# Exotic surfaces

Javier Reyes<EMAIL_ADDRESS>Department of Mathematics, University of
Maryland, 4176 Campus Drive - William E. Kirwan Hall, College Park, USA.  and
Giancarlo Urzúa<EMAIL_ADDRESS>Facultad de Matemáticas, Pontificia
Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna
4860, Santiago, Chile.

###### Abstract.

We construct the first exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ by
means of rational blowdowns. Similarly, we construct the first exotic
$3{\mathbb{C}}{\mathbb{P}}^{2}\allowbreak\\#\allowbreak
b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ for $b^{-}=9,8,7$. All of them
are minimal and symplectic, as they are produced from projective surfaces $W$
with Wahl singularities, and $K_{W}$ big and nef. In more generality, we
elaborate on the problem of finding exotic
$(2\chi({\mathcal{O}}_{W})-1){\mathbb{C}}{\mathbb{P}}^{2}\\#(10\chi({\mathcal{O}}_{W})-K^{2}_{W}-1)\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
from these Kollár–Shepherd-Barron–Alexeev surfaces $W$, obtaining explicit
geometric obstructions.

##  1\. Introduction

Let $b^{+},b^{-}$ be positive integers. An exotic
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
is a differentiable 4-manifold that is homeomorphic but not diffeomorphic to
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$.
There are many examples and constructions of exotic
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$,
with a particular emphasis on $b^{+}=1$, i.e. exotic blow-ups of
${\mathbb{P}}_{{\mathbb{C}}}^{2}$ (see e.g. [K89], [P05], [SS05], [PSS05],
[FS06], [P07], [A08], [AP08], [ABP08], [AP10], [FS11], [BK17], [BKS22]; with a
complex structure see e.g. [LP07], [PPS09a], [PPS09b], [PPS13], [RU21]).
Several of these constructions involve the rational blowdown technique of
Fintushel and Stern [FS97], generalized by Park [P97] which, in the context of
algebraic geometry, corresponds to ${\mathbb{Q}}$-Gorenstein smoothings of
Wahl singularities [W81, LW86, SSW08]. The rational blowdown surgery is
realized along the exceptional divisor of the minimal resolution of a Wahl
singularity. This divisor is called a Wahl chain, and it is a linear
configuration of ${\mathbb{P}}_{{\mathbb{C}}}^{1}$’s. Due to Symington [Sy98,
Sy01], the rational blowdown of a disjoint collection of Wahl chains on a
complex surface admits a symplectic structure.

Exotic
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
with small $b^{+}$ and $b^{-}$ (i.e. small topological Euler characteristic)
tend to be difficult to find, and it seems that there might not be a
satisfactory explanation for this. Constructions of exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
do exist for every $b^{-}\geq 2$, but it is an open question the existence of
an exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$.
Similarly, there are minimal symplectic examples of
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
for any $b^{-}\geq 4$, but it seems to be unknown for $b^{-}=3,2,1$. In these
two families, one can verify that the number of examples decreases rapidly as
$b^{-}$ gets smaller. If we only consider constructions for $b^{+}=1$ where
the rational blowdown is applied, then there are very few exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#5\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
[PPS09b, BKS22], and no examples of exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
for $b^{-}\leq 4$. For instance, in [BKS22, Question 2] it is asked: Is there
an exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
with $b^{-}<5$ that can be obtained from a standard rational surface via
rational blowdowns? If so, what is the smallest such $m$? In this paper we
show that $b^{-}=4$ is possible.

######  Theorem 1.1.

There exists a minimal, symplectic, and exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
which is constructed via rational blowdown from a rational surface.

Moreover, and in more generality, in this paper we elaborate on all the
details involved in the construction of such exotic 4-manifolds. Hopefully,
this can give us hints on why small exotic 4-manifolds are hard to find, and
how to deal with the corresponding difficulties. We study this problem in
connection with the Kollár–Shepherd-Barron–Alexeev (KSBA) surfaces, which
compactify the Gieseker moduli space of surfaces of general type [KSB88, K23].
To explain this connection concretely, let $X$ be a nonsingular projective
surface with a collection of disjoint Wahl chains. There exists a contraction
$X\to W$ of these Wahl chains, which produces a normal projective surface $W$
with only Wahl singularities [Art62]. In Theorem 2.3 and Corollary 2.5 we
prove a slightly more general version of the following.

######  Theorem 1.2.

Assume that $K_{W}$ is ample (i.e. $W$ is a KSBA surface). Then the rational
blowdown $Y$ of the Wahl chains in $X$ is a minimal symplectic 4-manifold. In
addition, if $Y$ is simply-connected and has an odd intersection form, then
$Y$ is an exotic
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
where $b^{+}=2\chi({\mathcal{O}}_{W})-1$, and
$b^{-}=10\chi({\mathcal{O}}_{W})-K^{2}_{W}-1>0$.

The possibilities for $X$ are limited. It must have irregularity equal to
zero, and it is birational to either ${\mathbb{P}}_{{\mathbb{C}}}^{2}$, a K3
surface, an Enriques surface, a proper elliptic surface, or some surface of
general type (see Proposition 3.1). Moreover, due to the positivity of
$K_{W}$, if $\sigma\colon X\to Z$ is some composition of blow-downs of
$(-1)$-curves, then the image under $\sigma$ of the collection of Wahl chains
is a non-empty configuration of rational curves. Thus, the big goal is to
identify conditions over such configurations in $Z$ that can guarantee the
existence of $W$.

We take the point of view of geography of configurations of curves, and the
invariants we study are their logarithmic Chern numbers. As the singularities
of a configuration of curves can become too complicated, we restrict our
configurations to the case of nodal rational curves with simple crossings
singularities (Definition 3.2). The study of these configurations, their log
Chern numbers, and the connection with the existence of $W$ is in Section 3,
whose main result is Theorem 3.8. It can be summarized as follows.

######  Theorem 1.3.

Let $W$ be a normal projective surface with $\ell$ Wahl singularities, and
$K_{W}$ big and nef. Let $\pi\colon X\to W$ be its minimal resolution. Assume
that there is a birational morphism $\sigma\colon X\to Z$ such that the image
under $\sigma$ of the Wahl chains is a nodal simple crossings configuration of
$r$ rational curves. Then,

$K_{W}^{2}\leq
12(1+p_{g}(Z))-\frac{1}{3}K_{Z}^{2}-\frac{2}{3}\ell-\frac{1}{3}r+\frac{1}{3}\sum_{k\geq
3}(k-2)t_{k}^{0}.$

Here $t_{k}^{0}$ is the number of exceptional curves over k-points with $k>2$
which are not part of the Wahl chains.

For example, if $Z={\mathbb{P}}_{{\mathbb{C}}}^{2}$, then $K_{W}^{2}\leq
9-\frac{2}{3}\ell-\frac{1}{3}r+\frac{1}{3}\sum_{k\geq 3}(k-2)t_{k}^{0}$.

Another way of obtaining constraints is via the ${\mathbb{Q}}$-Gorenstein
deformation space of $W$. To explain this briefly, let $T_{W}$ be the tangent
sheaf
$\mathcal{H}\text{om}_{{\mathcal{O}}_{W}}(\Omega_{W}^{1},{\mathcal{O}}_{W})$,
where $\Omega_{W}^{1}$ is the sheaf of differentials on $W$. Assume that $W$
is a normal projective surface with $\ell$ Wahl singularities, and $K_{W}$
big. Then one can show that (see Theorem 4.1)

$h^{2}(T_{W})=\ell+h^{1}(T_{W})+2K_{W}^{2}-10\chi({\mathcal{O}}_{W}).$

This dimension measures obstructions to deform $W$. If $h^{2}(T_{W})=0$, then
$W$ admits ${\mathbb{Q}}$-Gorenstein smoothings, and so the rational blowdown
$Y$ has a complex structure. Therefore if we have
$10\chi({\mathcal{O}}_{W})\leq 2K_{W}^{2}$, then $h^{2}(T_{W})>0$ and the
configuration of rational curves in $Z$ produces obstructions to deform $W$.
We elaborate on ${\mathbb{Q}}$-Gorenstein deformations of $W$ in Section 4.
For one of the constructions of an exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ we
compute that $H^{1}(W,T_{W})=0$ (so $W$ is equisingularly rigid), and
$H^{2}(T_{W})={\mathbb{C}}^{2}$ (see Proposition 5.4). We do not know if this
surface $W$, and any of the examples in this paper, admits a complex
structure.

Similarly, but starting with a particular configuration of rational curves in
a special K3 surface, we prove in Section 6 the following.

######  Theorem 1.4.

There exist minimal, symplectic, and exotic
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
for $b^{-}=9,8,7$ which are constructed via rational blowdown from a
composition of blow-ups starting with a K3 surface.

We point out that we have constructed many different surfaces $W$ that can be
used to produce small exotic 4-manifolds, and in this paper we present some of
them. For example, there are hundreds of surfaces $W$ that yield exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#5\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$, for
${\mathbb{C}}{\mathbb{P}}^{2}\\#6\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ we
can produce thousands, and so on. We may try to describe all of them in some
future work. In this paper, we exhibit 17 distinct surfaces $W$ to prove our
main theorems. The problem of finding the appropriate configuration of
rational curves involves a computer program, written from scratch in C++ by
the first-named author as part of his master’s thesis at UC Chile. This
program was fed with the data of configuration of curves to systematically
search for special Wahl chains among all possibilities. It was run on 80 cores
at the MiDaS cluster at UC. An interesting combinatorial problem here is: How
to classify the configurations of rational curves that produce KSBA surfaces
$W$?

### Acknowledgments

We thank R. İnanç Baykur, Jonny Evans, Paul Hacking, Pedro Montero, and Jenia
Tevelev for valuable discussions. The first-named author was funded by the
ANID scholarship 22201484. The second-named author was supported by the
FONDECYT regular grant 1190066. This work was also supported by ANID -
Millennium Science Initiative Program - grant NCN$17\textunderscore\,059$
“Millenium Nucleus Center for the Discovery of Structures in Complex Data”
MiDas.

###### Contents

  1. 1 Introduction
  2. 2 KSBA surfaces and exotic $4$-manifolds
  3. 3 KSBA surfaces and geography of rational configurations
  4. 4 Obstructions to complex smoothings
  5. 5 Exotic ${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
  6. 6 Exotic $3{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ for $b^{-}=9,8,7$

##  2\. KSBA surfaces and exotic $4$-manifolds

Let $0<q<m$ be coprime integers. A cyclic quotient singularity is a
singularity locally isomorphic to the germ at $(0,0)$ of the quotient of
${\mathbb{C}}^{2}$ by the action of $(x,y)\mapsto(\mu x,\mu^{q}y)$, where
$\mu$ is an $m$-th primitive root of $1$. The notation for this singularity is
$\frac{1}{m}(1,q)$.

Any normal two-dimensional singularity $(P\in W^{\prime})$ admits a minimal
resolution, this is, a birational morphism $\pi\colon X\to W^{\prime}$ such
that $X$ is nonsingular, the pre-image of $P$ is a collection of exceptional
curves $\operatorname{Exc}(\pi)$, none of them being a $(-1)$-curve 111In a
nonsingular surface a $(-p)$-curve is a ${\mathbb{P}}_{{\mathbb{C}}}^{1}$
whose self-intersection is equal to $-p$., and
$\pi|_{X\setminus\operatorname{Exc}(\pi)}$ is an isomorphism. If $(P\in
W^{\prime})$ is a cyclic quotient singularity of type $\frac{1}{m}(1,q)$, then
$\operatorname{Exc}(\pi)$ is a chain of ${\mathbb{P}}^{1}_{{\mathbb{C}}}$’s
$E_{1},\ldots,E_{s}$ such that $E_{i}^{2}=-e_{i}\leq-2$, where the $e_{i}$ are
obtained through the associated Hirzebruch-Jung continued fraction
$\frac{m}{q}=[e_{1},\ldots,e_{s}]$. We define its length as $s$. We have the
numerical relation
$K_{X}\equiv\pi^{*}(K_{W^{\prime}})+\sum_{i=1}^{s}d_{i}E_{i}$, where
$-1<d_{i}\leq 0$ are the discrepancies of the exceptional curves.

######  Definition 2.1.

A T-singularity is a cyclic quotient singularity of type
$\frac{1}{dn^{2}}(1,dna-1)$ where $d\geq 1$, and $0<a<n$ with gcd$(n,a)=1$. A
Wahl singularity is a T-singularity with $d=1$. Its exceptional divisor in its
minimal resolution will be called Wahl chain.

