}(-\log D)\to
T^{0}_{D}\to 0;$ $\displaystyle 0\to T_{\widetilde{Y}}(-\log D)\to
T_{\widetilde{Y}}\to\bigoplus_{i}N_{E_{i}/\widetilde{Y}}\to 0.$

Then the fact that
$H^{2}(D;T^{0}_{D})=\bigoplus_{i}H^{2}(E_{i};N_{E_{i}/\widetilde{Y}})=0$
immediately implies the lemma. ∎

The following is the key result linking the deformation theory of $Y$ to
certain vanishing statements:

######  Proposition 1.13.

  1. (i)

In the simple elliptic case, suppose that
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$. Then the map

$H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})$

is surjective and $H^{2}(Y;T^{0}_{Y})=\mathbb{T}^{2}_{Y}=0$. Thus
$\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is surjective.

  2. (ii)

In the cusp case, suppose that $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D))=0$. Then the map

$H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})$

is surjective and $H^{2}(Y;T^{0}_{Y})=\mathbb{T}^{2}_{Y}=0$. Thus
$\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is surjective.

###### Proof.

In the simple elliptic case, there is a commutative diagram

$\setcounter{MaxMatrixCols}{11}\begin{CD}H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D))@>{}>{}>H^{1}(\widetilde{Y};T_{\widetilde{Y}})@>{}>{}>H^{1}(D;N_{D/\widetilde{Y}})\\\
@V{}V{}V@V{}V{}V@V{}V{=}V\\\
0@>{}>{}>H^{1}(D;T_{D})@>{}>{}>H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}}<EMAIL_ADDRESS>
By Lemma 1.12, under the assumptions of (i),
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$ and hence
$H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to H^{1}(D;N_{D/\widetilde{Y}})$ is
surjective. Since $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$, the map
$H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))\to H^{1}(D;T_{D})$ is
surjective. Thus $H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})$ is surjective. Again by Lemma 1.12,
$H^{2}(\widetilde{Y};T_{\widetilde{Y}})=0$. The Leray spectral sequence then
gives an exact sequenece

$H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})\to H^{2}(Y;T^{0}_{Y})\to 0.$

Since $H^{1}(\widetilde{Y};T_{\widetilde{Y}})\to
H^{0}(Y;R^{1}\pi_{*}T_{\widetilde{Y}})$ is surjective, $H^{2}(Y;T^{0}_{Y})=0$.
By Lemma 1.11(ii), $\mathbb{T}^{2}_{Y}=0$ and the map $\mathbb{T}^{1}_{Y}\to
H^{0}(Y;T^{1}_{Y})$ is surjective. The proof in the cusp case is similar (and
simpler). ∎

######  Example 1.14.

Suppose that $D\in|-K_{\widetilde{Y}}|$ and $D$ is smooth. Then

$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=H^{2}(\widetilde{Y};T_{\widetilde{Y}}\otimes
K_{\widetilde{Y}})=H^{2}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}).$

It is a standard result (due in various forms to Kulikov-Persson-Roan) that
either $\widetilde{Y}$ is rational and $D$ is a single elliptic curve, or
$\widetilde{Y}$ is the blowup of a minimal elliptic ruled surface and
$D=D_{1}+D_{2}$ has two connected components $D_{1}$ and $D_{2}$, both
sections of the ruling. In case $\widetilde{Y}$ is rational,
$H^{2}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}})=0$. If $\widetilde{Y}$ is the
blowup of a minimal elliptic ruled surface, then $\dim
H^{2}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}})=1$. In this case, the exact
sequence $0\to T_{\widetilde{Y}}(-D)\to T_{\widetilde{Y}}(-\log D)\to T_{D}\to
0$ is identified with the Poincaré residue sequence

$0\to\Omega^{1}_{\widetilde{Y}}\to\Omega^{1}_{\widetilde{Y}}(\log(D_{1}+D_{2}))\to\mathcal{O}_{D_{1}}\oplus\mathcal{O}_{D_{2}}\to
0.$

The Gysin map $H^{1}(D_{i};\mathcal{O}_{D_{i}})\to
H^{2}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}})$ is an isomorphism for
$i=1,2$. Hence the image of $H^{1}(Y;T_{\widetilde{Y}}(-\log D))$ in
$H^{1}(D_{1};T_{D_{1}})\oplus H^{1}(D_{2};T_{D_{2}})$ has dimension one and
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$. Note that
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))\neq 0$; roughly speaking, this
represents the obstruction to deforming the two elliptic curves $D_{1}$ and
$D_{2}$ independently. However, $H^{2}(\widetilde{Y};T_{\widetilde{Y}})=0$
since this is a birational invariant of $\widetilde{Y}$ and $\widetilde{Y}$ is
a ruled surface. One can then check that $H^{0}(Y;T^{1}_{Y})\to
H^{2}(Y;T^{0}_{Y})$ is surjective. Thus $\mathbb{T}^{2}_{Y}=0$ in this case as
well. We will discuss this case in more detail below.

###  1.5. Some variations of the above

In some of our applications, Proposition 1.13 will not hold, and so we will
need some variations on it. For the first variation, assume that $Y$ has
simple elliptic singularities at the points $p_{1},\dots,p_{k}$ and is
otherwise smooth. We begin by fixing some notation:

######  Definition 1.15.

As always, let $\pi\colon\widetilde{Y}\to Y$ be the minimal resolution and let
$D_{i}=\pi^{-1}(p_{i})$. Set $D_{i}^{\prime}=\sum_{j\neq i}D_{j}$. We can form
partial resolutions in two different ways: first, let $\pi_{i}\colon Y_{i}\to
Y$ be a resolution at the singular point $p_{i}$ and an isomorphism elsewhere.
The exceptional set of $\pi_{i}$ is $D_{i}$. Then there is an induced morphism
$\psi_{i}\colon\widetilde{Y}\to Y_{i}$, with exceptional set $D_{i}^{\prime}$.
There is also the morphism $\pi_{i}^{\prime}\colon Y_{i}^{\prime}\to Y$, which
is a resolution at the points $p_{j},j\neq i$ and is an isomorphism over
$p_{i}$. Thus the exceptional set of $\pi_{i}^{\prime}$ is $D_{i}^{\prime}$.
Of course, if $k=2$, then $\pi_{1}\colon Y_{1}\to Y$ is identified with
$\pi_{2}^{\prime}\colon Y_{2}^{\prime}\to Y$. Let
$\psi_{i}^{\prime}\colon\widetilde{Y}\to Y_{i}^{\prime}$ be the corresponding
induced morphism, with exceptional set $D_{i}$.

We shall look at $\pi_{i}\colon Y_{i}\to Y$ more systematically in §2. At
present, we are concerned with the morphism $\pi_{i}^{\prime}\colon
Y_{i}^{\prime}\to Y$.

######  Definition 1.16.

Let $\Omega^{1}_{Y_{i}^{\prime}}(\log D_{i}^{\prime})$ be the sheaf equal to
$\Omega^{1}_{Y_{i}^{\prime}}$ on the singular surface
$Y_{i}^{\prime}-D_{i}^{\prime}$ and to $\Omega^{1}_{Y_{i}^{\prime}}(\log
D_{i}^{\prime})$ on the smooth surface $Y_{i}^{\prime}-\\{p_{i}\\}$, glued
together in the obvious way. Set $T^{i}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime})=\mathit{Ext}^{i}(\Omega^{1}_{Y_{i}^{\prime}}(\log
D_{i}^{\prime}),\mathcal{O}_{Y_{i}^{\prime}})$ and
$\mathbb{T}^{i}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime})=\operatorname{Ext}^{i}(\Omega^{1}_{Y_{i}^{\prime}}(\log
D_{i}^{\prime}),\mathcal{O}_{Y_{i}^{\prime}})$.

######  Theorem 1.17.

Let $\mathbf{Def}_{Y_{i}^{\prime},D_{i}^{\prime}}$ be the functor of
deformations of $Y_{i}^{\prime}$ keeping the effective Cartier divisor
$D_{i}^{\prime}$. Then the tangent space to
$\mathbf{Def}_{Y_{i}^{\prime},D_{i}^{\prime}}$ is
$\mathbb{T}^{1}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime})$ and
$\mathbb{T}^{2}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime})$ is an obstruction
space for $\mathbf{Def}_{Y_{i}^{\prime},D_{i}^{\prime}}$.

###### Proof.

This is a minor variation on the proof for smooth $Y$ (cf. e.g. Sernesi
[Ser06, 3.4.17]). ∎

######  Corollary 1.18.

In the above notation, suppose that
$H^{2}(Y_{i}^{\prime};T^{0}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime}))=0$ and
that $p_{i}$ is a local complete intersection. Then there exists a deformation
of $Y$ which smooths the singular point $p_{i}$ and for which the remaining
points $p_{j}$, $j\neq i$, remain simple elliptic singularities.

###### Proof.

In any case, identifying $H^{0}(Y_{i}^{\prime};T^{1}_{Y_{i}^{\prime}})$ with
$H^{0}(Y;T^{1}_{Y,p_{i}})$ and using the fact that
$H^{0}(Y;T^{2}_{Y,p_{i}})=0$ by our assumption, the local to global spectral
sequence for $\mathbb{T}^{i}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime})$ gives an
exact sequence

$\mathbb{T}^{1}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime})\to
H^{0}(Y;T^{1}_{Y,p_{i}})\to H^{2}(Y_{i}^{\prime};T^{0}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime}))\to\mathbb{T}^{2}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime})\to
0.$

Under the hypothesis that $H^{2}(Y_{i}^{\prime};T^{0}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime}))=0$, $\mathbb{T}^{2}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime})=0$, and hence $\mathbf{Def}_{Y_{i}^{\prime},D_{i}^{\prime}}$
is unobstructed. Moreover, the map $\mathbb{T}^{1}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime})\to H^{0}(Y;T^{1}_{Y,p_{i}})$ is surjective. We can then find a
deformation $\mathcal{Y}_{i}^{\prime}\to\Delta$ of $Y_{i}^{\prime}$ which
smooths the singular point $p_{i}$ and keeps the effective Cartier divisor
$D_{i}^{\prime}$, in the sense that there exists a relative Cartier divisor
$\mathcal{D}_{i}^{\prime}\subseteq\mathcal{Y}_{i}^{\prime}$ which restricts
over $0$ to $D_{i}^{\prime}$. By Wahl’s theory [Wah76, Proposition 2.7], we
can blow down $\mathcal{D}_{i}^{\prime}$ to get a flat family over $\Delta$
which is a deformation of $Y$ smoothing $p_{i}$ and which is equisingular (in
the sense of Remark 1.10) at the remaining points $p_{j}$ as desired. ∎

By analogy with Proposition 1.13(i), we then have:

######  Proposition 1.19.

(i) Suppose that $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D_{i}^{\prime})(-D_{i}))=0$. Then
$H^{2}(Y_{i}^{\prime};T^{0}_{Y_{i}^{\prime}}(-\log D_{i}^{\prime}))=0$ and
thus the conclusions of Corollary 1.18 hold.

(ii) If the natural map $H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))\to
H^{1}(D_{i};T_{D_{i}})$ is surjective and
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$, then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D_{i}^{\prime})(-D_{i}))=0$ and
the conclusions of Corollary 1.18 hold.

###### Proof.

(i) A minor variation on the proof of Proposition 1.13(i) shows that the Leray
map

$H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log D_{i}^{\prime}))\to
H^{0}(Y_{i}^{\prime};R^{1}(\psi_{i}^{\prime})_{*}T_{\widetilde{Y}}(-\log
D_{i}^{\prime}))$

is surjective and that $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D_{i}^{\prime}))=0$. Thus

$H^{2}(Y_{i}^{\prime};T^{0}_{Y_{i}^{\prime}}(-\log
D_{i}^{\prime}))=H^{2}(Y_{i}^{\prime};R^{0}(\psi_{i}^{\prime})_{*}T_{\widetilde{Y}}(-\log
D_{i}^{\prime}))=0.$

(ii) From the exact sequence

$0\to T_{\widetilde{Y}}(-\log D_{i}^{\prime})(-D_{i})\to
T_{\widetilde{Y}}(-\log D)\to T_{D_{i}}\to 0,$

we get a long exact sequence

$H^{1}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))\to H^{1}(D_{i};T_{D_{i}})\to
H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D_{i}^{\prime})(-D_{i}))\to
H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D)).$

Thus, if the first map is surjective and the last group is $0$, then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D_{i}^{\prime})(-D_{i}))=0$. ∎

######  Example 1.20.

Returning to Example 1.14, with $\widetilde{Y}$ the blowup of a minimal
elliptic ruled surface, $D_{1}$ and $D_{2}$ two disjoint sections with
negative self-intersection, and
$K_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(-D_{1}-D_{2})$,

$(T_{\widetilde{Y}}(-\log D_{1})(-D_{2}))^{\vee}\otimes
K_{\widetilde{Y}}=\Omega^{1}_{\widetilde{Y}}(\log
D_{1})\otimes\mathcal{O}_{\widetilde{Y}}(D_{2}-D_{1}-D_{2})=\Omega^{1}_{\widetilde{Y}}(\log
D_{1})(-D_{1}).$

Thus by Serre duality $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log
D_{1})(-D_{2}))$ is dual to
$H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}(\log D_{1})(-D_{1}))$. There
is an exact sequence

$0\to\Omega^{1}_{\widetilde{Y}}(\log
D_{1})(-D_{1})\to\Omega^{1}_{\widetilde{Y}}\to\Omega^{1}_{D_{1}}\to 0.$

Thus $H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}(\log D_{1})(-D_{1}))=0$
since $H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}})\to
H^{0}(D_{1};\Omega^{1}_{D_{1}})$ is an isomorphism. It follows that, if the
singularity at $p_{2}$ is a local complete intersection, then we can smooth
the point corresponding to $p_{2}$ while keeping a simple elliptic singularity
in the deformation corresponding to $p_{1}$. Such a deformation will have a
minimal resolution which is a rational surface.

The second variation is the following:

######  Proposition 1.21.

Assume that $Y$ has a simple elliptic singularity at $p_{1}$, a local complete
intersection cusp singularity at $p_{2}$, and is otherwise smooth. As before,
let $\pi\colon\widetilde{Y}\to Y$ be the minimal resolution and let
$D_{i}=\pi^{-1}(p_{i})$. Suppose that
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log(D_{1}+D_{2})))=0$. Then the map
$\mathbb{T}^{1}_{Y_{1}}(-\log D_{1})\to H^{0}(Y_{1};T^{1}_{Y_{1},p_{2}})$ is
surjective. In particular, there exists a deformation of $Y$ which is
equisingular at $p_{1}$ and deforms $p_{2}$ to a simple elliptic singularity
with the same multiplicity as $p_{2}$.

###### Proof.

As in Definition 1.15, let $\pi_{1}\colon Y_{1}\to Y$ be the minimal
resolution at $p_{1}$ and let $\psi_{1}\colon\widetilde{Y}\to Y_{1}$ be the
induced morphism. A Leray spectral sequence argument applied to the morphism
$\psi_{1}$ and the sheaf

$R^{0}\psi_{1}{}_{*}T_{\widetilde{Y}}(-\log D_{1})=T^{0}_{Y_{1}}(-\log D_{1})$

as in the proof of Proposition 1.13(ii) shows that
$H^{2}(Y_{1};T^{0}_{Y_{1}}(-\log D_{1}))=0$. Thus
$\mathbb{T}^{2}_{Y_{1}}(-\log D_{1})=0$ and the map
$\mathbb{T}^{1}_{Y_{1}}(-\log D_{1})\to H^{0}(Y_{1};T^{1}_{Y_{1},p_{2}})$ is
surjective, proving the first statement. An adjacency of $p_{2}$ to a simple
elliptic singularity with the same multiplicity gives a global $1$-parameter
deformation of the pair $(Y_{1},D_{1})$ keeping the Cartier divisor $D_{1}$
and deforms $p_{2}$ as desired. Blowing down the family of Cartier divisors
corresponding to $D_{1}$ as in the proof of Corollary 1.18 then gives a
deformation of $Y$ which is equisingular at $p_{1}$ and is as claimed at
$p_{2}$. ∎

###  1.6. Deformations of I-surfaces are unobstructed

We prove a very general result about weighted hypersurfaces of degree $10$ in
$\mathbb{P}(1,1,2,5)$. However, it is not of much use without further
information of the kind discussed above.

