# The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$

Klaus Hulek  Institut für Algebraische Geometrie, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany<EMAIL_ADDRESS>and  Carsten Liese  Institut für Algebraische Geometrie, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in
degree $2$ as well as the associated secondary fan. The main result is an
enumeration of all maximal dimensional cones of the two fans.

###### Contents

  1. 0 Introduction
  2. 1 The Dolgachev-Nikulin-Voisin mirror of degree $2d$: definition and construction
    1. 1.1 The Dolgachev-Nikulin-Voisin mirror of degree $2d$
    2. 1.2 Construction of Models
  3. 2 Relative Notions and the Mori fan
    1. 2.1 Relative Notions
      1. 2.1.1 Cones
    2. 2.2 The Mori fan
      1. 2.2.1 Interior facets
      2. 2.2.2 Action of $\operatorname{Bir}(\mathcal{Y}/S)$
  4. 3 $(-1)$-Curves and Curve Structures
    1. 3.1 $(-1)$-Curves
    2. 3.2 Curve Structures
  5. 4 Projective Models
    1. 4.1 Models of class $\mathscr{T}$
    2. 4.2 Models of class $\mathscr{P}$
  6. 5 Flops, Birational Automorphisms and Orbits
    1. 5.1 Flops
    2. 5.2 Birational Automorphisms
    3. 5.3 Orbits
  7. 6 Counting models
    1. 6.1 Automorphisms
    2. 6.2 Counting models of class $\mathscr{T}$
    3. 6.3 Counting models of class $\mathscr{P}$
      1. 6.3.1 Mori fan
  8. 7 The Secondary fan
    1. 7.1 Preliminaries
    2. 7.2 Flopping along a line
    3. 7.3 Reduction

##  0\. Introduction

To construct modular compactifications of the moduli space $\mathcal{F}_{2d}$
of polarized $K3$ surfaces of degree $2d$ is a notoriously difficult problem.
This has been studied from various aspects, such as Hodge theory, locally
symmetric domains, GIT and log-geometry. For small degree $d=1,2,3$ these $K3$
surfaces can be studied via concrete geometric models, namely $2:1$ covers of
the projective plane $\mathbb{P}^{2}$ branched along a sextic curve, degree
$4$ surfaces in $\mathbb{P}^{3}$ and complete $(2,3)$ intersections in
$\mathbb{P}^{4}$ respectively. Various authors have used this approach to
construct compactifications of $\mathcal{F}_{2d}$ in these degrees and to
relate the various models to each other. Here we would like to mention in
particular the work of Shah [Sha80]), Friedman [Fri83], [Loo86]), Laza-O’Grady
[LO19], [LO18], [LO17], and Laza [Laz16], Thompson [Tho14] and Alexeev-Engel-
Thompson [AET08].

Some years ago Gross, Hacking, Keel and Siebert [GHKS] introduced a new
approach. This is based on two main concepts: mirror symmetry and the minimal
model program (MMP). They start by considering the mirror family of
$\mathcal{F}_{2d}$. This is a $1$-dimensional family of lattice polarized $K3$
surfaces which in the literature is called the Dolgachev-Nikulin-Voisin family
of degree $2d$. The base of this family is a modular curve which, if $d$ is
squarefree, has exactly one cusp. The lattice polarization is given by the
lattice $\check{M}_{2d}=U\oplus 2E_{8}(-1)\oplus\langle-2d\rangle$ where $U$
is the hyperbolic plane and $E_{8}(-1)$ is the negative defnite $E_{8}$
lattice. The first step in their programme is to extend the Dolgachev-Nikulin-
Voisin family over the cusp and to consider the various models by which this
can be done. This allows them to define the Mori fan of the Dolgachev-Nikulin-
Voisin family, wich is a fan in
$\operatorname{N}^{1}(\mathcal{Y}/S)_{\mathbb{R}}$, where $\mathcal{Y}\to S$
is a model of the Dolgachev-Nikulin-Voisin family, a scheme of dimension
$19+d$, with base $S=\operatorname{Spec}\mathbb{C}[[t]]$. The second step is
to use a piecewise linear section of the restriction map
$r:\operatorname{Pic}(\mathcal{Y})\to\operatorname{Pic}(\mathcal{Y}_{\eta})$,
where $\mathcal{Y}_{\eta}$ the Dolgachev-Nikulin-Voisin family over a
punctured neighbourhood of the cusp, to obtain a fan in the hyperbolic space
$({\check{M}_{2d}})_{\mathbb{R}}$. An important aspect of their work is the
construction of a universal family (at least over a neighbourhood of the
$0$-dimensional cups). A detailed description of the GHKS programme for $K3$
surfaces is contained in [GHKS]. Yuecheng Zhu [Zhu18] has carried this out in
the case of polarized abelian varieties and has shown that this approach can
be used to recover the second Voronoi compactifiation of $\mathcal{A}_{g}$,
which is known to be a modular compactification by the work of Alexeev [Ale02]
and Olsson [Ols12].

Our aim is to start a concrete investigation of the GHKS approach for small
degree. To be precise, we want to understand the first step in the GHKS
programme in degree $2$. As it turns out this is a nontrivial problem in its
own right. The main result of our paper is a concrete description of the Mori
fan of the Dolgachev-Nikulin-Voisin family in degree $2$. Concretely, we
enumerate the maximal dimensional cones and prove the following:

######  Theorem 0.1.

Let $\mathcal{Y}\to S$ be a model of the Dolgachev-Nikulin-Voisin family. Then
$\operatorname{MF}(\mathcal{Y}/S)$ has $3460$ maximal cones. Of these $753$
are associated to a model of class $\mathscr{T}$ and $2707$ are associated to
a model of class $\mathscr{P}$. The number of orbits of maximal cones of
$\operatorname{MF}(\mathcal{Y}/S)$ under the natural action of the birational
group $\operatorname{Bir}(\mathcal{Y}/S)$ is $588$. The orbits decompose into
$457$ orbits of models of class $\mathscr{P}$ and $131$ orbits of models of
class $\mathscr{T}$.

This is a consequence of Theorems 6.7, 6.13, and 6.15. Here class
$\mathscr{P}$ and class $\mathscr{T}$ refer to the combinatorial structure of
the central fibre, which in turn correspond to the two possible triangulations
of the sphere $S^{2}$ into two triangles. The knowledge of the Mori fan is
necessary for the next step in the GHKS programme: it is used to construct
another fan in a certain $19$-dimensional hyperbolic space and it is this fan
which determines the toroidal compactification of the moduli space of
polarized of polarized $K3$ surfaces.

We also investigate the so called secondary fan which was used by by Hacking,
Keel and Yue [HKYY] and is a generalisation of the secondary fan for toric
varieties due to Gelfand-Kapranov-Zelevinskij [GKZ94]. This is a coarsening of
the Mori fan. Its relevance is that in the del Pezzo case the toric variety
defined by the secondary fan admits a finite morphism to the moduli space of
stable pairs, as was explained in [Kee17]. Note that for toric varieties, the
secondary fan and the Mori fan coincide.

As there is no published proof available that the secondary fan is indeed a
fan in the $K3$ setting, we will include a proof of this fact using the
techniques of our paper and we will also compute its maximal cones:

######  Theorem 0.2.

Let $\mathcal{Y}\to S$ be a model of the Dolgachev-Nikulin-Voisin family of
degree $2$. The secondary fan contains precisely $4$ maximal cones. There are
$2$ orbits of maximal cones under the natural action of
$\operatorname{Bir}(\mathcal{Y}/S)$.

This is Theorem 7.10 and Remark 7.12.

We shall now briefly describe the structure of this paper. We start in Section
1 by recalling the theory of mirror symmetry for lattice polarized $K3$
surfaces, which is due to Dolgachev, Nikulin and Pinkham. We then recapitulate
the basics of the degeneration theory of $K3$ surfaces as developed by
Carlson, Friedman, Kulikov, Person, Pinkham, Scattone and others. This allows
us to define the notion of the Dolgachev-Nikulin-Voisin family of degree
$2d>0$ and their models in Definitions 1.18 and 1.20, following [GHKS]. We
further recall the relation between triangulations of the sphere $S^{2}$ and
$d$-semistable models in $(-1)$-form of the Dolgachev-Nikulin-Voisin family of
degree $2d$ with maximal Picard rank for which we describe the geometry of the
special fibres in detail (Construction 1.23). We also prove that these
surfaces and their maximal smoothings are projective (Propositions 1.25 and
1.26). Finally we discuss the $(-1)$-models in degree $2$ in detail. In
Section 2 we start by recalling some basic facts of the minimal model program
and introduce the main object of this paper, the Mori fan of the Dolgachev-
Nikulin-Voisin family (Definition 2.2). We describe its main properties in
(Proposition 2.4), due to the work of Gross, Hacking, Keel and Siebert. Next
we explain the relationship between interior facets and flops (Proposition
2.9) and finally discuss the action of the group of birational automorphisms
on the Mori fan (Proposition 2.10).

In Section 3 we mostly specialise to degree $2$. Corresponding to the two
triangulations of the sphere $S^{2}$ with two triangles we have two possible
$d$-semistable $K3$ surfaces in $(-1)$-form, which we denote $Y_{\mathscr{P}}$
and $Y_{\mathcal{T}}$ respectively. We shall consider all models whose central
fibres can be transformed by a series of type I flops into $Y_{\mathscr{P}}$
or $Y_{\mathcal{T}}$, and call these models of type $\mathscr{P}$ and type
$\mathcal{T}$ respectively. As we shall see later, see Corollary 5.33, these
are all models in degree $2$. The main object of this section is a detailed
analysis of the configuration of certain curves forming an anticanonical
divisor on components of the central fibre of a given model of the Dolgachev-
Nikulin-Voisin family. This leads to the notion of curve structure, see
Definition 3.9 and will also provide us with a natural $\mathbb{Q}$-basis of
the Picard group of the normalisations of the central fibre (Proposition
3.16). The main application will be existence theorems of ample line bundles
of prescribed degree on the components of the anticanonical divisor
(Propositions 3.24, 3.25 and 3.25 ). This discussion will become vital in
Section 4 where we prove projectivity criteria for models of type
$\mathcal{T}$ (Proposition 4.1) and type $\mathscr{P}$ (Propositions 4.2, 4.4,
4.5 and 4.6 ).

Section 5 is in many ways the technical heart of the paper. Here we analyse
flops between models of the Dolgachev-Nikulin-Voisin family in some detail and
study the action of the birational automorphism group on the Mori fan. For
this we introduce various concepts describing (augmented) curve structures
(see Definitions 5.17, 5.18 and Construction 5.23). This allows us to
establish two crucial facts about models of the Dolgachev-Nikulin-Voisin
family in degree $2$. The first we have already mentioned above, namely that
any such model can be related by type I flops to a model of type $\mathscr{P}$
(or equivalently $\mathcal{T}$) (see Corollary 5.33). The second is that any
two models of the Dolgachev-Nikulin-Voisin family can be transformed into each
other using only type I and type II flops (Corollary 5.34). We note that these
results are specific to degree $2$. Finally, this enables us to study the
action of the birational automorphism group of a model of the Dolgachev-
Nikulin-Voisin family on the Mori fan. In Proposition 5.43 we determine the
possible orbits lengths of maximal cones of this fan under the birational
automorphism group, which can be $1$, $3$ or $6$.

In Section 6 we finally enumerate all models of the Dolgachev-Nikulin-Voisin
family in degree $2$ and determine the maximal cones in the Mori fan (Theorem
6.15). This section is rather combinatorial in nature. To obtain our result we
use the tools which we have developed before, in particular we use curve
structures: these allow us to characterise the isomorphism classes of the
central fibres of projective models of the Dolgachev-Nikulin-Voisin family.
This allows us an explicit enumeration of the models. The second ingredient is
the action of the birational automorphism group and its action on the set of
maximal cones in the Mori fan which we analysed in the previous section. The
main result then follows from a careful enumeration of all models, which we do
for types $\mathscr{P}$ (Theorem 6.8) and $\mathcal{T}$ (Theorem 6.7)
separately. In Section 7 we finally describe the secondary Mori fan. In this
section we give a fairly elementary proof that the secondary fan is indeed a
fan and compute its maximal cones (Theorem 7.10).

Throughout the paper we will work over the complex numbers $\mathbb{C}$.

######  Acknowledgement.

We have benefitted enormously from discussions with Christian Lehn. We thank
Mark Gross, Paul Hacking, Sean Keel, Bernd Siebert and Tony Yu Yue for sharing
drafts of [GHKS] and [HKYY] and for answering our questions. We are grateful
to DFG for financial support under grant Hu 337/7-1.

##  1\. The Dolgachev-Nikulin-Voisin mirror of degree $2d$: definition and
construction

###  1.1. The Dolgachev-Nikulin-Voisin mirror of degree $2d$

We first recall the mirror construction due to Dolgachev [Dol96] and Pinkham
[Pin77] for polarized $K3$ surfaces, using Nikulin’s theory of lattice
polarized $K3$ surfaces [Nik79]. For the basic facts about $K3$ surfaces and
their moduli which we will need, we refer the reader to e.g. [Huy16], [BHPV04,
Section VIII] and [GHS15]. The second cohomology group of a $K3$ surface,
together with the cup product (intersection pairing), define a lattice which
is isomorphic to the $K3$-lattice

$L_{K3}=2E_{8}(-1)\oplus 3U$

where $U$ is the hyperbolic plane and $E_{8}(-1)$ is the even negative
definite unimodular lattice of rank $8$. A polarization on $X$ is an ample
line bundle $\mathcal{L}$ and, since $K3$ surfaces are regular, we can
identify a polarization with its first Chern class $h=c_{1}(\mathcal{L})\in
H^{2}(X,\mathbb{Z})$. We assume $h$ to be primitive and of degree
$h^{2}=2d>0$. We note that the group of isometries of the $K3$ lattice
operates transitively on the set of primitive vectors of given positive
degree. Instead of working with the degree of a polarization we will often
also use its genus, by which we mean the genus of a general element in the
linear system defined by $\mathcal{L}$. The genus $g$ and the degree $d$ are
related by the adjunction formula $2g-2=2d$. Note that degree $2$ coincides
with genus $2$.

The moduli theory of $K3$ surfaces builds on the Torelli theorem. To describe
this, we first notice that the orthogonal complement of $h$ in $L_{K3}$
defines a lattice

$L_{2d}\cong 2E_{8}(-1)\oplus 2U\oplus\langle-2d\rangle.$

We obtain the period domain $\Omega_{2d}$ by

$\Omega_{2d}=\\{x\in\mathbb{P}({L_{2d}}\otimes\mathbb{C})\mid x^{2}=0,\langle
x\bar{x}\rangle>0\\}.$

This is a $19$-dimensional manifold which has two connected components of
which we fix one, say $D_{2d}$. A quasi-polarized $K3$ surface is a pair
$(X,\mathcal{L})$ where $\mathcal{L}$ is big and nef. Recall that a multiple
of $\mathcal{L}$ will embed $X$ as a $K3$ surface with ADE-singularities. It
is a classical application of the Torelli theorem that the moduli space of
degree $2d$ polarized $K3$ surfaces with ADE-singularities is isomorphic to
the quotient

$\mathcal{F}_{2d}=\Gamma_{2d}\backslash\Omega_{2d}=\Gamma^{+}_{2d}\backslash
D_{2d}$

where

$\Gamma_{2d}=\\{g\in\operatorname{O}(L_{K3})\mid g(h)=h\\}$

is the group of all isometries of $L_{K3}$ which fix the polarization $h$ and
$\Gamma^{+}_{2d}$ is the subgroup of elements of real spinor norm $1$ (which
is equivalent to the property that these elements fix the components of
$\Omega_{2d}$.) To obtain the moduli space of all _polarized_ $K3$ surfaces
one has to remove finitely many hyperplanes from $\mathcal{F}_{2d}$, see e.g.
[BHPV04, p. 355].