######  Remark 2.2.

Any T-singularity or Du Val singularity $(P\in W^{\prime})$ can be partially
resolved $\phi\colon W\to W^{\prime}$ so that $\phi^{*}(K_{W^{\prime}})\equiv
K_{W}$. We call them M-resolutions [BC94]. For a Du Val singularity, its
M-resolution is just the minimal resolution. For a T-singularity
$\frac{1}{dn^{2}}(1,dna-1)$, its M-resolution has as exceptional divisor a
chain of $d-1$ ${\mathbb{P}}_{{\mathbb{C}}}^{1}$’s passing through $d$ Wahl
singularities $\frac{1}{n^{2}}(1,na-1)$.

T-singularities and Du Val singularities are key to describing all
deformations of an arbitrary quotient singularity by means of its
P-resolutions. This was approached in the foundational work of Kollár and
Shepherd-Barron [KSB88], where a compactification for the moduli space of
surfaces of general type is introduced, nowadays known as the KSBA
compactification. Earlier than that work, Wahl introduces T-singularities in
[W81] as examples admitting a smoothing with constant $K^{2}$, and in [LW86,
Proposition 5.9] Looijenga and Wahl show that T-singularities (and Du Val type
A singularities) are the only cyclic quotient singularities admitting a
smoothing with that property. In [KSB88] it is proved that T-singularities and
Du Val singularities are precisely the quotient singularities that admit a
${\mathbb{Q}}$-Gorenstein smoothing, this is, a deformation over a disk with
nonsingular general fibers so that the underlying 3-fold has a
${\mathbb{Q}}$-Cartier canonical class.

One way to construct new complex surfaces is by considering
${\mathbb{Q}}$-Gorenstein smoothings of a surface $W$ with only Wahl
singularities. This approach has been successful in finding new surfaces of
general type with particular properties (see e.g. [LP07], [PPS09a], [PPS09b]).
To guarantee the existence of a smoothing, it is typically shown that there a
no local-to-global obstructions to deforming $W$. This is a strong condition
for surfaces in general, where it is not easy to prove the existence of
complex smoothings in other ways.

Instead, Fintushel and Stern [FS97] (see also Park [P97] for a complete
generality) introduced the rational blowdown technique. This is a surgical
procedure on a closed 4-manifold that contains disjoint Wahl chains, obtaining
a new smooth closed 4-manifold by removing a small neighborhood of the Wahl
chains and attaching rational homology balls instead. These are the Milnor
fibers of the ${\mathbb{Q}}$-Gorenstein smoothings of the corresponding Wahl
singularities. (See [SSW08] for more on the relation between these two
viewpoints.) In [Sy98, Sy01], Symington shows that one can give a symplectic
structure to the rational blowdown, provided that we start with a symplectic
4-manifold and the 2-spheres in the Wahl chains are symplectic.

A nonsingular complex projective surface $X$ admits a symplectic form
inherited from some embedding in projective space. Thus, if $X$ contains
disjoint Wahl chains, then we can apply Symington’s theorem to construct a
symplectic rational blowdown $Y$ along the Wahl chains in $X$.

We recall that a divisor $D$ on an irreducible projective surface $Z$ is nef
if $D\cdot\Gamma\geq 0$ for all curves $\Gamma\subset Z$. In addition $D$ is
big if and only if $D^{2}>0$ [L04, Theorem 2.2.16].

######  Theorem 2.3.

Let $W^{\prime}$ be a normal projective surface with only T-singularities and
Du Val singularities such that $K_{W^{\prime}}$ is ample. Let $\phi\colon W\to
W^{\prime}$ be the M-resolution of $W^{\prime}$, and let $\pi\colon X\to W$ be
the minimal resolution of $W$. Then the rational blowdown $Y$ of
$\operatorname{Exc}(\pi)$ is a (smoothly) minimal symplectic 4-manifold.

###### Proof.

Let $E_{i}$ be the curves in $\operatorname{Exc}(\pi)$, and let $F_{i}$ (resp.
$G_{i}$) be the proper transforms in $X$ of the exceptional curves of $\phi$
over T-singularities (resp. Du Val singularities). First we claim there is
$N>0$, and integers $n_{i},m_{i},s_{i}\geq 0$ such that

$H:=\pi^{*}(NK_{W})-\sum_{i}n_{i}E_{i}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i}$

is ample. We know that there are integers $n_{i},m_{i},s_{i}\geq 0$ such that

$\big{(}\sum_{i}n_{i}E_{i}+\sum_{i}m_{i}F_{i}+\sum_{i}s_{i}G_{i}\big{)}\cdot\Gamma<0,$

for all $\Gamma=E_{j},F_{j},G_{j}$ as the intersection matrices for
exceptional curves over singularities are negative definite.

We know that $K_{W}\equiv\phi^{*}(K_{W^{\prime}})$ is nef and big. Then by
[L04, Proposition 2.2.6] (Kodaira’s lemma), there is $M>0$ (sufficiently
divisible) such that

$H^{0}\big{(}X,\pi^{*}(MK_{W})-\sum_{i}n_{i}E_{i}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i}\big{)}\neq
0,$

and so we have that
$H_{0}=\pi^{*}(MK_{W})-\sum_{i}n_{i}E_{i}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i}$
is linearly equivalent to some fixed effective divisor. Thus there could be a
finite number of curves (in that fixed divisor) which may have a nonpositive
intersection with $H_{0}$. As $H_{0}\cdot\Gamma>0$ for all
$\Gamma=E_{j},F_{j},G_{j}$ and they are the only curves that intersect zero
with $\pi^{*}(MK_{W})$ (for the rest of curves it has to be positive as
$K_{W^{\prime}}$ is ample), we can find $N>M$ so that
$H=H_{0}+\pi^{*}((N-M)K_{W})$ intersects any curve in $X$ positively, and
$H^{2}>0$ (as $K_{W^{\prime}}$ is ample). Hence by the Nakai-Moishezon
criterion [L04, Theorem 1.2.23], we see that $H$ is ample.

Let us assume we took a higher multiple to make $H$ very ample. In this way,
we have an embedding of $X$ into projective space via $H$. Let $X^{0}$ be $X$
minus the Wahl chains in $\operatorname{Exc}(\pi)$, and let $W^{0}$ be $W$
minus the Wahl singularities. We have that $H|_{X^{0}}$ is
$\pi^{*}(NK_{W})|_{X^{0}}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i}$. We have the
symplectic form $\omega_{X}$ induced by the complex structure in the class of
$H$. As $Y$ is the rational blowdown of $X$ at the Wahl chains in
$\operatorname{Exc}(\pi)$, we have that $H_{2}(W^{0},{\mathbb{Z}})\subset
H_{2}(Y,{\mathbb{Z}})$, and we also have $H_{2}(Y,{\mathbb{Z}})\subset
H_{2}(W,{\mathbb{Z}})$ as the Milnor fibers a rational homology balls.
Therefore, the Symington’s symplectic form $\omega_{Y}$ on $Y$ is in the class
represented by $NK_{Y}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i}$.

By [Li06, Proposition 2.1], if $Y$ is not minimal (in a smooth sense), then
there is a symplectic $(-1)$-sphere $E$ in $Y$. But then

$0<\int_{E}\omega_{Y}|_{E}=E\cdot(NK_{Y}-\sum_{i}m_{i}F_{i}-\sum_{i}s_{i}G_{i})<0,$

as none of the $F_{i}$, $G_{i}$ are $(-1)$-spheres, which is a contradiction.
Therefore $Y$ is minimal.

∎

######  Remark 2.4.

We note that if $W^{\prime}=W$, i.e. only Wahl singularities on $W^{\prime}$,
then $\omega_{Y}$ is in the class of $NK_{Y}$. We can also get a symplectic
form $\omega_{Y}$ in the class of $NK_{Y}$ starting with the $W^{\prime}$ in
Theorem 2.3, by means of constructing $Y$ via cutting the T-singularities and
Du Val singularities from $W^{\prime}$, and then glueing the Milnor fibres
corresponding to the ${\mathbb{Q}}$-Gorenstein smoothings with their exact
symplectic forms.

Throughout this paper we will only consider those $Y$ that are simply
connected.

We recall that $\chi_{\text{top}}(Y)=\chi_{\text{top}}(W)$ and
$K_{Y}^{2}=K_{W}^{2}$ for the rational blowdown in Theorem 2.3. Let
$(b^{+},b^{-})$ be the signature of the unimodular quadratic form on
$H^{2}(Y,{\mathbb{Z}})$. Then we have the second Betti number
$b_{2}(Y)=b^{+}+b^{-}=\chi_{\text{top}}(W)-2$, and for its signature we have
$b^{+}-b^{-}=\frac{1}{3}(K^{2}_{W}-2\chi_{\text{top}}(W))$. Therefore

$b^{+}=2\chi({\mathcal{O}}_{W})-1\ \ \ \ \
b^{-}=10\chi({\mathcal{O}}_{W})-K^{2}_{W}-1,$

as the Noether’s formula
$12\chi({\mathcal{O}}_{W})=K_{W}^{2}+\chi_{\text{top}}(W)$ holds for $W$. In
particular, the intersection form cannot be negative definite. It cannot be
positive definite either as that would imply
$K_{W}^{2}=5\chi_{\text{top}}(W)-6$, and by the orbifold Bogomolov-Miyaoka-Yau
inequality (see e.g. [L03]) we would have $5\chi_{\text{top}}(W)-6\leq
3\chi_{\text{top}}(W)-t$, where $t>0$, and so $b^{+}<1$, a contradiction.
Therefore the quadratic form must be indefinite. By Freedman’s theorem, the
oriented homeomorphism type of $Y$ is completely determined by its
intersection form, and as it must be indefinite, it is determined by $b^{+}$,
$b^{-}$, and its parity (see e.g. [BHPV04, Chapter IX]).

In this paper, we will only consider the case when the intersection form is
odd, so $Y$ is homeomorphic to
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
with $b^{-}>0$. A consequence of [Li06, Proposition 2.1] is the following.

######  Corollary 2.5.

Let $Y$ be the minimal symplectic 4-manifold constructed in Theorem 2.3. If
$Y$ is simply-connected and it has an odd intersection form, then $Y$ is an
exotic
$b^{+}{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
with $b^{-}>0$.

##  3\. KSBA surfaces and geography of rational configurations

Let $W$ be a normal projective surface with $\ell$ Wahl singularities, and
$K_{W}$ big and nef. We will consider the following diagram of morphisms

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\pi}$$\textstyle{Z}$$\textstyle{W}$

where $\pi$ is the minimal resolution of $W$ (thus $\operatorname{Exc}(\pi)$
consists of $\ell$ Wahl chains), and $\sigma$ is some composition of blow-ups
at nonsingular points starting with a nonsingular projective surface $Z$.

######  Proposition 3.1.

The surface $Z$ satisfies one of the following:

  * •

It is a rational surface.

  * •

It is a blow-up of either a K3 surface, or an Enriques surface.

  * •

It has Kodaira dimension equal to $1$ and $b_{1}(Z)=0$.

  * •

It is of general type, $b_{1}(Z)=0$, and $K_{Z}^{2}<K_{W}^{2}$.

Moreover, the configuration of rational curves
$\sigma(\operatorname{Exc}(\pi))$ is non-empty.

###### Proof.

This is the content of [RU19, Proposition 2.3] when $W$ has one Wahl
singularity and $K_{W}$ is ample. But the same proof works for many Wahl
singularities, and $K_{W}$ big and nef except in the general type case. The
inequality is [R17, Lemma 2.2]. The claim $b_{1}(Z)=0$ can be shown via the
Albanese variety of $Z$. For a proof of the last statement see [RU21,
Proposition 2.3]. ∎

Evans and Smith [ES20] found a bound for the length of the Hirzebruch-Jung
continued fraction of each of the singularities in $W$ when $p_{g}(W)>0$ (see
also [RU19, Theorem 1.1] when $W$ has only one singularity). They prove that
if this length is $s$, then $s\leq 4K_{W}^{2}+7$. It turns out that one can
replace the hypothesis “$p_{g}>0$” with “rational blowup is not a rational
surface” in [ES20, Theorem 1.1]. This is done in [CZ20, Theorem A.3] by Chen
and Zhang. In addition, it is possible to improve that bound by modifying
[ES20, Propositions 8.3 and 8.4] following what is done in [RU19, Lemma 2.8],
so the bound ends up being

$s\leq 4K_{W}^{2}+1.$

This bound is optimal by [RU19], and it is not true for $W$ rational. Two
boundedness problems on these surfaces $W$ remain open:

  1. (1)

Find bounds depending on $K_{W}^{2}$ which involve all singularities at once,
and

  2. (2)

Find bounds depending on $K_{W}^{2}$ when $W$ is a rational surface.

Problem (1) involves hard combinatorics, as one has to deal with “bad”
configurations of $(-1)$-curves and Wahl chains in $X$. That analysis would
allow us to know and classify those bad configurations. Problem (2) was
partially worked out in [RU19, Theorem 1.3], but we do not have yet a real
bound depending only on $K_{W}^{2}$.