######  Theorem 1.22.

Let $Y$ be a weighted hypersurface of degree $10$ in $\mathbb{P}(1,1,2,5)$,
not passing through the singular locus. Then $\mathbb{T}^{2}_{Y}=0$, so that
the deformations of $Y$ are unobstructed.

###### Proof.

For simplicity of notation denote $\mathbb{P}(1,1,2,5)$ by $\mathbb{P}$.
Recall that
$\mathbb{T}^{2}_{Y}=\operatorname{Ext}^{2}(\Omega^{1}_{Y},\mathcal{O}_{Y})$.
The conormal sequence for $Y$ is

$0\to I_{Y}/I_{Y}^{2}\to\Omega^{1}_{\mathbb{P}}|Y\to\Omega^{1}_{Y}\to 0.$

In particular the long exact Ext sequence yields an exact sequence

$H^{1}(Y;N_{Y/\mathbb{P}})\to\operatorname{Ext}^{2}(\Omega^{1}_{Y},\mathcal{O}_{Y})\to
H^{2}(Y;T_{\mathbb{P}}|Y).$

Here $N_{Y/\mathbb{P}}\cong\mathcal{O}_{Y}(10)$. A standard argument shows
that
$H^{1}(\mathbb{P};\mathcal{O}_{\mathbb{P}}(a))=H^{2}(\mathbb{P};\mathcal{O}_{\mathbb{P}}(a))=0$
for all $a\in\mathbb{Z}$ and hence that $H^{1}(Y;\mathcal{O}_{Y}(a))=0$ by
using the long exact sequence associated to

$0\to\mathcal{O}_{\mathbb{P}}(-Y)(a)=\mathcal{O}_{\mathbb{P}}(a-10)\to\mathcal{O}_{\mathbb{P}}(a)\to\mathcal{O}_{Y}(a)\to
0.$

Thus $H^{1}(Y;N_{Y/\mathbb{P}})=0$. The remaining term
$H^{2}(Y;T_{\mathbb{P}}|Y)$ is Serre dual to
$H^{0}(\Omega^{1}_{\mathbb{P}}|Y\otimes\omega_{Y})$. The Euler exact sequence
for weighted projective space is

$0\to\Omega^{1}_{\mathbb{P}}\to\bigoplus_{i=1}^{4}\mathcal{O}_{\mathbb{P}}(-a_{i})\to\mathcal{O}_{\mathbb{P}}\to
0,$

with the weights $a_{i}=1,1,2,5$. Restricting to $Y$ and tensoring with
$\omega_{Y}$ gives an exact sequence

$0\to
H^{0}(Y;\Omega^{1}_{\mathbb{P}}|Y\otimes\omega_{Y})\to\bigoplus_{i=1}^{4}H^{0}(Y;\mathcal{O}_{\mathbb{P}}(1-a_{i}))\to
H^{0}(Y;\mathcal{O}_{Y}(1)).$

In our case, $1-a_{i}$ is nonnegative only for the two weights $a_{i}=1$,
$H^{0}(Y;\mathcal{O}_{Y}(1))$ has dimension $2$, and the map
$\bigoplus_{i=1}^{4}H^{0}(Y;\mathcal{O}_{\mathbb{P}}(1-a_{i}))\to
H^{0}(Y;\mathcal{O}_{Y}(1))$ is easily checked to be surjective and hence an
isomorphism. Thus $H^{0}(Y;\Omega^{1}_{\mathbb{P}}|Y\otimes\omega_{Y})=0$, and
so finally $\mathbb{T}^{2}_{Y}=0$. ∎

##  2\. Structure of I-surfaces with elliptic singularities

###  2.1. Some preliminary results

The first result in this section is a standard result about iterated blowups:

######  Lemma 2.1.

Let $\rho\colon S\to\overline{S}$ be the iterated blowup of a smooth surface
$\overline{S}$ and let $\bigcup_{i}C_{i}$ be the fiber of $\rho$ over a point
$p$. Let $K_{S}=\rho^{*}K_{\overline{S}}+\sum_{i}a_{i}C_{i}$ where the $a_{i}$
are nonnegative integers. Then $a_{j}>0$ for all $j$. If $C_{j}^{2}=-1$ for
some $j$ (i.e. $C_{j}$ is an exceptional curve) and $\rho^{-1}(p)$ is
reducible, then $a_{j}\geq 2$.

###### Proof.

The first statement is clear since every local section of
$\rho^{*}K_{\overline{S}}$ vanishes along $C_{j}$ as a section of $K_{S}$. If
$C_{j}^{2}=-1$, then by adjunction $C_{j}\cdot K_{S}=-1$. But
$K_{S}=\rho^{*}K_{\overline{S}}+\sum_{i}a_{i}C_{i}$. For $j\neq k$,
$C_{j}\cdot C_{k}\geq 0$, and by hypothesis $C_{j}\cdot C_{k}>0$ for some $k$
as the fiber is connected. Thus, for this choice of $k$,

$-1=a_{j}C_{j}^{2}+\sum_{i\neq j}a_{i}(C_{j}\cdot C_{i})\geq-a_{j}+a_{k}.$

Hence $a_{j}\geq a_{k}+1\geq 2$. ∎

######  Lemma 2.2.

Let $S$ be a smooth surface.

  1. (i)

If $S$ is the blowup of a geometrically ruled surface over a base curve of
genus at least $2$, then $S$ does not contain a smooth elliptic curve or a
cycle of rational curves.

  2. (ii)

If $S$ is the blowup of a geometrically ruled surface over a base curve of
genus at least $1$, then $S$ does not contain a cycle of rational curves.

  3. (iii)

If $S$ is the blowup of a surface with $\kappa(S)=0$ or $\kappa(S)=1$ and
$\chi(\mathcal{O}_{S})=0$, then $S$ does not contain a cycle of rational
curves.

###### Proof.

(i), (ii): Let $\rho\colon S\to C$ be the Albanese morphism and let
$\overline{S}$ be a relatively minimal model of $S$. Thus there is an induced
morphism $\bar{\rho}\colon\overline{S}\to C$ exhibiting $\overline{S}$ as a
geometrically ruled surface. Then $S$ is an iterated blowup of $\overline{S}$
and all fibers of $\rho$ are proper transforms of fibers of $\bar{\rho}$.
Hence the reduction of every fiber of $\rho$ is a tree of smooth rational
curves. In case (i), the image under $\rho$ of of a smooth elliptic curve on
$S$ would be a point, and hence the curve would have to be contained in the
blowup of a fiber, but this is impossible. Likewise, in case (i) or (ii), a
cycle of rational curves would have to be contained in the blowup of a fiber,
but this is again impossible.

(iii): First assume that $\kappa(S)=1$ and let $\overline{S}$ be the minimal
model of $S$ and let $\rho\colon S\to C$ be the canonical elliptic fibration.
Then, by the structure theory of elliptic surfaces, all fibers of $\rho$ have
smooth reduction. If there exists a cycle of rational curves on $S$, then
there exists a rational curve on $\overline{S}$, necessarily not contained in
a fiber, and hence $g(C)=0$. Moreover, $p_{g}(\overline{S})=0$ and
$q(\overline{S})=1$. Then there exists a finite étale cover of $\overline{S}$
which is biholomorphic to a product of two smooth nonrational curves (e.g.
[Bea78, Thm. VI.13] or [FM94, Chapter 2]). This contradicts the existence of a
rational curve on $\overline{S}$. The case where $\kappa(S)=0$ is similar (and
simpler). ∎

######  Lemma 2.3.

Let $Y$ be a normal projective surface with $k$ isolated minimally elliptic
singularities and let $\pi^{\prime}\colon Y^{\prime}\to Y$ be a partial
resolution of $Y$ at $k^{\prime}$ of the singularities. Then
$\chi(Y^{\prime};\mathcal{O}_{Y^{\prime}})=\chi(Y;\mathcal{O}_{Y})-k^{\prime}$.

###### Proof.

We have the Leray spectral sequence
$E_{2}^{p,q}=H^{p}(Y;R^{q}\pi^{\prime}_{*}\mathcal{O}_{Y^{\prime}})\implies
H^{p+q}(Y^{\prime};\mathcal{O}_{Y^{\prime}})$. Thus, since
$R^{0}\pi^{\prime}_{*}\mathcal{O}_{Y^{\prime}}=\mathcal{O}_{Y}$ and
$R^{1}\pi^{\prime}_{*}\mathcal{O}_{Y^{\prime}}$ is a skyscraper sheaf with
one-dimensional stalks at each of the $k^{\prime}$ singular points resolved by
$\pi^{\prime}$,
$\chi(Y^{\prime};\mathcal{O}_{Y^{\prime}})=\chi(Y;\mathcal{O}_{Y})-k^{\prime}$.
∎

######  Theorem 2.4.

Let $Y$ be a normal projective surface with $k>0$ isolated singularities which
are either simple elliptic or cusp singularities. Assume that
$h^{1}(Y;\mathcal{O}_{Y})=0$ and set $p_{g}=h^{2}(Y;\mathcal{O}_{Y})$. Then
$k\leq p_{g}+1$. If $k=p_{g}+1$, then all singularities of $Y$ are simple
elliptic and a smooth minimal model for $Y$ is either an elliptic ruled
surface, an elliptic surface such that all fibers have smooth reduction, or is
an abelian surface isogeneous to a product of two elliptic curves.

###### Proof.

Let $\pi\colon\widetilde{Y}\to Y$ be the minimal resolution. If $k>p_{g}+1$,
then

$\chi(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})=\chi(Y;\mathcal{O}_{Y})-k=1+p_{g}-k<0,$

by Lemma 2.3. Hence, by the Castelnuovo-de Franchis theorem, $\widetilde{Y}$
is the blowup of a geometrically ruled surface over a base curve of genus at
least $2$. This contradicts (i) of Lemma 2.2.

Now assume that $k=p_{g}+1$ and let $Y_{\text{\rm{min}}}$ be a smooth minimal
model of $\widetilde{Y}$. Then

$\chi(Y_{\text{\rm{min}}};\mathcal{O}_{Y_{\text{\rm{min}}}})=\chi(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})=\chi(Y;\mathcal{O}_{Y})-k=1+p_{g}-k=0.$

Thus
$c_{1}(Y_{\text{\rm{min}}})^{2}+c_{2}(Y_{\text{\rm{min}}})=12\chi(Y_{\text{\rm{min}}};\mathcal{O}_{Y_{\text{\rm{min}}}})=0$.
In particular, since $\widetilde{Y}$ is not ruled over a curve of genus at
least $2$, by classification $c_{1}(Y_{\text{\rm{min}}})^{2}\geq 0$ and
$c_{2}(Y_{\text{\rm{min}}})\geq 0$, hence
$c_{1}(Y_{\text{\rm{min}}})^{2}=c_{2}(Y_{\text{\rm{min}}})=0$. Thus, by the
classification of such surfaces, $Y_{\text{\rm{min}}}$ is either an elliptic
ruled surface, an elliptic surface over a curve $B$ such that all fibers have
smooth reduction, or is an abelian surface, necessarily isogeneous to a
product of two elliptic curves. In all cases, by Lemma 2.2, the blowup
$\widetilde{Y}$ of $Y_{\text{\rm{min}}}$ will not contain a cycle of rational
curves, so all singularities of $Y$ are simple elliptic. ∎

######  Remark 2.5.

With more effort, one can extend the proof of Lemma 2.2 and hence of Theorem
2.4 to cover the case where the singularities of $Y$ are just assumed to be
minimally elliptic.

###  2.2. Numerology of I-surfaces

We begin by defining more carefully the surfaces considered in this paper.

######  Definition 2.6.

An I-surface with simple elliptic or cusp singularities is a normal Gorenstein
surface $Y$ whose singularities are simple elliptic, cusp, or RDP
singularities, such that $\omega_{Y}$ is ample, $\omega_{Y}^{2}=1$,
$h^{1}(Y;\mathcal{O}_{Y})=0$, $h^{2}(Y;\mathcal{O}_{Y})=h^{0}(Y;\omega_{Y})=2$
and thus $\chi(\mathcal{O}_{Y})=3$. (In fact, the condition
$h^{1}(Y;\mathcal{O}_{Y})=0$ follows from the other two.) Let
$\pi\colon\widetilde{Y}\to Y$ be a minimal resolution of singularities, so in
particular no exceptional curve on $\widetilde{Y}$ is contained in a fiber of
$\pi$. Let $p_{1},\dots,p_{k}$ be the simple elliptic or cusp singularities,
let $D_{i}=\pi^{-1}(p_{i})$, and set $D=\sum_{i=1}^{k}D_{i}$. Thus each
$D_{i}$ is a smooth elliptic curve, an irreducible nodal rational curve, or or
a cycle of smooth rational curves, and $D$ is the exceptional set of $\pi$
$\iff$ there are no RDP singularities. Set $m_{i}=-D_{i}^{2}$ and
$m=-D^{2}=\sum_{i}m_{i}$, with $m_{i}>0$ for every $i$. In particular, $k\leq
m$. By Theorem 2.4, $k\leq 3$. Let $L$ be the nef and big line bundle
$\pi^{*}\omega_{Y}$.

######  Assumption 2.7.

As noted in the introduction, to keep the number of cases manageable we will
assume for the rest of this paper that $Y$ has no RDP singularities. Thus, in
the notation of Definition 2.6, $L\cdot C\geq 0$ for every irreducible curve
$C$ on $\widetilde{Y}$, and $L\cdot C=0$ $\iff$ $C$ is a component of $D_{i}$
for some $i$. In particular, $L\cdot C>0$ for every exceptional curve $C$ on
$\widetilde{Y}$.

######  Remark 2.8.

If there are some RDP singularities on $Y$, then it is still true that $L\cdot
C>0$ for every exceptional curve $C$ on $\widetilde{Y}$. Under this
assumption, the arguments in this section for the classification of the
possible $Y$ will carry over with minor modifications.

We have
$K_{\widetilde{Y}}=\pi^{*}\omega_{Y}\otimes\mathcal{O}_{\widetilde{Y}}(-D)$,
i.e. $L=K_{\widetilde{Y}}+D$. Since $L\cdot D_{i}=0$ for every $i$,

$K_{\widetilde{Y}}\cdot D_{i}+D_{i}^{2}=0.$

More precisely, by adjunction,
$(K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D_{i}))|D_{i}=\omega_{D_{i}}=\mathcal{O}_{D_{i}}$.
Hence $K_{\widetilde{Y}}\cdot D_{i}=m_{i}>0$. Also,

$\displaystyle L^{2}$ $\displaystyle=L\cdot(K_{\widetilde{Y}}+D)=L\cdot
K_{\widetilde{Y}}=1;$ $\displaystyle K_{\widetilde{Y}}^{2}$
$\displaystyle=(L-D)\cdot K_{\widetilde{Y}}=L\cdot K_{\widetilde{Y}}-m=1-m.$

We can also consider partial resolutions:

######  Definition 2.9.

As in Definition 1.15, let $\pi_{i}\colon Y_{i}\to Y$ be the minimal
resolution of the singular point $p_{i}$ and an isomorphism from $Y_{i}-D_{i}$
to $Y-p_{i}$. Note that, in case $k=1$, $Y_{i}=\widetilde{Y}$. Denote by
$\psi_{i}\colon\widetilde{Y}\to Y_{i}$ the induced morphism. Finally set
$M_{i}=\psi_{i}^{*}\omega_{Y_{i}}$.

######  Lemma 2.10.

In the above notation, $\chi(\mathcal{O}_{Y_{i}})=2$. Moreover,

  1. (i)

$M_{i}=L-D_{i}=K_{\widetilde{Y}}+D_{i}^{\prime}$, where
$D_{i}^{\prime}=\sum_{j\neq i}D_{j}$. Here, if $k=1$, then
$D_{i}^{\prime}=D_{1}^{\prime}=0$ and $M_{1}=K_{\widetilde{Y}}$.

  2. (ii)

$M_{i}$ is effective and $L\cdot M_{i}=1$. In particular, $M_{i}$ is not
numerically equivalent to $0$.