To simplify the following discussion we will now assume that $d$ is square
free. Then we have a well defined mirror moduli space which was described in
[Dol96, §6]. This parameterizes lattice polarized $K3$ surfaces of Picard rank
$19$ whose Picard lattice is isomorphic to

$\check{M}_{2d}=U\oplus 2E_{8}(-1)\oplus\langle-2d\rangle$

and we note that

(1.1) $L_{2d}=h^{\perp}_{L_{K3}}=\check{M}_{2d}\oplus U.$

By Nikulin’s theory the lattice $\check{M}_{2d}$ has a unique primitive
embedding into the $K3$ lattice $L_{K3}$ (up to isometries), and here we fix
once and for all the obvious embedding, which maps a generator of the summand
$\langle-2d\rangle$ to $e-df$ in a summand $U$, where $e,f$ are a basis of $U$
with $e^{2}=f^{2}=0$ and $e.f=1$. Similar to above, this leads to the moduli
space

$\check{\mathcal{F}}_{2d}=\check{\Gamma}_{2d}\backslash\Omega_{{(\check{M}_{2d})^{\perp}_{L_{K3}}}}$

where

$\check{\Gamma}_{2d}=\\{g\in\operatorname{O}(L_{K3})\mid
g|_{\check{M}_{2d}}=id\\}$

is now the group of all isometries of $L_{K3}$ which restrict to the identity
on $\check{M}_{2d}$. In our case

(1.2) ${{(\check{M}_{2d})^{\perp}_{L_{K3}}}}=U\oplus\langle 2d\rangle$

and this is dual to relation (1.1). The period domain
$\Omega_{{(\check{M}_{2d})^{\perp}_{L_{K3}}}}$ is $1$-dimensional, more
precisely it is two copies of the upper half plane $\mathbb{H}_{1}$ (which are
interchanged by the group $\check{\Gamma}_{2d}$). Hence
$\check{\mathcal{F}}_{2d}$ is a connected (non-compact) modular curve. Mirror
symmetry interchanges the roles of complex moduli and Kähler moduli. This
corresponds to the fact that the mirror moduli space
$\check{\mathcal{F}}_{2d}$ is one dimensional and that the very general $K3$
surface in $\check{\mathcal{F}}_{2d}$ has Picard group $\check{M}_{2d}$.

Since we assumed that $d$ is square-free it follows from Scattone’s
calculations in [Sca87, §4], that there is a unique $0$-dimensional boundary
component in the Baily-Borel compactification of $\mathcal{F}_{2d}$. By
[Dol96, Proposition 7.3] the same is true for $\check{\mathcal{F}}_{2d}$. The
mirror family which we are interested in is the universal family over
$\check{\mathcal{F}}_{2d}$ near the cusp. This requires an explanation. The
moduli spaces $\mathcal{F}_{2d}$ and $\check{\mathcal{F}}_{2d}$ do not carry
universal families in the category of schemes (due to the existence of non-
trivial automorphisms). Nevertheless, this concept can be made precise in the
neighbourhood of the cusp and we will do this below where we define the
Dolgachev-Nikulin-Voisin mirror family in a rigorous way, see Definition 1.18.
In what follows, we will typically work in the following situation. Let
$(R,m)$ be a local complete DVR with residue field $k=R/m$. We will always
assume here that $k=\mathbb{C}$. Let $K=\operatorname{Q}(R)$ be the field of
fractions of $R$. We set $S=\operatorname{Spec}(R)$. Typically we will work
with the completion $R=\hat{\mathcal{O}}_{C,p}$ of the local ring of an affine
curve. We denote by $0=\operatorname{Spec}(k)$ the closed point of $S$ and by
$\eta=\operatorname{Spec}(K)$ the generic point of $S$. It will essentially be
enough to consider the case $R=\mathbb{C}[[t]]$ of formal power series whose
field of fractions is the field $K=\mathbb{C}((t))$ of Laurent series. Indeed,
if $(C,p)$ is a curve germ, then we can choose a local parameter $\pi$, and
this defines an isomorphism of $k$-algebras $R\to\hat{\mathcal{O}}_{C,p}$. If
$\mathcal{Y}\to S$ is a scheme over $S$, then we denote the generic fibre by
${\mathcal{Y}}_{\eta}$ and the special (central) fibre by $\mathcal{Y}_{c}$.
Alternatively, we can also work in the analytic category and consider families
$\mathcal{Y}\to\mathbb{D}$ over the disc and their restriction to the origin
$0\in\mathbb{D}$ and the punctured disc $\mathbb{D}^{*}$ respectively. We will
sometimes use the analytic category in proofs.

In this paper we will use the term normal crossing to denote a scheme which is
locally (not necessarily globally) normal crossing with reduced components. We
say that a normal crossing scheme $Y$ is smoothable if there exists a regular
scheme $\mathcal{Y}$, a proper flat map $\mathcal{Y}\to S$ and an isomorphism
$\mathcal{Y}_{c}\cong Y$. In this case the restriction of the normal bundle of
$\mathcal{Y}_{c}$ in $\mathcal{Y}$ to the singular locus $D$ of
$\mathcal{Y}_{c}$ is trivial:

$N_{D}:=N_{\mathcal{Y}_{c}/\mathcal{Y}}|_{D}=\mathcal{O}_{\mathcal{Y}}(\mathcal{Y}_{c})|_{D}\cong{\mathcal{O}}_{D}.$

The line bundle $N_{D}$ is called the infinitesimal normal bundle and can also
be defined purely in terms of the singular scheme $Y$ by using the normal
bundle of the components of $D$ in the respective components of $Y$, see
[Fri83, §1]. We say that $Y$ is $d$-semistable if $N_{D}$ is trivial which is
a non-trivial condition if we just consider an abstract surface $Y$. The
triviality of $N_{D}$ is a necessary condition for smoothability of $Y$. It
was shown by Friedman that it is also sufficient [Fri83, Theorem 5.10].

We will now recall the basic facts about degenerations of $K3$ surfaces in so
far as they are relevant for us. We will denote by $Y$ a proper normal
crossing surface and by $Y_{i}$ the components of $Y$. If $Y$ is a simple
normal crossing (snc) surface, then the components $Y_{i}$ are smooth. We
will, however, also allow self-intersections of the components and we will
denote the normalisation of a component $Y_{i}$ by $Y^{\nu}_{i}$. As before,
we will denote the singular locus of $Y$ by $D$. We set $D_{ij}=Y_{i}\cap
Y_{j}$ and consider this as a curve on $Y_{i}$. We we also allow $i=j$ and in
this case we mean by $D_{ii}$ is the self-intersection of the component
$Y_{i}$. In our case the curves $D_{ij}$ will always be irreducible.
Intersection numbers $D^{2}_{ij}$ will always be calculated on the
normalisations $Y_{i}^{\nu}$. Note that this may depend on the ordering of
$\\{i,j\\}$.

There are three types of degenerations of $K3$ surfaces, classically called
type I, II and III. The type of a degeneration is a measure of how far the
Hodge structure degenerates. Type I are smooth $K3$ surfaces. The building
blocks of type II degenerations are rational surfaces and the curves $D_{ij}$
consist of elliptic curves (which still carry some Hodge structure). The
components of type III degenerations are rational surfaces, intersecting in
curves whose components are also rational. Another characterization of the
type can be given in terms of the nilpotency of the monodromy, see e.g.
[Sca87, §1.2]. Our interest will be in type III degenerations. The following
definitions are fundamental:

######  Definition 1.1.

[Fri83, 5.5] A $d$-semistable K3 surface of type III is a normal crossing
surface $Y$ such that

  * (i)

$Y$ is $d$-semistable

  * (ii)

$\omega_{Y}=\mathcal{O}_{Y}$

  * (iii)

$Y=\cup Y_{i}$ where each $Y_{i}$ is rational and the preimage of the double
curves $\sum_{j}D_{ij}$ form anticanonical cycles of rational curves on the
normalisation $Y^{\nu}_{i}$

  * (iv)

The dual intersection complex of $Y$ is a triangulation of the $2$-sphere
$S^{2}$.

In the projective situation we make the following

######  Definition 1.2.

A type III degeneration of $K3$ surfaces is a flat, projective scheme
$\mathcal{Y}\to S$ over the spectrum $S$ of a complete DVR, where
$\mathcal{Y}$ is a regular $3$-fold whose generic fibre $\mathcal{Y}_{\eta}$
is a $K3$ surface and whose central fibre $\mathcal{Y}_{c}$ is a type III
$d$-semistable $K3$ surface. We will also refer to such a family
$\mathcal{Y}\to S$ as a _Kulikov model_.

######  Remark 1.3.

Note that the total space $\mathcal{Y}$ is Calabi-Yau, i.e.
$\omega_{\mathcal{Y}}=\mathcal{O}_{\mathcal{Y}}$.

######  Remark 1.4.

By definition the central fibre of a Kulikov model is always projective. We
note, however, that a priori Definition 1.1 does not require $Y$ to be
projective.

######  Remark 1.5.

One can also consider (not necessarly projective) _analytic_ smoothings. In
the analytic category, a type III degeneration of $K3$ surfaces is a morphism
$\mathcal{X}\to\mathbb{D}$, with $\mathbb{D}$ a small disc, such that the
general fibres are smooth $K3$ surfaces and the central fibre is a type III
$d$-semistable $K3$ surface as above. In particular, such an $\mathcal{X}$ is
smooth. Every $d$-semistable surface admits an analytic smoothing by [Fri83,
Theorem 5.10].

Next, we discuss certain modifications of $d$-semistable $K3$ surfaces, the
_elementary modifications_. Recall that an _$F$ -flopping contraction_ of a
threefold $\mathcal{Y}\to S$ with trivial canonicial class, where $F$ is a
$\mathbb{Q}$-Cartier divisor, is a proper birational contraction
$f\colon\mathcal{Y}\to\mathcal{Z}$ to a normal scheme (or complex analytic
space) $\mathcal{Z}\to S$ such that the exceptional locus is of codimension at
least $2$ and the divisor $-F$ is $f$-ample. An _$F$ -flop_ of $\mathcal{Y}\to
S$ is a scheme $\mathcal{Y}^{+}\to S$ together with a proper birational
morphism $f^{+}\colon\mathcal{Y}^{+}\to\mathcal{Z}$ such that the birational
transform $F^{+}$ of $F$ is $\mathbb{Q}$-Cartier on $\mathcal{Y}^{+}$ and
$F^{+}$ is $f^{+}$-ample and the exceptional locus of $f^{+}$ has codimension
at least $2$. The induced birational map
$\phi\colon\mathcal{Y}\dashrightarrow\mathcal{Y}^{+}$ is, by abuse of
language, also called a flop of $\mathcal{Y}$. The situation can be summarized
by the following diagram

$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\textstyle{\mathcal{Y}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{+}}$$\textstyle{\mathcal{Z}.}$

If $\mathcal{Y}\to\mathcal{Z}$ is the contraction of an extremal ray, then the
$F$-flop is independent of the choice of $F$, see [KM98, §6.1]. In this paper,
we will always assume that a flop is given by a contraction of an extremal
ray.

One can also consider flops in the analytic category. Here we will recall
certain types of analytic flops, the elementary modifications. There are three
types of these, classically known as type 0,I and II [FM83b]. Here we will
only be concerned with elementary modifications of type I and II. For the
convenience of the reader we will recall these here. Let $Y$ be a
$d$-semistable $K3$ surface as above. Let $\mathcal{X}\to\mathbb{D}$ be an
_analytic_ smoothing over a small disc. Let $C\subset Y$ be a smooth rational
curve which intersects the double locus $D$ in exactly one point, more
precisely, transversally in a point of some $D_{ij}$ which is a smooth point
of $D$. The curve $C$ lies on a unique component $Y_{i}$ and we assume that
$C^{2}=-1$ on the normalisation $Y_{i}^{\nu}$. Then one can blow up
$\mathcal{X}$ in $C$ and the exceptional divisor will be isomorpic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ with normal bundle
$\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-1,-1)$. The blow-down map
contracts one ruling of the exceptional divisor to $C$. Contracting the other
ruling gives another model $\mathcal{X}^{\prime}\to\mathbb{D}$. The
exceptional divisor contracts to a curve $C^{\prime}$ on $Y_{j}$ and we have
flopped the curve $C$ to $Y_{j}$. The curve $C^{\prime}$ is again a $(-1)$
curve on $Y_{j}^{\nu}$. Here we allow $Y_{j}$ and $Y_{i}$ to coincide. This
defines an _elementary modification of type I_ , see Figure 1. This
construction induces a modification $\psi\colon Y\dashrightarrow Y^{\prime}$
of the central fibre $Y$. Following standard terminology we will also refer to
this induced map on the central fibre as an elementary modification of type
$I$.

$D_{ij}$$C$$C^{\prime}$ Figure 1. An elementary modification of type I

Alternatively, let $C$ be a smooth rational component of the double curve
$\sum_{j}D_{ij}$ with $C^{2}=-1$ on both $Y_{i}^{\nu}$ and $Y_{j}^{\nu}$,
where we again allow the components to coincide. Blow up $\mathcal{X}$ in $C$;
the resulting exceptional divisor will again be
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. As before, we can contract the other
ruling to obtain a degeneration $\mathcal{X}^{\prime}\to\mathbb{D}$, yielding
an _elementary modification of type II_ , see Figure 2. Again, one obtains a
modification $Y^{\prime}$ of the surface $Y$ to which we will also refer to as
elementary modification of type II.

$Y_{j}$$Y_{i}$$Y_{j}$$Y_{i}$$C$$C^{\prime}$ Figure 2. An elementary
modification of type II

By the nature of birational geometry for threefolds, for each Kulikov model,
there are in general many birational Kulikov models. One can, however, pick
out a special class of such models, namely the Kulikov models in $(-1)$-form.

######  Definition 1.6.

Let $Y$ be a type III $d$-semistable $K3$ surface. Then we say that $Y$ is in
$(-1)$-form if for each smooth double curve $D_{ij}=Y_{i}\cap Y_{j}$, we have
$D^{2}_{ij}=D^{2}_{ji}=-1$ and if $D_{ij}$ is singular (and hence a nodal
rational curve) and $Y_{i}$ is singular, then $D^{2}_{ij}=1$ and
$D^{2}_{ji}=-1$.

We will call a Kulikov model $\mathcal{Y}\to S$ with central fibre $Y$ in
$(-1)$-form a _Kulikov model in $(-1)$-form_. Note that by a theorem of
Miranda and Morrison [MM83, Main Theorem 1.2] any _analytic_ type III Kulikov
model can, by a series of elementary modifications of type I and II, be
brought into $(-1)$-form, but that $(-1)$-forms are still not unique. As
discussed above, one can also think of this sequence of modifications as a
sequence of modifications on the central fibre. Thus one may interpret the
theorem of Miranda and Morrison as a result on type III $d$-semistable $K3$
surfaces.

######  Theorem 1.7.

[MM83, Main Theorem 1.2] Let $Y$ be a $d$-semistable $K3$ surface of type III.
Then there is a sequence $Y\dashrightarrow Z$ of elementary modifications of
type I and II such that $Z$ is a $d$-semistable $K3$ surface in $(-1)$-form.

For future use we also want to recall the index of the monodromy. Let $Y$
denote a $d$-semistable $K3$ surface in $(-1)$-form, let $\Gamma$ be its dual
graph, a triangulation of the sphere $S^{2}$. We say $Y$ has _special
$n$-bands of hexagons_ if $\Gamma$ is a refinement of another triangulation
$\Gamma^{\prime}$ of $S^{2}$, and is in fact obtained from $\Gamma^{\prime}$
by subdividing each triangle of $\Gamma^{\prime}$ into $n^{2}$ triangles, see
Figure 3 for $n=4$.

Figure 3.  The refinement of a triangle of $\Gamma^{\prime}$.

Now let $Y$ be any $d$-semistable $K3$ surface of type III. Let $Y^{\prime}$
denote a $d$-semistable $K3$ surface in $(-1)$-form that is obtained from $Y$
by a sequence of elementary modifications. Let $k$ be the largest integer such
that $Y^{\prime}$ has special $k$-bands of hexagons. Note that this number is
independent of the choice of $Y^{\prime}$ by [FS86, Theorem 0.5].

######  Definition 1.8.

The integer $k$ is called the _index_ of $Y$. If $k=1$, $Y$ is called
primitive.

######  Remark 1.9.

The index is an invariant of $Y$ that may also be defined in terms of
monodromy in the analytic setting. For details see [FS86, p. 4].