We are going to focus on the configuration of rational curves
$\sigma(\operatorname{Exc}(\pi))$. The objective is to find a connection
between the problem of the existence of a $W$ with the required properties,
and log Chern numbers of $\sigma(\operatorname{Exc}(\pi))$. In this way, by
applying the log BMY inequality, we will produce constraints for the existence
of $W$.

We recall the log Chern numbers of a configuration of curves
$\\{C_{1},\ldots,C_{r}\\}$ in a surface $Z$; Cf. [Urz10]. Let
$\phi\colon\tilde{Z}\to Z$ be a log resolution of $D=\sum_{i=1}^{r}C_{i}$, in
the sense that $\phi$ is a birational morphism such that the (reduced) pull-
back $\tilde{D}$ of $D$ has only simple normal crossings, and any exceptional
$(-1)$-curve in $\tilde{D}$ intersects at least two other curves in
$\tilde{D}$. Then the log Chern numbers of $(Z,D)$ are defined via the Chern
classes of $\Omega_{\tilde{Z}}^{1}(\log\tilde{D})$ as
$\bar{c}_{1}^{2}(Z,D):=c_{1}(\Omega_{\tilde{Z}}^{1}(\log\tilde{D})^{*})^{2}$
and $\bar{c}_{2}(Z,D):=c_{2}(\Omega_{\tilde{Z}}^{1}(\log\tilde{D})^{*})$. We
have $\bar{c}_{1}^{2}(Z,D)=(K_{\tilde{Z}}+\tilde{D})^{2}$ and
$\bar{c}_{2}(Z,D)=\chi_{\text{top}}(\tilde{Z}\setminus\tilde{D})$. It can be
proved that they do not depend on the log resolution. (See [EFU22, Urz22] for
a systematic study of these invariants when $Z={\mathbb{P}}^{2}$ and $D$ is a
configuration of lines.) We will only consider the following type of
configurations.

######  Definition 3.2.

Let $Z$ be a nonsingular projective surface. A nodal simple crossings
configuration is a collection of projective curves $C_{1},C_{2},\ldots,C_{r}$
such that:

  * •

The curve $C_{i}$ is nodal and rational with $\nu_{i}$ nodes, and

  * •

At a common point of $C_{i}$ and $C_{j}$ there are different tangent
directions.

Hence, the divisor $D=\sum_{i=1}^{r}C_{i}$ has only simple crossings
singularities, which are locally of the form
$(x-a_{1}y)(x-a_{2}y)\cdots(x-a_{k}y)=0$ at $(0,0)\in{\mathbb{C}}^{2}$ with
$a_{i}\neq a_{j}$. Such a point will be called k-point. The number of k-points
in $D$ will be denoted by $t_{k}$. We denote the number of nodes as

$\nu:=\sum_{i}\nu_{i}.$

The numbers $\nu$ and $t_{2}$ are not necessarily comparable.

######  Proposition 3.3.

We have $\bar{c}_{1}^{2}(Z,D)=K_{Z}^{2}-\sum_{i=1}^{r}C_{i}^{2}+\sum_{k\geq
2}(3k-4)t_{k}-4r+2\nu,$ and $\bar{c}_{2}(Z,D)=\chi_{\text{top}}(Z)+\sum_{k\geq
2}(k-1)t_{k}-2r.$

###### Proof.

Let $Z^{\prime}\to Z$ be the blow-up at all nodes from the $C_{i}$. These
nodes could be isolated, or they could be at k-points with $k>2$. Let
$D^{\prime}$ be the reduced pull-back of $D$ in $Z^{\prime}$. By [Urz10],
There are formulas for the log Chern numbers of $(Z^{\prime},D^{\prime})$, and
they translate to

$\bar{c}_{1}^{2}(Z,D)=K_{Z}^{2}-\sum_{i=1}^{r}C_{i}^{2}+\sum_{k\geq
2}(3k-4)t_{k}+4\sum_{i=1}^{r}(p_{a}(C_{i})-1)-2\nu,$

and

$\bar{c}_{2}(Z,D)=\chi_{\text{top}}(Z)+\sum_{k\geq
2}(k-1)t_{k}+2\sum_{i=1}^{r}(p_{a}(C_{i})-1)-2\nu.$

But the curves $C_{i}$ are rational, and so $p_{a}(C_{i})=\nu_{i}$ for all
$i$. ∎

######  Definition 3.4.

Let $\\{{C^{\prime}}_{1},\ldots,{C^{\prime}}_{r^{\prime}}\\}$ be any
configuration of curves on a nonsingular surface $Z^{\prime}$, and let
$D^{\prime}=\sum_{i=1}^{r^{\prime}}{C^{\prime}}_{i}$. We define

$P(Z^{\prime},D^{\prime}):=\sum_{i=1}^{r^{\prime}}\left({C^{\prime}}_{i}^{2}+5-\sum_{P\in\operatorname{Sing}(D^{\prime})}m_{P}({C^{\prime}}_{i})^{2}\right),$

where $m_{P}(C)$ is the multiplicity of $P\in C$, and

$K(Z^{\prime},D^{\prime}):=K_{Z^{\prime}}^{2}+2r^{\prime}-|\operatorname{Sing}(D^{\prime})|-P(Z^{\prime},D^{\prime}).$

Note that if $D^{\prime}$ is a collection of $\ell$ disjoint Wahl chains, then
$P(Z^{\prime},D^{\prime})=\ell$ and
$K(Z^{\prime},D^{\prime})=K_{Z^{\prime}}^{2}+r^{\prime}$ by the particular
properties of Wahl chains [KSB88, Proposition 3.11].

######  Proposition 3.5.

Let $D^{\prime}=C^{\prime}_{1}+\ldots+C^{\prime}_{r^{\prime}}$ be any
configuration of curves on a nonsingular surface $Z^{\prime}$. If $E$ is a
nonsingular curve not in $D^{\prime}$, intersecting it at $n$ nonsingular
points of $D^{\prime}$ and $m$ singular points of $D^{\prime}$, then
$P(Z^{\prime},D^{\prime}+E)=P(Z^{\prime},D^{\prime})+E^{2}+5-2n-m$, and
$K(Z^{\prime},D^{\prime}+E)=K(Z^{\prime},D^{\prime})-E^{2}-3+n+m$.

######  Proposition 3.6.

Let $D^{\prime}=C^{\prime}_{1}+\ldots+C^{\prime}_{r^{\prime}}$ be any
configuration of curves on a nonsingular surface $Z^{\prime}$. Let
$Z^{\prime\prime}\to Z^{\prime}$ be the blow-up at a point $P$ in
$\operatorname{Sing}(D^{\prime})$, and let $D^{\prime\prime}$ be the strict
transform of $D^{\prime}$. Then
$P(Z^{\prime\prime},D^{\prime\prime})=P(Z^{\prime},D^{\prime})$ and
$K(Z^{\prime\prime},D^{\prime\prime})=K(Z^{\prime},D^{\prime})$. In
particular, if the singularity of $D^{\prime}$ at $P$ is a node and $E$ is the
exceptional $(-1)$-curve, then
$P(Z^{\prime\prime},D^{\prime\prime}+E)=P(Z^{\prime},D^{\prime})$ and
$K(Z^{\prime\prime},D^{\prime\prime}+E)=K(Z^{\prime},D^{\prime})$.

###### Proof.

Both propositions are direct from the formulas in Definition 3.4. ∎

From now on, let $D=\sum_{i=1}^{r}C_{i}$ be a nodal simple crossings
configuration on $Z$.

######  Proposition 3.7.

We have

$P(Z,D)=\sum_{i=1}^{r}C_{i}^{2}+5r-\sum_{k\geq 2}kt_{k}-2\nu,$

and

$K(Z,D)=K_{Z}^{2}-3r-\sum_{i=1}^{r}C_{i}^{2}+\sum_{k\geq 2}(k-1)t_{k}+2\nu.$

Let $\sigma^{\prime}\colon Z^{\prime}\to Z$ be the blow-up at all the k-points
of $D$ with $k>2$. Let us choose $t_{k}^{0}$ of the $t_{k}$ exceptional curves
for each $k>2$. Let $D^{\prime}$ be the (reduced) pull-back of $D$ minus the
$\sum_{k>2}t_{k}^{0}$ chosen exceptional curves. Then, by Propositions 3.5 and
3.6, we can compute

$P(Z^{\prime},D^{\prime})=P(Z,D)-\sum_{k\geq 3}(2k-4)(t_{k}-t_{k}^{0}),$

and $K(Z^{\prime},D^{\prime})=K(Z,D)+\sum_{k\geq 3}(k-2)(t_{k}-t_{k}^{0})$.

We note that further blow-ups at the notes of $D^{\prime}$ and additions of
the corresponding exceptional curves will not change $P$ and $K$.

######  Theorem 3.8.

Let $W$ be a normal projective surface with $\ell$ Wahl singularities, and
$K_{W}$ big and nef. Let $\pi\colon X\to W$ be its minimal resolution. Assume
that there is a birational morphism $\sigma\colon X\to Z$ such that
$\sigma(\operatorname{Exc}(\pi))$ is a nodal simple crossings configuration of
rational curves $D=C_{1}+\ldots+C_{r}$. Then,

$K_{W}^{2}\leq
12(1+p_{g}(Z))-\frac{1}{3}K_{Z}^{2}-\frac{2}{3}\ell-\frac{1}{3}r+\frac{1}{3}\sum_{k\geq
3}(k-2)t_{k}^{0}.$

Moreover,

$\bar{c}_{1}^{2}(Z,D)=K_{Z}^{2}-\ell+r+2\sum_{k\geq 3}(k-2)t_{k}^{0},$
$\bar{c}_{2}(Z,D)=12(1+p_{g}(Z))-\ell-K_{W}^{2}+\sum_{k\geq 3}(k-2)t_{k}^{0},$

and

$2r=K_{W}^{2}-K_{Z}^{2}+\ell+\sum_{k\geq 2}(k-1)t_{k}-\sum_{k\geq
3}(k-2)t_{k}^{0}.$

As we defined earlier, here $t_{k}^{0}$ is the number of exceptional curves
over $k$-points with $k>2$ which are not part of $\operatorname{Exc}(\pi)$.

###### Proof.

As before, let $\sigma^{\prime}\colon Z^{\prime}\to Z$ be the blow-up at all
the k-points of $D$ with $k>2$. We have already computed
$P(Z^{\prime},D^{\prime})$ and $K(Z^{\prime},D^{\prime})$ where $D^{\prime}$
is the pull-back of $D$ taking out the exceptional curves of $\sigma^{\prime}$
that will not be part of the Wahl chains in $X$. As we want to calculate the
log Chern numbers of the pair $(Z,D)$, we will consider all exceptional
divisors of the log-resolution $Z^{\prime}\to Z$ of $D$. It turns out that
including them all imposes more restrictions on $K_{W}^{2}$ than not
considering some of them.

On the other hand, we know that $P(Z^{\prime},D^{\prime})=\ell$ and
$K(Z^{\prime},D^{\prime})=K_{W}^{2}$ by Proposition 3.6, and the observation
at the end of Definition 3.4. We can isolate the terms $\sum_{k}kt_{k}$ and
$\sum_{k}t_{k}$ from the formulas for $P(Z^{\prime},D^{\prime})$ and
$K(Z^{\prime},D^{\prime})$, and evaluate them on the formulas for
$\bar{c}_{1}^{2}(Z,D)$ and $\bar{c}_{2}(Z,D)$ to obtain the last three claims
in the theorem. Here we are using that $b_{1}(Z)=0$, and Noether’s formula
$12\chi({\mathcal{O}}_{Z})=K_{Z}^{2}+\chi_{\text{top}}(Z)$ as well.

For the first claim, we apply the log Bogomolov-Miyaoka-Yau inequality as
expressed by Langer in [L03, Theorem 0.1]. For that we need to have a log
canonical pair $(X,D^{\prime\prime})$ so that a multiple of
$K_{X}+D^{\prime\prime}$ is effective. We are taking as $D^{\prime\prime}$ the
(reduced) pull-back of $D$ by $\sigma$. Hence
$D^{\prime\prime}=\operatorname{Exc}(\pi)+\sum_{i}G_{i}$ where $G_{i}$ are the
exceptional curves of $\sigma$. We also know that
$K_{X}\equiv\pi^{*}(K_{W})+\sum_{i}d_{i}C_{i}$, where $-1<d_{i}<0$,
$C_{i}\in\operatorname{Exc}(\pi)$, and by assumption $\pi^{*}(K_{W})$ is big
and nef, so it is a sum of an ample divisor plus some effective divisor.
Therefore some multiple of $K_{X}+D^{\prime\prime}$ is indeed effective. ∎

######  Corollary 3.9.

With the same hypothesis of Theorem 3.8, assume in addition that the image of
$\operatorname{Exc}(\pi)$ under $\sigma$ has only nodes as singularities. Then

$7K_{W}^{2}\leq 12+6b_{2}(Z)+5K_{Z}^{2}-5\ell-t_{2}.$

######  Corollary 3.10.

With the same hypothesis of Theorem 3.8, assume in addition that $t_{k}^{0}=0$
for all $k>2$ and $K_{Z}^{2}=0$. Then

$7K_{W}^{2}\leq 12+6b_{2}(Z)-5\ell-\sum_{k\geq 2}(k-1)t_{k}.$

##  4\. Obstructions to complex smoothings

In this section, we analyze the potential complex deformations of a normal
projective surface $W$ with $\ell$ Wahl singularities, and $K_{W}$ big and nef
based on [H04, H12]. As in the introduction, let
$T_{W}:=\mathcal{H}\text{om}_{{\mathcal{O}}_{W}}(\Omega_{W}^{1},{\mathcal{O}}_{W})$
where $\Omega_{W}^{1}$ is the sheaf of differentials on $W$.