  3. (iii)

For $i\neq j$, $M_{i}\cdot M_{j}=1$. In case $i=j$,
$\omega_{Y_{i}}^{2}=M_{i}^{2}=1-m_{i}$. (Here $\omega_{Y_{i}}^{2}$ means the
self-intersection of the Cartier divisor on $Y_{i}$.)

  4. (iv)

$M_{i}\cdot D_{i}^{\prime}=0$, $M_{i}\cdot D_{i}=m_{i}$, and $M_{i}\cdot
K_{\widetilde{Y}}=1-m_{i}$ .

  5. (v)

If $k>1$ and either $h^{1}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})=0$ or
$\widetilde{Y}$ is a blown up elliptic ruled surface, then

$\dim H^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(M_{i}))=1.$

###### Proof.

The first statement follows from Lemma 2.3. Then (i) is clear since
$\widetilde{Y}$ is a resolution of $Y_{i}$ at the points $p_{j},j\neq i$.
Since $\chi(\mathcal{O}_{Y_{i}})=2$, $h^{2}(Y_{i};\mathcal{O}_{Y_{i}})\neq 0$
and hence $h^{0}(Y_{i};\omega_{Y_{i}})\neq 0$. Thus $M_{i}$ is effective.
Since $\omega_{Y_{i}}$ is trivial in a neighborhood of $D_{i}^{\prime}$,
$M_{i}\cdot D_{i}^{\prime}=0$. Moreover, $L\cdot
M_{i}=L\cdot(L-D_{i})=L^{2}=1$. We have

$M_{i}\cdot M_{j}=(L-D_{i})\cdot(L-D_{j})=L^{2}+(D_{i}\cdot
D_{j})=\begin{cases}1,&\text{if $i\neq j$;}\\\ 1-m_{i},&\text{if
$i=j$.}\end{cases}$

Then $M_{i}\cdot D_{i}=(L-D_{i})\cdot D_{i}=-D_{i}^{2}=m_{i}$. Likewise
$M_{i}\cdot
K_{\widetilde{Y}}=M_{i}\cdot(M_{i}-D_{i}^{\prime})=M_{i}^{2}=1-m_{i}$.

To see (v), note that $\dim
H^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(M_{i}))=h^{2}(Y_{i};\mathcal{O}_{Y_{i}})$.
Since $\chi(\mathcal{O}_{Y_{i}})=2$, it suffices to prove that
$h^{1}(\mathcal{O}_{Y_{i}})=0$. The Leray spectral sequence gives an exact
sequence

$0\to H^{1}(Y_{i};\mathcal{O}_{Y_{i}})\to
H^{1}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})\to
H^{0}(R^{1}\psi_{i}{}_{*}\mathcal{O}_{\widetilde{Y}}).$

Thus, if $H^{1}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})=0$, then
$H^{1}(Y_{i};\mathcal{O}_{Y_{i}})=0$. If $\widetilde{Y}$ is the blowup of an
elliptic ruled surface with base an elliptic curve $B$, then all of the
singularities on $Y_{i}$ are simple elliptic, and
$H^{0}(R^{1}\psi_{i}{}_{*}\mathcal{O}_{\widetilde{Y}})\cong\bigoplus_{j\neq
i}H^{1}(\mathcal{O}_{D_{j}})$. Choosing a $j\neq i$, the map
$H^{1}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})\to
H^{1}(\mathcal{O}_{D_{j}})$ is an isomorphism since $D_{j}\to B$ is an
isogeny. Thus $H^{1}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}})\to
H^{0}(R^{1}\psi_{i}{}_{*}\mathcal{O}_{\widetilde{Y}})$ is injective so that
$H^{1}(Y_{i};\mathcal{O}_{Y_{i}})=0$ in this case as well. ∎

######  Remark 2.11.

The passage from $Y$ or $Y_{i}$ to $\widetilde{Y}$ is reversible, in the
following sense. Suppose that $\widetilde{Y}$ is a smooth surface and that
$D=\sum_{i}D_{i}$ is an effective divisor on $\widetilde{Y}$, where the
$D_{i}$ are the connected components of $D$ and each $D_{i}$ is a smooth
elliptic curve, an irreducible nodal rational curve or a cycle of smooth
rational curves (one could also allow configurations corresponding to the
triangle singularity case). By adjunction,
$K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D)|D_{i}\cong\omega_{D_{i}}\cong\mathcal{O}_{D_{i}}$.
Suppose in addition that the intersection matrix of $D$ is negative definite,
or equivalently that there exists a contraction $\pi\colon\widetilde{Y}\to Y$,
where $Y$ is a normal analytic surface and $\pi(D_{i})=p_{i}\in Y$. In
particular, the singular points are simple elliptic or cusp singularities.
Thus $\widetilde{Y}$ is the minimal resolution of a Gorenstein surface $Y$
with elliptic singularities. There exists a contractible Stein neighborhood
$U_{i}$ of $p_{i}$ in $Y$. Let $\widetilde{U}_{i}=\pi^{-1}(U_{i})$. A standard
argument using the Grauert-Riemenschneider theorem shows that restriction of
line bundles induces an isomorphism
$\operatorname{Pic}\widetilde{U}_{i}\to\operatorname{Pic}D_{i}$. Let
$L=K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D)$. By the above,
$L|\widetilde{U}_{i}=K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D)|\widetilde{U}_{i}\cong\mathcal{O}_{\widetilde{U}_{i}}$.
Hence $L$ is the pullback of a line bundle on $Y$, necessarily $\omega_{Y}$. A
similar statement holds for the line bundles
$M_{i}=K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D_{i}^{\prime})$
and the morphism $\psi_{i}\colon\widetilde{Y}\to Y_{i}$. Note that
$\omega_{Y}$ is nef and big $\iff$ $L$ is nef and big, and in this case
$\omega_{Y}$ is ample $\iff$ the irreducible curves $C$ on $\widetilde{Y}$
such that $L\cdot C=0$ are exactly the components of $D_{i}$ for some $i$.

###  2.3. The main geometric result

We begin with the following, due to Franciosi-Pardini-Rollenske [FPR15,
Theorem 4.1], [FPR17, Proposition 4.3], which deals with the case where
$\kappa(\widetilde{Y})\geq 0$, i.e. $\widetilde{Y}$ is not rational or ruled
(and does not require Assumption 2.7):

######  Theorem 2.12.

Suppose that $L^{2}=1$ and that $\kappa(\widetilde{Y})\geq 0$. Let
$Y_{\text{\rm{min}}}$ be the minimal model of $\widetilde{Y}$.

  1. (i)

If $\kappa(\widetilde{Y})=2$, i.e. $Y_{\text{\rm{min}}}$ is of general type,
then $\widetilde{Y}=Y=Y_{\text{\rm{min}}}$ and $k=0$, i.e. there are no simple
elliptic or cusp singularities.

  2. (ii)

If $\kappa(\widetilde{Y})=1$, i.e. $Y_{\text{\rm{min}}}$ is properly elliptic,
then $\widetilde{Y}=Y_{\text{\rm{min}}}$, $k=m=1$, $\widetilde{Y}$ is a
multiplicity two logarithmic transform of an elliptic $K3$ surface at a single
fiber, one component of $D=D_{1}$ is a bisection of the elliptic fibration,
and either $D$ is irreducible with $D^{2}=-1$ or $D$ consists of a bisection
of square $-3$ plus smooth components of square $-2$ lying in a fixed fiber.
In this case, $K_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(F)$, where $F$ is
the multiple fiber, $D\cdot F=1$, and $L=F+D$.

  3. (iii)

If $\kappa(\widetilde{Y})=0$, then either $k=m=2$ and $Y_{\text{\rm{min}}}$ is
an Enriques surface or $k=1$, $m=2$ and $Y_{\text{\rm{min}}}$ is a $K3$
surface. In this case, $\widetilde{Y}$ is the blowup of $Y_{\text{\rm{min}}}$
at one point. If $C$ is the exceptional curve, then $C\cdot D=2$ if
$Y_{\text{\rm{min}}}$ is a $K3$ surface and $C\cdot D_{1}=C\cdot D_{2}=1$ if
$Y_{\text{\rm{min}}}$ is an Enriques surface. Finally,
$K_{\widetilde{Y}}\equiv C$ and $L\equiv D+C$.

###### Proof.

Let $\rho\colon\widetilde{Y}\to Y_{\text{\rm{min}}}$ be the canonical
morphism. Then
$K_{\widetilde{Y}}=\rho^{*}K_{Y_{\text{\rm{min}}}}+\sum_{i}a_{i}C_{i}$, where
the $C_{i}$ are the positive dimensional fibers of $\rho$ and $C_{j}$ is an
exceptional curve for at least one $j$. Since $\kappa(\widetilde{Y})\geq 0$,
$K_{Y_{\text{\rm{min}}}}$ is nef and $K_{Y_{\text{\rm{min}}}}^{2}\geq 0$.
Since $L$ is nef and big, $L\cdot\rho^{*}K_{Y_{\text{\rm{min}}}}\geq 0$, with
equality $\iff$ $K_{Y_{\text{\rm{min}}}}\equiv 0$ by the Hodge index theorem.
Hence, if $\kappa(\widetilde{Y})\geq 1$, from

$1=L^{2}=L\cdot
K_{\widetilde{Y}}=L\cdot\left(\rho^{*}K_{Y_{\text{\rm{min}}}}+\sum_{i}a_{i}C_{i}\right),$

and the fact that $L\cdot C_{j}>0$ for at least one $j$, the sum
$\sum_{i}a_{i}C_{i}$ must be empty, i.e. $\widetilde{Y}=Y_{\text{\rm{min}}}$.
If $\kappa(\widetilde{Y})=0$ and $\widetilde{Y}\neq Y_{\text{\rm{min}}}$, then
$\sum_{i}a_{i}(L\cdot C_{i})=1$. By Lemma 2.1 and the fact that $L\cdot C>0$
for every exceptional curve $C$ on $\widetilde{Y}$, there is a unique
connected component of the fiber of $\rho$ and it is an irreducible
exceptional curve, i.e. $\rho\colon\widetilde{Y}\to Y_{\text{\rm{min}}}$ is a
single blowup.

The case $\kappa(\widetilde{Y})=2$: Then $K_{\widetilde{Y}}^{2}>0$. By the
Hodge index theorem,

$1\leq(L^{2})(K_{\widetilde{Y}}^{2})=K_{\widetilde{Y}}^{2}\leq(L\cdot
K_{\widetilde{Y}})^{2}=L^{2}=1.$

Since equality holds, $L$ and $K_{\widetilde{Y}}$ are numerically equivalent,
hence $D\equiv 0$ and $k=0$.

The case $\kappa(\widetilde{Y})=1$: Since $\widetilde{Y}$ is minimal,
$K_{\widetilde{Y}}^{2}=0$ and $1=L^{2}=K_{\widetilde{Y}}^{2}+m$, so $m=k=1$
and $K_{\widetilde{Y}}\cdot D=1$. In particular $D=D_{1}$ is connected. By
Lemma 2.3, $\chi(\mathcal{O}_{\widetilde{Y}})=3-k=2$. Since $D^{2}<0$ and
$p_{a}(D)=1$, $D$ cannot be a union of fiber components (it would have to be a
complete fiber). Hence $D$ contains a multisection of the elliptic fibration.
By the canonical bundle formula,

$K_{\widetilde{Y}}\equiv(2+2g-2)f+\sum_{\alpha}(m_{\alpha}-1)F_{\alpha}=2gf+\sum_{\alpha}(m_{\alpha}-1)F_{\alpha},$

where the $F_{\alpha}$ are the multiple fibers, of multiplicity
$m_{\alpha}\geq 2$. Since $D\cdot F_{\alpha}\geq 1$ for all $\alpha$ and
$K_{\widetilde{Y}}\cdot D=1$, the only possibility is that $g=0$, there is a
unique multiple fiber $F_{\alpha}$ with $m_{\alpha}=2$ and $D\cdot
F_{\alpha}=1$. Hence $D$ contains a bisection of the elliptic fibration, with
the other components equal to components of a single reducible fiber. In
particular, the Jacobian elliptic surface corresponding to $\widetilde{Y}$ is
an elliptic surface over $\mathbb{P}^{1}$ with a section and holomorphic Euler
characteristic equal to $2$, hence is an elliptic $K3$ surface. Then
$K_{\widetilde{Y}}\equiv F$. The remaining statements are clear or follow from
adjunction: either $D$ is an irreducible bisection, or $D$ is reducible and
one component $E$ is a bisection which is smooth rational and the other
components are smooth components of a fiber. In this case
$-2=K_{\widetilde{Y}}\cdot E+E^{2}=1+E^{2}$, so that $E$ is the unique
component of $D$ of self-intersection $-3$ and all other components have self-
intersection $-2$.

The case $\kappa(\widetilde{Y})=0$: In this case, $\widetilde{Y}\neq
Y_{\text{\rm{min}}}$, because otherwise

$0=2p_{a}(D_{i})-2=D_{i}^{2}+K_{\widetilde{Y}}\cdot D_{i}=D_{i}^{2},$

contradicting $D_{i}^{2}<0$. By the remarks at the beginning of the proof,
$\widetilde{Y}$ is then the blowup of $Y_{\text{\rm{min}}}$ at one point.
Thus,

$1=L^{2}=K_{\widetilde{Y}}^{2}+m=-1+m,$

so that $m=2$. Either $k=m=2$ and $D=D_{1}+D_{2}$ with $D_{i}^{2}=-1$ or $k=1$
and $D=D_{1}$ is connected with $D^{2}=-2$. In the first case,
$\chi(\mathcal{O}_{\widetilde{Y}})=3-2=1$, so $\widetilde{Y}$ is an Enriques
surface. In the second case, $\chi(\mathcal{O}_{\widetilde{Y}})=3-k=2$, so
$\widetilde{Y}$ is a $K3$ surface.

Let $C$ be the unique exceptional curve on $\widetilde{Y}$. In the $K3$ case,
$K_{\widetilde{Y}}=C$ and $C\cdot D=K_{\widetilde{Y}}\cdot D=-D^{2}=2$. In the
Enriques case, $K_{\widetilde{Y}}\equiv C$ and $C\cdot
D_{i}=K_{\widetilde{Y}}\cdot D_{i}=-D_{i}^{2}=1$. In both cases,
$L=K_{\widetilde{Y}}+D\equiv C+D$. ∎

We are thus reduced to analyzing the case $\kappa(\widetilde{Y})=-\infty$.
First note the following special case of [FPR15, Lemma 4.5]:

######  Lemma 2.13.

If $\kappa(\widetilde{Y})=-\infty$, then there are two possibilities:

  1. (i)

$\widetilde{Y}$ is a rational surface and $k=2$.

  2. (ii)

$\widetilde{Y}$ is a ruled surface over an elliptic curve $B$ and $k=3$. In
this case all singularities are simple elliptic.

###### Proof.

By Lemma 2.2, if $\kappa(\widetilde{Y})=-\infty$, then either $\widetilde{Y}$
is rational and hence $\chi(\mathcal{O}_{\widetilde{Y}})=1$, or
$\widetilde{Y}$ is the blowup of an elliptic ruled surface and hence
$\chi(\mathcal{O}_{\widetilde{Y}})=0$ and all singularities are simple
elliptic. By Lemma 2.3, $k=2$ in the first case and $k=3$ in the second case.
∎

The strategy now is to see what prevents the divisors $M_{i}$ introduced in
Definition 2.9 from being nef.

######  Lemma 2.14.

If $C$ is an irreducible curve on $\widetilde{Y}$ such that $M_{i}\cdot C<0$,
then $C$ is an exceptional curve, $M_{i}\cdot C=-1$, and $D_{i}^{\prime}\cdot
C=0$.