We finally have to recall some facts about the Picard group of $d$-semistable
$K3$ surfaces and their smoothings. We first start with a normal crossing
surface $Y=\cup Y_{i}$ fulfilling the conditions of Definition 1.1 with
possibly the exception that $Y$ is $d$-semistable. Then there is an exact
sequence

(1.3) $0\to
L\to\bigoplus_{i}H^{2}(Y^{\nu}_{i},\mathbb{Z})\to\bigoplus_{C}H^{2}(C,\mathbb{Z})$

where $C$ runs through all components of preimages of $D$ under the
normalisation $Y^{\nu}\to Y$ and $L$ is defined as the kernel of the right
hand map given by the differences of the restrictions, see e.g. [KK89, p.
151]. By [Laz08, Section 3.1], $\operatorname{rank}L=18+n$, with $n$ the
number of components of $Y$. Obviously, $\operatorname{Pic}(Y)\subset L$.
Recall also from [Car80, §4] the Carlson homomorphism

$c_{Y}:L\to\mathbb{C}^{*}.$

One way to define it as follows, see [FS86, §3]: Let $W_{0}=W_{1}\subset
W_{2}$ be the weight filtration of the natural mixed Hodge structure on
$H^{2}(Y)$. We have $L=(W_{2}/W_{0})_{\mathbb{Z}}$. Choose a section $s$ of
the projection $W_{2}\to W_{2}/W_{0}$ preserving the Hodge filtration and a
retraction $r\colon W_{2}\to W_{0}$ of the inclusion $W_{0}\subset W_{2}$.
Then $c_{Y}=r\circ s\text{ mod }(W_{0})_{\mathbb{Z}}\colon
L\to({W_{0}})_{\mathbb{C}}/({W_{0}})_{\mathbb{Z}}$.

Its significance is that

(1.4) $\operatorname{Pic}(Y)=\operatorname{ker}(c_{Y}).$

We note that for each component $Y_{i}$, there is an element
$\xi_{i}=\sum_{j}D_{ij}-D_{ji}\in L$. By results of Friedman and Scattone
[FS86, p. 25], $Y$ is smoothable if and only if
$\xi_{i}\in\operatorname{ker}(c_{Y})$ for all $i$.

We can then consider locally trivial deformations of $Y$, i.e. deformations
$\mathfrak{X}\to B$, with $B$ an analytic set, such that each point of
$\mathfrak{X}$ has a neighbourhood $U$ such that $\mathfrak{X}_{|U}$ is a
product. By [FS86, §4] these are parameterized by the Carlson map $c_{Y}$. The
$d$-semistable deformations define a divisor in this $20$-dimensional family.
From the relation (1.4) we obtain in particular the following result.

######  Lemma 1.10.

Let $Y=\cup Y_{i}$ be a $d$-semistable $K3$ surface of type III with
$c_{Y}=1$. Then

$\operatorname{Pic}(Y)\cong\\{(\mathcal{L}_{i})_{i}\in\bigoplus_{i}\operatorname{Pic}({Y_{i}^{\nu}})\mid\deg\mathcal{L}_{i_{|C}}=\deg\mathcal{L}_{j_{|C}},C\subset
D^{\nu}\\}$

where $D^{\nu}$ denotes the preimage of $D$ under the normalisation map
$\nu\colon Y^{\nu}\to Y$ and $C$ runs through all components of $D$.

The following Lemma is an application of [FS86, Lemma 5.5].

######  Lemma 1.11.

Let $Y$ be a $d$-semistable K$3$ surface with $n$ components. There is a
unique locally trivial $d$-semistable deformation $Y_{c}$ of $Y$ such that
$\text{rank }\operatorname{Pic}(Y_{c})=18+n$. The Carlson homomorphism
$c_{Y_{c}}$ is trivial, i.e. $c_{Y_{c}}=1$ and hence
$\operatorname{Pic}(Y_{c})=L$.

###### Proof.

A detailed proof is in [GHKS, Section 10.4]. If $Y$ is such that its Picard
group has rank $18+n$, one can in particular choose $19$ linearly independent
divisors $L_{i}$ different from the $\xi_{i}$. The result then follows by
iterating the proof of [FS86, Lemma 5.5]. ∎

######  Remark 1.12.

We will now consider deformations $\mathcal{Y}\to S$ of $d$-semistable K$3$
surfaces. The relevant relative notions of divisors and cones are recalled in
Section 2. Here we simply remark that under our assumptions on $S$, linear
equivalence over the base coincides with the usual linear equivalence, in
particular
$\operatorname{Pic}(\mathcal{Y}/S)=\operatorname{Pic}(\mathcal{Y})$.

The following will play an important role for us.

######  Proposition 1.13.

Let $Y$ be a projective type III $d$-semistable $K3$ surface with $c_{Y}=1$.
Then there is an up to isomorphism unique Kulikov model $\mathcal{Y}\to S$ of
$Y$ such that

(1.5) $r_{c}\colon\operatorname{Pic}(\mathcal{Y}/S)\cong\operatorname{Pic}(Y)$

with $r_{c}$ being the restriction map. Also, if $k$ is the index and $t$ is
the number of triple points of $Y$ then

$\operatorname{Pic}(\mathcal{Y}_{\eta})\cong 2E_{8}(-1)\oplus
U\oplus\langle\frac{-t}{k}\rangle.$

###### Proof.

This goes back to [FS86]. If $c_{Y}=1$, then $\operatorname{Pic}(Y)\cong L$.
The surface $Y$ also determines the invariants $t$,$k$. Let $n$ be the number
of components of $Y$. Consider the divisors $\xi_{i}=\sum_{j}D_{ij}-D_{ji}$,
$i=1,\dots,n$ defined above. Note that $\sum_{i}\xi_{i}=0$. The lattice
$\langle\xi_{1},\dots\xi_{n}|\sum_{i}\xi_{i}=0\rangle$ is a primitive
sublattice of $L$ by [FS86, (4.13)]. Hence we can pick linearly independent
divisors $L_{1},\dots L_{19}$ that generate $\operatorname{Pic}(Y)\text{ mod
}K$. If $\mathcal{Y}\to S$ is a deformation with
$\operatorname{Pic}(\mathcal{Y}/S)\cong\operatorname{Pic}(Y)$ via restriction,
then by definition $\mathcal{Y}\to S$ is a deformation of the tuple
$(Y;L_{1},\dots,L_{19})$. We shall show that there is a unique such
1-parameter deformation.

Let $\mathfrak{X}\to V$ be the semiuniversal analytic deformation of $Y$ as
defined in in the proof of [Fri83, Theorem 5.10]. By the arguments in the
proof of [FS86, Lemma 5.5], the locus $V^{\prime}$ in the smoothing component
of $V$ where the $L_{i}$ deform is $1$-dimensional and smooth. Let
$\mathfrak{X}^{\prime}\to V^{\prime}$ be the restriction of the semiuniversal
family. By [Fri83, Theorem 5.10], this is a smoothing of $Y$. Let $W^{\prime}$
be the analytic algebra defining the germ $V^{\prime}$, and let $W$ be the
completion of $W^{\prime}$ with respect to the maximal ideal. Then
$W\cong\mathbb{C}[[t]]$. This defines a formal scheme
$\hat{\mathcal{Y}}\to\operatorname{Spf}\mathbb{C}[[t]]$, and by the condition
that all $L_{i}$ deform, there is an
$\mathcal{L}\in\operatorname{Pic}(\hat{\mathcal{Y}})$ restricting to an ample
line bundle on $Y$ and thus by Grothendieck’s existence theorem a deformation
$\mathcal{Y}\to S$ with $S=\operatorname{Spec}\mathbb{C}[[t]]$ such that
$\hat{\mathcal{Y}}$ is the completion of $\mathcal{Y}\to S$ along $Y$. By
construction, $\operatorname{Pic}(\mathcal{Y}/S)\cong\operatorname{Pic}(Y)$
via restriction.

We show that $\mathcal{Y}\to S$ is a smoothing of $Y$. The total space of the
deformation $\mathfrak{X}^{\prime}\to V^{\prime}$ is smooth, as follows from
[Fri83, Theorem 5.10]. In particular, its local rings in closed points are
regular, and thus by [Mat89, Theorem 23.7] the local rings
$\mathcal{O}_{\hat{\mathcal{Y}},y}$ for $y\in Y$ of the formal smoothing are
regular. By the same theorem, this implies that the stalks of
$\mathcal{O}_{{\mathcal{Y}},y}$ at closed points of the central fibre of
$\mathcal{Y}\to S$ are regular local rings. This implies that $\mathcal{Y}$ is
regular by [GW10, Remark 6.25]. In particular, the generic fibre is a smooth
K$3$ surface. Also, by adjunction, $\mathcal{Y}$ has trivial canonical bundle.
So $\mathcal{Y}\to S$ is indeed a semistable model.

Also, for the degeneration $\mathcal{Y}\to S$, it follows from [Kaw97, Lemma
4.2], using the fact that $S$ is a $DVR$, that we have an exact sequence

(1.6)
$0\to\mathbb{Z}^{\mathcal{Y}}\to\operatorname{Pic}(\mathcal{Y}/S)\to\operatorname{Pic}(\mathcal{Y}_{\eta})\to
0$

with $\mathbb{Z}^{\mathcal{Y}}$ the abelian group generated by the components
$Y_{i}$ of the central fibre modulo the relation $\sum_{i}Y_{i}=0$. The
statement about the Picard group of the generic fibre then follows from
[Laz08, Prop. 4.3] and [Laz08, Corollary 4.6] together with Sequence 1.6. So
$\mathcal{Y}\to S$ is a model as claimed.

Now, suppose $\mathcal{Y}^{\prime}\to S$ is a second model . By formal
semiuniversality, $\mathcal{Y}^{\prime}\to S$ is pulled back from
$\mathcal{Y}\to S$ via a homomorphism $\mathbb{C}[[t]]\to\mathbb{C}[[t]]$.
Because $\mathcal{Y}^{\prime}\to S$ is regular, the uniformizing parameter $t$
maps to $at$ with $a$ a unit. Hence $\mathcal{Y}^{\prime}\to S$ is canonically
isomorphic to $\mathcal{Y}\to S$. This proves the result. ∎

######  Definition 1.14.

We shall call a $d$-semistable K$3$ surface $Y$ with $c_{Y}=1$ maximal and a
degeneration $\mathcal{Y}\to S$ of a maximal K$3$ surface with
$r_{c}\colon\operatorname{Pic}(\mathcal{Y}/S)\cong\operatorname{Pic}(Y)$ a
maximal degeneration.

The next proposition says that maximal degenerations behave well under flops.

######  Proposition 1.15.

Let $\mathcal{Y}\to S$ be a maximal degeneration with central fibre
$Y=\mathcal{Y}_{c}$ and let $\mathcal{Y}^{+}\to S$ be a flop of
$\mathcal{Y}\to S$. Then $\mathcal{Y}^{+}\to S$ is again a maximal
degeneration and the dual graph of the central fibre
$Y^{+}=\mathcal{Y}^{+}_{c}$ is a triangulation of $S^{2}$ with the same number
of triangles as the dual graph of $Y$.

###### Proof.

We shall prove this result using the analytic theory. For this we first note
that any flop factors into flops given by contractions of extremal rays, see
[KM98, §6.4]. Thus we can assume the flop is given by a small contraction
$\operatorname{contr}_{R}$ with $R$ an extremal ray of $\mathcal{Y}\to S$. Let
$\pi\colon\mathcal{Y}\to\bar{\mathcal{Y}}$ be the flopping contraction over
$S$, $F$ be a divisor that is anti-ample on the fibres of $\pi$. The morphism
$\pi$ is given by a divisor $G$ on $\mathcal{Y}$ with restriction $G_{c}$ to
$Y$. This defines a divisor $G^{\prime}$ on the maximal analytic smoothing
$\mathcal{X}\to\mathbb{D}$ of $Y$ and a contraction
$\mathcal{X}\to\bar{\mathcal{X}}$ of the extremal ray $R$. Hence
$\mathcal{X}\to\bar{\mathcal{X}}$ is a small contraction. Restricting the
divisor $F$ to the central fibre and extending by maximality to $\mathcal{X}$
we obtain the induced divisor $F^{\prime}$ on $\mathcal{X}$. This is anti-
ample on the fibres of $\mathcal{X}\to\bar{\mathcal{X}}$. Hence there is a
flop $\mathcal{X}^{+}\to\bar{\mathcal{X}}$. By a result of Kulikov,
$\mathcal{X}^{+}\to\mathbb{D}$ is semistable, see e.g. [Cor95, Corollary 3.7].
Completion along the central fibre gives morphisms of the corresponding formal
schemes, and by Grothendieck’s existence theorem, we get proper morphisms
$f\colon\mathcal{Y}\to\mathcal{Z}$ and
$f^{+}\colon\mathcal{Y}^{+}\to\mathcal{Z}$. We claim that the morphism
$f\colon\mathcal{Y}\to\mathcal{Z}$ is given by the contraction of $R$. Indeed,
we have $f_{*}\mathcal{O}_{\mathcal{Y}}\cong\mathcal{O}_{\mathcal{Z}}$: the
restriction to the central fibres
$f_{c}\colon\mathcal{Y}_{c}\to\mathcal{Z}_{c}$ is a proper surjective
birational morphism with connected fibres. It is is straightforward to check
that $(f_{c})_{*}\mathcal{O}_{Y}\cong\mathcal{O}_{Z}$, using that every
regular section of $\mathcal{O}_{\mathcal{Y}}$ is constant on the fibres. By
the global version of [Wah76, Lemma 1.2], we have
$(f_{n})_{*}\mathcal{O}_{\mathcal{Y}_{n}}\cong\mathcal{O}_{\mathcal{Z}_{n}}$
for the truncations of order $n$ and thus
$\hat{f}_{*}\widehat{\mathcal{O}_{\mathcal{Y}}}\cong\widehat{\mathcal{O}_{\mathcal{Z}}}$,
by [Ill05, Theorem 8.2.2]. By the same theorem,
$\hat{f}_{*}\widehat{\mathcal{O}_{\mathcal{Y}}}\cong\widehat{f_{*}\mathcal{O}_{\mathcal{Y}}}$,
so
$\widehat{\mathcal{O}_{\mathcal{Z}}}\cong\widehat{f_{*}\mathcal{O}_{\mathcal{Y}}}$
and from [Ill05, Theorem 8.4.2], it follows that
$f_{*}\mathcal{O}_{\mathcal{Y}}\cong\mathcal{O}_{\mathcal{Z}}$. As
$\mathcal{Z}$ is a Nagata scheme, its normalisation is finite over
$\mathcal{Z}$. Because of the universal property of normalisation and
connectedness of the fibres of $f$, it follows from finiteness that
$\mathcal{Z}$ is normal. By uniqueness of contractions, $f=\pi$ and, in
particular $\bar{\mathcal{Y}}=\mathcal{Z}$.

Hence $f\colon\mathcal{Y}^{+}\to\mathcal{Z}$ is the flop of $\pi$. It has
central fibre $Y^{+}\cong(\mathcal{X}^{+})_{c}$, which is a $d$-semistable K3
surface of type III. As $Y^{+}$ is the central fibre of a degeneration, it
follows that the $\xi_{i}$ classes from above are in fact Cartier, so the dual
graph is indeed a triangulation of the sphere $S^{2}$ by Kulikov’s theorem
[Kul77, Theorem II], with the same number of triangles as the dual graph of
$Y$, because the number of components is the same. It has trivial Carlson
extension since it follows from $\text{rank
}\operatorname{Pic}(\mathcal{Y}^{+}/S)=18+n$ that its Picard group has rank
$18+n$, with $n$ the number of components of $Y^{+}$, so by Lemma 1.11 it
follows that $c_{Y^{+}}=1$. This proves the proposition. ∎

We shall now consider the special case arising from the mirror families of
$2d$-polarized $K3$ surfaces. We will also construct explicit models. Here we
first state the more general

######  Proposition 1.16.

Let $d>0$ and $\check{M}_{2d}=U\oplus 2E_{8}(-1)\oplus\langle-2d\rangle$. Then
there exists a primitive maximal Kulikov model $\mathcal{Y}\to S$ such that
$\operatorname{Pic}(\mathcal{Y}_{\eta})\cong\check{M}_{2d}$. Any two such
Kulikov models are related by a sequence of flops.

###### Proof.

The existence follows from Proposition 1.13, all we require is the existence
of a primitive type III $d$-semistable K3 surface with $t=2d$ triple points
and primitivity $k=1$, which exists by [FS86, Theorem 0.6]. We now show that
two such models are related by flops. For this let $\mathcal{Y}\to S$ and
$\mathcal{Y}^{\prime}\to S$ be two distinct deformations with the properties
stated and central fibres $Y$ and $Y^{\prime}$ respectively. It follows from
Sequence (1.6) and e.g. [Laz08, Proposition 4.3] that both central fibres have
exactly $2d$ triple points. Also, both degenerations are primitive. As above,
we have the maximal analytic family $\mathcal{X}\to\mathbb{D}$ over a small
disc with central fibre $Y$,
$\operatorname{Pic}(\mathcal{X})\cong\operatorname{Pic}(Y)=L$ and smooth
fibres K$3$ surfaces with Picard rank $19$. Similarly we have
$\mathcal{X}^{\prime}\to\mathbb{D}$ with central fibre $Y^{\prime}$.