A deformation (proper, flat morphism)
$(W\subset{\mathscr{W}})\to(0\in{\mathbb{D}})$ over a nonsingular curve germ
${\mathbb{D}}$ is ${\mathbb{Q}}$-Gorenstein if $K_{{\mathscr{W}}}$ is
${\mathbb{Q}}$-Cartier. Hence, for each singularity we have either a locally
trivial deformation, or a smoothing with Milnor number zero. It is called
${\mathbb{Q}}$-Gorenstein smoothing if in addition $W_{t}$ (the fiber over
$t$) is nonsingular for $t\neq 0$. We assume that $W_{0}$ is a reduced fiber
isomorphic to $W$. We know that there is a versal deformation space
$\operatorname{Def}_{QG}(W)$ which parametrizes all ${\mathbb{Q}}$-Gorenstein
deformations of $W$. A main question for us is whether this space contains
smoothings. For this, we want to study ${\mathbb{T}}_{QG,W}^{0}$, the vector
space of infinitesimal automorphisms of $W$; ${\mathbb{T}}_{QG,W}^{1}$, the
vector space of ${\mathbb{Q}}$-Gorenstein first order deformations of $W$; and
${\mathbb{T}}_{QG,W}^{2}$, the vector space of obstructions for
${\mathbb{Q}}$-Gorenstein deformations.

In general, it is a theorem that for proper schemes $W$ over a field $k$,
Aut$(W)$ is a group scheme locally of finite type over $k$, and its tangent
space at the identity is $H^{0}(W,T_{W})$. In our case, since
$k={\mathbb{C}}$, Aut$(W)$ is reduced, so if Aut$(W)$ is finite, then
$H^{0}(W,T_{W})=0$. But Aut$(W)$ is indeed finite when $K_{W}$ is big by a
well-known theorem of Iitaka [Ii82, Section 11.1 for definitions, Theorem
11.12 for the result]. On the other hand, it is shown in [H04, Lemma 3.8] that
${\mathbb{T}}_{QG,W}^{0}=H^{0}(W,T_{W})$, as there is an isomorphism of
sheaves $T_{W}\simeq\mathcal{T}_{QG,W}^{0}$. Therefore
${\mathbb{T}}_{QG,W}^{0}=0$.

For the analysis of the two other vector spaces, we first consider the minimal
resolution of singularities $\pi\colon X\to W$. We have that
$\pi_{*}T_{X}=T_{W}$ (see [BW74, Proposition 1.2]). By the Leray spectral
sequence of low-degree terms (see [NSW08, Lemma (2.1.3)], [H12, Section 3]),
we have the exact sequence

$0\to H^{1}(W,T_{W})\to{\mathbb{T}}^{1}_{QG,W}\to
H^{0}(W,\mathcal{T}^{1}_{QG,W})\to H^{2}(W,T_{W})\to{\mathbb{T}}^{2}_{QG,W}\to
0.$

This happens because $\mathcal{T}^{i}_{QG,W}$ is supported on the isolated
singularities of $W$ for $i>0$, and $\mathcal{T}^{2}_{QG,W}=0$ as the local
canonical covers of $W$ are complete intersections (see [H04, p.227]). The
general set-up is described in [H12, Section 3]. The vector space
$H^{0}(W,\mathcal{T}^{1}_{QG,W})$ is the direct sum of the one-dimensional
tangent spaces of the ${\mathbb{Q}}$-Gorenstein deformations of each Wahl
singularity, i.e. $H^{0}(W,\mathcal{T}^{1}_{QG,W})={\mathbb{C}}^{\ell}$. If
${\mathbb{T}}^{2}_{QG,W}=0$, then $\operatorname{Def}_{QG}(W)$ is nonsingular
(this happens, for example, when $H^{2}(W,T_{W})=0$). The vector space
$H^{1}(W,T_{W})$ parametrizes equisingular deformations of $W$. These
dimensions are related to the invariants of $W$ as follows.

######  Theorem 4.1.

Let $W$ be a normal projective surface with $\ell$ Wahl singularities, and
$K_{W}$ big. Then

$\dim_{{\mathbb{C}}}{\mathbb{T}}^{1}_{QG,W}=10\chi({\mathcal{O}}_{W})-2K_{W}^{2}+\dim_{{\mathbb{C}}}{\mathbb{T}}^{2}_{QG,W},$

and so $h^{2}(T_{W})=\ell+h^{1}(T_{W})+2K_{W}^{2}-10\chi({\mathcal{O}}_{W})$.

###### Proof.

Let us consider the minimal resolution $\pi\colon X\to W$ with exceptional
divisor $E=\sum_{i=1}^{\ell}E_{i}$, where $E_{i}$ is the exceptional Wahl
chain over each Wahl singularity. In particular, $E$ is a simple normal
crossings divisor. We have the short exact sequence

$0\to T_{X}(-\log E)\to T_{X}\to\bigoplus_{C\in E}N_{C/X}\to 0,$

where $N_{C/X}\simeq{\mathcal{O}}_{C}(C)$ is the normal sheaf of $C$ in $X$.
As in [LP07, Lemma 1], we have $R^{i}\pi_{*}T_{X}(-\log E)=0$ for $i=1,2$, and
we know that $\pi_{*}T_{X}=T_{W}$. This implies

$h^{2}(T_{W})-h^{1}(T_{W})=\chi(W,T_{W})=\chi(X,T_{X}(-\log E)),$

as $h^{0}(T_{W})=0$. We now use the Hirzebruch-Riemann-Roch theorem for the
locally free sheaf $T_{X}(-\log E)$. Its exponential Chern character is

$\text{ch}(T_{X}(-\log(E)))=2-(K_{X}+E)+\frac{1}{2}\big{(}(K_{X}+E)^{2}-2\chi_{\text{top}}(X\setminus
E)\big{)}.$

From this we compute
$\chi(X,T_{X}(-\log(E)))=2K_{W}^{2}-10\chi({\mathcal{O}}_{W})+\ell$. The
result follows from the Leray exact sequence above. ∎

If $\text{dim}_{{\mathbb{C}}}H^{2}(W,T_{W})=0$, then
$\operatorname{Def}_{QG}(W)$ is nonsingular and any local deformations of the
singularities in $W$ may be glued to obtain a global deformation of $W$. We
say in this case that there are no local-to-global obstructions to deform $W$.
For instance, if $K_{W}^{2}=5$ and $p_{g}(W)=q(W)=0$, then we have
$h^{2}(W,T_{W})>0$ and
$\text{dim}_{{\mathbb{C}}}{\mathbb{T}}^{1}_{{\text{QG}},W}=\text{dim}_{{\mathbb{C}}}{\mathbb{T}}^{2}_{{\text{QG}},W}$.

For the examples in the following sections, we will start with an elliptic
fibration, and we will consider singular fibers and multi-sections as part of
the configurations of rational curves. We also want to concretely compute
$H^{2}(T_{W})$, which is the same as computing $H^{2}(X,T_{X}(-\log E))$ as we
just saw.

Let $Z\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$ be an elliptic fibration with
sections, and $D$ a configuration of rational curves $D$ in $Z$. Consider a
composition of blow-ups

$\sigma\colon X=Z_{n}\to Z_{n-1}\to\ldots\to Z_{0}=Z,$

as in the previous section, where each intermediate $Z_{i+1}\to Z_{i}$ is the
blow-up at a single point. We successively construct a divisor $D_{i+1}$ in
$Z_{i+1}$ with curves in the proper transform of $D_{i}$ plus some exceptional
curves. We will keep track of $H^{2}(Z_{i},T_{Z_{i}}(-\log D_{i}))$ as we add
or erase curves, applying the following principles. The dimension of
$H^{2}(Z_{i+1},T_{Z_{i+1}}(-\log D_{i+1}))$ is equal to the dimension of
$H^{2}(Z_{i},T_{Z_{i}}(-\log D_{i}))$ when $D_{i}$ has simple normal crossings
and $D_{i+1}$ is the reduced pullback of $D_{i}$. Furthermore, the dimension
of $H^{2}(Z_{i},T_{Z_{i}}(-\log D_{i}^{\prime}))$ is equal to the dimension of
$H^{2}(Z_{i},T_{Z_{i}}(-\log D_{i}))$ if

  * (-1)

$D_{i}^{\prime}$ is obtained from $D_{i}$ by adding or erasing $(-1)$-curves
that are transversal to the rest of $D_{i}$. This applies to removing the
$(-1)$-curve of the reduced pullback of $D_{i-1}$, if the blow-up was made at
a node. (see e.g. [PSU13, Proposition 4.2 and 4.3].

  * (-2)

$D_{i}^{\prime}$ is obtained from $D_{i}$ by adding any Du Val configuration
of $(-2)$-curves that are disjoint from $D_{i}$ [PSU13, Theorem 4.4].

######  Lemma 4.2.

Let $f\colon Z\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$ be an elliptic fibration
with at least one section. Assume that it has $n$ fibers $F_{i}$ of type
$I_{1}$ and $m$ fibers $G_{j}$ of type $I_{k}$ with $k>1$. (They may not be
all the singular fibers of $f$.) Let $\sigma^{\prime}\colon Z^{\prime}\to Z$
be the blow-up at the nodes of the $F_{i}$. Let $F_{i}$ (resp. $G_{i}$) be the
proper transform of $F_{i}$ (resp. $G_{i}$), and let $H_{i}$ be the
exceptional curves of $\sigma^{\prime}$. Then

$H^{2}\Big{(}Z^{\prime},T_{Z^{\prime}}\big{(}-\log(\sum_{i=1}^{n}(F_{i}+H_{i})+\sum_{i=1}^{m}G_{i}\big{)}\Big{)}\geq
m+n-2+p_{g}(Z).$

###### Proof.

This follows [LP07, Proposition 7]. We have
$K_{Z^{\prime}}\sim(p_{g}(Z)-1)F+\sum_{i=1}^{n}H_{i}$, where $F$ is a general
fiber of $Z^{\prime}\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$. Consider the
inclusion

$f^{*}(\Omega_{{\mathbb{P}}_{{\mathbb{C}}}^{1}}^{1}(n+m))\otimes
K_{Z^{\prime}}\to\Omega_{Z^{\prime}}^{1}\big{(}\log(\sum_{i=1}^{n}(F_{i}+H_{i})+\sum_{i=1}^{m}G_{i}\big{)}\otimes
K_{Z^{\prime}}.$

The inequality follows by Serre duality. ∎

######  Lemma 4.3.

With the same hypothesis as in the previous lemma, let us assume that
$p_{g}(Z)=0$. Then

$H^{2}\Big{(}Z^{\prime},T_{Z^{\prime}}\big{(}-\log(\sum_{i=1}^{n}(F_{i}+H_{i})+\sum_{i=1}^{m}G_{i}\big{)}\Big{)}=m+n-2.$

###### Proof.

First, by the $(-1)$ principle, we have

$H^{2}\big{(}Z^{\prime},T_{Z^{\prime}}\big{(}-\log(\sum_{i=1}^{n}(F_{i}+H_{i})\big{)}\big{)}=H^{2}\big{(}Z^{\prime},T_{Z^{\prime}}\big{(}-\log(\sum_{i=1}^{n}F_{i}\big{)}\big{)}.$

We have the inclusions

$H^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=1}^{n}F_{i}\big{)}\otimes
K_{Z^{\prime}})\subset
H^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=2}^{n}F_{i}\big{)}\otimes
K_{Z^{\prime}}+F_{1})\ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \subset
H^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=2}^{n}F_{i}\big{)}\otimes\sum_{i=2}^{n}H_{i})\subset
H^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=2}^{n}F_{i}+\sum_{i=2}^{n}H_{i}\big{)}),$

and by the long exact sequence in cohomology applied to the residue sequence

$0\to\Omega_{Z^{\prime}}^{1}\to\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=2}^{n}F_{i}+\sum_{i=2}^{n}H_{i}\big{)}\to\bigoplus{\mathcal{O}}_{F_{i}}\oplus\bigoplus{\mathcal{O}}_{H_{i}}\to
0,$

we obtain that
$H^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=2}^{n}F_{i}+\sum_{i=2}^{n}H_{i}\big{)})$
is the kernel of the Chern map

$H^{0}({\mathcal{O}}_{F_{i}}\oplus{\mathcal{O}}_{H_{i}})\to
H^{1}(\Omega_{Z^{\prime}}^{1}).$

This kernel has dimension $n-2$, as we have $n-1$ fibers with two components
each.