###### Proof.

By Lemma 2.10, $M_{i}$ is effective. Thus a curve $C$ such that $M_{i}\cdot
C<0$ must satisfy $C^{2}<0$. If $C$ is a component of $D_{i}^{\prime}$, then
$M_{i}\cdot C=0$ since $M_{i}$ is trivial in a neighborhood of
$D_{i}^{\prime}$. It follows that $C$ is not a component of $D_{i}^{\prime}$
so that $D_{i}^{\prime}\cdot C\geq 0$. Then

$0>M_{i}\cdot C=(K_{\widetilde{Y}}+D_{i}^{\prime})\cdot C\geq
K_{\widetilde{Y}}\cdot C.$

Thus $C$ is exceptional, $K_{\widetilde{Y}}\cdot C=-1$, and therefore
$D_{i}^{\prime}\cdot C=0$ and $M_{i}\cdot C=-1$. ∎

The main result is then the following, which generalizes Theorem 2.12 to the
case where $\kappa(\widetilde{Y})=-\infty$:

######  Theorem 2.15.

With notation as above, suppose that $\kappa(\widetilde{Y})=-\infty$ and in
particular that $k>1$. Then exactly one of the following holds:

  1. (i)

$M_{i}$ is not nef. Then there exists an exceptional curve $C$ such that
$C\cdot D_{i}^{\prime}=0$. In this case, if $\gamma\colon\widetilde{Y}\to
Y_{0}$ is the surface obtained by blowing down $C$, then $M_{i}=C$, $C\cdot
D_{i}=2$ and $m_{i}=2$. Finally, identifying $D_{i}^{\prime}$ with its image
in $Y_{0}$, $K_{Y_{0}}\equiv-D_{i}^{\prime}$, $\widetilde{Y}$ is the blowup of
$Y_{0}$ at a point not on $D_{i}^{\prime}$, and $D_{i}$ is the proper
transform of a divisor $\Gamma_{i}$ on $Y_{0}$ disjoint from $D_{i}^{\prime}$
and containing $p$ as a point of multiplicity $2$ with $p_{a}(\Gamma_{i})=2$.

  2. (ii)

$M_{i}$ is nef. Then $M_{i}^{2}=0$ and $m_{i}=1$. Moreover, $|2M_{i}|$ is a
base point free linear system defining an elliptic fibration
$\widetilde{Y}\to\mathbb{P}^{1}$ (not necessarily relatively minimal) with at
most $3$ multiple fibers, all of multiplicity $2$. More precisely, there is a
single multiple fiber if $\widetilde{Y}$ is rational and there are $2$ or $3$
multiple fibers if $\widetilde{Y}$ is an elliptic ruled surface. Moreover
$M_{i}$ is a multiple fiber. Finally, either $D_{i}$ is an irreducible
bisection or one irreducible component of $D_{i}$ is a smooth bisection of the
fibration, and the remaining components are curves of self-intersection $-2$
contained in a single reducible, non-multiple fiber.

###### Proof.

Case (i): $M_{i}$ is not nef. Let $C$ be as in the statement of Lemma 2.14.
Contracting $C$ gives a morphism $\gamma_{1}\colon\widetilde{Y}\to Y_{1}$,
where $Y_{1}$ is a smooth surface containing a union of curves isomorphic to
$D_{i}^{\prime}$, which we will continue to denote by $D_{i}^{\prime}$. Let
$N_{1}=\gamma_{1}{}_{*}M_{1}$, so that
$N_{1}=\gamma_{1}{}_{*}(K_{\widetilde{Y}}+D_{i}^{\prime})=K_{Y_{1}}+D_{i}^{\prime}$
and $M_{i}=\gamma_{1}^{*}N_{1}+C=\gamma_{1}^{*}K_{Y_{1}}+C+D_{i}^{\prime}$.
The argument of Lemma 2.14 shows that, if $N_{1}$ is not nef, then there
exists an exceptional curve $C^{\prime}$ on $\overline{Y}$ such that
$N_{1}\cdot C^{\prime}<0$ and $C^{\prime}\cdot D_{i}^{\prime}=0$. Then we can
keep repeating this argument, which cannot continue indefinitely. Thus we
eventually find a birational morphism $\gamma\colon\widetilde{Y}\to Y_{0}$,
where $Y_{0}$ is smooth, and a nef divisor $N_{0}$ on $Y_{0}$ such that
$M_{i}=\gamma^{*}N_{0}+\sum_{j}a_{j}C_{j}=\gamma^{*}K_{Y_{0}}+\sum_{j}a_{j}C_{j}+D_{i}^{\prime}$,
where the $C_{j}$ are the components of the one-dimensional fibers of the
morphism $\gamma\colon\widetilde{Y}\to Y_{0}$ and $a_{j}>0$. Then
$L\cdot\gamma^{*}N_{0}\geq 0$ since $L$ and $\gamma^{*}N_{0}$ are nef, and
$L\cdot a_{j}C_{j}\geq a_{j}$ if $C_{j}$ is exceptional. Since at least one
curve in every connected positive dimensional fiber of $\gamma$ is exceptional
and

$1=L\cdot M_{i}=(L\cdot\gamma^{*}N_{0})+\sum_{j}a_{j}(L\cdot C_{j}),$

the only possibility via Lemma 2.1 is that there is just one exceptional curve
$C=C_{j}$, $a_{j}=1$, and $L\cdot\gamma^{*}N_{0}=0$. Since $N_{0}$ is nef,
$N_{0}^{2}=(\gamma^{*}N_{0})^{2}\geq 0$. By the Hodge index theorem,
$\gamma^{*}N_{0}\equiv 0$ and hence $N_{0}\equiv 0$. Thus
$M_{i}=K_{\widetilde{Y}}+D_{i}^{\prime}\equiv C$. Since $M_{i}$ is effective
and $C$ is irreducible with $C^{2}<0$, $M_{i}\equiv C$. Hence
$L=M_{i}+D_{i}\equiv D_{i}+C$ and $K_{\widetilde{Y}}\equiv C-D_{i}^{\prime}$.
Then $K_{\widetilde{Y}}=\gamma^{*}K_{Y_{0}}+C\equiv C-D_{i}^{\prime}$, so that
$\gamma^{*}K_{Y_{0}}\equiv-D_{i}^{\prime}\equiv-\gamma^{*}D_{i}^{\prime}$
where we view the divisor $D_{i}^{\prime}$ as living on $Y_{0}$. Thus
$K_{Y_{0}}\equiv-D_{i}^{\prime}$ since
$\gamma^{*}\colon\operatorname{Num}Y_{0}\to\operatorname{Num}\widetilde{Y}$ is
injective. Next we claim that $C\cdot D_{i}=2$, as follows from the equalities

$1=L^{2}=L\cdot(D_{i}+C)=L\cdot C=(D_{i}+C)\cdot C=(D_{i}\cdot
C)+C^{2}=(D_{i}\cdot C)-1.$

The statement that $m_{i}=2$ follows from the equalities

$0=L\cdot D_{i}=(M_{i}+D_{i})\cdot D_{i}=(D_{i}+C)\cdot D_{i}=-m_{i}+2.$

If $\Gamma_{i}=\gamma(D_{i})$, then $D_{i}=\gamma^{*}\Gamma_{i}-2C$ because
$D_{i}\cdot C=2$. This proves the remaining statements in Case (i).

Case (ii): $M_{i}$ is nef. Then $M_{i}^{2}=1-m_{i}\geq 0$. Hence $m_{i}=1$ and
$M_{i}^{2}=0$. Next we claim that $\dim|2M_{i}|\geq 1$ and in fact that
$|2M_{i}|$ is a base point free linear system. This is a minor variation on a
standard argument in classification theory (cf. e.g. [Fri98, Lemma 24 p.
295]): First,
$h^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(nM_{i}))=h^{0}(Y_{i};\omega_{Y_{i}}^{\otimes
n})$. Lemma 2.10 implies that $\chi(Y_{i};\mathcal{O}_{Y_{i}})=2$. By an easy
extension of the Riemann-Roch theorem to a projective Gorenstein surface with
isolated singularities,

$\chi(Y_{i};\omega_{Y_{i}}^{\otimes
n})=\frac{n^{2}M_{i}^{2}-nM_{i}^{2}}{2}+\chi(Y_{i};\mathcal{O}_{Y_{i}})=2.$

For $n\geq 2$, $h^{2}(Y_{i};\omega_{Y_{i}}^{\otimes
n})=h^{0}(Y_{i};\omega_{Y_{i}}^{\otimes
1-n})=h^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}((1-n)M_{i}))=0$ since
$M_{i}$ is effective and nonzero. It follows that

$h^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(nM_{i}))=h^{0}(Y_{i};\omega_{Y_{i}}^{\otimes
n})\geq 2$

for all $n\geq 2$, in particular for $n=2$. Thus $\dim|2M_{i}|\geq 1$. Write
$2M_{i}=G_{f}+G_{m}$, where $G_{f}$ is the fixed curve and $G_{m}$ the moving
part of the linear system $|2M_{i}|$. Since $M_{i}$ is nef, $M_{i}\cdot
G_{f}\geq 0$ and $M_{i}\cdot G_{m}\geq 0$, and hence $M_{i}\cdot
G_{f}=M_{i}\cdot G_{m}=0$. By the Hodge index theorem, since $M_{i}$ is not
numerically equivalent to $0$, $G_{f}$ and $G_{m}$ are rational multiples of
$M_{i}$ and hence $G_{m}^{2}=G_{f}^{2}=0$. Thus $|2M_{i}|=|G_{m}|$ defines a
fibration $\varphi\colon\widetilde{Y}\to B$ over some base curve $B$. Let $G$
be a general fiber of $\varphi$, so that $G$ is smooth and $G^{2}=0$. Since
$G\cdot M_{i}=G_{f}\cdot M_{i}=G_{m}\cdot M_{i}=0$, every effective curve
$\Gamma\in|M_{i}|$ is contained in fibers of the morphism $\varphi$ as is
$G_{f}$. Hence $\Gamma$ and $G_{f}$ are numerically equivalent to positive
rational multiples of $G$ since $M_{i}^{2}=G_{f}^{2}=0$. Also, $M_{i}$ is a
primitive class since $L\cdot M_{i}=1$. Since $2M_{i}=G_{f}+G_{m}$, either
$G_{f}=0$ and $G_{m}=2M_{i}$ or $G_{f}\equiv G_{m}\equiv M_{i}$. In any case,
$G_{m}\equiv aG$ for some positive integer $a$ and hence $L\cdot G\leq 2$. By
Lemma 2.10(iv), $G\cdot K_{\widetilde{Y}}=0$, hence $G$ is an elliptic curve
and $\varphi\colon\widetilde{Y}\to B$ is a not necessarily relatively minimal
elliptic fibration.

Let $\varphi_{0}\colon Y_{0}\to B$ be the relatively minimal model of
$\varphi$. Then $Y_{0}$ is a (relatively minimal) elliptic surface. By the
canonical bundle formula, after identifying $G$ with a general fiber of
$\varphi_{0}$,

$K_{Y_{0}}=(2g-2+\chi)G+\sum_{\alpha}(m_{\alpha}-1)F_{\alpha},$

where $g$ is the genus of $B$,
$\chi=\chi(\mathcal{O}_{Y_{0}})=\chi(\mathcal{O}_{\widetilde{Y}})\geq 0$, and
the $F_{\alpha}$ are the multiple fibers of multiplicities $m_{\alpha}>1$.
Since $L\cdot G=m_{\alpha}(L\cdot F_{\alpha})\leq 2$, $m_{\alpha}\leq 2$ as
well, and hence $m_{\alpha}=2$ for every $\alpha$. Because
$\kappa(Y_{0})=\kappa(\widetilde{Y})=-\infty$, $g=0$, $B\cong\mathbb{P}^{1}$,
$\chi=0$ (the elliptic ruled case) or $1$ (the rational case), and

$\chi-2+\sum_{\alpha}\left(1-\frac{1}{m_{\alpha}}\right)=\chi-2+\frac{\\#(\alpha)}{2}<0.$

In particular, in the rational case there is at most one multiple fiber. In
the elliptic ruled case, $Y_{0}$ is a logarithmic transform of a product
elliptic surface $G\times\mathbb{P}^{1}$ and there are either $0$, $2$ or $3$
multiple fibers, because a logarithmic transform of a product surface at a
single fiber is never algebraic (cf. [FM94, I.6.12]). Then we have the
following formula for the canonical bundle of $\widetilde{Y}$:

$K_{\widetilde{Y}}=(\chi-2)G+\sum_{\alpha}F_{\alpha}+\sum_{t}a_{t}C_{t},$

where the $C_{t}$ are the exceptional curves in the blowup $\widetilde{Y}\to
Y_{0}$. Since $M_{i}\cdot D_{i}^{\prime}=0$, the connected components of
$D_{i}^{\prime}$ are contained in fibers of $\varphi$ and have arithmetic
genus one. Hence $D_{i}^{\prime}$ is a union of proper transforms of fibers of
$\varphi_{0}$, and so

$D_{i}^{\prime}=nG+\sum_{\alpha}n_{\alpha}F_{\alpha}-\sum_{t}b_{t}C_{t},$

where $n,n_{\alpha},b_{t}$ are nonnegative integers and $n_{\alpha}$ is either
$0$ or $1$. Clearly $n+\sum_{\alpha}n_{\alpha}=k-1$. Since
$M_{i}=K_{\widetilde{Y}}+D_{i}^{\prime}$ and $M_{i}^{2}=0$, $a_{t}=b_{t}$ for
all $t$. Thus

$M_{i}=(n+\chi-2)G+\sum_{\alpha}(n_{\alpha}+1)F_{\alpha}.$

First suppose that $\widetilde{Y}$ is rational. Then $k=2$ and hence $n\leq
1$. Since $M_{i}$ is effective and nonzero and
$M_{i}=(n-1)G+\sum_{\alpha}(n_{\alpha}+1)F_{\alpha}$, there has to exist a
multiple fiber. But $\widetilde{Y}$ is rational, so there is exactly one
multiple fiber $F=F_{\alpha}$, necessarily of multiplicity $2$. Since
$n+n_{\alpha}=1$, the case $n=0$, $n_{\alpha}=1$ is not possible since then
$M_{i}=-G+2F_{\alpha}=0$. Hence $n=1$, $n_{\alpha}=0$ and
$M_{i}=F_{\alpha}=F$.

Now suppose that $\widetilde{Y}$ is elliptic ruled. Then $k=3$ and
$n+\sum_{\alpha}n_{\alpha}=2$. In particular, $n\leq 2$. Here
$M_{i}=(n-2)G+\sum_{\alpha}(n_{\alpha}+1)F_{\alpha}$. As in the rational case,
there must exist a multiple fiber, and hence there are $2$ or $3$ multiple
fibers. Note that $n=2$ is impossible, since then
$M_{i}=\sum_{\alpha}(n_{\alpha}+1)F_{\alpha}$ is a sum of at least two
multiple fibers, hence is not primitive. There are two cases:

Case (1): $n=1$, $n_{\alpha}=1$ for exactly one $\alpha$, and $n_{\beta}=0$
for $\beta\neq\alpha$. In this case,

$M_{i}=-G+2F_{\alpha}+\sum_{\beta\neq\alpha}F_{\beta}=\sum_{\beta\neq\alpha}F_{\beta}.$

Since $M_{i}$ is primitive, there are exactly two multiple fibers
$F_{\alpha},F_{\beta}$, and $M_{i}=F_{\beta}$.

Case (2): $n=0$, $n_{\alpha_{1}}=n_{\alpha_{2}}=1$ for two multiple fibers
$F_{\alpha_{1}}$, $F_{\alpha_{2}}$, and $n_{\beta}=0$ for a potential third
multiple fiber $F_{\beta}\neq F_{\alpha_{i}}$. In this case,

$M_{i}=-2G+2F_{\alpha_{1}}+2F_{\alpha_{1}}+\sum_{\beta\neq\alpha_{1},\alpha_{2}}F_{\beta}=\sum_{\beta\neq\alpha_{1},\alpha_{2}}F_{\beta}.$

Using again the fact that $M_{i}$ is primitive and effective, there are
exactly $3$ multiple fibers $F_{\alpha_{1}},F_{\alpha_{2}},F_{\beta}$, and
$M_{i}=F_{\beta}$.