By [FS86, Theorem 0.6] there is a sequence of type I and type II modifications
$\mathcal{X}\dashrightarrow\mathcal{X}^{\prime}$. As both models are
projective, this sequence factors into a sequence of projective flops given by
contractions of extremal rays, by [KM98, Remark 6.37]. Let $F$ be an effective
divisor defining the first flop, say
$\mathcal{X}\dashrightarrow\mathcal{X}^{+}$ in this sequence. Restriction to
the central fibre and then lifting to $\mathcal{Y}\to S$ via maximality yields
an effective divisor $F^{\prime}$ inducing a contraction of an extremal ray.
As the exceptional locus on $Y$ does not deform to $\mathcal{Y}$ (because it
does not deform on $\mathcal{X}$, so it does not fullfill the numerical
conditions for lifting to the family), we obtain a flop $\mathcal{Y}^{+}$ of
$\mathcal{Y}$ with $\mathcal{Y}^{+}_{c}=\mathcal{X}^{+}_{c}$. Proceeding in
this way, one obtains a sequence of flops

$\mathcal{Y}\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}\mkern
12.0mu}\dots\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}\mkern
12.0mu}\mathcal{Y}^{{}^{\prime\prime}}$

such that the central fibre of $\mathcal{Y}^{{}^{\prime\prime}}$ is
$Y^{\prime}$. By uniqueness of maximal smoothings, see Proposition 1.13 ,
$\mathcal{Y}^{\prime}\cong\mathcal{Y}^{{}^{\prime\prime}}$ over $S$. ∎

######  Remark 1.17.

Proposition 1.15 shows that the generic fibre $\mathcal{Y}_{\eta}$ of a
maximal primitive degeneration $\mathcal{Y}\to S$ with
$\operatorname{Pic}(\mathcal{Y}_{\eta})=\check{M}_{2d}$ is independent of
$\mathcal{Y}\to S$.

This allows us to make the following definition, where we follow the
established terminology in the existing literature:

######  Definition 1.18.

Fix $2d>0$ with $d$ square free. The (unique) $K3$ surface
$\mathcal{Y}_{\eta}\to\operatorname{Spec}(\mathbb{C}((t)))$ with
$\operatorname{Pic}(\mathcal{Y}_{\eta})=\check{M}_{2d}$ which is the generic
fibre of some maximal primitive smoothing $\mathcal{Y}\to S$ is called the
_Dolgachev-Nikulin-Voisin family of degree $2d$_.

######  Remark 1.19.

Here, we require $d$ to be square free to obtain the Dolgachev-Nikulin-Voisin
family as in [GHKS]. For $d$ not square free, there are several distinct local
models.

######  Definition 1.20.

We fix $2d>0$ with $d$ square free. Then any primitive projective type III
degeneration $\mathcal{Y}\to S$ that is maximal and whose generic fibre has
Picard group $\check{M}_{2d}$ is called a model of the Dolgachev-Nikulin-
Voisin family of degree $2d$.

We will abbreviate Dolgachev-Nikulin-Voisin family by DNV family. We note the
following result.

######  Proposition 1.21.

Let $Y$ be a maximal projective $d$-semistable $K3$ surface. Then there exists
a sequence of elementary modifications of type I and II

$Y\dashrightarrow Y_{1}\dashrightarrow\dots\dashrightarrow
Y_{i}\dashrightarrow\dots\dashrightarrow Y_{n}$

such that all $Y_{i}$ are maximal $d$-semistable $K3$ surfaces and $Y_{n}$ is
in $(-1)$-form.

###### Proof.

Let $\mathcal{X}\to\mathbb{D}$ be the maximal analytic smoothing of $Y$. By
the $(-1)$-Theorem, see Theorem 1.7, there is a sequence of type I and type II
modifications

(1.7)
$\mathcal{X}\dashrightarrow\mathcal{X}_{1}\dashrightarrow\dots\dashrightarrow\mathcal{X}_{i}\dashrightarrow\dots\dashrightarrow\mathcal{X}_{n}$

such that the central fibre of $\mathcal{X}_{n}$ is in $(-1)$-form. The Picard
groups of the $\mathcal{X}_{i}$ all have the same rank and hence the central
fibres are maximal. Thus restricting the sequence (1.7) to the central fibre
gives the result. ∎

###  1.2. Construction of Models

There is a bijection between triangulations $\mathscr{G}$ of the sphere such
that no vertex has valency greater than $6$ and locally trivial deformation
classes $[Y]_{\mathscr{G}}$ of d-semistable K3 surfaces of type III in
$(-1)$-form, see [Laz08, §5.1]. By [FS86, §3.9], there is a unique element
$Y_{\mathscr{G}}\in[Y]_{\mathscr{G}}$ with trivial Carlson extension. Here we
describe the possible components of this surface. Each vertex $v$ of
$\mathscr{G}$ corresponds to a component with normalisation a (weak) del Pezzo
surface of degree the valency of $v$. The edges of $\mathscr{G}$ then
determine which components are glued. We will describe the gluing later. We
emphasize that this construction is at least implicit in [GHKS].

We first make the following definition.

######  Definition 1.22.

Let $(Y,D)$ be an anticanonical pair. Let $D=\sum D_{i}$ and $p$ be a smooth
point of $D$ lying on the component $D_{i}$. If $n=1$, the _ $n$-fold blow-up
_ of $Y$ in $p$ is the usual blow-up, if $n>1$, the n-fold blow-up of $Y$ in
$p$ is the blow-up of the $n-1$-fold blow-up $\pi\colon Y^{\prime}\to Y$ in
the point $\text{ex}(\pi)\cap D_{i}^{\prime}$, where $D_{i}^{\prime}$ is the
strict transform of $D_{i}$ on $Y^{\prime}$. More generally, if
$(p_{1},\dots,p_{k})$ is an ordered set of points $p_{j}\in Y$, which lie on
the smooth locus of $D$ such that each $D_{i}$ contains at most one point
$p_{j}$, we define by the obvious generalisation the _$(n_{1},\dots,n_{k})$
-blow-up of $Y$ in $(p_{1},\dots,p_{k})$_.

######  Construction 1.23.

We now describe the possible anticanonical pairs $(Y,D)$ that appear as
(normalisations) of components of $Y_{\mathscr{G}}$. By [Laz08, Proposition
5.2], we know the description in general terms. The (normalisations of the)
components are (weak) del Pezzo surfaces of degree $d=1,\dots,6$ and the
double locus of the central fibre gives rise to an anticanonical cycle on the
normalisations of the components. The pairs $(Y,D)$ we are looking for are
rigid pairs. To obtain rigidity we first demand that $D$ be a cycle of
$(-1)$-curves if $d>1$. In degree $d=5,6$ this leads to a unique pair $(Y,D)$
where $Y$ is a del Pezzo surface. For $d=1,\ldots,4$ we obtain weak del Pezzo
surfaces and we ensure rigidity by asking that $Y$ contains a maximal number
of $(-2)$-curves. This leads to unique pairs $(Y,D)$, see [Laz08, Lemma 5.14].
Note that the configurations of $(-2)$-curves defines a root lattices.
Contracting the $(-2)$-curves one obtains singular del Pezzo surfaces with a
unique maximal ADE-singularity. We further specify _special_ points on each
component of the anticanonical cycle $D$. We will call a special point
_interior_ if it is not a node of $D$. The special points will be used to
define the gluing of $Y_{\mathscr{G}}$. We will now describe the pairs $(Y,D)$
and the special points explicitly.

  * d=1:

Let $Q=\mathbb{P}^{1}\times\mathbb{P}^{1}$ with toric boundary
$\tilde{D}=\tilde{D}_{1}+\tilde{D}_{2}+\tilde{D}_{3}+\tilde{D}_{4}$, ordered
cyclically. Let $p_{i}\in\tilde{D}_{i}$, $i=1\dots 4$, be points in the smooth
part of $\tilde{D}$ such that $p_{i},p_{i+2}$ are in the same fibre of one of
the two rulings, see the right hand side of Figure 4. Let $\tilde{Q}$ be the
$(1,5,1,3)$-blow-up of $Q$ in $(p_{1},p_{2},p_{3},p_{4})$. The strict
transforms of $\tilde{D}_{1},\tilde{D}_{3}$ have self-intersection $-1$ on
$\tilde{Q}$. Blowing down these yields a surface such that the strict
transform of $\tilde{D}_{4}$ has self intersection $(-1)$. Then, blowing this
down gives a surface $\mathfrak{Y}_{1}$ with an anticanonical cycle ${D}$ of
self intersection $1$ and an $E_{8}$ root system of effective $(-2)$-curves.
This is a weak del Pezzo of degree 1, see Figure 5. There is a unique
$(-1)$-curve $E$ meeting ${D}$. The special points are the node of ${D}$ and
the point $E\cap{D}$ (which is an interior special point).

$\small{p_{1}}$$\small{p_{2}}$$\small{p_{3}}$$\small{p_{4}}$ Figure 4. The
$(1,1,1,1)$\- blow-up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ in
$(p_{1},p_{2},p_{3},p_{4})$.

  * d=2:

For degree $2$, we take $\mathbb{P}^{2}$ together with its toric boundary
$(xyz=0)$. We can fix three collinear points, one on each boundary divisor,
say $p,q,r$. Let $Y^{\prime}$ be the $(3,3,2)$-blow-up of $\mathbb{P}^{2}$ in
$(p,q,r)$. This yields a weak del Pezzo surface of degree $1$, as we have
blown up $8$ points that are not on $(-2)$ curves. Now, blow down the strict
transform of the toric divisor that is a $(-1)$-curve. The resulting surface
$\mathfrak{Y}_{2}$ is a weak del Pezzo surface of degree $2$ with
anticanonical cycle $D=D_{1}+D_{2}$. It carries an $E_{6}$ configuration of
effective $(-2)$-curves by construction. There are also 2 exceptional curves
$E_{1},E_{2}$ of the first kind each meeting a long end of the root system and
a component of the anticanonical divisor, see Figure 6. The special points of
$D_{i}$ are the points $D_{i}\cap E_{i}$ and the two points of the
intersection $D_{1}\cap D_{2}$.

$-1$$1$ Figure 5. The surface $\mathfrak{Y}_{1}$.
$E_{1}$$2$$3$$4$$1$$5$$6$$E_{2}$ Figure 6. The $E_{6}$ root system and the
exceptional curves $E_{1},E_{2}$.

  * d=3:

For degree $3$, we take again $\mathbb{P}^{2}$ together with its toric
boundary $(xyz=0)$. As above, we fix three collinear points, one on each
boundary divisor, say $p,q,r$. Let $Y$ be the $(2,2,2)$-blow-up of
$\mathbb{P}^{2}$ in $(p,q,r)$, a weak del Pezzo surface of degree $3$. It
carries a $D_{4}$ root system of effective $(-2)$-curves, given by the strict
transform of the line through the points $p,q,r$ together with the strict
transforms of the first blow-ups in each point. The anticanonical cycle
$D=D_{1}+D_{2}+D_{3}$ is given by the strict transform of the toric boundary.
There are also three irreducible $(-1)$-curves $E_{i}$ in the exceptional
locus of the blow-up, each meeting a unique component $D_{i}$. The special
points are again the nodes of the anticanonical cycle and the points
$E_{i}\cap D_{i}$, see Figure 7.

$E_{1}$$2$$3$$1$$4$$E_{2}$$E_{3}$ Figure 7. The $D_{4}$ root system and the
exceptional curves $E_{1},E_{2}$, $E_{3}$.

  * d=4:

Again, let $Q=\mathbb{P}^{1}\times\mathbb{P}^{1}$ with toric boundary
$\tilde{D}=\tilde{D}_{1}+\tilde{D}_{2}+\tilde{D}_{3}+\tilde{D}_{4}$. Let
$p_{i}$, $i=1\dots 4$, be points on the intersection of the fibres of the two
ruling with the toric boundary components. Blow up $Q$ once in each $p_{i}$,
the resulting surface $\mathfrak{Y}_{4}$ is a weak del Pezzo of degree $4$,
with an $A_{2}$ root system of effective $(-2)$-curves and an anticanonical
cycle ${D}={D}_{1}+{D}_{2}+{D}_{3}+{D}_{4}$ where the ${D}_{i}$ are the strict
transforms of the $\tilde{D}_{i}$. There are $4$ $(-1)$-curves $E_{i}$,
$i=1\dots 4$ on $\mathfrak{Y}_{4}$ that are not components of ${D}$, each
meeting exactly one of the $D_{i}$ transversally. The special points of ${D}$
are the points ${D}_{i}\cap E_{i}$ and ${D}_{i}\cap{D}_{i+1}$, where the
indices are considered cyclically, see Figure 4.

  * d=5:

Pick $4$ points $\\{p,q,r,s\\}$ in $\mathbb{P}^{2}$, no three on a line. Then
$\text{Bl}_{\\{p,q,r,s\\}}(\mathbb{P}^{2})$ is the del Pezzo surface of degree
$5$. The anticanonical divisor $D=D_{1}+D_{2}+D_{3}+D_{4}+D_{5}$ is a cycle of
$(-1)$-curves forming a pentagon, which we obtain as follows: we choose the
strict transforms of the lines spanned by $(p,q)$, $(p,r)$ and $(q,s)$,
together with the exceptional lines which arise form blowing up $p$ and $q$.
Note that permutations of the points $p,q,r,s$ gives rise to isomorphic pairs
$(Y,D)$. There are $5$ more $(-1)$-curves $E_{i}$, $i=1\dots 5$. Each $E_{i}$
meets exactly one of the $D_{i}$ transversally. The special points of $D_{i}$
are the points $D_{i}\cap E_{i}$ and $D_{i}\cap D_{i+1}$.

  * d=6:

Pick the three coordinate points $\\{p,q,r\\}$ in $\mathbb{P}^{2}$ which are
torus orbits. Then $\mathfrak{Y}_{6}=\text{Bl}_{\\{p,q,r\\}}(\mathbb{P}^{2})$
is the del Pezzo surface of degree $6$. Let
$D=D_{1}+D_{2}+D_{3}+D_{4}+D_{5}+D_{6}$ be the anticanonical divisor which
consists of the strict transforms of the coordinate lines and the exceptional
lines. This is a cycle of $(-1)$-curves forming a hexagon. The toric structure
of $\mathfrak{Y}_{6}$ indentifies a copy of $\mathbb{G}_{m}\subset D_{i}$ for
each $i$. The special points of $D_{i}$ are the points $-1\in\mathbb{G}_{m}$,
and $D_{i}\cap D_{i+1}$.

######  Definition 1.24.

We will denote the (weak) del Pezzo surfaces of degree $d$ constructed here by
$\mathfrak{Y}_{d}$.

Now, the triangulation $\mathscr{G}$ defines a locally trivial deformation
class $[Y]_{\mathscr{G}}$ of $d$-semistable K3 surfaces and we can take the
member $Y_{\mathscr{G}}$ of $[Y]_{\mathscr{G}}$ such that the (normalisation)
of each component is isomorphic (as an anticanonical pair) to a surface from
the above list, by [Laz08, Lemma 5.14], where we recall that the valency of a
vertex is equal to the degree of the corresponding (weak) del Pezzo surface.
The gluing is such that the special points are identified pairwise where, in
particular, interior special points are identified with interior special
points.

We learned the next result from [GHKS]:

######  Proposition 1.25.

Let $\mathscr{G}$ be a triangulation with valency at most $6$ and no $k$-bands
of hexagons. Then the surfaces $Y_{\mathscr{G}}$ are maximal, i.e. have
trivial Carlson map $c_{Y_{\mathscr{G}}}=1$.