By the (-2) principle, we can add the configurations $G^{\prime}_{i}$ which
are the $G_{i}$ minus one $(-2)$-curve without altering the dimension, so

$h^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=1}^{n}F_{i}+\sum_{i=1}^{m}G^{\prime}_{i}\big{)}\otimes
K_{Z^{\prime}})\leq n-2,$

and by the (-1) principle we can add the $H_{i}$, so

$h^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=1}^{n}F_{i}+\sum_{i=1}^{n}H_{i}+\sum_{i=1}^{m}G^{\prime}_{i}\big{)}\otimes
K_{Z^{\prime}})\leq n-2.$

As the normal bundle of a $(-2)$-curve has a one-dimensional first cohomology,
when we add the missing $(-2)$-curves from the $G_{i}$, the dimension
increases by at most $1$ for each one, so

$h^{0}(\Omega_{Z^{\prime}}^{1}\big{(}\log\sum_{i=1}^{n}F_{i}+\sum_{i=1}^{n}H_{i}+\sum_{i=1}^{m}G_{i}\big{)}\otimes
K_{Z^{\prime}})\leq m+n-2.$

Finally, by Lemma 4.2, this dimension is also greater than $m+n-2$, so the
claim follows. ∎

In what follows, we will again use the fact that adding a transversal
$(-2)$-curve to a SNC configuration increases the dimension by at most $1$. As
in the previous proof, this claim follows from the standard residue sequences
(see [PSU13, Section 4]).

##  5\. Exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$

There are several constructions of exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#5\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ (see
[A08, ABP08, FS11, BK17, BKS22]), but the only examples in the literature
obtained via the rational blowdown technique are in [PPS09b] –where the
authors found two examples that admit a complex structure, from distinct Wahl
singularities– and [BKS22]. It turns out that it is possible to construct
hundreds of pairs of Wahl chains in suitable rational surfaces which produce
exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#5\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ by
means of Theorem 2.3 and Corollary 2.5. Among them, we have found examples
with a Wahl chain of length $r$ for each $1\leq r\leq 18$, in particular, the
bound $4K_{W}^{2}+1$ on lengths for nonrational surfaces does not hold in the
rational case. Typically $H^{2}(T_{W})=0$ for these surfaces, so they admit a
complex structure. The local dimension at $W$ for the KSBA compactification of
the moduli space is $2$, so these examples correspond to hundreds of KSBA
boundary curves.

######  Question 5.1.

Is the moduli space of simply-connected $p_{g}=0$ surfaces of general type
with $K^{2}=4$ an irreducible surface? Is it a ruled surface?

We will describe these examples with $K^{2}=4$ and connections between them
somewhere else. In this section we will focus on constructions for the
$K^{2}=5$ case, i.e. exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$.

Let us consider the pencil of cubics in ${\mathbb{P}}^{2}_{{\mathbb{C}}}$

$\\{\mu xyz+\nu(x-z)(x+y+z)(x-y+z)=0\colon\
[\mu,\nu]\in{\mathbb{P}}^{1}_{{\mathbb{C}}}\\}.$

Its base points are $[0,1,0]$ (double), $[-1,0,1]$ (double), $[0,1,1]$,
$[0,-1,1]$, $[1,-1,0]$, and $[1,1,0]$. Let us name the lines
$L_{x}=\\{x=0\\}$, $L_{y}=\\{y=0\\}$, $L_{z}=\\{z=0\\}$, $A=\\{x-z=0\\}$,
$B=\\{x+y+z=0\\}$, and $C=\\{x-y+z=0\\}$. The blow-up at the nine base points
defines an elliptic fibration $f\colon Z\to{\mathbb{P}}^{1}_{{\mathbb{C}}}$
with precisely four singular fibers: two of type $I_{4}$ and two of type
$I_{2}$. Let us denote the exceptional curves of the blow-up as: $E_{1},E_{2}$
for $[0,1,0]$, $E_{3},E_{4}$ for $[-1,0,1]$, $E_{6}$ for $[0,1,1]$, $E_{7}$
for $[0,-1,1]$, $E_{8}$ for $[1,1,0]$, and $E_{9}$ for $[1,-1,0]$. Consider
the curves $Q_{1}=\\{(x+z)^{2}-y(x-z)=0\\}$, $L_{1}=\\{x+y-z=0\\}$,
$Q_{2}=\\{(x+z)^{2}+y(x-z)=0\\}$, and $L_{2}=\\{x-y-z=0\\}$. We have that

$Q_{1}\cap L_{1}=\\{B_{1}:=[i,-i+1,1],T_{1}:=[-i,i+1,1]\\},\quad\text{and}$
$Q_{2}\cap L_{2}=\\{B_{2}:=[i,i-1,1],T_{2}:=[-i,-i-1,1]\\},$

where $i^{2}=-1$. The singular fibers of $f$ are

$E_{1}+L_{z}+L_{y}+L_{x},\quad E_{3}+B+A+C,\quad
Q_{1}+L_{1},\quad\text{and}\quad Q_{2}+L_{2}.$

The fibration $f$ has precisely eight disjoint sections:
$E_{2},E_{4},E_{5},E_{6},E_{7},E_{8},E_{9}$, and $H$, the last being the
strict transform of the line $\\{x+z=0\\}$. We will denote by the same symbols
the strict transforms of curves under birational morphisms.

We also consider the following double sections of $f$:

  * •

$BT:=\\{x-iy+z=0\\}$, it passes through $[-1,0,1]$, $B_{1}$, $T_{2}$.

  * •

$TB:=\\{x+iy+z=0\\}$, it passes through $[-1,0,1]$, $T_{1}$, $B_{2}$.

  * •

$BB:=\\{x-iz=0\\}$, it passes through $[0,1,0]$, $B_{1}$, $B_{2}$.

  * •

$TT:=\\{x+iz=0\\}$, it passes through $[0,1,0]$, $T_{1}$, $T_{2}$.

  * •

$N:=\\{(2i-1)x-y-z=0\\}$, it passes through $[0,-1,1]$, $T_{1}$.

In Figure 1, we show how these curves intersect in $Z$. This is a simple
crossings configuration $D$ of $25$ curves with 12 $(-2)$-curves, 8
$(-1)$-curves, and 5 $(0)$-curves (double sections). Moreover $t_{2}=62$,
$t_{4}=3$, and $t_{5}=1$ and $t_{k}=0$ for other $k$. In this way, by
Proposition 3.3, we have $\bar{c}_{1}^{2}=82$, $\bar{c}_{2}=37$, and so
$\bar{c}_{1}^{2}/\bar{c}_{2}=2.\overline{216}$.

Figure 1. The $25$ relevant ${\mathbb{P}}^{1}_{{\mathbb{C}}}$ in $Z$

Below we will consider subconfigurations $D_{0}$ from $D$ to construct
surfaces $W$ with two Wahl singularities and $K_{W}$ big and nef, so that the
corresponding rational blowdown is a minimal symplectic exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$. For
this and the next section, we adopt the following way to express the
information necessary to construct $X$ (that is, a choice of blow-ups of $Z$
at points in $D_{0}$), and $W$ (contraction of two Wahl chains in $X$). For
this, it indicates $K_{W}^{2}$, $D_{0}$, the points to be blown-up, and the
Wahl chains together with their $(n,a)$:

$K_{W}^{2}$ \- configuration $D_{0}$ \- points where blow-ups happen and in
the order shown - $(n_{1},a_{1}):[b_{1},\ldots,b_{\ell_{1}}]$ \-
$(n_{2},a_{2}):[c_{1},\ldots,c_{\ell_{2}}]$

There are two different notations for points where blow-ups happen. If there
is a single blow-up at an intersection of curves $L$ and $R$, then it is
denoted by $L\cap R$. If there are several infinitely near blow-ups over that
intersection, then it is denoted by $[a_{1},\ldots,a_{n}]\times L\cap R$,
where $n$ blow-ups are done at successive intersections, with a
$(-a_{1})$-curve intersecting $L$ and a $(-a_{n})$-curve intersecting $R$.

If there is a $k$-point, then we denote its blow-up by $L_{1}\cap\ldots\cap
L_{k}$, where $L_{i}$ are the curves passing through that point, counted with
multiplicities. If another blow-up is realized over the exceptional curve,
then we also give a name to it as $E:=L_{1}\cap\ldots\cap L_{k}$.

We will elaborate on all details for the first example, and then we only
comment on particular characteristics for the rest.

(1) $K_{W}^{2}=5$ \- $\\{L_{z},~{}\allowbreak L_{y},~{}\allowbreak
L_{x},~{}\allowbreak E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak
E_{3},~{}\allowbreak Q_{1},~{}\allowbreak L_{1},~{}\allowbreak
Q_{2},~{}\allowbreak L_{2},~{}\allowbreak E_{5},~{}\allowbreak
E_{6},~{}\allowbreak BT\\}$ \- $~{}E_{17}:=Q_{1}\cap L_{1}\cap BT$,
$~{}Q_{2}\cap L_{2}\cap BT$, $~{}L_{z}\cap E_{1}$, $~{}L_{y}\cap E_{5}$,
$~{}L_{x}\cap E_{6}$, $~{}L_{x}\cap BT$, $~{}E_{3}\cap BT$, $~{}L_{1}\cap
E_{6}$, $~{}Q_{1}\cap E_{17}$, $[2,1]\times~{}Q_{1}\cap L_{1}$ \-
$(3,1):[5,\allowbreak 2]$, $(700,257):[3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 6,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 2]$

Let us consider the subconfiguration

$D_{0}:=\\{L_{z},~{}\allowbreak L_{y},~{}\allowbreak L_{x},~{}\allowbreak
E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
Q_{1},~{}\allowbreak L_{1},~{}\allowbreak Q_{2},~{}\allowbreak
L_{2},~{}\allowbreak E_{5},~{}\allowbreak E_{6},~{}\allowbreak BT\\}.$

It has $r=14$ curves: 11 $(-2)$-curves, 2 $(-1)$-curves, and 1 $(0)$-curve. It
also has $t_{2}=18$ and $t_{3}=2$, and so $\bar{c}_{1}^{2}=14$ and
$\bar{c}_{2}=6$. We now consider particular blow-ups at singular points of
$D_{0}$, as indicated in Figure 2.

Figure 2. The blow-up $\sigma\colon X\to Z$

In $X$ we can find two Wahl chains corresponding to the Hirzebruch-Jung
continued fractions $[5,\allowbreak 2]$, and $[3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 6,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 2]$. Let $\pi\colon X\to W$ the contraction of these two Wahl
chains, so $W$ is a normal projective surface with two Wahl singularities
[Art62].

######  Proposition 5.2.

The canonical divisor $K_{W}$ is ample, and $K_{W}^{2}=5$.

###### Proof.

We have $K_{X}^{2}+2+14=K_{W}^{2}$, $K_{X}^{2}=K_{Z}^{2}-11=-11$, so
$K_{W}^{2}=5$.

Moreover, $K_{X}\sim\sigma^{*}(-F)+\sum_{i=10}^{20}E_{i}+E_{16}+E_{18}$, where
$F$ is a fiber of $Z\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$. We note that

$-\frac{Q_{1}}{2}-\frac{L_{1}}{2}-E_{17}-\frac{3E_{18}}{2}-E_{15}-\frac{3E_{16}}{2}-\frac{E_{19}}{2}\equiv\sigma^{*}\left(-\frac{F}{2}\right)\equiv-\frac{Q_{2}}{2}-\frac{L_{2}}{2}-E_{20},$

and so

$K_{X}\equiv-\frac{Q_{1}}{2}-\frac{L_{1}}{2}-\frac{Q_{2}}{2}-\frac{L_{2}}{2}+\sum_{i=10}^{14}E_{i}+\frac{E_{16}}{2}+\frac{E_{18}}{2}+\frac{E_{19}}{2}.$

On the other hand, we have

$\displaystyle\pi^{*}(K_{W})$ $\displaystyle\equiv
K_{X}+\frac{2}{3}Q_{1}+\frac{1}{3}E_{17}+\frac{443}{700}E_{1}+\frac{629}{700}L_{x}+\frac{673}{700}L_{y}+\frac{690}{700}L_{z}+\frac{697}{700}BT$
$\displaystyle\quad+\frac{698}{700}E_{15}+\frac{699}{700}L_{1}+\frac{696}{700}E_{5}+\frac{693}{700}L_{2}+\frac{683}{700}Q_{2}+\frac{656}{700}E_{6}+\frac{585}{700}C$
$\displaystyle\quad+\frac{514}{700}E_{3}+\frac{257}{700}B,$

and so $\pi^{*}(K_{W})$ can be written as a nonnegative rational combination
of curves. By the Nakai-Moishezon criterion, for $K_{W}$ to be ample we need
to check that for each irreducible curve $\Gamma$ in $X$, not exceptional for
$\pi$, we have that $\sigma^{*}(K_{W})\cdot\Gamma>0$. Hence this condition
needs to be checked for curves in the support of $\sigma^{*}(K_{W})$ and we
also must verify that there are no curves disjoint from that support. One
easily verifies the first explicitly. For the second, we point out that this
support of $\sigma^{*}(K_{W})$ contains the curves
$L_{1},E_{15},E_{16},Q_{1},E_{17},E_{18},E_{19}$, and so a $\Gamma$ disjoint
from this support must be contained in a fiber of
$X\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$. But this is not possible in our
example. Therefore $K_{W}$ is ample.