In all cases, $G=2M_{i}$, hence $G_{f}=0$ and $G_{m}=G$. Note that $M_{i}$ is
effective and $M_{i}=F$ for a multiple fiber $F$. In particular, the elliptic
fibration cannot have a section. Since $M_{i}\cdot D_{i}=1$, $D_{i}$ is a
bisection of the fibration. More precisely, since $M_{i}$ is nef, exactly one
component of $D_{i}$ is a bisection (because there are no sections of the
fibration) and the remaining components are fiber components. In particular,
if $D_{i}$ is reducible, then $\widetilde{Y}$ is a rational surface. Since
$D_{i}$ is then both a bisection and a cycle of rational curves, it is clear
that the remaining components must lie in a single reduced fiber. ∎

If $\widetilde{Y}$ is a rational surface, the possibilities for the
multiplicities $(m_{1},m_{2})$ up to order are $(2,2)$, $(2,1)$, or $(1,1)$,
and we will show in §5 that all such possibilities occur. For the case where
$\widetilde{Y}$ is a blown up elliptic ruled surface, the situation is quite
different:

######  Proposition 2.16.

Suppose that $\widetilde{Y}$ is a blown up elliptic ruled surface and that
$\rho\colon\widetilde{Y}\to B$ is the Albanese map. Then the possibilities for
$(m_{1},m_{2},m_{3})$ up to order are either $(2,1,1)$ or $(1,1,1)$. More
precisely, if $m_{1}=2$, then $m_{2}=m_{3}=1$ and $D_{2}$, $D_{3}$ are
sections of $\rho$.

###### Proof.

By Theorem 2.15(i), if $m_{1}=2$, then there exists an exceptional curve $C$
such that $M_{1}=C$ and $D_{1}\cdot C=2$. If $f$ is a general fiber of the
Albanese map $\rho\colon\widetilde{Y}\to B$, then $f\cong\mathbb{P}^{1}$,
$f^{2}=0$, and $K_{\widetilde{Y}}\cdot f=-2$. Also, $D_{i}$ is not contained
in a fiber of $\rho$ and hence $f\cdot D_{i}\geq 1$. Since $C$ is contained in
a (reducible) fiber, $f\cdot C=0$. From

$0=f\cdot C=f\cdot(K_{\widetilde{Y}}+D_{2}+D_{3})=-2+(f\cdot D_{2})+(f\cdot
D_{3}),$

it follows that $(f\cdot D_{2})=(f\cdot D_{3})=1$, i.e. $D_{2}$ and $D_{3}$
are sections. If $m_{i}=2$ for $i=2$ or $3$, the above would then imply that
$D_{1}\cdot f=1$. But $D_{1}\cdot C=2$, and hence $D_{1}\cdot f\geq 2$, a
contradiction. Thus $m_{2}=m_{3}=1$. ∎

We will show in §7 that both possibilities $(2,1,1)$ and $(1,1,1)$ occur.

##  3\. The case $\kappa(\widetilde{Y})=1$

###  3.1. The cotangent bundle

We begin with a fairly general result:

######  Theorem 3.1.

Suppose that $\pi\colon X\to\mathbb{P}^{1}$ is an elliptic surface with
$p_{g}(X)=p_{g}$ and that the only non-reduced fibers of $\pi$ are multiple
fibers $F_{i}$ with multiplicities $m_{i}$. Then there is an exact sequence

$0\to\mathcal{O}_{X}(-2f+\sum_{i}(m_{i}-1)F_{i})\to\Omega^{1}_{X}\to
I_{Z}\otimes\mathcal{O}_{X}((p_{g}+1)f)\to 0,$

where $Z$ is a zero-dimensional subscheme of length $\ell(Z)=12(1+p_{g})$
supported at the singular points of the reductions of the fibers, and at a
point $x$ of a fiber whose reduction has a normal crossing singularity,
$I_{Z,x}=\mathfrak{m}_{x}$ is the maximal ideal at $x$.

###### Proof.

The natural homomorphism $\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}\to\Omega^{1}_{X}$
vanishes along the multiple fibers, to order exactly $m_{i}-1$ along $F_{i}$.
Thus there is an induced homomorphism

$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(\sum_{i}(m_{i}-1)F_{i})=\mathcal{O}_{X}(-2f+\sum_{i}(m_{i}-1)F_{i})\to\Omega^{1}_{X}$

whose cokernel is torsion free and hence of the form $I_{Z}\otimes\lambda$ for
a line bundle $\lambda$. By taking determinants and using the the canonical
bundle formula,

$\displaystyle\lambda$
$\displaystyle=K_{X}\otimes(\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(\sum_{i}(m_{i}-1)F_{i}))^{-1}$
$\displaystyle=\mathcal{O}_{X}((p_{g}-1)f+\sum_{i}(m_{i}-1)F_{i})\otimes\mathcal{O}_{X}(2f-\sum_{i}(m_{i}-1)F_{i})=\mathcal{O}_{X}((p_{g}+1)f).$

It follows that

$c_{2}(X)=c_{2}(T_{X})=c_{2}(\Omega^{1}_{X})=\ell(Z),$

and hence that $\ell(Z)=12(1+p_{g})$. A local calculation then establishes
that $I_{Z,x}=\mathfrak{m}_{x}$ in case the reduction of the fiber containing
$x$ is nodal. ∎

We will need a somewhat specialized variant of Theorem 3.1 for logarithmic
$1$-forms:

######  Theorem 3.2.

Suppose that $\pi\colon X\to\mathbb{P}^{1}$ is an elliptic surface such that
all non-multiple fibers are nodal and there is exactly one multiple fiber $F$
of $\pi$, of multiplicity $2$. Let $D=E_{1}+\cdots+E_{r}$ be a reduced normal
crossing cycle of rational curves, where $E_{1}$ is a smooth bisection and,
for $i>1$, the $E_{i}$ are components of a non-multiple Ia fiber $f_{t}$, with
$E_{1}\cdot E_{2}=E_{1}\cdot E_{r}\neq 0$ and $E_{i}\cdot E_{i+1}=1$, $2\leq
i\leq r-1$. Then there is an exact sequence

$0\to\mathcal{O}_{X}(-2f+F+\sum_{i>1}E_{i})\to\Omega^{1}_{X}(\log D)\to
I_{W}\otimes\mathcal{O}_{X}((p_{g}+1)f+E_{1})\to 0,$

where $W$ is a zero-dimensional subscheme of length $\ell(W)=12(1+p_{g})-r+1$.
Moreover, $x\in\operatorname{Supp}W$ $\iff$ either $x$ is a singular point of
a fiber not equal to $f_{t}$, $x$ is a singular point of $f_{t}-D$, or $x\in
E_{1}-F$ and $x$ is a ramification point for the induced double cover morphism
$E_{1}\to\mathbb{P}^{1}$, necessarily not on $f_{t}$. Finally,

  1. (i)

If $x$ is a point of a fiber different from $f_{t}$ whose reduction has a
normal crossing singularity at $x$ and $x\notin E_{1}$, then
$I_{W,x}=\mathfrak{m}_{x}$ is the maximal ideal at $x$.

  2. (ii)

If $x\in E_{1}-F$ is a ramification point for the induced double cover
morphism $E_{1}\to\mathbb{P}^{1}$ and $x$ is not a singular point of the
corresponding fiber, then $I_{W,x}=\mathfrak{m}_{x}$.

  3. (iii)

If $x\in E_{1}-F$ is a ramification point for the induced double cover
morphism $E_{1}\to\mathbb{P}^{1}$ and $x$ is a singular point of the
corresponding fiber, then $\ell(I_{W,x})=2$.

  4. (iv)

If $x\in f_{t}-D$ is a singular point of $f_{t}$, then
$I_{W,x}=\mathfrak{m}_{x}$.

###### Proof.

Note that $\det\Omega^{1}_{X}(\log
D)=K_{X}\otimes\mathcal{O}_{X}(D)=\mathcal{O}_{X}((p_{g}-1)f+F+D)$. As in the
proof of Theorem 3.1, the natural map
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}\to\Omega^{1}_{X}(\log D)$ vanishes along
$F$ and has a torsion free cokernel away from $F+D$. If $x\notin D$, then the
analysis of the proof of Theorem 3.1 shows that the cokernel of
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(F)\to\Omega^{1}_{X}(\log D)$ is locally
free at $x$ if $x$ is not a singular point of a fiber and is locally
isomorphic to $\mathfrak{m}_{x}$ if $x$ is a singular point of a fiber.

Next suppose that $x\in E_{1}-f_{t}$. If the reduced fiber containing $x$
meets $E_{1}$ transversally (including the case where $x\in E_{1}\cap F$),
then the cokernel of
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(F)\to\Omega^{1}_{X}(\log D)$ is locally
free at $x$. Otherwise, $x$ is a branch point of the morphism
$E_{1}\to\mathbb{P}^{1}$ and a local calculation shows that $\ell(I_{W,x})=1$
if $x$ is a smooth point of the fiber and $\ell(I_{W,x})=2$ otherwise.

Thus we may assume that $x\in f_{t}\cap D=\sum_{i>1}E_{i}$. If $x\in
E_{i}-E_{1}$ is a smooth point of $f_{t}$, then, locally around $x$, there
exist local coordinates $z_{1},z_{2}$ such that $E_{i}$ is defined by
$z_{1}=0$ and the morphism $X\to\mathbb{P}^{1}$ is given by $z_{1}$. A local
basis for $\Omega^{1}_{X}(\log D)$ is
$\displaystyle\frac{dz_{1}}{z_{1}},dz_{2}$ and the image of
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}$ is the span of $dz_{1}$. Thus the
cokernel of the induced morphism
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(E_{i})\to\Omega^{1}_{X}(\log D)$ is
locally free at $x$, i.e. $x$ is not in the support of the scheme $W$. Next
suppose that $x\in E_{i}\cap E_{i+1}$ is a singular point of the fiber
$f_{t}$. Then there are local coordinates $z_{1},z_{2}$ such that the fiber is
defined at $x$ by $z_{1}z_{2}$. A local basis for $\Omega^{1}_{X}(\log D)$ is
$\displaystyle\frac{dz_{1}}{z_{1}},\frac{dz_{2}}{z_{2}}$ and the image of
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}$ is

$z_{2}dz_{1}+z_{1}dz_{2}=(z_{1}z_{2})\left(\frac{dz_{1}}{z_{1}}+\frac{dz_{2}}{z_{2}}\right).$

This vanishes to order one along $E_{i}+E_{i+1}$ and the cokernel of the
induced morphism

$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(E_{i}+E_{i+1})\to\Omega^{1}_{X}(\log D)$

is locally free at $x$, i.e. $x$ is not in the support of the scheme $W$. If
$x\in E_{1}\cap E_{2}$ or $E_{r}\cap E_{1}$, then $x$ is not a singular point
of $f_{t}$ and it is not a branch point of $E_{1}\to\mathbb{P}^{1}$. Then
there are local coordinates $z_{1},z_{2}$ such that $E_{1}$ is defined by
$z_{1}=0$, $E_{2}$ by $z_{2}=0$, and the image of
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}$ in $\Omega^{1}_{X}(\log D)$ is spanned by
$\displaystyle dz_{2}=z_{2}\cdot\frac{dz_{2}}{z_{2}}$. Thus the cokernel of
the induced homomorphism
$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(E_{2})\to\Omega^{1}_{X}(\log D)$ is
locally free at $x$ in this case as well and $x$ is not in the support of the
scheme $W$. A similar argument shows that, if $x$ is a singular point of
$f_{t}$ of the form $E_{2}\cap C$ or $E_{r}\cap C$, where $C$ is not a
component of $D$, then $x$ is likewise not in the support of the scheme $W$.

Thus, $\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(F)\to\Omega^{1}_{X}(\log D)$
vanishes along $E_{i}$, $i>1$, and extends to a homomorphism

$\pi^{*}\Omega^{1}_{\mathbb{P}^{1}}(F+\sum_{i>1}E_{i})=\mathcal{O}_{X}(-2f+F+\sum_{i>1}E_{i})\to\Omega^{1}_{X}(\log
D)$

with torsion free cokernel. Computing determinants then gives the exact
sequence in the statement of the theorem. Finally, a straightforward argument
shows that $c_{2}(\Omega^{1}_{X}(\log D))=12(1+p_{g})-r$. By the Whitney
product formula,

$c_{2}(\Omega^{1}_{X}(\log D))=12(1+p_{g})-r=\ell(W)-4+1+2=\ell(W)-1,$

and hence $\ell(W)=12(1+p_{g})-r+1$. ∎

###  3.2. Multiple fibers of multiplicity $2$ and double covers

Let $\pi\colon X\to\mathbb{P}^{1}$ be a relatively minimal elliptic surface
with a single multiple fiber, of multiplicity $2$, with reduction $F$,
together with an irreducible, not necessarily smooth bisection $\Sigma$. There
is an involution on the generic fiber $X_{\eta}$, defined by: if $p$ is a
smooth point of a fiber $f$, then $\iota(p)=p^{\prime}$, where
$\mathcal{O}_{X}(\Sigma)|f\cong\mathcal{O}_{f}(p+p^{\prime})$. Hence (by
relative minimality) there is an induced involution $\iota\colon X\to X$,
whose fixed locus is a $4$-section together with some isolated fixed points,
in particular two isolated fixed points on $F$. Note that $\iota$ only depends
on the restriction of $\Sigma$ to the generic fiber. Also,
$\iota(\Sigma)=\Sigma$ and $\iota|\Sigma$ is the involution corresponding to
the double cover $\Sigma\to\mathbb{P}^{1}$. We make the following additional
assumption on $X$, which will be satisfied in §3.3, and in particular in the
hypotheses of Lemma 3.5 and Theorem 3.6:

######  Assumption 3.3.

The multiple fiber $F$ is irreducible, the non-multiple fibers of $\pi$ have
at most two components, and, in case there are two components, $\Sigma$ has
intersection number one with each component.

Assumption 3.3 has the following consequences: Every non-multiple fiber is
reduced, and is either a nodal or a cuspidal rational curve, if irreducible,
or a I2 fiber or a tacnodal fiber, if there are two components. The involution
$\iota$ has exactly two isolated fixed points and they both lie on $F$.
Finally, if the fiber has two components, they are exchanged by $\iota$. Thus,
all fibers of the induced morphism $X/\iota\to\mathbb{P}^{1}$ are irreducible,
but $X/\iota$ has two ordinary double points at the fixed points of $\iota$
and is otherwise smooth.

Then as in [Hor78], [Fri95], the blowup of $X$ at the two fixed points is
obtained as follows: let $\mathbb{F}_{N}$ be a Hirzebruch surface, $N\geq 0$,
and let $\sigma_{0}$ be the negative section if $N>0$ and any section of
square $0$ in case $N=0$. Blow up a point $p$ in a fiber $f$, such that
$p\notin\sigma_{0}$, and denote the exceptional curve by $d_{1}^{\prime}$ and
the proper transform $f-d_{1}^{\prime}$ by $d_{2}^{\prime}$. Note that we can
always assume that we are in the case $p\notin\sigma_{0}$, because if
$p\in\sigma_{0}$ then the proper transform of $\sigma_{0}$ has self-
intersection $-N-1$ and the proper transform of the fiber is a disjoint
exceptional curve. Thus we could have started instead with the elementary
transform $\mathbb{F}_{N+1}$ of $\mathbb{F}_{N}$ and a point not in the
negative section. In particular, in case $N=0$, there is always a section
$\sigma_{0}$ of square $0$ passing through a point $p$. Hence we can in fact
always assume that $N\geq 1$ in what follows. Finally blow up the intersection
point of $d_{1}^{\prime}$ and $d_{2}^{\prime}$ and let
$\widetilde{\mathbb{F}}_{N}$ be the resulting surface. Denote the new
exceptional curve by $e$ and the proper transforms of
$d_{1}^{\prime},d_{2}^{\prime}$ by $d_{1}$ and $d_{2}$. Then
$d_{1}^{2}=d_{2}^{2}=-2$, $e^{2}=-1$ and $f=d_{1}+2e+d_{2}$, where $f$ is the
class of a general fiber of $\widetilde{\mathbb{F}}_{N}\to\mathbb{P}^{1}$.
Identifying $\sigma_{0}$ with its preimage in $\widetilde{\mathbb{F}}_{N}$,
$d_{2}\cdot\sigma_{0}=1$, $d_{1}\cdot\sigma_{0}=e\cdot\sigma_{0}=0$, and
$e\cdot d_{1}=e\cdot d_{2}=1$. Also,

$K_{\widetilde{\mathbb{F}}_{N}}=-2\sigma_{0}-(N+2)f+d_{1}+2e.$

Note: the labeling of the curves $d_{1},d_{2}$ is the opposite of that in
[Fri95].