###### Proof.

A proof can be found in [GHS16, Construction 10.14]. Here we sketch a proof in
the case of simple normal crossing. Each of the surfaces $\mathfrak{Y}_{i}$,
for $i=1\dots 5$, has a $\mathbb{Q}$-basis $B_{i}$ of
$\operatorname{Pic}(\mathfrak{Y}_{i})$ given by the $(-2)$-curves and the
interior $(-1)$-curves, i.e. those that are not components of the
anticanonical divisor. Let $Y_{\mathscr{G}}=\cup_{j\in J}Y_{j}$. We shall
choose an order on $J$. The Picard group of $Y_{\mathscr{G}}$ is given by the
kernel of the Carlson map. Let $Y_{\operatorname{simp}}$ be a semi-simplicial
resolution of $Y_{\mathscr{G}}$, see [Car80],[KK89, §4.2.2]. Then one defines
$Y_{p}$ to be

$Y_{p}=\coprod Y_{j_{0}}\cap\dots\cap Y_{j_{p}}\qquad(j_{0}<\dots<j_{p})$

for $p=0,1,2$. Face maps are given by maps $\delta_{p}\colon Y_{p}\to Y_{p-1}$
such that $\delta_{p}$ is the inclusion on the components of $Y_{p}$, see
[KK89, §4.2.2]. We start with a collection of line bundles $L_{j},j\in J$ on
the components $Y_{j}$ such that the degrees on the double curves $D_{ij}$
coincide. We want to show that these glue to a line bundle $L$ on
$Y_{\mathscr{G}}$. For $i(j)\leq 5$ we write $L_{j}$ in the basis $B_{i(j)}$.
Also, given a line bundle $L_{j}$ on a hexagonal component $\mathfrak{Y}_{6}$,
we can find a linearly equivalent divisor on $Y_{j}$ whose restriction to a
component $D_{k}$ of the anticanonical divisor of $\mathfrak{Y}_{6}$ is $(\deg
L_{j|D_{k}})p_{k}$ where $p_{k}$ denotes the interior special point on
$D_{k}$. This follows from (the proof of) [GHK15, Lemma 2.8]. The line bundles
$L_{j}$ can therefore be represented by divisors whose restrictions to the
double curves coincide. It follows that the collection $L=\\{L_{j},j\in J\\}$
is in the kernel of the Carlson map as follows from [Car80, p. 277 ff] applied
to $Y_{\operatorname{simp}}$. This implies the result. ∎

We also obtain that maximal $d$-semistable $K3$ surfaces in $(-1)$-form are
always projective.

######  Proposition 1.26.

Let $Y=\cup Y_{i}$ be a $d$-semistable $K3$ surface of type III with
$c_{Y}=1$. If $Y$ is in $(-1)$-form, then $Y$ is projective.

###### Proof.

Let $Y$ be a surface as in the proposition. By [Laz08, Proposition 5.2], the
normalisations $Y^{\nu}_{i}$ of the components $Y_{i}$ of $Y$ are weak del
Pezzo surfaces of degree $d_{i}\leq 6$ and if $D=\sum D_{i}$ is the
anticanonical cycle on $Y^{\nu}_{i}$, the orthogonal complement of the
sublattice generated by the $D_{i}$ in $\operatorname{Pic}(Y_{i})$ is a root
lattice $E_{8}$, $E_{6}$, $D_{4}$ or $A_{2}$ for $d_{i}=1,2,3$ and $4$,
respectively and empty for $d_{i}=5,6$ (see our above discussion). These
classes are effective by construction. We show that $Y$ is projective by
giving an ample bundle $A_{i}$ on each $Y^{\nu}_{i}$ such that $A_{i}.D_{ij}=\
A_{i}.D_{ik}$ for all $j,k$, i.e. such that the $A_{i}$ have the same degree
on all components of the anticanonical cycle on $Y^{\nu}_{i}$. By Lemma 1.10,
after taking suitable multiples, one can then glue these line bundles to
obtain a bundle $A$ on $Y$. The bundle $A$ is ample as its restriction to each
irreducible component is ample, by [Laz04, Proposition 1.2.16]. As the
structure morphism is proper, Y is projective. To simplify notation, we
taciturnly assume $Y_{i}^{\nu}=Y_{i}$.

For $d_{i}=1$ there is nothing to show. For $d_{i}=5,6$ the claim is obvious,
we can take the anticanonical divisor. We consider the cases $d_{i}=2,3$. For
the root lattices $R_{i}$ described above we have a natural root basis given
by the set $B(Y_{i})$ of $(-2)$-curves constructed in the blow-up procedure.
Note also that for all such $Y_{i}$ by construction we have $(-1)$-curves as
in Figures 6 and 7 connecting the root system to the boundary. We denote the
set of these curves by $E(Y_{i})$.

Suppose there is an integer $e>0$ and a divisor

$A_{i}=\sum_{B_{j}\in B(Y_{i})}b_{j}B_{j}+\sum_{E_{k}\in E(Y_{i})}eE_{k}$

in $\operatorname{Pic}(Y_{i})$ such that $A_{i}.C>0$ for all $C\in
B(Y_{i})\cup E(Y_{i})$. Then $A_{i}$ defines an ample bundle on $Y_{i}$: being
a (weak) del Pezzo surface, $Y_{i}$ is a Mori Dream space by [TVV11, Theorem
2.9], so in particular, the cone of curves of $Y_{i}$ is rational polyhedral.
The Picard rank is greater than $3$ and thus, by [AL11, Proposition 1],
$\operatorname{Pic}(Y_{i})$ generated by curves $C$ with $C^{2}<0$, i.e. by
$(-2)$ and $(-1)$-curves. Let $C$ be such a curve. If $C\in B(Y_{i})\cup
E(Y_{i})$, then $A_{i}.C>0$ by assumption. If $C$ is not in $B(Y_{i})\cup
E(Y_{i})$, $C$ is a component of $D$, as follows from Proposition 3.5 below
(which is independent of this result). Then $C.D=e$. Hence $A_{i}$ is strictly
positive on the (polyhedral) cone of curves, and hence ample by Kleiman’s
criterion. We define such $A_{i}$’s as follows: for $E_{6}$, take the numbers
$(7,11,13,16,13,11)$ for the roots, where the number in the $i$-th position is
the coefficient of the $i$-th root, with indexing as in the figures, and set
$e=10$. For $D_{4}$ we can take $(11,11,13,11)$ and $e=10$.

It remains to consider the case $d_{i}=4$. In this case
$Y_{4}=\mathfrak{Y}_{4}$ with the $A_{2}$ root system and the same
construction as before with numbers $(3,3)$ on the root system and $e=2$
defines an ample bundle $A_{i}$. ∎

We have the following proposition.

######  Proposition 1.27.

Let $\mathscr{G}$ be a triangulation of $S^{2}$ with $2d$ triangles with
valency at most $6$ and without $k$-bands of hexagons for any $k>1$.

The surface $Y_{\mathscr{G}}$ is a maximal primitive $d$-semistable K3
surface. The associated maximal smoothing $\mathcal{Y}_{\mathscr{G}}\to S$ is
a model in $(-1)$-form of the Dolgachev-Nikulin-Voisin family of degree $2d$,
i.e. $\operatorname{Pic}(\mathcal{Y}_{\mathscr{G}}{{}_{\eta}})=\check{M}_{2d}$
and
$\operatorname{Pic}(\mathcal{Y}_{\mathscr{G}}/S)\cong\operatorname{Pic}(Y_{\mathscr{G}})$
by restriction. The threefold $\mathcal{Y}_{\mathscr{G}}$ is Calabi-Yau with
$H^{1}(\mathcal{Y}_{\mathscr{G}},\mathcal{O}_{\mathcal{Y}_{\mathscr{G}}})=0$
and projective over the base $S$.

###### Proof.

$Y_{\mathscr{G}}$ is a $d$-semistable K3 surface in $(-1)$-form by
construction. It is primitive because the dual graph does not have $k$-bands
of hexagons. It is maximal, i.e. $c_{Y_{\mathscr{G}}}=1$, due to our choice of
the gluing. It is also projective by Proposition 1.26. Hence there is a
deformation with the claimed properties by Proposition 1.13. Letting
$f\colon\mathcal{Y}_{\mathscr{G}}\to S$ denote the structure morphism, we
prove the vanishing of
$H^{1}(\mathcal{Y}_{\mathscr{G}},\mathcal{O}_{\mathcal{Y}_{\mathscr{G}}})=0$
as follows: We have $H^{1}(Y_{\mathscr{G}},\mathcal{O}_{Y_{\mathscr{G}}})=0$
by [Fri83, Lemma 5.7] so by semi-continuity it follows from a result of
Grauert [Har77, Cor 12.9] that
$R^{1}f_{*}\mathcal{O}_{\mathcal{Y}_{\mathscr{G}}}=0$. As we work over an
affine base,
$H^{1}(\mathcal{Y}_{\mathscr{G}},\mathcal{O}_{\mathcal{Y}_{\mathscr{G}}})=0$
follows from [Har77, Proposition 8.5]. ∎

We give examples for low degree.

######  Example 1.28 ([Lie17]).

Specialising to genus $2$, there are precisely two different triangulations of
$S^{2}$ with two triangles, corresponding to dual intersection complexes of
degenerations with central fibres having three components. By e.g. [ES18],
these are given by two triangles glued along the boundary and two triangles
glued along one side to each other, with the remaining sides identified. We
shall denote the first of these triangulations by $\mathscr{P}$ and the latter
by $\mathscr{T}$, see Figure 8.

feef$\mathscr{P}$ffee$\mathscr{T}$ Figure 8. The triangulations $\mathscr{P}$
and $\mathscr{T}$.

Let $Y_{\mathscr{P}}$ be the surface obtained by gluing three copies of
$\mathfrak{Y}_{2}$ by identifying components of the boundary cycle such that
interior special points are identified. For the triangulation $\mathscr{T}$,
the associated maximal surface is a copy of $\mathfrak{Y}_{4}$, with two
opposite components, say ${D}_{1}$ and ${D}_{3}$, of the anticanonical curve
${D}$ identified, and two copies of $\mathfrak{Y}_{1}$ glued to the (images)
of ${D}_{2}$ and $D_{4}$, such that the images of the special points match.

######  Example 1.29.

There are $4$ combinatorially distinct triangulations of $S^{2}$ with $4$
triangles, see [ES18, Example 2.5] and Figure 9. We collect the valencies of
the vertices in a tuple. The tuples are $(3,3,3,3),(3,2,1,6),(5,5,1,1)$ and
$(4,4,2,2)$. None of these triangulations have $k$-bands of hexagons for
$k>1$, because the least number of triangles in such a triangulation is $8$.
To take an example let $\mathscr{G}_{1}$ denote the triangulation such that
every vertex has valency $3$. The resulting surface $Y_{\mathscr{G}_{1}}$ are
three copies of $\mathfrak{Y}_{3}$ glued according to the triangulation. Here,
this means that each component meets each of the remaining components in a
curve $C\cong\mathbb{P}^{1}$ in such a way that the special points are
identified.

egfgfe$\mathscr{G}_{1}$eefgfg$\mathscr{G}_{2}$eefggf$\mathscr{G}_{3}$geegff$\mathscr{G}_{4}$
Figure 9. The triangulations with $4$ triangles.

##  2\. Relative Notions and the Mori fan

###  2.1. Relative Notions

As we will make extensive use of various concepts of the minimal model program
we will recall the relevant basic concepts here, following [Kaw97]. Let
$\pi\colon\mathcal{X}\to U$ be a projective morphism of normal schemes. An
$\mathbb{R}$-Cartier divisor $L$ on $\mathcal{X}$ is an $\mathbb{R}$-linear
combination of Cartier divisors. A $\mathbb{Q}$-Cartier divisor $L$ on
$\mathcal{X}$ is a $\mathbb{Q}$-linear combination of Cartier divisors.
Similarly, an $\mathbb{R}$-Weil divisor is an $\mathbb{R}$-linear combination
of prime divisors. Two $\mathbb{R}$-Cartier divisors $L,L^{\prime}$ are
linearly equivalent over $U$ if their difference is an $\mathbb{R}$-linear
combination of principal divisors and an $\mathbb{R}$-Cartier divisor pulled
back from $U$. Two $\mathbb{R}$-Cartier diviors $L,L^{\prime}$ are numerically
equivalent over $U$, denoted by $L\equiv_{U}L^{\prime}$, if $L.C=L^{\prime}.C$
for all curves -integral $1$-dimensional closed subschemes - $C$ in fibres of
$\pi$.

We define

$\operatorname{Pic}(\mathcal{X}/U)_{\mathbb{R}}=\\{\mathbb{R}\text{-Cartier
divisors on }X\\}/\text{linear equivalence over }U$

and

$\operatorname{N}^{1}(\mathcal{X}/U)=\\{\mathbb{R}\text{-Cartier divisors on
}X\\}/\text{numerical equivalence over }U.$

The latter is a finite dimensional vector space [Kle66, Proposition IV.4.3].

In this context, we call an element of $\operatorname{N}^{1}(\mathcal{X}/U)$ a
$\mathbb{Z}$-divisor if we want to emphasise that it is the class of a Cartier
divisor.

######  Remark 2.1.

For a type III degeneration $\mathcal{Y}\to S$ of K$3$ surfaces, we have
$\operatorname{Pic}(\mathcal{Y}/S)_{\mathbb{R}}\cong\operatorname{N}^{1}(\mathcal{Y}/S)$:
indeed, by maximality, it is enough to show this on the central fibre $Y$,
where it follows from the fact that any divisor on the central fibre is given
by an element of the lattice $L$ defined in Sequence 1.3 and all (normalised)
components are smooth rational surfaces.

We say that a Cartier divisor $L$ is _nef over $U$_, or _$\pi$ -nef_, if
$L.C\geq 0$ for all curves $C$ in $\mathcal{X}$ mapping to a point. A Cartier
divisor $L$ is _ample over $U$_, or _$\pi$ -ample_, if the restriction $L_{u}$
is ample on the fibre $\mathcal{X}_{u}$ for all $u\in U$. A Cartier divisor
$L$ is _semiample over U_ , or _semiample_ , if there is a projective morphism
$g\colon\mathcal{Z}\to U$, a $U$-morphism $\mathcal{X}\to\mathcal{Z}$ and a
$g$-ample divisor $A$ on $\mathcal{Z}$ such that $L^{\otimes n}=g^{*}A$ for
some integer $n>0$. A Cartier divisor $L$ is _$\pi$ -movable_ (or _$\pi$
-moving_) if
$\dim\operatorname{Supp}\operatorname{Coker}(\pi^{*}\pi_{*}\mathcal{O}_{X}(L)\to\mathcal{O}_{X}(L))\geq
2$.

####  2.1.1. Cones

The _$\pi$ -nef cone_ $\operatorname{Nef\textbullet}(\mathcal{X}/U)$ is the
closure of the $\pi$-ample cone in $\operatorname{N}^{1}(\mathcal{X}/U)$, the
_closed $\pi$-movable cone_ $\bar{\mathcal{M}}(\mathcal{X}/U)$ is the closure
of the convex cone generated by $\pi$-movable divisors. A Cartier divisor $L$
is _ $\pi$-effective _if $\pi_{*}\mathcal{O}(L)\neq 0$. Let
$\mathcal{B}^{e}(\mathcal{X}/U)$ be the convex cone generated by all
$\pi$-effective divisors, and let $\overline{\mathcal{B}^{e}}(\mathcal{X}/U)$
denote its closure, the cone of _pseudo-effective_ divisors. Set

$\operatorname{Mov}(\mathcal{X}/U)=\bar{\mathcal{M}}(\mathcal{X}/U)\cap\mathcal{B}^{e}(\mathcal{X}/U).$

We will refer to $\operatorname{Mov}(\mathcal{X}/U)$ as the _moving cone_.

###  2.2. The Mori fan

The Mori fan was first introduced in [HK00]. Here we shall recall the
definition in the situation which we consider, namely for models of the DNV
family.

First however, we recall the pullback under dominant rational maps in general.
For this we fix a dominant rational map
$f\colon\mathcal{X}\dashrightarrow\mathcal{Z}$ of projective
$\mathbb{Q}$-factorial schemes over $U$. Let $L$ be an effective Cartier
divisor on $\mathcal{Z}$. We define $f^{*}(L)$ to be the unique Weil divisor
that is equal to the pullback of $L$ by $f$ on the open set of codimension $2$
where $f$ is a morphism. The divisor $f^{*}(L)$ is $\mathbb{Q}$-Cartier by
$\mathbb{Q}$-factorality of $\mathcal{X}$. In this way we obtain a linear map
$f^{*}:\operatorname{N}^{1}(\mathcal{Z}/U)\to\operatorname{N}^{1}(\mathcal{X}/U)$.

Now we specialize to the case where $\mathcal{Y}\to S$ is a model of the DNV
family, of degree $2d$, see also [GHKS]. Let
$f\colon\mathcal{Y}\dashrightarrow\mathcal{Y}^{\prime}$ be a _small_
modification over $S$, i.e. a birational map which is an isomorphism in
codimension $1$ such that $\mathcal{Y}^{\prime}$ is projective over $S$. In
this situation, we will also write $(\mathcal{Y}^{\prime},f)$ for
$f\colon\mathcal{Y}\dashrightarrow\mathcal{Y}^{\prime}$ and call it a _marked
minimal model_ of the DNV family (where we have fixed $\mathcal{Y}\to S$ as a
reference model).