∎

######  Theorem 5.3.

The rational blowdown of $X$ at these two Wahl chains is an exotic (minimal,
symplectic)
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$.

###### Proof.

As $K_{W}$ is ample by the previous proposition, by Theorem 2.3, the rational
blowdown $Y$ of $\operatorname{Exc}(\pi)$ is a minimal symplectic 4-manifold.
By Corollary 2.5, $Y$ is an exotic
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
provided that the intersection form is odd and $Y$ is simply-connected. The
intersection form is indeed odd as $K_{Y}^{2}=5$. For the fundamental group we
apply an analogous computation with Seifert-Van Kampen’s theorem as in [LP07]:
Consider some simple loops $\alpha$ around $E_{17}$, and $\beta$ around
$L_{1}$. They are homotopic via the sphere $E_{18}$. As gcd$(3,700)=1$, we see
that $\alpha$ and $\beta$ are trivial in the complement of the Wahl chains. As
$\alpha$ is a generator for the complement of $Q_{1}\cup E_{17}$, the only
thing left is to check that a generator for the other Wahl chain is also
trivial. Let $\gamma$ be the generator defined by a loop around $E_{1}$. Then,
following Mumford’s computation [Mum61], we obtain that $\beta$ and
$\gamma^{493}$ are homotopic. But gcd$(493,700)=1$, so $\beta$ is also a
generator, but we already saw that it is trivial in the complement of the Wahl
chains. ∎

######  Proposition 5.4.

The surface $W$ satisfies $H^{1}(W,T_{W})=0$ and
$H^{2}(T_{W})={\mathbb{C}}^{2}$.

###### Proof.

To compute $H^{2}(T_{W})$, we first start with the elliptic fibration
$Z\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$ and consider the singular fibers
$E_{1}+L_{x}+L_{y}+L_{z}$, $Q_{1}+L_{1}$, and $Q_{2}+L_{2}$. We have that
$H^{2}(Z,T_{Z}(-\log(E_{1}+L_{x}+L_{y}+L_{z}+Q_{1}+L_{1}+Q_{2}+L_{2})))={\mathbb{C}}$
by Lemma 4.3. By the (-2) principle we can add the curves $C+E_{3}+B$ keeping
this dimension; by the (-1) principle we can add $E_{5}+E_{6}$ keeping again
the dimension. We now blow-up $B_{1}$ and $T_{2}$. The curve $BT$ becomes a
$(-2)$-curve. Thus when we add $BT$ we may keep the dimension or increase it
by $1$. By Theorem 4.1, we have that $h^{2}(T_{W})=h^{1}(T_{W})+2$, and so
$h^{2}(T_{W})\geq 2$. Therefore $h^{2}(T_{W})=2$ (in particular, the addition
of $BT$ indeed increases the dimension by $1$), and $h^{1}(T_{W})=0$. ∎

In this case, we have the exact sequence

$0\to{\mathbb{T}}^{1}_{QG,W}\to
H^{0}(W,\mathcal{T}^{1}_{QG,W})\simeq{\mathbb{C}}^{2}\to
H^{2}(W,T_{W})\simeq{\mathbb{C}}^{2}\to{\mathbb{T}}^{2}_{QG,W}\to 0,$

and ${\mathbb{T}}^{1}_{QG,W}={\mathbb{T}}^{2}_{QG,W}={\mathbb{C}}^{m}$ with
$m=0,1$ or $2$. We do not know if $W$ admits a ${\mathbb{Q}}$-Gorenstein
smoothing. If it did, then it would be the first example of a simply-connected
surface of general type with $p_{g}=0$ and $K^{2}=5$.

The following examples also define exotic (minimal, symplectic)
${\mathbb{C}}{\mathbb{P}}^{2}\\#4\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ from
surfaces with two Wahl chains, one of them equal to $[3,\allowbreak
4,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak
6,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 3,\allowbreak 2]$. Those examples suggest that there should be
at least a one dimensional equisingular family of such surfaces.

(2) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{X},~{}\allowbreak
B,~{}\allowbreak E_{3},~{}\allowbreak Q_{1},~{}\allowbreak
L_{1},~{}\allowbreak Q_{2},~{}\allowbreak L_{2},~{}\allowbreak
E_{2},~{}\allowbreak E_{5},~{}\allowbreak E_{8},~{}\allowbreak
BT,~{}\allowbreak TB,~{}\allowbreak BB,~{}\allowbreak TT,~{}\allowbreak N\\}$
\- $~{}Q_{1}\cap L_{1}\cap TB\cap TT\cap N$, $~{}B_{1}:=Q_{1}\cap L_{1}\cap
BT\cap BB$, $~{}Q_{2}\cap L_{2}\cap BT\cap TT$, $~{}Q_{2}\cap L_{2}\cap TB\cap
BB$, $~{}L_{Z}\cap E_{8}$, $~{}L_{Z}\cap BT$, $~{}B\cap E_{3}$, $~{}Q_{1}\cap
B_{1}$, $~{}Q_{2}\cap N$, $~{}B_{1}\cap BT$, $~{}L_{X}\cap TB$,
$[2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 2,\allowbreak 1,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 3]\times~{}Q_{2}\cap E_{2}$ \- $(700,257):[3,\allowbreak
4,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak
6,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 3,\allowbreak 2]$, $(493,181):[3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 5,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 2]$

(3) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{Y},~{}\allowbreak
L_{X},~{}\allowbreak E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak
E_{3},~{}\allowbreak Q_{1},~{}\allowbreak L_{1},~{}\allowbreak
Q_{2},~{}\allowbreak L_{2},~{}\allowbreak E_{5},~{}\allowbreak
E_{6},~{}\allowbreak BT\\}$ \- $~{}B_{1}:=Q_{1}\cap L_{1}\cap BT$,
$~{}Q_{2}\cap L_{2}\cap BT$, $~{}L_{Z}\cap E_{1}$, $~{}L_{Y}\cap E_{5}$,
$~{}L_{X}\cap E_{6}$, $~{}L_{X}\cap BT$, $~{}E_{3}\cap BT$, $~{}L_{1}\cap
E_{6}$, $~{}L_{1}\cap B_{1}$, $[2,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 2,\allowbreak 1,\allowbreak
3,\allowbreak 4,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak
2]\times~{}Q_{1}\cap L_{1}$ \- $(700,257):[3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 6,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 2]$, $(700,257):[3,\allowbreak 4,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 6,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 2]$

Moreover, we have found 5 other examples from other subconfigurations of $D$.

(4) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{Y},~{}\allowbreak
L_{X},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
Q_{1},~{}\allowbreak L_{1},~{}\allowbreak Q_{2},~{}\allowbreak
L_{2},~{}\allowbreak E_{2},~{}\allowbreak E_{5},~{}\allowbreak
E_{6},~{}\allowbreak BT,~{}\allowbreak BB\\}$ \- $~{}Q_{1}\cap L_{1}\cap
BT\cap BB$, $~{}Q_{2}\cap L_{2}\cap BT$, $~{}Q_{2}\cap L_{2}\cap BB$,
$~{}L_{Y}\cap E_{5}$, $~{}L_{Y}\cap BB$, $~{}L_{X}\cap BT$, $~{}L_{1}\cap
E_{6}$, $~{}C\cap BB$, $[2,\allowbreak 2,1]\times~{}BT\cap E_{3}$,
$[2,\allowbreak 2,1]\times~{}C\cap E_{6}$ \- $(256,75):[4,\allowbreak
2,\allowbreak 4,\allowbreak 2,\allowbreak 3,\allowbreak 5,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 4,\allowbreak 2,\allowbreak 2]$,
$(17,5):[4,\allowbreak 2,\allowbreak 5,\allowbreak 4,\allowbreak 2,\allowbreak
2]$

(5) $K^{2}=5$ \- $\\{L_{Y},~{}\allowbreak E_{1},~{}\allowbreak
A,~{}\allowbreak E_{3},~{}\allowbreak Q_{1},~{}\allowbreak
L_{1},~{}\allowbreak Q_{2},~{}\allowbreak L_{2},~{}\allowbreak
E_{6},~{}\allowbreak E_{7},~{}\allowbreak E_{8},~{}\allowbreak
BT,~{}\allowbreak TB,~{}\allowbreak BB,~{}\allowbreak TT,~{}\allowbreak N\\}$
\- $~{}Q_{1}\cap L_{1}\cap TB\cap TT\cap N$, $~{}B_{1}:=Q_{1}\cap L_{1}\cap
BT\cap BB$, $~{}Q_{2}\cap L_{2}\cap BT\cap TT$, $~{}Q_{2}\cap L_{2}\cap TB\cap
BB$, $~{}L_{Y}\cap N$, $~{}A\cap BT$, $~{}Q_{1}\cap E_{7}$, $~{}B_{1}\cap BT$,
$~{}L_{Y}\cap BB$, $~{}Q_{2}\cap N$, $[2,\allowbreak 2,1]\times~{}Q_{1}\cap
B_{1}$, $[2,\allowbreak 2,1]\times~{}Q_{2}\cap N$ \- $(82,25):[4,\allowbreak
2,\allowbreak 2,\allowbreak 3,\allowbreak 6,\allowbreak 2,\allowbreak
3,\allowbreak 5,\allowbreak 2,\allowbreak 2]$, $(59,18):[4,\allowbreak
2,\allowbreak 2,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak
5,\allowbreak 2,\allowbreak 2]$

(6) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{X},~{}\allowbreak
E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
Q_{1},~{}\allowbreak L_{1},~{}\allowbreak Q_{2},~{}\allowbreak
L_{2},~{}\allowbreak E_{2},~{}\allowbreak E_{5},~{}\allowbreak
TB,~{}\allowbreak TT,~{}\allowbreak N\\}$ \- $~{}Q_{1}\cap L_{1}\cap TB\cap
TT\cap N$, $~{}Q_{2}\cap L_{2}\cap TT$, $~{}Q_{2}\cap L_{2}\cap TB$,
$~{}L_{Z}\cap N$, $~{}E_{1}\cap E_{2}$, $~{}C\cap N$, $~{}E_{3}\cap TB$,
$~{}B\cap TT$, $[2,1]\times~{}L_{X}\cap E_{1}$, $[2,1]\times~{}N\cap Q_{2}$ \-
$(89,34):[3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2]$,
$(26,11):[3,\allowbreak 2,\allowbreak 3,\allowbreak 2,\allowbreak
6,\allowbreak 4,\allowbreak 2]$

(7) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{X},~{}\allowbreak
E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
Q_{1},~{}\allowbreak L_{1},~{}\allowbreak Q_{2},~{}\allowbreak
L_{2},~{}\allowbreak E_{5},~{}\allowbreak TB,~{}\allowbreak BB,~{}\allowbreak
N\\}$ \- $~{}Q_{1}\cap L_{1}\cap TB\cap N$, $~{}B_{1}:=Q_{1}\cap L_{1}\cap
BB$, $~{}Q_{2}\cap L_{2}\cap TB\cap BB$, $~{}L_{Z}\cap N$, $~{}L_{X}\cap
E_{1}$, $~{}C\cap N$, $~{}E_{3}\cap TB$, $~{}B_{1}\cap BB$,
$[2,1]\times~{}B\cap BB$, $[2,\allowbreak 2,1]\times~{}N\cap Q_{2}$ \-
$(89,34):[3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2]$,
$(37,11):[4,\allowbreak 2,\allowbreak 3,\allowbreak 2,\allowbreak
6,\allowbreak 4,\allowbreak 2,\allowbreak 2]$

(8) $K^{2}=5$ \- $\\{L_{Z},~{}\allowbreak L_{X},~{}\allowbreak
E_{1},~{}\allowbreak B,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
Q_{1},~{}\allowbreak L_{1},~{}\allowbreak Q_{2},~{}\allowbreak
L_{2},~{}\allowbreak E_{2},~{}\allowbreak E_{5},~{}\allowbreak
TB,~{}\allowbreak TT,~{}\allowbreak N\\}$ \- $~{}Q_{1}\cap L_{1}\cap TB\cap
TT\cap N$, $~{}Q_{2}\cap L_{2}\cap TT$, $~{}Q_{2}\cap L_{2}\cap TB$,
$~{}L_{Z}\cap N$, $~{}E_{1}\cap E_{2}$, $~{}E_{1}\cap TT$, $~{}C\cap E_{3}$,
$~{}C\cap N$, $[2,\allowbreak 2,1]\times~{}L_{X}\cap TB$, $[2,1]\times~{}N\cap
Q_{2}$ \- $(111,31):[4,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak 2,\allowbreak 2]$,
$(26,11):[3,\allowbreak 2,\allowbreak 3,\allowbreak 2,\allowbreak
6,\allowbreak 4,\allowbreak 2]$

##  6\. Exotic
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
for $b^{-}=9,8,7$

In [RU21], we constructed exotic
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#b^{-}\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$
for $b^{-}=18,17,\ldots,10$ (see also [PPS13]). They are actually (simply-
connected) complex surfaces of general type with $K^{2}=19-b^{-}$, and
$p_{g}=1$. In this section, we show how to go below $b^{-}=10$, providing
several examples for $b^{-}=9,8,7$. We do not see any constraints to getting
below $7$, but we have not yet been able to produce those examples. In
addition, we can find many surfaces $W$ for $b^{-}\geq 10$ (thousands), but
very few for $b^{-}<10$. As a matter of fact, we have found thousands of
surfaces $W$ for $b^{-}\geq 10$, but only a few for $b^{-}<10$. We note that
there exist exotic examples for $b^{-}=6,5,4$ by other means (see e.g. [AP08,
ABP08, AP10, BK17]), but this seems to be an open question for $b^{-}\leq 3$.