Choose a smooth
$B_{0}\in|4\sigma_{0}+(2k+1)f-4e-2d_{1}|=|4\sigma_{0}+(2k-1)f+2d_{2}|$ not
containing $\sigma_{0}$ as a component and let $B=B_{0}+d_{1}+d_{2}$ as
effective curves. Thus as divisor classes

$\displaystyle B$
$\displaystyle=B_{0}+d_{1}+d_{2}=4\sigma_{0}+2kf-4e-2d_{1}+f+d_{1}+d_{2}$
$\displaystyle=4\sigma_{0}+2kf-2e+2d_{2}.$

Note that $B_{0}\cdot e=2$, $B_{0}\cdot d_{1}=B_{0}\cdot d_{2}=0$, and that

$\displaystyle B\cdot f$ $\displaystyle=B\cdot e=4;$ $\displaystyle
B\cdot\sigma_{0}$ $\displaystyle=-4N+2k+2.$

Then $(B_{0}-d_{i})\cdot d_{i}=-d_{i}^{2}>0$. Since $B_{0}$ is smooth, $d_{1}$
and $d_{2}$ are not components of $B_{0}$ and hence are disjoint from $B_{0}$.
Also, since $\sigma_{0}$ is not a component of $B_{0}$,

$B_{0}\cdot\sigma_{0}=B\cdot\sigma_{0}-1=-4N+2k+1\geq 0,$

and hence $k\geq 2N-\frac{1}{2}$. Since $k$ is an integer, $k\geq 2N$.

The divisor class $B$ is uniquely divisible by $2$, with
$\frac{1}{2}B=2\sigma_{0}+kf-e+d_{2}$. Thus

$K_{\widetilde{\mathbb{F}}_{N}}+\textstyle\frac{1}{2}B=(k-N-2)f+d_{1}+e+d_{2}=(k-N-1)f-e.$

We can then form the double cover
$\nu\colon\widetilde{X}\to\widetilde{\mathbb{F}}_{N}$. The inverse images
$e_{i}$ of the $d_{i}$ become exceptional curves, since $\nu^{*}d_{i}=2e_{i}$
and hence $4e_{i}^{2}=2(-2)=-4$. Contracting these, we get an elliptic surface
$X$ with a multiple fiber $F$. Here, letting $f$ denote both a general fiber
on $\mathbb{F}_{N}$, $\widetilde{\mathbb{F}}_{N}$, $X$, or $\widetilde{X}$,
$\nu^{*}f=f$. Moreover, $f\equiv 2F=2F^{\prime}+2e_{1}+2e_{2}$ and
$\nu^{*}e=F^{\prime}$. Since
$p_{g}(X)=h^{0}(\widetilde{\mathbb{F}}_{N};\mathcal{O}_{\widetilde{\mathbb{F}}_{N}}(K_{\widetilde{\mathbb{F}}_{N}}+\frac{1}{2}B))$,

$p_{g}(X)=\begin{cases}k-N-1,&\text{if $k-N\geq 1$};\\\
0,&\text{otherwise}.\end{cases}$

Note however that if $k-N=0$, i.e. $k=N$, then since $k\geq 2N$, we must have
$k=N=0$. So we can ignore this case in what follows since we are only
interested in the case $N\geq 1$. Then $p_{g}=p_{g}(X)=k-N-1$ and $k\geq 2N$,
so $N\leq p_{g}+1$ and $k=p_{g}+1+N$. This is in agreement with:

$\displaystyle K_{\widetilde{X}}$
$\displaystyle=\nu^{*}(K_{\widetilde{\mathbb{F}}_{N}}+\textstyle\frac{1}{2}B)=(k-N-1)\nu^{*}f-F^{\prime}$
$\displaystyle=(k-N-2)f+2F^{\prime}+2e_{1}+2e_{2}-F^{\prime}=(p_{g}-1)f+F^{\prime}+2e_{1}+2e_{2}$
$\displaystyle=(p_{g}-1)f+F+e_{1}+e_{2}.$

Also,

$B\cdot\sigma_{0}=-4N+2k+2=2(k-N)-2N+2=2p_{g}-2N+4.$

In particular, if $\widetilde{\Sigma}$ is the preimage of $\sigma_{0}$ on the
double cover $\widetilde{X}$, then $\widetilde{\Sigma}^{2}=-2N$ and
$p_{a}(\widetilde{\Sigma})=p_{g}-N+1$. Since $\sigma_{0}\cdot d_{2}=1$ and
$\sigma_{0}\cdot d_{1}=0$, $\widetilde{\Sigma}$ meets the exceptional curve
$e_{2}$ transversally at a smooth point and is disjoint from $e_{1}$. Thus,
setting $\Sigma$ to be the image of $\widetilde{\Sigma}$ in $X$,
$\Sigma\cong\widetilde{\Sigma}$. Hence $p_{a}(\Sigma)=p_{g}-N+1$, and

$\Sigma^{2}=\widetilde{\Sigma}^{2}+1=-2N+1.$

It is easy to see that a divisor class $\Sigma$ is automatically effective as
soon as $\Sigma^{2}\geq-1$. From our point of view, we want to see when there
is an effective bisection $\Sigma$ with $\Sigma^{2}<-1$ as well. The above
shows that, as $N$ increases by $1$, $\Sigma^{2}$ drops by $2$ and
$p_{a}(\Sigma)$ by $1$. In particular, as $N$ increases from $1$ to $p_{g}+1$,
$\Sigma^{2}$ decreases from $-1$ to $-2p_{g}-1$ and $p_{a}(\Sigma)$ decreases
from $p_{g}$ to $0$. Note that the case $N=1$ is the generic case. More
generally, for $1\leq N\leq p_{g}+1$, the above constructs a stratum of the
moduli space which has codimension one in the stratum for $N-1$. It would be
interesting to find an intrinsic characterization of the strata.

In the cases of interest, $p_{g}=1$ or $0$. In case $p_{g}=1$, the only
possibilities for $\mathbb{F}_{N}$ are $N=1$, $k=3$, $\Sigma^{2}=-1$,
$p_{a}(\Sigma)=1$, or $N=2$, $k=4$, $\Sigma^{2}=-3$, $p_{a}(\Sigma)=0$ (and
hence $\Sigma$ is smooth). In case $p_{g}=0$, we must have $N=1$, $k=2$,
$\Sigma^{2}=-1$, and $p_{a}(\Sigma)=0$.

A similar description applies if we allow more complicated non-multiple
fibers. We shall just describe the following situation which will arise in
§3.4 (per the statement of Theorem 3.9 below):

######  Assumption 3.4.

The multiple fiber $F$ is irreducible, there is a nonmultiple fiber
$f_{t}\subseteq X$ which is a Ia fiber, $a\geq 3$ and the bisection $\Sigma$
meets two different components of $f_{t}$, necessarily transversally. Every
other nonmultiple fiber of $\pi$ has at most two components, and, in case
there are two components, $\Sigma$ has intersection number one with each
component. Finally, assume that $\Sigma$ is rational, so that the map
$\Sigma\to\mathbb{P}^{1}$ is branched at the point corresponding to $F$ and at
one other point.

Under this assumption, the involution $\iota$ then exchanges the two
components of $f_{t}$ meeting $\Sigma$, as well as all of the others except
possibly one or two “end” components where it induces a degree $2$ map onto a
quotient $\mathbb{P}^{1}$. If $\overline{X}$ is the surface obtained by
contracting all of the components of $f_{t}$ not meeting $\Sigma$ and
$\widetilde{\overline{X}}$ is the blowup at the two isolated fixed points on
$F$, $\iota$ induces a finite morphism
$\nu\colon\widetilde{\overline{X}}\to\widetilde{\mathbb{F}}_{N}$, where
$\widetilde{\mathbb{F}}_{N}$ is the blowup of $\mathbb{F}_{N}$ at the two
points corresponding to the isolated fixed points of $\iota$. Let
$\overline{B}_{0}$ be the branch locus of $\nu$. Then $\overline{B}_{0}$ has
two curve singularities of type $A_{k_{1}}$, $A_{k_{2}}$ lying over the point
$t\in\mathbb{P}^{1}$, with $k_{1},k_{2}\geq 1$ since $\Sigma$ meets two
distinct components of the fiber on $X$. An embedded resolution of the
singularities of $\overline{B}_{0}$ leads to a surface
$\widetilde{\widetilde{\mathbb{F}}}_{N}$ with two chains of rational curves,
of lengths $[k_{1}+1/2]$ and $[k_{2}+1/2]$ respectively. If $B_{0}$ is the
proper transform of $\overline{B}_{0}$, then $\widetilde{X}$ is the double
cover of $\widetilde{\widetilde{\mathbb{F}}}_{N}$ branched along
$B_{0}+d_{1}+d_{2}$, and the inverse images of the two chains of rational
curves on $\widetilde{\widetilde{\mathbb{F}}}_{N}$ are the minimal resolutions
of (surface) singularities of types $A_{k_{1}}$ and $A_{k_{2}}$. The fiber
over $t$ is a Ia fiber, with $a=k_{1}+k_{2}+2$.

In this case, we identify $\sigma_{0}$ with its the proper transform on
$\widetilde{\widetilde{\mathbb{F}}}_{N}$. Then
$\widetilde{\Sigma}=\nu^{*}\sigma_{0}$ and $\Sigma$ is the image of
$\widetilde{\Sigma}$ in $X$. In the application to the proof of Theorem 3.9,
$\Sigma$ will be a smooth rational curve with $\Sigma^{2}=-3$. Thus the degree
two morphism $\Sigma\to\mathbb{P}^{1}$ induced by $\pi$ is branched at
$\Sigma\cap F$ and at one other point. Hence $\sigma_{0}\cdot B_{0}=1$.
Moreover, $\overline{B}_{0}$ is a $4$-section, so that $f\cdot
B_{0}=f\cdot\overline{B}_{0}=4$. Finally, by construction $e\cdot
B_{0}=e\cdot\overline{B}_{0}=2$.

###  3.3. A vanishing theorem: the case where $D$ is irreducible

For the rest of this section, we assume that $\widetilde{Y}$ is the minimal
resolution of an I-surface with $\kappa(\widetilde{Y})=1$. In particular,
$\widetilde{Y}$ is minimal and either there exists an irreducible bisection
$D$ with $D^{2}=-1$ and $p_{a}(D)=1$ or there exists a smooth rational
bisection $E_{1}$ with $E_{1}^{2}=-3$. Thus we can apply the previous results
to $X=\widetilde{Y}$. First we note the following, recalling that Assumption
2.7, that there are no rational double points on $Y$, is always in force:

######  Lemma 3.5.

  1. (i)

If $D$ is irreducible, then $F$ is irreducible and every fiber of the morphism
$\widetilde{Y}\to\mathbb{P}^{1}$ has at most two components. In particular,
every non-multiple fiber is reduced, and is either a nodal or a cuspidal
rational curve, if irreducible, or an I2 fiber or a tacnodal fiber, if there
are two components.

  2. (ii)

If $D$ is reducible, then exactly one component of $D$ is an irreducible
bisection and the remaining components are contained in a single fiber
$f_{t}$. In this case, $F$ is irreducible and all nonmultiple fibers except
for $f_{t}$ have at most two components and are nodal, cuspidal, or a tacnode.
Moreover, all but at most two components of $f_{t}$ are contained in $D$.
Finally, if there are two such components of $f_{t}$, they meet at one point,
they are exchanged by the involution $\iota$ on $\widetilde{Y}$, and the point
where they meet is a fixed point of $\iota$.

###### Proof.

By assumption, if $\Gamma$ is an irreducible curve on $\widetilde{Y}$ and
$\Gamma$ is not a component of $D$, then $(F+D)\cdot\Gamma>0$. However, if
$\Gamma$ is a component of a reducible fiber, then $F\cdot\Gamma=0$ so that
$D\cdot\Gamma>0$. But $D\cdot F=1$ and $D\cdot f=2$, where $f$ is any non-
multiple fiber. It follows that $F$ has just one component and every non-
multiple fiber not containing a component of $D$ has at most two. By the
classification of singular fibers, every non-multiple fiber with at most two
components is nodal, cuspidal, or a tacnode, and in particular is reduced. If
$D$ is reducible, then exactly one component is a bisection, necessarily
smooth rational of square $-3$, and the remaining components therefore form a
chain of $-2$-curves. In particular, since the chain is connected and has
intersection number $0$ with a general fiber, it must be contained in a single
fiber $f_{t}$. Since $D\cdot f=2$ and $D\cdot C>0$ for every component of
$f_{t}$ which is not a component of $D$, it is clear that there can be at most
two such components. If there are two such, they form a connected component of
$f_{t}-D$ and the one or two other components which meet them are the
components meeting the bisection $E_{1}$. Thus they are exchanged by $\iota$
and their intersection point is a fixed point of $\iota$. ∎

In this subsection, we consider the irreducible case and show:

######  Theorem 3.6.

If $D$ is irreducible and there are no cusp or tacnodal fibers, then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$.

###### Proof.

Since
$K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D)=\mathcal{O}_{\widetilde{Y}}(F+D)$,
by duality it suffices to show that

$H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}\otimes
K_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(D))=H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}\otimes\mathcal{O}_{\widetilde{Y}}(F+D))=0.$

By Theorem 3.1, there is an exact sequence

$0\to\mathcal{O}_{\widetilde{Y}}(-2f+F)\to\Omega^{1}_{\widetilde{Y}}\to
I_{Z}\otimes\mathcal{O}_{\widetilde{Y}}(2f)\to 0,$

where $Z$ is the scheme defined by the singular points of the singular, non-
multiple fibers. Hence $\ell(Z)=24$ and by our assumptions $Z$ consists of
$24$ distinct points. Thus it suffices to show that

$H^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(-2f+2F+D))=H^{0}(\widetilde{Y};I_{Z}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+D))=0.$

First, since $-2f+2F+D=D-f$,
$H^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(-2f+2F+D))=0$ as $D-f$ is not
effective (because $D^{2}<0$). So it is enough to show that
$H^{0}(\widetilde{Y};I_{Z}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+D))=0$.
Referring to §3.2, there is a blowup $\widetilde{\widetilde{Y}}$ of
$\widetilde{Y}$ at two smooth points of $F$ and a degree $2$ morphism
$\nu\colon\widetilde{\widetilde{Y}}\to\widetilde{\mathbb{F}}_{1}$, branched
along a divisor in $\widetilde{\mathbb{F}}_{1}$ of the form
$B_{0}+d_{1}+d_{2}$, where $B_{0}\in|4\sigma_{0}+5f+2d_{2}|$. Here, if
$e_{1},e_{2}$ are the exceptional curves on $\widetilde{\widetilde{Y}}$ and
$F^{\prime}$ is the proper transform of the fiber, then $\nu^{*}\sigma_{0}=D$,
$\nu^{*}d_{i}=2e_{i}$, $\nu^{*}e=F^{\prime}$, and (using $f$ to denote a
general fiber of $\widetilde{\widetilde{Y}}\to\mathbb{P}^{1}$ as well as
$\widetilde{\mathbb{F}}_{1}\to\mathbb{P}^{1}$), $\nu^{*}f=f$. Also, we can
identify $Z$ with the corresponding subscheme of $\widetilde{\widetilde{Y}}$.
The total transform of $F$ on $\widetilde{\widetilde{Y}}$ is
$F^{\prime}+e_{1}+e_{2}$, and hence

$H^{0}(\widetilde{Y};I_{Z}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+D))=H^{0}(\widetilde{\widetilde{Y}};I_{Z}\otimes\mathcal{O}_{\widetilde{\widetilde{Y}}}(2f+F^{\prime}+e_{1}+e_{2}+D)).$

By the above remarks, since $\nu^{*}d_{i}=2e_{i}$, there is an inclusion

$\mathcal{O}_{\widetilde{\widetilde{Y}}}(2f+F^{\prime}+e_{1}+e_{2}+D)\subseteq\nu^{*}\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(2f+e+d_{1}+d_{2}+\sigma_{0}).$

In what follows, we will repeatedly use the following for divisor classes on
$\widetilde{\mathbb{F}}_{1}$:

$2f+e+d_{1}+d_{2}+\sigma_{0}=\sigma_{0}+3f-e.$

For every line bundle $M$ on $\widetilde{\mathbb{F}}_{1}$, we have
$\nu_{*}\nu^{*}M=M\oplus(M\otimes\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(-\frac{1}{2}B))$,
where $\frac{1}{2}B$ is the divisor $2\sigma_{0}+3f-e+d_{2}$. Applying this to
$\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(\sigma_{0}+3f-e)$, first note that

$\sigma_{0}+3f-e-(2\sigma_{0}+3f-e+d_{2})=-\sigma_{0}-d_{2}.$

Since this last divisor is not effective, every section of
$\nu^{*}\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(\sigma_{0}+3f-e)$ is the
pullback of a section $s$ of
$\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(\sigma_{0}+3f-e)$, and we need to
find sections vanishing on $\nu(Z)$. If $x\in Z$, then $x$ is singular point
of the fiber of $\widetilde{Y}\to\mathbb{P}^{1}$ which contains it. Viewing
$x$ as a point on $\widetilde{\widetilde{Y}}$, $x$ is the preimage of a point
on a fiber of $\widetilde{\mathbb{F}}_{1}$ which is tangent to $B_{0}$ and the
point $\nu(x)$ is a point of tangency. In particular, $\nu(Z)$ consists of
$24$ points and the section $s$ has to vanish at those points of $B_{0}$.
(This includes the case where $F$ is singular; since $B_{0}\cdot e=2$ and
$d_{1}$, $d_{2}$ meet $e$ transversally, $F$ is singular $\iff$ $B_{0}$ meets
$e$ doubly at a single point.) Note that

$(\sigma_{0}+3f-e)-B_{0}=-3\sigma_{0}-2f-2d_{2}-e$

is not effective, and so a section $s$ of
$\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(\sigma_{0}+3f-e)$ vanishing along
$B_{0}$ has to vanish identically. Conversely, if $s\neq 0$, then the curve
$(s)$ defined by $s$ satisfies: $\\#((s)\cap B_{0})$ is finite and $\geq 24$.
Hence $B_{0}\cdot(\sigma_{0}+3f-e)\geq 24$. But

$\displaystyle
B_{0}\cdot(\sigma_{0}+3f-e)=(4\sigma_{0}+5f+2d_{2})\cdot(\sigma_{0}+3f-e)$
$\displaystyle=-4+12+5+2-2=13<24.$

Thus, a section $\sigma$ of
$\mathcal{O}_{\widetilde{\mathbb{F}}_{1}}(\sigma_{0}+3f-e)$ vanishing at the
$24$ points of $\nu(Z)$ must be identically $0$. Hence
$H^{0}(\widetilde{Y};I_{Z}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+D))=0$, so
that $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$. ∎

######  Remark 3.7.

The proof still works if there are a small number of cuspidal or tacnodal
fibers. Concretely, let $c$ be the number of cuspidal fibers and $t$ the
number of tacnodal fibers. Then the above proof works as long as $c+2t\leq
10$. It is possible that this, or a more detailed analysis, is enough to
handle the general case.

######  Corollary 3.8.

Under the assumptions of Theorem 3.6, the deformations of $Y$ are versal for
the deformations of the simple elliptic singularity or cusp singularity, i.e.
the homomorphism $\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is surjective. ∎

###  3.4. A vanishing theorem: the case where $D$ is reducible

We prove an analogue of Theorem 3.6 in the case of a cusp singularity with
$r>1$, again under some mild general position assumptions.

######  Theorem 3.9.

Suppose that all fibers of $\widetilde{Y}$ are nodal. Let
$D=E_{1}+\cdots+E_{r}$ correspond to a cusp singularity, where $E_{1}$ is a
smooth bisection with $E_{1}^{2}=-3$ and the remaining components are fiber
components of the fiber $f_{t}$, with $r\leq 14$. Then
$H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$.

###### Proof.

The proof is similar to that of Theorem 3.6. We must show that
$H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}(\log D)\otimes
K_{\widetilde{Y}})=0$. Let
$\nu\colon\widetilde{\widetilde{Y}}\to\widetilde{\mathbb{F}}_{2}$ be the
double cover morphism of §3.2, where $\widetilde{\widetilde{\mathbb{F}}}_{2}$
is the blowup of $\mathbb{F}_{2}$ described at the end of §3.2. The exact
sequence of Theorem 3.2 gives an exact sequence

$0\to\mathcal{O}_{\widetilde{Y}}(-2f+2F+\sum_{i>1}E_{i})\to\Omega^{1}_{\widetilde{Y}}(\log
D)\otimes K_{\widetilde{Y}}\to
I_{W}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+E_{1})\to 0,$

with $\ell(W)=25-r$. Moreover, if $x\in\operatorname{Supp}W$, then either $x$
is a singular point of a fiber, not on $D$, or $x$ is a ramification point for
the morphism $E_{1}\to\mathbb{P}^{1}$ not on $F$. Since
$E_{1}\to\mathbb{P}^{1}$ is a double cover with $E_{1}$ a smooth rational
curve, the morphism $E_{1}\to\mathbb{P}^{1}$ ramifies at the intersection
$E_{1}\cap F$ and at one other point. If $x\in\operatorname{Supp}W$, $x$ is a
singular point of its fiber, and $x\notin D$, then $x\in B_{0}$ by Lemma
3.5(ii). If $x$ is the unique ramification point of $E_{1}\to\mathbb{P}^{1}$
not on $F$, then again $x\in B_{0}$, and $W$ has length at most $2$ at $x$.
Thus, if $\overline{W}=\nu(W)$, then $\overline{W}\subseteq B_{0}$ and

$\\#(\overline{W})\geq\ell(W)-1=24-r.$

As $-2f+2F+\sum_{i>1}E_{i}=-f+\sum_{i>1}E_{i}$ is the negative of an effective
divisor,

$H^{0}(\widetilde{Y};\mathcal{O}_{\widetilde{Y}}(-2f+2F+\sum_{i>1}E_{i}))=0.$

We must show that
$H^{0}(\widetilde{Y};I_{W}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+E_{1}))=0$
as well. It suffices to show that

$H^{0}(\widetilde{\widetilde{Y}};I_{W}\otimes\mathcal{O}_{\widetilde{\widetilde{Y}}}(2f+F^{\prime}+e_{1}+e_{2}+E_{1}))=0,$

which will follow if we show that

$H^{0}(\widetilde{\widetilde{Y}};I_{W}\otimes\nu^{*}\mathcal{O}_{\widetilde{\widetilde{\mathbb{F}}}_{2}}(2f+e+d_{1}+d_{2}+\sigma_{0}))=0.$

We again use the equality of divisor classes

$2f+e+d_{1}+d_{2}+\sigma_{0}=\sigma_{0}+3f-e.$

Arguing as in the proof of Theorem 3.6,

$H^{0}(\widetilde{\widetilde{Y}};\nu^{*}\mathcal{O}_{\widetilde{\widetilde{\mathbb{F}}}_{2}}(\sigma_{0}+3f-e))=H^{0}(\widetilde{\widetilde{\mathbb{F}}}_{2};\mathcal{O}_{\widetilde{\widetilde{\mathbb{F}}}_{2}}(\sigma_{0}+3f-e)).$

It suffices to show that
$H^{0}(\widetilde{\widetilde{\mathbb{F}}}_{2};I_{\overline{W}}\otimes\mathcal{O}_{\widetilde{\widetilde{\mathbb{F}}}_{2}}(\sigma_{0}+3f-e))=0$.
Suppose by contradiction that there is a nonzero section $s$ of
$\mathcal{O}_{\widetilde{\widetilde{\mathbb{F}}}_{2}}(\sigma_{0}+3f-e)$
vanishing along $\overline{W}$. As in the proof of Theorem 3.6, such a section
cannot vanish identically along $B_{0}$. Now we compute as before, using the
computations at the end of §3.2:

$B_{0}\cdot(\sigma_{0}+3f-e)=(B_{0}\cdot\sigma_{0})+3(B_{0}\cdot
f)-2(B_{0}\cdot e)=1+12-4=9.$

In particular, we must have $\\#(\overline{W})\leq 9$. But we have seen that
$\\#(\overline{W})\geq 24-r$, hence $r\geq 15$. Conversely, if $r\leq 14$,
then
$H^{0}(\widetilde{Y};I_{W}\otimes\mathcal{O}_{\widetilde{Y}}(2f+F+E_{1}))=0$.
Thus $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$. ∎

######  Corollary 3.10.

Under the assumptions of Theorem 3.9, the deformations of $Y$ are versal for
the deformations of the cusp singularity, i.e. the homomorphism
$\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is surjective. ∎

##  4\. The case $\kappa(\widetilde{Y})=0$

###  4.1. Some examples of lattices

We begin by defining various lattices which will arise in what follows.

######  Definition 4.1.

(i) Let $\Lambda_{0}(n)$ be the lattice with basis vectors
$e_{0},\dots,e_{n}$, satisfying: $e_{0}^{2}=0$, $e_{i}^{2}=-2$ for $i>0$,
$e_{i}\cdot e_{j}=0$ for $i\neq j\pm 1\pmod{n}$, and $e_{i}\cdot e_{i+1}=1$
(including the case $e_{n}\cdot e_{0}=1$).

(ii) Let $\Lambda_{1}(n,m)$ be the lattice spanned by vectors $e_{1},\dots
e_{n},f_{1},\dots,f_{m},g_{1},g_{2}$ with $e_{i}^{2}=f_{i}^{2}=g_{i}^{2}=-2$,
$e_{1}\cdot e_{2}=\dots=e_{n-1}\cdot e_{n}=1$, $f_{1}\cdot
f_{2}=\dots=f_{m-1}\cdot f_{m}=1$, $e_{1}\cdot g_{1}=f_{1}\cdot g_{1}=1$,
$e_{n}\cdot g_{2}=f_{m}\cdot g_{2}=1$, $g_{1}\cdot g_{2}=1$, and all other
intersections are $0$.

(iii) Let $\Lambda_{2}(n,m)$ be the lattice spanned by vectors
$e_{0},\dots,e_{n},f_{0},\dots,f_{m}$ with $e_{i}^{2}=f_{j}^{2}=-2$,
$e_{i}\cdot e_{j}=0$ for $i\neq j\pm 1\pmod{n}$, and $e_{i}\cdot e_{i+1}=1$
(including the case $e_{n}\cdot e_{0}=1$), similarly $f_{i}\cdot f_{j}=0$ for
$i\neq j\pm 1\pmod{m}$, and $f_{i}\cdot f_{i+1}=1$ (including the case
$f_{n}\cdot f_{0}=1$), $e_{0}\cdot f_{0}=1$, and $e_{i}\cdot f_{j}=0$ if
$(i,j)\neq(0,0)$.

A straightforward argument shows:

######  Lemma 4.2.

The lattices $\Lambda_{0}(n)$, $\Lambda_{1}(n,m)$, and $\Lambda_{2}(n,m)$ are
nondegenerate of signatures $(1,n)$, $(1,n+m+1)$, and $(1,n+m+1)$
respectively. ∎

###  4.2. A lemma on double covers

######  Lemma 4.3.

Suppose that $\nu\colon W\to Z$ is a double cover branched along $\Sigma$,
where $Z$, $W$, and $\Sigma$ are all smooth. Let $\lambda$ be the line bundle
defining the double cover, i.e. $\lambda^{\otimes
2}\cong\mathcal{O}_{Z}(\Sigma)$. Then there is an exact sequence

$0\to\Omega^{1}_{Z}\to\nu_{*}\Omega^{1}_{W}\to\Omega^{1}_{Z}(\log\Sigma)\otimes\lambda^{-1}\to
0.$

###### Proof.

Choose local holomorphic coordinates $z_{1},\dots,z_{n}$ near a point of
$\Sigma$ so that $\Sigma$ is defined by $z_{n}=0$ and the map $\nu$ is given
by $z_{i}=w_{1}$, $i<n$, and $z_{n}=w_{n}^{2}$, i.e. $w_{n}=\sqrt{z_{n}}$.
Thus $z_{n}$ is a local section of $\mathcal{O}_{Z}(-\Sigma)$ and
$\sqrt{z_{n}}$ is a local section of $L^{-1}$. Then $dz_{n}=2w_{n}\,dw_{n}$,
i.e. $\displaystyle dw_{n}=\frac{dz_{n}}{2\sqrt{z_{n}}}$. Thus a basis for
$\nu_{*}\Omega^{1}_{W}$ as an $\mathcal{O}_{Z}$-module is given locally by

$dz_{1},\sqrt{z_{n}}dz_{1},\dots,dz_{n-1},\sqrt{z_{n}}dz_{n-1},dz_{n},\frac{dz_{n}}{\sqrt{z_{n}}}.$

Hence the quotient of $\nu_{*}\Omega^{1}_{W}$ by the subspace $\Omega^{1}_{Z}$
has as a basis

$\sqrt{z_{n}}dz_{1},\dots,\frac{dz_{n}}{\sqrt{z_{n}}}=\sqrt{z_{n}}\cdot\frac{dz_{n}}{z_{n}}.$

This says invariantly that the quotient is
$\Omega^{1}_{Z}(\log\Sigma)\otimes\lambda^{-1}$. ∎

######  Remark 4.4.

In fact, the natural involution on $\nu_{*}\Omega^{1}_{W}$ coming from the
double cover involution induces a splitting

$\nu_{*}\Omega^{1}_{W}\cong\Omega^{1}_{Z}\oplus\Big{(}\Omega^{1}_{Z}(\log\Sigma)\otimes\lambda^{-1}\Big{)}.$

###  4.3. The case where $\widetilde{Y}$ is the blowup of a $K3$ surface

Throughout this subsection, we assume that the minimal model $Y_{0}$ of
$\widetilde{Y}$ is a (smooth) $K3$ surface. By Theorem 2.12(iii), there exists
a morphism $\rho\colon\widetilde{Y}\to Y_{0}$, where $\rho$ is the blowup of
$Y_{0}$ at a single point $p\in Y_{0}$, and the exceptional curve $C$ on
$\widetilde{Y}$ satisfies: $C\cdot D=2$. In particular,
$K_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(C)$.

First, we consider the case where $D$ is irreducible, and is thus either an
elliptic curve or a nodal rational curve.

######  Theorem 4.5.

If $D$ is irreducible, then $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))=0$.
Hence the homomorphism $\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is
surjective.

###### Proof.

With notation as above, let $\overline{D}=\rho(D)$ be the image of $D$ in
$Y_{0}$. Then $\overline{D}$ is an irreducible curve with either a node or a
cusp, and $p_{a}(\overline{D})=2$. Thus $\overline{D}^{2}=2$ and the complete
linear system $|\overline{D}|$ defines a morphism $\varphi\colon
Y_{0}\to\mathbb{P}^{2}$, for which
$\mathcal{O}_{Y_{0}}(\overline{D})=\varphi^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$.
By hypothesis, $L=\mathcal{O}_{\widetilde{Y}}(D+C)$ induces an ample divisor
on $Y$. Thus, if $\Gamma$ is an irreducible curve in $\widetilde{Y}$ not equal
to $C$ or $D$, then either $D\cdot\Gamma>0$ or $C\cdot\Gamma>0$, so for every
irreducible curve $\Gamma_{0}$ on $Y_{0}$,

$\Gamma_{0}\cdot\overline{D}=\Gamma_{0}\cdot\rho_{*}D=\rho^{*}\Gamma_{0}\cdot(D+2C)>0.$

Hence $\varphi\colon Y_{0}\to\mathbb{P}^{2}$ is a finite degree $2$ morphism
and its branch locus is a smooth sextic curve $\Sigma\subseteq\mathbb{P}^{2}$.
Thus the square root $\lambda$ of $\mathcal{O}_{\mathbb{P}^{2}}(\Sigma)$ is
$\mathcal{O}_{\mathbb{P}^{2}}(3)$.