Define the cone

$C(f):=f^{*}\operatorname{Nef\textbullet}(\mathcal{Y}^{\prime}/S)\subset\operatorname{N}^{1}(\mathcal{Y}/S).$

######  Definition 2.2.

Let $\mathcal{Y}\to S$ be a model of the DNV family of degree $2d$. The set of
all cones $C(f)$ and their faces, with $f$ a small modification, is the _Mori
fan_ of $\mathcal{Y}\to S$, denoted by $\operatorname{MF}(\mathcal{Y}/S)$.

######  Remark 2.3.

If $(\mathcal{Y}^{\prime},f)$ and $(\mathcal{Y}^{\prime\prime},g)$ are two
marked minimal models and $C(f)\cap C(g)$ has codimension $0$, then by [Kaw97,
Lemma 1.5] there is an isomorphism
$\beta\colon\mathcal{Y}^{\prime\prime}\to\mathcal{Y}^{\prime}$ with
$f=\beta\circ g$, and hence $C(f)=C(g)$.

Note that if $\mathcal{Y}^{\prime}\to S$ is another model,
$\operatorname{MF}(\mathcal{Y}^{\prime}/S)$ is canonically identified with
$\operatorname{MF}(\mathcal{Y}/S)$ via the isomorphism
$\operatorname{N}^{1}(\mathcal{Y}/S)\cong\operatorname{N}^{1}(\mathcal{Y}^{\prime}/S)$
induced by taking strict transforms. In particular, the Mori fan only depends
on the DNV family, not on a specific model.

We recall the following result of [GHKS], which implies that
$\operatorname{MF}(\mathcal{Y}/S)$ is indeed a fan, and, in particular, is
closed under intersection and taking faces. If $\Sigma$ is a fan in a vector
space $V$, then we denote its support by $|\Sigma|$.

######  Theorem 2.4 ([GHKS], Thm 6.5).

The following holds:

  * (i)

Let $\triangle\subset\operatorname{Nef\textbullet}(\mathcal{Y}_{\eta})$ be a
rational polyhedral cone. Let $r_{\eta}$ be the restriction map
$\operatorname{Pic}(\mathcal{Y}/S)\to\operatorname{Pic}(\mathcal{Y}_{\eta})$.
Then $r^{-1}_{\eta}(\triangle)$ and
$r^{-1}_{\eta}(\triangle)\cap\operatorname{Mov}(\mathcal{Y}/S)$ are rational
polyhedral cones and

$\\{r_{\eta}^{-1}(\triangle)\cap\gamma|\gamma\in\operatorname{MF}(\mathcal{Y}/S)\\}$

is a finite set of rational polyhedral cones, with support
$r^{-1}_{\eta}(\triangle)\cap\operatorname{MF}(\mathcal{Y}/S)$.

  * (ii)

The support of $\operatorname{MF}(\mathcal{Y}/S)$ is
$\operatorname{Mov}(\mathcal{Y}/S)$.

###### Proof.

We sketch the approach of [GHS16]. The crucial point is that one can construct
models of the DNV family not only over $S$, but over the spectrum of the local
ring of a smooth algebraic curve, namely the local ring of a cusp of the
compactified Dolgachev mirror space which, in this case, is a modular curve.
For these models one can run MMP, see [KM98, §3.6]. Then, invoking [Sho96],
one concludes by [Kaw11, Theorems 3 and 4]. ∎

######  Remark 2.5.

The fact that one can run MMP for $\mathcal{Y}\to S$ implies that a small
modification $f\colon\mathcal{Y}\dashrightarrow\mathcal{Y}^{\prime}$ factors
into flops. Hence $\mathcal{Y}^{\prime}\to S$ is also a model of the DNV
family, which justifies the use of the name marked model.

######  Remark 2.6.

The proof we have just sketched also shows that log-abundance holds for models
$\mathcal{Y}\to S$ of the DNV family. In particular nef line bundles are semi-
ample.

######  Remark 2.7.

The nef cone of $\mathcal{Y}_{\eta}$ is finitely polyhedral if and only if
$d=1$. To see this we argue by means of the analytic category. Using the
sequence (1.6) and its analytic analogue we can identify the nef cone of
$\mathcal{Y}_{\eta}$ with the nef cone of a very general fibre of an analytic
Kulikov model $\mathcal{X}\to\mathbb{D}$. The claim then follows from
Nikulin’s classification in [Nik81], as the Picard lattice of
$\mathcal{Y}_{\eta}$ is $U\oplus 2E_{8}(-1)\oplus\langle-2d\rangle$. See
[Nik14, Theorem 1] for a more explicit statement of Nikulin’s classification.
Hence the Mori fan of a model $\mathcal{Y}\to S$ of degree $2$ is a finite
collection of rational polyhedral cones. Note that this is special to the case
$d=1$.

####  2.2.1. Interior facets

It is well known that the codimension $1$ faces separating maximal
(dimensional) cones of the Mori fan correspond to flopping contractions. We
recall the details here. Let $\mathcal{Y},\mathcal{Y}^{\prime}$ be models of
the DNV family of degree $2d$. Let
$\phi\colon\mathcal{Y}\mathrel{-\mkern-6.0mu{\to}\mkern-20.0mu{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}\mkern
12.0mu}\mathcal{Y}^{\prime}$ be a flop. Then there exists a divisor $F$ on
$\mathcal{Y}$ and morhpisms $\psi,\psi^{\prime}$ with $\psi$ a contraction of
an $F$-negative extremal ray such that the diagram

$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\textstyle{\mathcal{Y}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$$\textstyle{\mathcal{Z}}$

commutes. Write $\phi_{*}F=F^{\prime}$ for the birational transform. By the
contraction theorem, we have

$\operatorname{N}^{1}(\mathcal{Y}/S)=\psi^{*}\operatorname{N}^{1}({\mathcal{Z}}/S)\oplus\mathbb{R}[F]$

and

$\operatorname{N}^{1}(\mathcal{Y}^{\prime}/S)={(\psi^{\prime})}^{*}\operatorname{N}^{1}({\mathcal{Z}}/S)\oplus\mathbb{R}[F^{\prime}].$

Using the isomorphism
$\operatorname{N}^{1}(\mathcal{Y}/S)\cong\operatorname{N}^{1}(\mathcal{Y}^{\prime}/S)$
given by taking strict transforms under $\phi$, we can identify
$\psi^{*}\operatorname{N}^{1}(\mathcal{Z}/S)={(\psi^{\prime})}^{*}\operatorname{N}^{1}(\mathcal{Z}/S)$.
This cone spans a hyperplane in $\operatorname{N}^{1}(\mathcal{Y}/S)$ and
$\mathbb{R}[F]=\mathbb{R}[F^{\prime}]$. Hence the cones
$\operatorname{Nef\textbullet}(\mathcal{Y}/S)$, which we can think of as
$C(\operatorname{id}_{\mathcal{Y}/S})$, and $C(\phi)$ in the Mori fan meet in
codimension $1$, compare [Mat02, Proposition 12.2.2].

Conversely, we now assume that $\sigma,\sigma^{\prime}$ are two maximal cones
of $\operatorname{MF}(\mathcal{Y}/S)$ with $\sigma\cap\sigma^{\prime}=\tau$ a
codimension $1$ face. Without loss of generality we can assume
$\sigma=\operatorname{Nef\textbullet}(\mathcal{Y})$. By the definition of the
Mori fan there is a small modification
$f\colon\mathcal{Y}\dashrightarrow\mathcal{Y}^{\prime}$ of projective minimal
models with
$f^{*}(\operatorname{Nef\textbullet}(\mathcal{Y}^{\prime}/S))=\sigma^{\prime}$.
Let $\psi\colon\mathcal{Y}\to\mathcal{Z}$ be the contraction defined by the
linear system of a suitable multiple of a very general effective divisor $L$
in $\tau$. Note that this is indeed a morphism as $L$ is semiample by
abundance, see Remark 2.6. Then
$\tau=\psi^{*}(\operatorname{Nef\textbullet}(\mathcal{Z}))$. The facet $\tau$
defines an extremal ray $R$ in $\operatorname{N}^{1}(\mathcal{Y}/S)$ that is
cut out by $L$, i.e. $\psi=\operatorname{contr}_{R}$. Also, any divisor in the
interior of $\sigma^{\prime}$ is negative on $R$. Let
$F\in\operatorname{MF}(\mathcal{Y}/S)$ be such a divisor. Then $\psi$ is an
$F$-flopping contraction. Thus the diagram

$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\psi}$$\textstyle{\mathcal{Y}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$$\textstyle{\mathcal{Z}}$

exhibits $f$ as an $F$-flop.

######  Definition 2.8.

We say that a codimension $1$ cone $\tau$ of
$\operatorname{MF}(\mathcal{Y}/S)$ is an _interior facet_ if there are two
marked models $(\mathcal{Y}^{\prime},f),(\mathcal{Y}^{\prime\prime},g)$ of the
DNV family such that $C(f)\cap C(g)=\tau$.

The above discussion can now be summarized as follows:

######  Proposition 2.9.

Every facet of
$\operatorname{Nef\textbullet}(\mathcal{Y}/S)\in\operatorname{MF}(\mathcal{Y}/S)$
that is an interior facet of $\operatorname{MF}(\mathcal{Y}/S)$ defines a flop
$\mathcal{Y}\dashrightarrow\mathcal{Y}^{\prime}$ and conversely any such flop
is defined by an interior facet.

####  2.2.2. Action of $\operatorname{Bir}(\mathcal{Y}/S)$

The birational $S$-automorphisms $\operatorname{Bir}(\mathcal{Y}/S)$ do not
contract divisors, as $\mathcal{Y}\to S$ is a minimal model. Hence we can
define a representation

$\sigma\colon\operatorname{Bir}(\mathcal{Y}/S)\to\operatorname{GL}(\operatorname{N}^{1}(\mathcal{Y}/S))$

by $\sigma(\theta)(D)=\theta_{*}(D)$ for
$\theta\in\operatorname{Bir}(\mathcal{Y}/S)$. Since $\theta$ is a small
modification, this defines a permutation action on the set of maximal cones of
$\operatorname{MF}(\mathcal{Y}/S)$. The following result, which will be
crucial for us, follows directly as in [Kaw97, Lemma 1.5].

######  Proposition 2.10.

Let $f_{i}\colon\mathcal{Y}\dashrightarrow\mathcal{Y}_{i}$, $i=1,2$ be small
modifications of $\mathcal{Y}$ over $S$, with $\mathcal{Y}_{1}$ and
$\mathcal{Y}_{2}$ models of the DNV family (of a given degree). Suppose that
the associated Mori cones are in the same orbit under the action of
$\operatorname{Bir}(\mathcal{Y}/S)^{\operatorname{op}}$, i.e. that there is a
a birational automorphism $\theta\in\operatorname{Bir}(\mathcal{Y}/S)$ such
that

$C(f_{1})=\theta^{*}C(f_{2}).$

Then there is an isomorphism $\beta:\mathcal{Y}_{1}\to\mathcal{Y}_{2}$ such
that $\beta\circ f_{1}=f_{2}$.

##  3\. $(-1)$-Curves and Curve Structures

From this section on, $Y_{c}$ will denote a $d$-semistable $K3$ surfaces of
type III while $Y$ will denote a component of $Y_{c}$. For any degree $2d$,
let $\operatorname{DNV}_{2d}$ be the set of $d$-semistable $K3$ surfaces of
type III in $(-1)$-form with $t=2d$ triple points. For $d=1$, these are the
surfaces $Y_{\mathscr{P}}$ and $Y_{\mathscr{T}}$ described in Example 1.28.
Let $\operatorname{Mod}_{2d}$ denote the set of surfaces $Y_{c}$ such that
there is $Y_{0}\in\operatorname{DNV}_{2d}$ and a sequence of elementary
modifications of type I

$Y_{0}\dashrightarrow Y_{1}\dashrightarrow\dots\dashrightarrow
Y_{i}\dashrightarrow\dots\dashrightarrow Y_{n}=Y_{c}.$

This also defines an anticanonical divisor on each component of $Y_{c}$. We
denote by $\operatorname{PMod}_{2d}$ the subset of projective surfaces in
$\operatorname{Mod}_{2d}$. Each of these surfaces determines a model of the
Dolgachev family of degree $2d$. In general, one does not obtain the full set
of models in this way as, usually, a central fibre of a model of the DNV
family of degree $2d$ cannot be obtained by type I modifications alone.
However, we will later see that for $d=1$ the set $\operatorname{PMod}_{2}$
parametrizes the isomorphism classes of models of the Dolgachev-Nikulin-Voisin
family of degree $2$.

######  Definition 3.1.

Let $Y_{c}\in\operatorname{Mod}_{2}$. We say $Y_{c}$ is of _class_
$\mathscr{T}$ (resp. $\mathscr{P}$) if the dual intersection graph of $Y_{c}$
is given by $\mathscr{T}$ (resp. $\mathscr{P}$).

We denote the set of $Y_{c}\in\operatorname{Mod}_{2}$ of class $\mathscr{T}$
(resp. $\mathscr{P}$) by $\operatorname{Mod}_{2}(\mathscr{T})$ (resp.
$\operatorname{Mod}_{2}(\mathscr{P})$), and similarly we define
$\operatorname{PMod}_{2}(\mathscr{T})$ and
$\operatorname{PMod}_{2}(\mathscr{P})$. Note that this defines decompositions
into disjoint unions
$\operatorname{Mod}_{2}=\operatorname{Mod}_{2}(\mathscr{T})\sqcup\operatorname{Mod}_{2}(\mathscr{P})$
and
$\operatorname{PMod}_{2}=\operatorname{PMod}_{2}(\mathscr{T})\sqcup\operatorname{PMod}_{2}(\mathscr{P})$
respectively.

We recall the following elementary facts about anticanonical pairs, i.e. pairs
$(Y,D)$ with $Y$ a smooth rational surface and $D$ an effective anticanonical
cycle. We will use these without further mention.

######  Proposition 3.2.

Let $(Y,D)$ be an anticanonical pair. Let $C$ be an irreducible curve that is
not a component of $D$.

  * (i)

If $C^{2}=-1$, then $C.D$=1 and $C$ is smooth rational.

  * (ii)

If $C^{2}=0$, then $C.D=2$ or $C.D=0$. In the first case, $C$ is smooth
rational, in the second case $p_{a}(C)=1$.

  * (iii)

If $C^{2}=-2$, then $C$ is smooth rational and $C.D=0$.

###  3.1. $(-1)$-Curves

In order to control flops, we need to control exceptional curves of the first
kind on the components of the central fibres. We show that the set of possible
elementary modifications of a surface in $\operatorname{Mod}_{2}$ is quite
small. Recall the following terminology. A _cycle_ is a graph whose vertices
and edges can be ordered $C_{1},\dots C_{n}$ and $e_{1},\dots,e_{n}$ such that
$e_{i}$ connects $C_{i}$ and $C_{i+1}$ (where the indices have to be read
cyclically). A _tree_ is a connected graph not containing a subgraph that is a
cycle. A vertex $v$ of a graph is called a _fork_ if there are at least three
edges from $v$. If $G$ is a tree with a unique fork $v\in G$, the connected
components of $G\backslash\\{v\\}$ are the _branches_ of $v$.

######  Construction 3.3.

Let $\Gamma_{1}^{\mathbf{0}}$ be a tree with a unique fork $v$ such that there
are $3$ branches $B_{i}$, $i=1,2,3$, consisting of $1$, $2$ and $4$ vertices,
i.e. the graph underlying the $E_{8}$ diagram. Let $\mathbf{n}=n_{1}\geq 0$.
Let $\Gamma_{1}^{\mathbf{n}}$ be the tree with a unique fork containing
$\Gamma_{1}^{\mathbf{0}}$ such that
$\Gamma_{1}^{\mathbf{n}}\backslash\\{B_{1}\cup B_{2}\\}$ has a unique
connected component given by a chain of length $n_{1}+4$. We label the end of
this tree which is not is not a fork of $\Gamma_{1}^{\mathbf{n}}$ with $-1$,
all other vertices are labelled with $-2$, see Figure 10. (We shall later
interpret these labels as intersection numbers.)