All of our examples will be constructed from a single nodal simple crossings
configuration $D$ in a particular K3 surface $Z$, which admits an elliptic
fibration $Z\to{\mathbb{P}}_{{\mathbb{C}}}^{1}$ with singular fibers of type
$2I_{8}+2I_{1}+I_{2}+I_{4}$. We will construct $Z$ via a degree two base
change of a rational elliptic fibration
$Z_{0}\to{\mathbb{P}}_{\mathbb{C}}^{1}$ with singular fibers
$I_{8}+2I_{1}+I_{2}$, branched at an $I_{1}$ and the $I_{2}$. the
configuration $D$ will then consist of $36$ rational curves in $Z$.

Let us consider the pencil of cubics in ${\mathbb{P}}^{2}_{{\mathbb{C}}}$

$\\{4\mu xyz+\nu(z-y)(x^{2}-yz)=0\colon\
[\mu,\nu]\in{\mathbb{P}}^{1}_{{\mathbb{C}}}\\}.$

Its base points consist of $[0,0,1]$ (triple), $[1,0,0]$ (double), $[0,1,1]$,
and $[0,1,0]$ (triple). Let us name: $L_{1}=\\{y=0\\}$, $L_{2}=\\{x=0\\}$,
$L_{3}=\\{z=0\\}$, $C=\\{x^{2}-yz=0\\}$, and $L=\\{z-y=0\\}$. The blow-up at
the nine base points defines an elliptic fibration $f_{0}\colon
Z_{0}\to{\mathbb{P}}^{1}_{{\mathbb{C}}}$. Denote the exceptional curves of the
blow-up as: $E_{1},E_{2},E_{3}$ for $[0,0,1]$, $E_{4},E_{5}$ for $[1,0,0]$,
$E_{6}$ for $[0,1,1]$, and $E_{7},E_{8},E_{9}$ for $[0,1,0]$. The elliptic
fibration $f_{0}$ has singular fibers:

  * $I_{8}$:

$L_{2}+E_{7}+E_{8}+L_{3}+E_{4}+L_{1}+E_{2}+E_{1}$,

  * $I_{2}$:

$C+L$,

  * $I_{1}$:

$F_{1}$ defined by $4xyz+(z-y)(x^{2}-yz)=0$ with node at $[1,-1,1]$,

  * $I_{1}$:

$F_{2}$ defined by $4xyz-(z-y)(x^{2}-yz)=0$ with node at $[1,1,-1]$.

It has four sections: $E_{3},E_{6},E_{5},E_{9}$. Let us also consider the
double sections of $f_{0}$ (which are $0$-curves): $M$: $z+y=0$, $N$:
$x^{2}+yz=0$, $MR_{1}$: $x-y=0$, $MR_{2}$: $x+y=0$, $ML_{1}$: $x-z=0$, and
$ML_{2}$: $x+z=0$. The intersections among all of these curves are shown in
Figure 3.

Figure 3. A configuration of 22 rational curves in $Z_{0}$

Let $D_{0}$ be the configuration formed by the 12 singular fibers, the 4
sections, and the 6 double sections defined above. It is a nodal simple
crossings configuration with $r=22$, $\nu=2$, $t_{2}=48$, $t_{4}=4$,
$t_{6}=2$, and $t_{k}=0$ for all other $k$. We compute $\bar{c}_{1}^{2}=96$,
$\bar{c}_{2}=38$, and so $\bar{c}_{1}^{2}/\bar{c}_{2}=48/19\approx 2.53$.

Let us consider the base change of $f_{0}$ of order $2$ branched over the
singular fibers $F_{2}$ and $C+L$. Let $g\colon Z\to Z_{0}$ be the
corresponding composition of the double cover and the minimal resolution over
the nodes at $F_{2}$ and at $L+C$. Then we obtain an elliptic fibration
$f\colon Z\to Z_{0}$ (where $Z$ is a K3 surface) with singular fibers 2
$I_{8}$ (pre-image of $I_{8}$), $I_{2}$ (pre-image of $F_{2}$), $I_{4}$ (pre-
image of $L+C$), and 2 $I_{1}$ (pre-image of $F_{1}$). Let $D$ be the pre-
image of $D_{0}$ under $g$, still a nodal simple crossings configuration. In
Figure 4, we give notations for all the 36 rational curves in $D$ and we show
how they intersect.

Figure 4. A configuration of 36 rational curves in $Z$

Note that the pre-image of $MR_{1}$ and $ML_{2}$ split into two sections each.
We denote those pre-images by $MR_{1},MR_{1}^{\prime}$ and
$ML_{2},ML_{2}^{\prime}$ respectively. A similar convention is used for
components of the $I_{8}$’s fibers. We use $F_{3}$ and $F_{4}$ for the pre-
image of $F_{1}$. The rest of the curves keep the same names. The
configuration $D$ has $r=36$, $\nu=2$, $t_{2}=102$, $t_{6}=2$, and $t_{k}=0$
for other $k$. The Chern numbers of $(Z,D)$ are $\bar{c}_{1}^{2}=160$, and
$\bar{c}_{2}=64$, so $\bar{c}_{1}^{2}/\bar{c}_{2}=2.5$.

In what follows, we will show subconfigurations of $D$ which produce the
claimed examples for $K^{2}=10,11,12$. As in the previous section, we will
present each of them by means of a box with the necessary information to
construct it. We will develop details for some of them.

$K^{2}=10$

(9) $K^{2}=10$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
F_{3},~{}\allowbreak E_{3},~{}\allowbreak E_{9},~{}\allowbreak
ML_{1},~{}\allowbreak MR_{1}\\}$ \- $~{}F_{3}\cap F_{3}\cap ML_{1}$,
$~{}E_{7}\cap ML_{1}$, $~{}E_{8}\cap E_{9}$, $~{}L_{1}\cap ML_{1}$,
$~{}E_{2}\cap E_{1}$, $~{}E^{\prime}_{8}\cap L^{\prime}_{3}$,
$~{}L^{\prime}_{1}\cap E^{\prime}_{2}$, $~{}F_{3}\cap E_{3}$, $~{}E_{1}\cap
MR_{1}$, $[2,\allowbreak 2,1]\times~{}E^{\prime}_{8}\cap E^{\prime}_{7}$,
$[3,\allowbreak 5,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak
2,\allowbreak 1,\allowbreak 4,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 2,\allowbreak 3]\times~{}E^{\prime}_{4}\cap
L^{\prime}_{3}$ \- $(19843,5873):[4,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak 5,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 5,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$, $(571,169):[4,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak
3,\allowbreak 5,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak
4,\allowbreak 2,\allowbreak 2]$

We will explain the configuration defined in box (9). It consists of the 2
$I_{8}$ fibers, the $I_{1}$ fiber $F_{3}$, the sections $E_{3},MR_{1},E_{9}$,
and the double section $ML_{1}$. It has $r=21$, $\nu=1$, $t_{2}=29$,
$t_{3}=1$, and $t_{k}=0$ for other $k$. We have Chern numbers
$\bar{c}_{1}^{2}=21$ and $\bar{c}_{2}=13$, and so
$\bar{c}_{1}^{2}/\bar{c}_{2}=21/13\approx 1.62$. All curves and intersections
are shown in Figure 5.

Figure 5. The blow-up $\sigma\colon X\to Z$

Let $\sigma\colon X\to Z$ be the composition of blow-ups shown in Figure 5.
The places an numbers of blow-ups are indicated in the picture. We note that
in $X$ there are two Wahl chains:

$[4,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 2,\allowbreak 5,\allowbreak 5,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak 4,\allowbreak
2,\allowbreak 2],$

and

$[4,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
2,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak 5,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2].$

Let $\pi\colon X\to W$ be the contraction of them.

######  Theorem 6.1.

The rational blowdown of $X$ at these two Wahl chains is an exotic (minimal,
symplectic)
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#9\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$.

###### Proof.

We start computing $K_{W}^{2}=-26+22+14=10$. We can easily write
$\pi^{*}(K_{W})$ as a positive rational sum of rational curves in a similar
way to (1). Therefore, to show that $K_{W}$ is nef, we just intersect
$\pi^{*}(K_{W})$ with all the relevant $(-1)$-curves, which are the red dashed
curves in Figure 5, and check that the intersection is positive. This shows
that $K_{W}$ is big and nef, so we can apply Theorem 2.3 (see Remark 6.2). To
show that it is an exotic
$3{\mathbb{C}}{\mathbb{P}}^{2}\\#9\overline{{\mathbb{C}}{\mathbb{P}}^{2}}$ via
Corollary 2.5, we need to check that the rational blowdown is simply-
connected. For this, we can use the $(-1)$-curve over the intersection of
$E^{\prime}_{4}$ and $L^{\prime}_{3}$ and the fact that the indices of both
Wahl singularities have gcd$(19843,571)=1$. This $(-1)$-curve is joining the
ends of both Wahl chains, so here we can apply the Seifert-Van Kampen strategy
directly. ∎

Below we have two more examples. Details are left to the reader.

(10) $K^{2}=10$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
F_{3},~{}\allowbreak E_{3},~{}\allowbreak E_{9},~{}\allowbreak
ML_{1},~{}\allowbreak MR_{1}\\}$ \- $~{}F_{3}\cap F_{3}\cap ML_{1}$,
$~{}E_{7}\cap ML_{1}$, $~{}E_{4}\cap L_{1}$, $~{}E_{2}\cap E_{3}$,
$~{}E_{1}\cap MR_{1}$, $~{}L^{\prime}_{2}\cap E^{\prime}_{7}$,
$~{}E^{\prime}_{8}\cap L^{\prime}_{3}$, $~{}L^{\prime}_{1}\cap ML_{1}$,
$~{}F_{3}\cap E_{9}$, $[2,1]\times~{}E_{7}\cap E_{8}$,
$[2,1]\times~{}L^{\prime}_{1}\cap E^{\prime}_{2}$ \- $(513,212):[3,\allowbreak
2,\allowbreak 4,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
4,\allowbreak 2]$, $(121,50):[3,\allowbreak 2,\allowbreak 4,\allowbreak
3,\allowbreak 5,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
4,\allowbreak 2]$

(11) $K^{2}=10$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
F_{3},~{}\allowbreak E_{3},~{}\allowbreak E_{9},~{}\allowbreak
ML_{1},~{}\allowbreak MR_{1}\\}$ \- $~{}F_{3}\cap F_{3}\cap ML_{1}$,
$~{}E_{7}\cap E_{8}$, $~{}L_{1}\cap ML_{1}$, $~{}E_{1}\cap MR_{1}$,
$~{}E^{\prime}_{7}\cap ML_{1}$, $~{}E^{\prime}_{8}\cap L^{\prime}_{3}$,
$~{}L^{\prime}_{1}\cap E^{\prime}_{2}$, $~{}F_{3}\cap E_{9}$, $~{}E_{2}\cap
E_{1}$, $[2,\allowbreak 2,1]\times~{}E_{2}\cap E_{3}$, $[2,\allowbreak
2,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 2,\allowbreak
1,\allowbreak 4,\allowbreak 2,\allowbreak 3,\allowbreak 5]\times~{}LL_{1}\cap
EE_{4}$ \- $(139,41):[4,\allowbreak 2,\allowbreak 3,\allowbreak 5,\allowbreak
5,\allowbreak 2,\allowbreak 2,\allowbreak 3,\allowbreak 4,\allowbreak
2,\allowbreak 2]$, $(19309,5695):[4,\allowbreak 2,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 5,\allowbreak
5,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$

######  Remark 6.2.

We point out that examples (9), (10), and (11) come from the same
configurations of curves! We do not know how to explain this. On the other
hand, the canonical class $K_{W}$ is not ample for the 3 examples. But it can
be proved that the proper transforms of $L$ and $P_{2}$ are the only curves
which intersect zero with $K_{W}$. The contraction of this $A_{2}$
configuration $L+P_{1}$ in $W$ gives the canonical model $W^{\prime}$ of $W$.
Therefore $K_{W^{\prime}}$ is ample, and $W^{\prime}$ has 2 Wahl singularities
and 1 Du Val singularity.