By Serre duality, $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D))$ is dual to
$H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}(D)\otimes\mathcal{O}_{\widetilde{Y}}(C))$.
A section of
$\Omega^{1}_{\widetilde{Y}}(D)\otimes\mathcal{O}_{\widetilde{Y}}(C)$ defines a
section of $\Omega^{1}_{Y_{0}}(\overline{D})$ over $Y_{0}-\\{p\\}$ which then
extends to a holomorphic section of $\Omega^{1}_{Y_{0}}(\overline{D})$ over
$Y_{0}$ by Hartogs’ theorem. Thus it will suffice to prove that
$H^{0}(Y_{0};\Omega^{1}_{Y_{0}}(\overline{D}))=0$. Equivalently, since
$\mathcal{O}_{Y_{0}}(\overline{D})=\varphi^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$,
it suffices to show that
$H^{0}(\varphi_{*}\Omega^{1}_{Y_{0}}\otimes\mathcal{O}_{\mathbb{P}^{2}}(1))=0$.
By Lemma 4.3, since $\lambda^{-1}=\mathcal{O}_{\mathbb{P}^{2}}(-3)$, it is
enough to show that

$H^{0}(\mathbb{P}^{2};\Omega^{1}_{\mathbb{P}^{2}}\otimes\mathcal{O}_{\mathbb{P}^{2}}(1))=H^{0}(\mathbb{P}^{2};\Omega^{1}_{\mathbb{P}^{2}}(\log\Sigma)\otimes\mathcal{O}_{\mathbb{P}^{2}}(-2))=0.$

The fact that
$H^{0}(\mathbb{P}^{2};\Omega^{1}_{\mathbb{P}^{2}}\otimes\mathcal{O}_{\mathbb{P}^{2}}(1))=0$
is well-known and follows from looking at the global sections coming from the
Euler exact sequence

$0\to\Omega^{1}_{\mathbb{P}^{2}}\otimes\mathcal{O}_{\mathbb{P}^{2}}(1)\to\left(\mathcal{O}_{\mathbb{P}^{2}}\right)^{3}\to\mathcal{O}_{\mathbb{P}^{2}}(1)\to
0.$

As for the remaining equality
$H^{0}(\mathbb{P}^{2};\Omega^{1}_{\mathbb{P}^{2}}(\log\Sigma)\otimes\mathcal{O}_{\mathbb{P}^{2}}(-2))=0$,
the Poincaré residue sequence gives

$0\to\Omega^{1}_{\mathbb{P}^{2}}(-2)\to\Omega^{1}_{\mathbb{P}^{2}}(\log\Sigma)(-2)\to\mathcal{O}_{\Sigma}(-2)\to
0.$

On global sections, $H^{0}(\mathbb{P}^{2};\Omega^{1}_{\mathbb{P}^{2}}(-2))=0$
via the Euler exact sequence, and $H^{0}(\mathcal{O}_{\Sigma}(-2))=0$ because
$\mathcal{O}_{\mathbb{P}^{2}}(-2)|\Sigma$ is a line bundle of negative degree.
Thus $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-D)=0$ as claimed. ∎

In case $D$ is reducible, we have the following which uses a mild general
position assumption.

######  Theorem 4.6.

Suppose that $D$ is reducible, and assume that, if there is a component $E$ of
$D$ of self-intersection $-4$, then the exceptional curve $C$ meets $E$ in $2$
distinct points. Then $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))=0$, and
hence the homomorphism $\mathbb{T}^{1}_{Y}\to H^{0}(Y;T^{1}_{Y})$ is
surjective.

###### Proof.

Since all components of $D$ are smooth rational, $C$ is disjoint from every
component of square $-2$. If there is a component $E$ of self-intersection
$-4$, then $C\cdot E=2$, and if there are two components of square $-3$, then
$C$ meets each of them transversally at one point. Thus, the hypothesis of the
theorem is exactly that the curve $C+D$ is nodal on $\widetilde{Y}$, or
equivalently that $\overline{D}=\rho(D)$ is nodal.

The group $H^{2}(\widetilde{Y};T_{\widetilde{Y}}(-\log D))$ is dual to
$H^{0}(\widetilde{Y};\Omega^{1}_{\widetilde{Y}}(\log
D)\otimes\mathcal{O}_{\widetilde{Y}}(C))$. As in the proof of Theorem 4.5, it
suffices to prove that $H^{0}(Y_{0};\Omega^{1}_{Y_{0}}(\log\overline{D}))=0$.
Thus it suffices to prove that the fundamental classes of the components of
$\overline{D}$ are independent in cohomology. This follows from Lemma 4.2. ∎

######  Corollary 4.7.

Suppose that there is a component $E$ of $D$ of self-intersection $-4$ with
$r\geq 2$ or two components of self-intersection $-3$ with $r\geq 3$. Then
there is a deformation of $Y$ to an I-surface $Y_{t}$ with a cusp or simple
elliptic singularity of multiplicity $1$. Hence the minimal resolution
$\widetilde{Y}_{t}$ is a surface with $\kappa(\widetilde{Y}_{t})=1$.

###### Proof.

This follows immediately from Theorem 4.6 and Theorem 1.5(iii). ∎

Before we turn to an example showing that there are indeed examples satisfying
the hypothesis of Corollary 4.7, we note the following standard result:

######  Lemma 4.8.

Let $X$ be a $K3$ surface and let $C_{1},\dots,C_{k}$ be smooth rational
curves on $X$ meeting transversally whose fundamental classes are linearly
independent in cohomology. Let $C=\sum_{i}C_{i}$. Then $H^{2}(X;T_{X}(-\log
C))=0$ and hence the map
$H^{1}(X;T_{X})\to\bigoplus_{i}H^{1}(C_{i};N_{C_{i}/X})$ is surjective.

###### Proof.

By Serre duality, we have to show that $H^{0}(X;\Omega^{1}_{X}(\log C))=0$,
which is an easy consequence of the Poincaré residue sequence. ∎

######  Example 4.9.

Let $Q$ be a smooth plane quartic curve and let $L$ be a tangent line to $Q$
which is not a bitangent, so that $L$ meets $Q$ doubly at $p_{0}$ and
transversally at $p_{1},p_{2}$. Then there exists a conic $C$ which passes
through $p_{1}$ and $p_{2}$ and meets $Q$ transversally at $6$ remaining
points $p_{3},\dots,p_{8}$. If $Y_{0}$ is the minimal resolution of the double
cover of $\mathbb{P}^{2}$ branched along $Q+C$, then $X$ contains $8$ $(-2)$
curves $E_{1},\dots,E_{8}$ corresponding to the $8$ points of $Q\cap C$. The
line $L$ splits into two $(-2)$-curves $G_{2},G_{2}$, which are disjoint from
$E_{3},\dots,E_{8}$. Moreover $G_{1}\cdot G_{2}=1$, corresponding to the point
$p_{0}$, $G_{i}\cdot E_{1}=G_{i}\cdot E_{2}=1$, and, for $i=1,2$, the
intersection points $G_{1}\cap E_{i}$ and $G_{2}\cap E_{i}$ are distinct since
$L$ is not tangent to $Q$ or $C$ at $p_{i}$.

Blowing up $Y_{0}$ at the point $G_{1}\cap G_{2}$ yields a surface
$\widetilde{Y}$ which is the minimal resolution of an I-surface $Y$ with a
cusp singularity with intersection sequence $(-3,-2,-3,-2)$ as well as an
exceptional curve $C$ and $6$ ordinary double points corresponding to the
curves $E_{3},\dots,E_{8}$. Here $L=G_{1}+G_{2}+E_{1}+E_{2}+C$, so that $L$ is
nef and big and contracts the resolution $D=G_{1}+G_{2}+E_{1}+E_{2}$ as well
as the curves $E_{3},\dots,E_{8}$. Using Lemma 4.8, we can deform $Y_{0}$ and
hence $\widetilde{Y}$ keeping the same configuration for the curves
$E_{1},E_{2},G_{1},G_{2}$ but losing the remaining curves $E_{3},\dots,E_{8}$.
The resulting I-surface $Y_{t}$ will have a cusp singularity with intersection
sequence $(-3,-2,-3,-2)$ but no rational double point singularities.

######  Remark 4.10.

Let $\Lambda_{1}(n,m)$ be the lattice of Definition 4.1(ii). If $n+m$ is not
too large, there is a primitive embedding of $\Lambda_{1}(n,m)$ in the $K3$
lattice. Thus there exist $K3$ surfaces whose Picard lattice is isomorphic to
$\Lambda_{1}(n,m)$. The key question is whether the classes corresponding to
$e_{i},f_{i},g_{1},g_{2}$ are represented by irreducible curves. By standard
arguments, this reduces to the following question: Is every $\alpha\in\Lambda$
with $\alpha^{2}=-2$ in the $W$-orbit of $e_{1}$, where $W$ is the reflection
group generated by the classes $e_{i},f_{i},g_{1},g_{2}$?

######  Remark 4.11.

Suppose that $E$ is irreducible of self-intersection $-4$ as above. The
condition in Theorem 4.6 that the exceptional curve $C$ meets $E$ in $2$
distinct points is exactly the condition that, on the $K3$ surface $Y_{0}$,
the irreducible curve $\overline{E}=\rho(E)$ of arithmetic genus one has a
node, not a cusp. If instead $\overline{E}$ has a cusp, it is likely that a
small deformation of $Y_{0}$, keeping the curve $\overline{E}$ as well as the
remaining components of $\overline{D}$ as effective divisors, will have the
property that $\overline{E}$ is nodal. Compare [CGL22] for related results.

###  4.4. A remark on semistable smoothings

The method of proof above can be applied to show the existence of semistable
smoothings. This picture is essentially the semistable version of one of
Wahl’s “exotic” adjacencies (Theorem 1.5(iii)(b)). First we recall the
terminology of [Fri83b]:

######  Definition 4.12.

Let $Z=Z_{1}\cup_{D}Z_{2}$ be the simple normal crossing surface obtained by
gluing the two smooth surfaces $Z_{1}$ and $Z_{2}$ along a smooth divisor
$D\subseteq Z_{i}$, where $D\subseteq Z_{1}$ is identified with $D\subseteq
Z_{2}$ by the choice of an isomorphism $\gamma$. Then $Z$ is $d$-semistable if
$N_{D/Z_{1}}\otimes\gamma^{*}N_{D/Z_{2}}\cong\mathcal{O}_{D}$.

######  Theorem 4.13.

Let $X$ be a $K3$ surface with an irreducible nodal rational curve
$\overline{D}$ with $p_{a}(\overline{D})=1$, so that $\overline{D}^{2}=0$. Let
$\widetilde{X}$ be the blowup of $X$ at the singular point of $\overline{D}$
and let $D$ be the proper transform of $\overline{D}$, so that $D^{2}=-4$.
Then:

  1. (i)

$H^{2}(\widetilde{X};T_{\widetilde{X}}(-\log D))=0$.

  2. (ii)

The $d$-semistable surface $Z$ obtained by gluing $\widetilde{X}$ to
$\mathbb{P}^{2}$ by identifying $D$ with a smooth conic in $\mathbb{P}^{2}$ is
smoothable, and its smoothings are elliptic surfaces with $p_{g}=1$ and a
single multiple fiber, of multiplicity $2$.

###### Proof.

(i) The arguments used in the proof of Theorem 4.6 show that the dual of
$H^{2}(\widetilde{X};T_{\widetilde{X}}(-\log D))$ injects into
$H^{0}(X;\Omega^{1}_{X}(\log\overline{D}))$, and this is $0$ as the
fundamental class of $\overline{D}$ is nonzero in $H^{1}(X;\Omega^{1}_{X})$.

(ii) Let $\nu\colon\widetilde{X}\amalg\mathbb{P}^{2}\to Z$ be the
normalization. By a local calculation (cf. [FL23, Lemma 4.6(iv)], there is an
exact sequence

$0\to T^{0}_{Z}\to\nu_{*}(T_{\widetilde{X}}(-\log D)\oplus
T_{\mathbb{P}^{2}}(-\log D))\to i_{*}T_{D}\to 0,$

where $i\colon D\to Z$ is the inclusion. Then $H^{1}(D;T_{D})=0$ since
$D\cong\mathbb{P}^{1}$, $H^{2}(\widetilde{X};T_{\widetilde{X}}(-\log D))=0$ by
(i), and $H^{2}(\mathbb{P}^{2};T_{\mathbb{P}^{2}}(-\log D))=0$ by an easy
direct argument. Thus $H^{2}(Z;T^{0}_{Z})=0$. By construction,
$T^{1}_{Z}=\mathcal{O}_{D}$. Since $D\cong\mathbb{P}^{1}$,
$H^{1}(Z;T^{1}_{Z})=H^{1}(D;\mathcal{O}_{D})=0$. Thus $\mathbb{T}_{Z}^{2}=0$
and $\mathbb{T}_{Z}^{1}\to H^{0}(Z;T^{1}_{Z})=H^{0}(Z;\mathcal{O}_{D})$ is
surjective. In particular, there is a deformation of $Z$ over the disk whose
image in $H^{0}(Z;T^{1}_{Z})$ is an everywhere generating section. By [Fri83b,
Proposition 2.5], $Z$ is smoothable. We omit the straightforward argument that
the smoothings are as claimed. ∎

######  Remark 4.14.

(i) The point in Wahl’s construction is that the deformation space of the
rational singularity whose minimal resolution is a smooth rational curve of
self-intersection $-4$ has two different components, one corresponding to
rational ruled surfaces of degree $4$ in $\mathbb{P}^{5}$ and the other to the
Veronese surface (the “non-Artin component”).

(ii) A slightly weaker version of Theorem 4.13 is already present in [Fri83a,
Theorem 3.1]. Here, the result is stated under the somewhat loosely worded
hypothesis that “the number of moduli of $\widetilde{X}$ keeping the curve $D$
is $-\chi(T_{\widetilde{X}})-3$,” i.e. it is exactly $3$ conditions in moduli
to keep the smooth rational curve $D$ with self-intersection $-4$. For the
blown up $K3$ surface $\widetilde{X}$, there are $22$ moduli: $20$ for the
$K3$ and $2$ for the point to blow up. To keep the curve $D$ on the blowup, it
is one condition to keep the elliptic pencil defined by the nodal curve
$\overline{D}$ (which will necessarily have nodal singular fibers near
$\overline{D}$ and then two conditions because the point to blow up has to be
the singular point for a nearby fiber. Of course, a more convincing way to
express the above condition about the number of moduli is to note that, since
$H^{2}(\widetilde{X};T_{\widetilde{X}}(-\log D))=0$, the map
$H^{1}(\widetilde{X},T_{\widetilde{X}})\to H^{1}(D;N_{D/\widetilde{X}})$ is
surjective. As $\dim H^{1}(D;N_{D/\widetilde{X}})=\dim
H^{1}(\mathbb{P}^{1};\mathcal{O}_{\mathbb{P}^{1}}(-4))=3$, it follows that
$\dim H^{1}(\widetilde{X},T_{\widetilde{X}}(-\log D))=\dim
H^{1}(\widetilde{X},T_{\widetilde{X}})-3$.

(iii) Following the method of [Fri83a], one can explicitly construct
degenerations of an elliptic surface with $\kappa=1$ to the normal crossing
surface $Z$ by degenerating the branch locus.

(iv) The degeneration of a $\kappa=1$ elliptic surface with a multiplicity $2$
fiber to the union of a blown up $K3$ surface and a $\mathbb{P}^{2}$ glued
along a conic has appeared previously in the literature, cf. [Usu89].

###  4.5. Blown up Enriques surfaces

In this subsection, we assume that the minimal model $Y_{0}$ of
$\widetilde{Y}$ is a (smooth) Enriques surface. By Theorem 2.12(iii), there
exists a morphism $\rho\colon\widetilde{Y}\to Y_{0}$, where $\rho$ is the
blowup of $Y_{0}$ at a single point $p\in Y_{0}$, and the exceptional curve
$C$ on $\widetilde{Y}$ satisfies: $C\cdot D=2$. In particular,
$K_{\widetilde{Y}}\equiv\mathcal{O}_{\widetilde{Y}}(C)$. If $D_{1}$ and
$D_{2}$ are the two connected components of $D$, then $C\cdot D_{1}=C\cdot
D_{2}=1$. Thus, if $\overline{D}_{1}$ and $\overline{D}_{2}$ are images of
$D_{1}$ and $D_{2}$ in $Y_{0}$, then $\overline{D_{1}}$ and $\overline{D_{2}}$