Let $\Gamma_{2}^{\mathbf{0}}$ be the tree with a unique fork $f$ such that
there are $3$ branches $B_{i}$, $i=1,2,3$, consisting of $1$, $3$ and $3$
vertices. Assume that $B_{1}$ is the branch that is a singleton. There are
exactly two vertices $v_{1},v_{2}$ \- the ends of the branches - in
$\Gamma_{2}^{\mathbf{0}}\backslash B_{1}$ that are connected to a unique
vertex $\Gamma_{2}^{\mathbf{0}}\backslash\\{v_{1},v_{2}\\}$. Let
$\mathbf{n}=(n_{1},n_{2})$. Let $\Gamma_{2}^{\mathbf{n}}$ be the tree with a
unique fork containing $\Gamma_{2}^{\mathbf{0}}$ such that
$\Gamma_{2}^{\mathbf{n}}\backslash\Gamma_{2}^{\mathbf{0}}$ has two connected
components $C_{1},C_{2}$ such that $C_{i}$ contains a vertex connected to
$v_{i}$ and $C_{i}$ has $n_{i}$ vertices. The ends of the branches $C_{i}$ are
labelled with $-1$, all other vertices are labelled with $-2$, see Figure 11.

Finally, let $\Gamma_{4}^{\mathbf{0}}$ be the unique tree with $2$ forks and
six vertices. Label the vertices that are not forks from $1$ to $4$ such that
one fork is connected to the vertices labelled $1$ and $2$. Let
$\mathbf{n}=(n_{1},n_{2},n_{3},n_{4})$. Let $\Gamma^{\mathbf{n}}_{4}$ be the
tree with $2$ forks, containig $\Gamma^{\mathbf{0}}_{4}$ such that
$\Gamma^{\mathbf{n}}_{4}$ minus the forks has $4$ connected components $B_{i}$
$i=1\dots 4$ such that if $v$ is the unique labelled vertex contained in
$B_{i}$, $B_{i}\backslash\\{v\\}$ contains $n_{j}$ vertices, with $j$ the
label of $v$. Again, we label the ends of the branches with $-1$ and all other
vertices with $-2$, see Figure 12.

$-1$ Figure 10. The intersection graph $\Gamma_{1}^{\bf n}$. All vertices with
label not displayed are labelled with $-2$.  ${-1}$${-1}$ Figure 11. The
intersection graph $\Gamma_{2}^{\bf n}$. All vertices with label not displayed
are labelled with $-2$.  $-1$$-1$$-1$$-1$ Figure 12. The intersection graph
$\Gamma_{4}^{\bf n}$. All vertices with label not displayed are labelled with
$-2$.

######  Lemma 3.4.

Let $n\in\mathbb{N}$, $n\geq 2$. Let $C_{i}$, $i=1,\dots n$ be a collection of
curves on a smooth surface $Y$. Suppose the dual graph is $A_{n}$, with
labelling of the vertices $v_{i}$ as in Figure 13. Assume the labelling is
such that $v_{i}$ corresponds to the curve $C_{i}$. Moreover, assume
$C_{i}^{2}=-1$ if $i=1$ and $-2$ if $1<i<n$. Let $\mathcal{H}$ be a set of
curves on $Y$ such for all $h\in\mathcal{H}$, $h.c_{i}=0$ for $i=1,\dots,n-1$.
Set $A=\sum_{i=1}^{n}a_{i}C_{i}+\sum_{h\in\mathcal{H}}\gamma_{h}h$ for
$a_{i},\gamma_{h}\in\mathbb{Q}$ and assume $A.C_{i}\geq 0$ for $i=1,\dots,n-1$
and $a_{1}\geq 0$. Then $a_{i+1}\geq a_{i}$. If $A.C_{i}>0$ for
$i=1,\dots,n-1$ and $a_{1}\geq 0$, it follows $a_{i+1}>a_{i}$.

###### Proof.

We have $0\leq A.C_{1}=-a_{1}+a_{2}$, so $a_{2}\geq a_{1}$. Suppose $a_{i}\geq
a_{i-1}$ for some $1<i<n-1$. Then $0\leq A.C_{i}=a_{i+1}-2a_{i}+a_{i-1}$, so
$a_{i+1}\geq 2a_{i}-a_{i-1}\geq a_{i}$. The claim follows. Replacing weak by
strict inequalites shows the second claim. ∎

$1$$2$$3$$n$ Figure 13. The numbering of the vertices of the $A_{n}$ graph in
Lemma 3.4.

Let $\mathfrak{Y}_{i}$ be as in Construction 1.23, with anticanonical cycle
$D$, and let $p$ be a special point. We recall that $p$ is an _interior_
special point if $p$ is a smooth point of $D$.

######  Proposition 3.5.

Consider $\mathfrak{Y}_{i}$ for $i=1,2$ or $4$, with anticanonical divisor
$D$. Let $(p_{1},\dots p_{i})$ denote the tuple of interior special points.
Let $Y$ denote the ${\mathbf{n}}=(n_{1},\dots n_{i})$ blow-up of
$\mathfrak{Y}_{i}$ in $(p_{1},\dots p_{i})$.

Then there are exactly $i$ exceptional curves $C_{k}\subset Y$, $k=1\dots i$
of the first kind such that $C_{k}$ is not a component of $D$. More precisely,
the intersection graph of the negative curves that are not components of $D$
is given by the graph $\Gamma_{i}^{\mathbf{n}}$.

###### Proof.

Let $\psi\colon Y\to\mathfrak{Y}_{i}$ be the $(n_{1},\dots,n_{i})$ blow-up of
$\mathfrak{Y}_{i}$ in the interior special points $(p_{1},\dots,p_{i})$.

Let $C(Y)$ be the set consisting of strict transforms of the $(-1)$ curves
$E_{i}$ from the construction of the $\mathfrak{Y}_{i}$ in Construction 1.23
and the $(-2)$-curves of $\mathfrak{Y}_{i}$ together with the irreducible
components of the exceptional locus of $\psi$, see also Figures 6 and 7. By
the results of [Loo81, Section $2$ ], the set $C(Y)$ defines a
$\mathbb{Q}$-basis of $\operatorname{Pic}(Y)$, with intersection graph
$\Gamma_{i}^{\mathbf{n}}$ and $\mathbf{n}=(n_{1},\dots n_{i})$. We can assume
$i=4$ as it is straightforward to check, using the construction of the
$\mathfrak{Y}_{i}$, that the result for $i=4$ implies the result for $i=1,2$.

Let $C_{k}$, $k=1,\dots,4$ denote the integral $(-1)$-curves in
$\operatorname{Ex}(\psi)$, corresponding to the vertices labelled with $(-1)$
in the diagrams. Let $D_{1},\dots,D_{4}$ denote the components of the
anticanonical divisor $D$ given as birational transforms of the anticanonical
cycle from the construction of $\mathfrak{Y}_{4}$. Suppose there is an
integral curve $C$ with $C^{2}<0$ not in
$\mathcal{B}:=\\{D_{1},\dots,D_{4}\\}\cup C(Y)$. We have $C.B\geq 0$ for all
curves $B$ in $\mathcal{B}$. Write

(3.1) $mC=\sum_{B\in C(Y)}\beta_{B}B.$

with $m,\beta_{B}\in\mathbb{Z}$ and $m>0$. From Lemma 3.4, it follows that all
coefficients $\beta_{B}$ are non-negative.

We find that

(3.2) $0>mC^{2}=\sum_{B\in C(Y)}\beta_{B}(C.B)\geq 0,$

where the last inquality follows since $\beta_{B}\geq 0$ and $C.B\geq 0$.
Hence a curve such as $C$ cannot exist. ∎

######  Remark 3.6.

The same proof works for $i=3$, the case $i=5$ also holds. We restrict to
$i=1,2,4$ as these are the relevant surfaces in the $d=1$ case.

We have the the following result.

######  Corollary 3.7.

Let $Y_{c}=Y_{1}\cup Y_{2}\cup Y_{3}$ be in $\operatorname{Mod}_{2}$. Then the
cones of curves $\operatorname{NE}(Y_{i}^{\nu})$ on the normalisations of the
components are finitely generated. If the Picard rank of $Y_{i}$ is at least
$3$, a generating set is given by the curves $C$ with $C^{2}<0$. For smaller
Picard rank, either $Y_{i}$ is a Hirzebruch surface or $\mathbb{P}^{2}$.

###### Proof.

The only thing that remains to show is the statement on the generators of
cones of curves. By Proposition 3.5, there are finitely many curves $C$ on
$Y_{i}$ with $C^{2}<0$, so, if the Picard rank is at least $3$, the corollary
follows from [AL11]. ∎

###  3.2. Curve Structures

Let $Y$ be a component of a surface $Y_{c}\in\operatorname{Mod}_{2}$. By
definition, there is a sequence of type I flops
$Y_{\mathscr{G}}\dashrightarrow Y_{c}$ whose inverse connects $Y$ to a
component of a model $Y_{\mathscr{G}}\in\operatorname{DNV}_{2}$ in
$(-1)$-form. Using this sequence, we will recursively define a set of curves
$C(Y)$ on the normalisation $Y^{\nu}$. Note that there is an induced sequence
of blow-ups and blow-downs $\psi\colon Y^{\nu}\dashrightarrow\mathfrak{Y}_{i}$
for some $i\in\\{1,2,4\\}$.

Let $D$ denote the anticanonical divisor of $\mathfrak{Y}_{i}$ as discussed in
Construction 1.23. Set

$C(\mathfrak{Y}_{i}):=\\{C\subset\mathfrak{Y}_{i}\mid C\text{ is an integral
curve with }C^{2}<0,C\not\subset D\\}.$

To start the induction, we factor $\psi$ as $Y^{\nu}\dashrightarrow
W\dashrightarrow\mathfrak{Y}_{i}$ with $Y^{\nu}\dashrightarrow W^{\nu}$
corresponding to an elementary modification of type I. Suppose we have defined
$C(W)$. If $Y^{\nu}=\text{Bl}_{p}(W^{\nu})$ for some $p$, we take $C(Y)$ to be
the strict transforms of curves in $C(W)$ together with the exceptional curve
of $Y^{\nu}\to W^{\nu}$. If $Y^{\nu}\dashrightarrow W^{\nu}$ is the inverse of
$\pi\colon W^{\nu}=\text{Bl}_{p}(Y^{\nu})\to Y^{\nu}$, we set

$C(Y):=\\{C\subset Y^{\nu}\mid C=\pi(C^{\prime})\text{ for }C^{\prime}\in
C(W)\\}.$

Because the normalisation morphism $Y^{\nu}\to Y$ identifies components of the
singular locus of $Y_{c}$, which are part of the anticanonical cycle, and thus
never curves in $C(Y)$, we can also interpret $C(Y)$ as a collection of curves
on $Y$. We then proceed by induction. The sequence
$Y_{\mathscr{G}}\dashrightarrow Y_{c}$ is not uniquely defined, but the set
$C(Y)$ and its intersection graph is independent of the choice such a
sequence, as the following lemma shows.

######  Lemma 3.8.

Let $Y$ be a component of $Y_{c}\in\operatorname{Mod}_{2}$. The collection
$C(Y)$ and the intersection graph are independent of the sequence
$Y_{\mathscr{G}}\dashrightarrow Y_{c}$.

###### Proof.

First, the surface $\mathfrak{Y}_{i}$ is uniquely determined by the number of
components of the double locus $D$ of $Y$. Let $D=D_{1}+\dots+D_{i}$ and
$p_{1},\dots,p_{i}$ be the interior special points. Let
$\bar{D}=\bar{D}_{1}+\dots+\bar{D}_{i}$ be the anticanonical cycle on
$\mathfrak{Y}_{i}$.

We can write the induced sequence $Y^{\nu}\dashrightarrow\mathfrak{Y}_{i}$ as

$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$$\textstyle{Y^{\nu}}$$\textstyle{\mathfrak{Y}_{i}}$

with $\pi$ the blow-up in the interior special points on components
$\bar{D}_{i}$ of $\bar{D}$ such that their strict transform $D_{i}$ under
$Y\dashrightarrow\mathfrak{Y}_{i}$ has $D_{i}^{2}<-1$. Up to trivial
modifications, i.e. blowing up a special point and then blowing down the
resulting exceptional curve, this morphism is uniquely defined by the self-
intersection numbers of $Y^{\nu}$. Let $J=\\{j_{1},\dots,j_{k}\\}$ be the set
of indices of components $D_{j}$ of $D$ with $D_{j}^{2}>-1$, where we
calculate intersection numbers on the normalisation. Then
$\psi\colon\tilde{Y}\to Y^{\nu}$ is the
$(n_{j_{1}},\dots,n_{j{{}_{k}}})$-blow-up of $Y^{\nu}$ in
$(p_{j_{1}},\dots,p_{j{{}_{k}}})$. This uniquely defines the curves that are
contracted. ∎

######  Definition 3.9.

Let $Y$ be as above. We define $\Gamma_{Y}$ to be the dual graph of $C(Y)$ and
label its vertices with the self-intersection numbers. We call $\Gamma_{Y}$
the _curve structure_ of $Y$. We say that a curve structure has _type_ $d_{i}$
if $Y$ maps to $\mathfrak{Y}_{i}$ under a sequence of type I flops. This is
well defined as the number of components of $D$ is fixed under type I flops.
Note that the chosen anticanonical divisor of $\mathfrak{Y}_{i}$ has $i$
components.

######  Remark 3.10.

We will usually consider $C(Y)$ as a set of curves on $Y^{\nu}$. In
particular, intersection numbers will always be calculated on $Y^{\nu}$.

######  Remark 3.11.

Note that for surfaces $Y$ which are $(n,m)$-blow-ups of some
$\mathfrak{Y}_{i}$ this coincides with the set $C(Y)$ which we have used in
the proof of Proposition 3.5.

We will often interpret $\Gamma_{Y}$ as the underlying set of vertices. Also,
for any vertex $v\in\Gamma_{Y}$, we will denote the underlying curve in $C(Y)$
by $C_{v}$, but abusing notation, we will often just write $v$. Note that if
$v,w$ are two distinct vertices of $\Gamma_{Y}$, the number of edges between
$v$ and $w$ is either $1$ or $0$ and this is the same as the intersection
number $C_{w}.C_{v}$. Two vertices $v,w$ with $v.w=1$ for the underlying
curves are called _adjacent_.

Given $Y$ with normalisation $Y^{\nu}$ and with curve structure $\Gamma_{Y}$,
let $D=\sum D_{i}$ be the anticanonical divisor of $Y^{\nu}$. For each
$D_{i}$, append a vertex $v_{D_{i}}$ to $\Gamma_{Y}$ and for each
$v\in\Gamma_{Y}$ such that $C_{v}$ is not a component of $D$, add an edge
joining $v_{D_{i}}$ and $v$ if $D_{i}.C_{v}\neq 0$. This defines the
_augmented_ curve structure $\Gamma_{Y}^{a}$. See Figure 14 for an example.

Note that the existence of an edge between a vertex $v\in\Gamma_{Y}$ and a
vertex corresponding to some $D_{i}$ does not necessarily imply that the
intersection number of the corresponding curves is exactly $1$, it simply
means that the intersection is non-empty.

If $\Gamma_{Y}$ and $\Gamma_{Y^{\prime}}$ are two curve structures, we then
have an obvious notion of an isomorphism $\Gamma_{Y}\to\Gamma_{Y^{\prime}}$: a
bijection of graphs $\Gamma_{Y}\to\Gamma_{Y^{\prime}}$ preserving the
intersection numbers that extends to a bijection of graphs
$\Gamma_{Y}^{a}\to\Gamma_{Y^{\prime}}^{a}$.

$-1$$-1$ Figure 14. The augmented curve structure of a component $Y$ of
$Y_{\mathscr{P}}$. The black vertices correspond to the components of the
anticanonical divisor.