$K^{2}=11$

Here we are able to describe 2 examples from the configuration $D$. The two
subconfigurations found in (12) and (13) have no nodal curves, and their
singularities are only nodes. Thus, both of them must satisfy $r=24$ and
$t_{2}=35$, as we require $K_{W}^{2}=11$ and two Wahl chains (see also [RU21,
Proposition 3.1]). Therefore, for both of them $\bar{c}_{1}^{2}=22$,
$\bar{c}_{2}=11$, so $\bar{c}_{1}^{2}/\bar{c}_{2}=2$.

One verifies that $K_{W}$ is ample using the Nakai-Moishezon criterion as
before. We can also easily compute $\pi_{1}(W\setminus\operatorname{Sing}W)=1$
as the indices of the Wahl singularities are coprime, and because there is a
${\mathbb{P}}_{{\mathbb{C}}}^{1}$ connecting the ends of the Wahl chains. In
addition, we can also compute the dimension of obstruction. By [RU21,
Proposition 2.8], the dimension of $H^{2}(W,T_{W})$ is equal to the dimension
of the kernel of the intersection matrix of the subconfiguration. It can be
checked then, that $h^{2}(T_{W})=4$, and so $h^{1}(T_{W})=0$ by Theorem 4.1
and $W$ is equisingularly rigid.

(12) $K^{2}=11$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
B_{1},~{}\allowbreak B_{2},~{}\allowbreak L,~{}\allowbreak
E_{3},~{}\allowbreak E_{5},~{}\allowbreak E_{6},~{}\allowbreak
MR_{1},~{}\allowbreak ML_{2}\\}$ \- $~{}L_{2}\cap E_{1}$, $~{}E_{7}\cap
ML_{2}$, $~{}E_{4}\cap E_{5}$, $~{}E_{2}\cap E_{3}$, $~{}L^{\prime}_{3}\cap
E^{\prime}_{4}$, $~{}L^{\prime}_{1}\cap E^{\prime}_{2}$, $~{}B_{1}\cap B_{2}$,
$~{}B_{1}\cap E_{5}$, $~{}B_{1}\cap E_{6}$, $~{}B_{2}\cap MR_{1}$, $~{}L\cap
E_{6}$, $[2,1]\times~{}E^{\prime}_{7}\cap L^{\prime}_{2}$, $[3,\allowbreak
3,\allowbreak 2,\allowbreak 2,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 2,\allowbreak 1,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak 5,\allowbreak
3,\allowbreak 3]\times~{}L^{\prime}_{2}\cap E_{6}$ \-
$(58441,21457):[3,\allowbreak 4,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 5,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 6,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
2,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 2]$,
$(42249,15512):[3,\allowbreak 4,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 5,\allowbreak 3,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 2]$

(13) $K^{2}=11$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
B_{1},~{}\allowbreak E_{3},~{}\allowbreak E_{5},~{}\allowbreak
E_{9},~{}\allowbreak M,~{}\allowbreak MR_{1},~{}\allowbreak
MR^{\prime}_{1},~{}\allowbreak ML_{2}\\}$ \- $~{}L_{2}\cap E_{7}$,
$~{}E_{8}\cap L_{3}$, $~{}E_{4}\cap L_{1}$, $~{}E_{4}\cap M$, $~{}E_{2}\cap
E_{1}$, $~{}L^{\prime}_{2}\cap E^{\prime}_{1}$, $~{}E^{\prime}_{8}\cap
L^{\prime}_{3}$, $~{}E^{\prime}_{4}\cap E_{5}$, $~{}E^{\prime}_{4}\cap M$,
$~{}E^{\prime}_{2}\cap E_{3}$, $~{}B_{1}\cap E_{9}$, $[2,1]\times~{}ML_{2}\cap
L^{\prime}_{1}$, $[2,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 2,\allowbreak 1,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 3]\times~{}E^{\prime}_{4}\cap
L^{\prime}_{3}$ \- $(88889,33952):[3,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 5,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2]$,
$(51584,19703):[3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 6,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
4,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 2]$

$K^{2}=12$

Surprisingly, the next examples for $K^{2}=12$ (i.e. $b^{-}=7$) come from
nodal configurations of ${\mathbb{P}}_{{\mathbb{C}}}^{1}$’s as well. Moreover,
the first two (14), (15) share the same subconfiguration of $D$, and the same
happens with (16), (17). As we said before, this phenomenon seems to happen
frequently and we have no explanation for it. Since these are nodal
configurations of ${\mathbb{P}}_{{\mathbb{C}}}^{1}$’s in a K3 surface,
$K_{W}^{2}=12$, and there are two Wahl chains, they must have $r=26$,
$t_{2}=38$, $\bar{c}_{1}^{2}=24$, and $\bar{c}_{2}=10$, so
$\bar{c}_{1}^{2}/\bar{c}_{2}=2.4$ for each of them. One can easily check that
$K_{W}$ is ample in all cases, and that the rational blowdown is indeed
simply-connected, as we have coprime indices and the usual Seifert-Van Kampen
tricks work out. Moreover, as in the previous $K^{2}=11$ case, we can compute
$h^{2}(T_{W})=6$ and $h^{1}(T_{W})=0$. Hence these $W$ are equisingularly
rigid.

(14) $K^{2}=12$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
B_{2},~{}\allowbreak L,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
E_{5},~{}\allowbreak E_{6},~{}\allowbreak E_{9},~{}\allowbreak
MR_{2},~{}\allowbreak MR_{1},~{}\allowbreak ML_{2}\\}$ \- $~{}L_{2}\cap
E_{6}$, $~{}E_{7}\cap ML_{2}$, $~{}L_{3}\cap E_{4}$, $~{}E_{2}\cap E_{1}$,
$~{}E_{1}\cap MR_{2}$, $~{}L^{\prime}_{2}\cap E^{\prime}_{1}$,
$~{}E^{\prime}_{8}\cap E_{9}$, $~{}L^{\prime}_{3}\cap MR_{2}$,
$~{}L^{\prime}_{3}\cap MR_{1}$, $~{}E^{\prime}_{4}\cap E_{5}$,
$~{}E^{\prime}_{2}\cap E_{3}$, $~{}E_{4}\cap E_{5}$, $[2,\allowbreak
2,1]\times~{}E_{7}\cap E_{8}$, $[2,\allowbreak 2,1]\times~{}E^{\prime}_{2}\cap
L^{\prime}_{1}$ \- $(2687,795):[4,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$, $(436,129):[4,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak 4,\allowbreak
2,\allowbreak 2]$

(15) $K^{2}=12$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak
B_{2},~{}\allowbreak L,~{}\allowbreak C,~{}\allowbreak E_{3},~{}\allowbreak
E_{5},~{}\allowbreak E_{6},~{}\allowbreak E_{9},~{}\allowbreak
MR_{2},~{}\allowbreak MR_{1},~{}\allowbreak ML_{2}\\}$ \- $~{}L_{2}\cap
E_{6}$, $~{}E_{7}\cap ML_{2}$, $~{}E_{8}\cap E_{9}$, $~{}L_{3}\cap E_{4}$,
$~{}E_{2}\cap E_{1}$, $~{}E_{1}\cap MR_{2}$, $~{}L^{\prime}_{2}\cap
E^{\prime}_{1}$, $~{}E^{\prime}_{8}\cap E_{9}$, $~{}L^{\prime}_{3}\cap
MR_{2}$, $~{}L^{\prime}_{3}\cap MR_{1}$, $~{}E^{\prime}_{4}\cap E_{5}$,
$~{}E^{\prime}_{2}\cap E_{3}$, $[2,\allowbreak 2,1]\times~{}E^{\prime}_{2}\cap
L^{\prime}_{1}$, $[3,\allowbreak 4,\allowbreak 3,\allowbreak 4,\allowbreak
2,\allowbreak 2,\allowbreak 1,\allowbreak 4,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 2,\allowbreak 3]\times~{}E_{7}\cap E_{8}$ \-
$(263303,77905):[4,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 2,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 4,\allowbreak 3,\allowbreak 4,\allowbreak 3,\allowbreak
4,\allowbreak 2,\allowbreak 2]$, $(436,129):[4,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak 5,\allowbreak
3,\allowbreak 4,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$

(16) $K^{2}=12$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak L,~{}\allowbreak
P_{1},~{}\allowbreak C,~{}\allowbreak P_{2},~{}\allowbreak
E_{3},~{}\allowbreak E_{5},~{}\allowbreak E_{6},~{}\allowbreak
MR_{1},~{}\allowbreak MR^{\prime}_{1},~{}\allowbreak ML_{2}\\}$ \-
$~{}L_{2}\cap E_{7}$, $~{}L_{3}\cap E_{4}$, $~{}E_{2}\cap E_{3}$,
$~{}E_{2}\cap E_{1}$, $~{}L^{\prime}_{2}\cap E_{6}$, $~{}L^{\prime}_{3}\cap
MR_{1}$, $~{}E^{\prime}_{4}\cap L^{\prime}_{1}$, $~{}E^{\prime}_{2}\cap
E^{\prime}_{1}$, $~{}L\cap P_{1}$, $~{}L\cap E_{5}$, $~{}P_{1}\cap
MR^{\prime}_{1}$, $~{}C\cap P_{2}$, $[2,\allowbreak
2,1]\times~{}L^{\prime}_{1}\cap ML_{2}$, $[3,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2,\allowbreak 1,\allowbreak
4,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
3]\times~{}MR_{1}\cap P_{1}$ \- $(266348,78757):[4,\allowbreak 2,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak
2,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
2,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak 4,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$,
$(487,144):[4,\allowbreak 2,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 5,\allowbreak 3,\allowbreak 3,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$

(17) $K^{2}=12$ \- $\\{L_{2},~{}\allowbreak E_{7},~{}\allowbreak
E_{8},~{}\allowbreak L_{3},~{}\allowbreak E_{4},~{}\allowbreak
L_{1},~{}\allowbreak E_{2},~{}\allowbreak E_{1},~{}\allowbreak
L^{\prime}_{2},~{}\allowbreak E^{\prime}_{7},~{}\allowbreak
E^{\prime}_{8},~{}\allowbreak L^{\prime}_{3},~{}\allowbreak
E^{\prime}_{4},~{}\allowbreak L^{\prime}_{1},~{}\allowbreak
E^{\prime}_{2},~{}\allowbreak E^{\prime}_{1},~{}\allowbreak L,~{}\allowbreak
P_{1},~{}\allowbreak C,~{}\allowbreak P_{2},~{}\allowbreak
E_{3},~{}\allowbreak E_{5},~{}\allowbreak E_{6},~{}\allowbreak
MR_{1},~{}\allowbreak MR^{\prime}_{1},~{}\allowbreak ML_{2}\\}$ \-
$~{}E_{7}\cap ML_{2}$, $~{}L_{3}\cap MR^{\prime}_{1}$, $~{}L_{3}\cap E_{4}$,
$~{}E_{2}\cap E_{3}$, $~{}E_{1}\cap MR_{1}$, $~{}L^{\prime}_{2}\cap
E^{\prime}_{1}$, $~{}L^{\prime}_{3}\cap E^{\prime}_{4}$,
$~{}L^{\prime}_{1}\cap E^{\prime}_{2}$, $~{}L\cap E_{5}$, $~{}L\cap P_{1}$,
$~{}P_{1}\cap C$, $~{}L\cap E_{6}$, $[2,\allowbreak 2,1]\times~{}L\cap P_{2}$,
$[2,\allowbreak 2,1]\times~{}E_{6}\cap L_{2}$ \- $(4,1):[6,\allowbreak
2,\allowbreak 2]$, $(267721,78962):[4,\allowbreak 2,\allowbreak 3,\allowbreak
5,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 3,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 5,\allowbreak
3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak 3,\allowbreak
4,\allowbreak 3,\allowbreak 3,\allowbreak 2,\allowbreak 2,\allowbreak
3,\allowbreak 4,\allowbreak 2,\allowbreak 2]$

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