######  Definition 3.12.

Let $v_{e}\in\Gamma_{Y}$ be a vertex with $v_{e}^{2}=-1$. Let $D_{0}$ be the
unique component of the anticanonical cycle met by $v_{e}$. The vertex $v_{e}$
is called _exceptional_ if

$\\{v\in\Gamma_{Y}\mid v.D_{0}\neq 0\\}=\\{v_{e}\\}\text{ and
}|\\{v\in\Gamma_{Y}\mid v.v_{e}=1\\}|=1.$

Let $Y$ be a component of a surface $Y_{c}\in\operatorname{Mod}_{2}$ with
curve structure $\Gamma_{Y}$. We will also denote the preimage of the double
locus of $Y_{c}$ on $Y$ under the normalisation map $Y^{\nu}\to Y$ by $D$. Let
$v_{e}\in\Gamma_{Y}$ be an exceptional vertex. Starting with this exceptional
vertex we now define a subgraph of $\Gamma_{Y}$ inductively. Since $v_{e}$ is
exceptional, there is a unique vertex $v_{1}$ with $v_{1}.v_{e}=1$. Set
$L_{1}(v_{e}):=(v_{e},v_{1}).$ Now suppose that we have already defined the
ordered tuple of vertices $L_{n}(v_{e})=(v_{e},v_{1},\dots,v_{n})$. If there
is a unique vertex $v\in\Gamma_{Y}\backslash\\{v_{e},\dots v_{n}\\}$ adjacent
to $v_{n}$ and $v_{n}.D=0$, set $v_{n+1}:=v$ and
$L_{n+1}(v_{e})=L_{n}(v_{e})\cup\\{v_{n+1}\\}$. Else set
$L(v_{e}):=L_{n}(v_{e})$. There is a unique connected subgraph of $\Gamma_{Y}$
whose vertices are given by the vertices in $L(v_{e})$. By abuse of notation
we consider $L(v_{e})$ as this subgraph.

######  Definition 3.13.

Let $v_{e}$ be an exceptional vertex. The graph $L(v_{e})$ is the _leg_
defined by $v_{e}$. The unique vertex $v$ of $L(v_{e})$ not equal to $v_{e}$
meeting precisely one vertex of $L(v_{e})$ is the _end_ of the leg. In this
situation we also say that $L(v_{e})$ ends in $v$.

######  Definition 3.14.

A curve structure $\Gamma_{Y}$ is called _degenerate_ if there is no
exceptional vertex, or if for some exceptional vertex $v_{e}$ the leg
$L(v_{e})$ ends in a vertex $v$ with $v.D_{i}=1$ for some component
$D_{i}\subset D$ such that $D_{i}$ is in the preimage of a smooth component of
the restriction of the double locus of $Y_{c}$ to $Y$.

The degenerate curve structures of type $d_{2}$ are displayed in Figure 15.
Curve structures that are not degenerate will be called _non-degenerate_.

######  Example 3.15.

A non-degenerate curve structure $\Gamma_{Y}$ has at least one exceptional
vertex. If $\Gamma_{Y}$ is of type $d_{2}$, then $Y$ is obtained from
$\mathfrak{Y}_{2}$ by a sequence of blow-ups and blow-downs. It follows from
the definitions that $\Gamma_{Y}$ has two exceptional vertices. More
precisely, $\Gamma_{Y}$ has two legs that both end on the same vertex $v$.
Moreover, there is a unique vertex $v^{\prime}$ that is on none of the legs
and also meets $v$. In particular, we have $D_{1}^{2}\leq 1$ and
$D_{2}^{2}\leq 1$ for the two components $D_{1}$ and $D_{2}$ of the anti-
canonical divisor.

$-2$$0$$m$$0$$1$ Figure 15. The possible (augmented) degenerate curve
structures of type $d_{2}$ without exceptional vertex. The numbers give the
self intersection of the underling curve, with $m\geq-1$.

######  Proposition 3.16.

Let $\Gamma_{Y}$ be a curve structure of type $d_{i}$ with $i\in\\{1,2,4\\}$.
Then

$\\{C_{v}\mid v\in\Gamma_{Y}\\}$

is a $\mathbb{Q}$-basis of $\operatorname{Pic}(Y^{\nu})$.

###### Proof.

The statement is true for $\mathfrak{Y}_{i}$: as mentioned earlier, by the
results in [Loo81, Section $2$], one obtains a $\mathbb{Q}$-basis of
$\operatorname{Pic}(\mathfrak{Y}_{i})$. In the general case, the curve
structures in question are obtained via blow-ups and blow-downs from the
$\mathfrak{Y}_{i}$, $i=i\in\\{1,2,4\\}$. Arguing inductively, we need to show
that given a curve structure $C(Y_{0})$, the structure $C(Y)$ obtained on $Y$
under one of these operations is also a $\mathbb{Q}$-basis. Obviously, a blow-
up $Y^{\nu}\to Y^{\nu}_{0}$ induces a curve structure on $Y^{\nu}$ that is
also a $\mathbb{Q}$-basis, as it adds the exceptional curve. So let $g\colon
Y^{\nu}_{0}\to Y^{\nu}$ be the blow-down in an exceptional curve corresponding
to a type I flop, i.e. the contraction of an extremal ray spanned by a curve
$E$. We have the exact sequence

$0\to\operatorname{Pic}(Y^{\nu})\to\operatorname{Pic}(Y^{\nu}_{0})\to\mathbb{Z}\to
0$

with first map pullback via $g$ and second map evaluation on $E$. Let
$A=\\{H_{i}\\}$ denote the collection of elements of $C(Y_{0})$ disjoint from
$E$. They induce a collection $\\{L_{i}\\}$ with $H_{i}=g^{*}L_{i}$. If
$C_{j}$ is in $C(Y_{0})$ but not in $A$ and not equal to $E$, then
$C_{j}.E=1$. Hence $C_{j}+E=g^{*}(F_{j})$ for some $F_{j}$. The collection
$\\{L_{i},F_{j}\\}$ is linearly independent as
$C(Y_{0})=\\{H_{i},C_{i}+E,E\\}$ is a spanning set of
$\operatorname{Pic}(Y^{\nu}_{0})_{\mathbb{Q}}$ with
$\operatorname{rank}\operatorname{Pic}(Y^{\nu}_{0})$ elements, and thus
linearly independent. Also, note that $g_{*}H_{i}=g_{*}g^{*}L_{i}=L_{i}$ and
$g_{*}(C_{j})=g_{*}(C_{j}+E)=g_{*}g^{*}(F_{j})=F_{j}$. ∎

We now specialise the discussion to curve structures of type $d_{2}$. These
are the ones appearing on components $Y$ of surfaces $Y_{c}$ in
$\operatorname{Mod}_{2}(\mathscr{P})$. Note that in this case all components
of $Y_{c}$ are smooth. Our goal in the remainder of this section is to express
properties of the ample cone of such $Y$ in terms of curve structures. In the
next section, we will see how these properties give criteria for the
projectivity of $Y_{c}$. Note that all such $Y$ are smooth rational surfaces
and the anticanonical divisor $D$ is the restriction of the double locus of
$Y_{c}$ to $Y$.

######  Definition 3.17.

Let $\Gamma_{Y}$ be a curve structure of type $d_{2}$. We say $\Gamma_{Y}$ is
_regular_ if $|\Gamma_{Y}|>1$ and no leg $L(e)$ of an exceptional vertex $e$
ends in a vertex $v$ with $v^{2}=0$.

A curve structure that is not regular is called non-regular. In particular, a
non-degenerate curve structure is regular and a non-regular curve structure is
degenerate. Let $\Gamma_{Y}$ be a non-regular curve structure. Then for each
component $D_{i}$ of the anticanonical divisor $D$ there is a unique vertex
$v_{D_{i}}$ of $\Gamma_{Y}$ with $D_{i}.v_{D_{i}}>0$.

Let $\Gamma_{Y}$ be a curve structure. Consider a map

$f\colon\Gamma_{Y}\to\mathbb{Z}.$

By abuse of notation we mean here that the map is defined on the underlying
set of $\Gamma_{Y}$, i.e. its vertices.

This defines a divisor

$\Gamma_{f}=\sum_{v\in\Gamma_{Y}}f(v)C_{v}.$

Rephrasing Lemma 3.4, we obtain the following lemma.

######  Lemma 3.18.

Let $\Gamma_{f}.v>0$ for all $v\in\Gamma_{Y}$. Suppose there is an exceptional
vertex $v_{0}\in\Gamma_{Y}$ and suppose $f(v_{0})>0$. Assume the leg
$L(v_{0})$ defined by $v_{0}$ is given by the graph
$(v_{0},v_{1},v_{2},\dots,v_{n})$. Then $f(v_{i+1})>f(v_{i})$ for
$i=0,\dots,n-1$.

######  Lemma 3.19.

Let $\Gamma_{Y}$ be of type $d_{2}$. Let $\Gamma_{f}$ be a divisor defined by
a map $f\colon\Gamma_{Y}\to\mathbb{Z}$. Suppose $\Gamma_{f}.v>0$ for all
$v\in\Gamma_{Y}$ and assume that $f(v_{e})>0$ for all exceptional vertices
$v_{e}$ of $\Gamma_{Y}$ or, if there is no exceptional vertex, $f(c)>0$ for
the unique vertex meeting both components of $D$. Then $\Gamma_{f}$ is an
ample divisor.

###### Proof.

If $\Gamma_{Y}$ does not have an exceptional vertex, it is degenerate. It
follows from the description of possible degenerate curve structures without
exceptional vertices, compare Figure 15, that all $f(v)$ are strictly
positive. The same statement holds if $Y$ has an exceptional vertex. This
follows from Lemma 3.18. We next claim that for any curve $C$ not in $C(Y)$
there exists some $C_{v},v\in\Gamma_{Y}$ such that $C.C_{v}>0$. To see this,
let $A$ be an ample divisor on $Y$. Then $A.C>0$. As $C(Y)$ is a basis of
$\operatorname{Pic}(Y)_{\mathbb{Q}}$, we can, replacing $A$ by a multiple if
necessary, write $A=\sum_{C_{v}\in C(Y)}\beta_{v}C_{v}$ with
$\beta_{v}\in\mathbb{Z}$. Then $C.C_{v}>0$ for some $v\in C(Y)$, as else
$A.C=0$. It follows that $\Gamma_{f}.C>0$ for all integral curves, so
$\Gamma_{f}$ is positive on $\operatorname{NE}(Y)$, as the latter is finitely
polyhedral by Corollary 3.7. By Kleiman’s criterion, $\Gamma_{f}$ is ample. ∎

######  Proposition 3.20.

Let $Y$ be a component of a surface
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$. Suppose $\Gamma_{Y}$ is non-
degenerate. Let $f_{0},g_{0}$ be the exceptional vertices and let
$k_{1},k_{2}$ be non-negative integers. Then, there is a positive integer
$\triangle$ such that for any $e>\triangle$, there is a map
$f\colon\Gamma_{Y}\to\mathbb{Z}$ such that $\Gamma_{f}$ is an ample divisor
with $f(f_{0})=e+k_{1}$ and $f(g_{0})=e+k_{2}$.

###### Proof.

Let $c$ be the fork of $\Gamma_{Y}$ and let $(f_{0},\dots,f_{n},c)$ and
$(g_{0},\dots,g_{m},c)$ denote the legs associated to the exceptional
vertices. Let $y$ denote the unique vertex not appearing on any of the legs.
We can assume

$k_{1}+\frac{1}{2}n(n+1)\geq k_{2}+\frac{1}{2}m(m+1).$

Set
$\triangle^{\prime}:=k_{1}+\frac{1}{2}n(n+1)-k_{2}-\frac{1}{2}m(m+1)+2\max\\{n,m\\}+2$.
Let $e$ be any integer such that $e>\triangle:=2\triangle^{\prime}+2$.

Define a map $f\colon\Gamma_{Y}\to\mathbb{Z}$ by

$\displaystyle f(f_{i})=e+k_{1}+\frac{1}{2}i(i+1)$ $\displaystyle
f(g_{i})=e+k_{2}+\frac{1}{2}i(i+1)$ $\displaystyle
f(c)=\max\\{f(f_{n}),f(g_{m})\\}+\max\\{n,m\\}+1$ $\displaystyle
f(y)=\left\lceil{\frac{1}{2}e}\right\rceil-1.$

Note that by assumption $f(f_{n})\geq f(g_{m})$. Also, we have
$f(f_{1})-f(f_{0})=1$ and

$f(f_{i-1})-2f(f_{i})+f(f_{i+1})=1$

for $1\leq 1\leq n-1$. Also, $f(f_{i})-f(f_{i-1})=i$, so

$\displaystyle-2f(f_{n})+f(c)+f(f_{n-1})=\max\\{n,m\\}+1-n>0.$

Analoguously, we find $f(g_{1})-f(g_{0})=1$,
$f(g_{i-1})-2f(g_{i})+f(g_{i+1})=1$ for $1\leq 1\leq m-1$ and

$\displaystyle-2f(g_{m})+f(c)+f(g_{m-1})\geq\max\\{n,m\\}+1-m>0.$

We also have

$\displaystyle-2f(c)+f(f_{n})+f(g_{m})+f(y)$
$\displaystyle=-2\max\\{f_{n},g_{m}\\}-2\max\\{n,m\\}-2+f(f_{n})+f(g_{m})+f(y)$
$\displaystyle=-f(f_{n})+f(g_{m})+f(y)-2\max\\{n,m\\}-2$
$\displaystyle=f(y)-\triangle^{\prime}>0.$

Finally, $\max\\{f(f_{n}),g(g_{m})\\}\geq e$, so
$-2f(y)+f(c)=-e+2+\max\\{f_{n},g_{m}\\}+\max\\{n,m\\}+1>0.$ By Lemma 3.19,
$\Gamma_{f}$ is an ample divisor with properties as desired. ∎

######  Proposition 3.21.

Let $Y$ be a component of a surface
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$. Suppose $\Gamma_{Y}$ is non-
degenerate. Write $D=D_{1}+D_{2}$ for the anticanonical cycle. Assume
$D_{1}^{2}\leq 0$. Let $k_{1},k_{2}$ and $\gamma$ be integers with
$k_{1},k_{2}\geq 0$ and $\gamma>1$. Then there are positive integers
$\triangle$ and $\alpha$ such that for any $e>\triangle$ divisible by
$\alpha$, there is an ample divisor $A$ with $A.D_{1}=e+k_{1}$ and
$A.D_{2}=\gamma e+k_{2}$.

###### Proof.

Assume first that $D_{1}^{2}<0$. Take an ample divisor $L$ on $Y$ and set
$M=-D_{1}^{2}L+(L.D_{1})D_{1}$. Let $C$ be an integral curve on $Y$. If $C$ is
distinct form $D_{1}$ then $M.C=(-D_{1}^{2})L.C+(L.D_{1})D_{1}.C>0$. Also,
$M.D_{1}=-D_{1}^{2}L.D_{1}+(L.D_{1})D_{1}^{2}=0$. If $D_{1}^{2}=0$, then set
$M=D_{1}$. In any case there is a nef divisor $M$ with $M.D_{2}=\alpha>0$. and
$M.D_{1}=0$. Now, by Proposition 3.20, there is an integer $\triangle$ and
such that for any integer $e>\triangle$ there is an ample divisor $\Gamma_{f}$
such that $\Gamma_{f}.D_{1}=e+k_{1}$ and $\Gamma_{f}.D_{2}=e+k_{2}$. Fix
$e=\alpha r$ for some integer $r>0$ such that $e>\triangle$. Then
$A:=\Gamma_{f}+(\gamma-1)rM$ is ample. Also, $A.D_{1}=e+k_{1}$ and
$A.D_{2}=k_{2}+\gamma e$, as desired. ∎

######  Example 3.22.

Proposition 3.21 fails if $D_{1}^{2}>0$, as the legs of the curve structure
are then too short: Let $D^{\prime}=D_{1}^{\prime}+D_{2}^{\prime}$ be the
anticanonical divisor of $\mathfrak{Y}_{2}$, $p_{i}\in D^{\prime}_{i}$,
$i=1,2$ be the interior special points of $\mathfrak{Y}_{2}$. Assume that $C$
is the $(-1)$-curve meeting $D^{\prime}_{1}$ and $F$ the $(-2)$-curve with
$C.F=1$. For $n\in\mathbb{N}$ let $Y(n)$ be the surface obtained from
$\mathfrak{Y}_{2}$ obtained by blowing up $n$ times in $p_{2}$ and contracting
$C+F$. Note that $\Gamma_{Y(n)}$ is non-degenerate of type $d_{2}$. Let $k$
and $e$ be positive integers. Suppose $A$ is an ample divisor with $A.D_{1}=e$
and $A.D_{2}=\gamma e+k$ with some integer $\gamma>1$, and define
$f\colon\Gamma_{Y(n)}\to Z$ to be the map with $A=\Gamma_{f}$. Let $v_{e_{i}}$
be the exceptional vertex meeting $D_{i}$. We have $L(v_{e_{1}})=(e_{1},c)$
with $c$ the fork of $\Gamma_{Y(n)}$. Also, write
$L(v_{e_{2}})=(v_{e_{2}},\dots,w,c)$ and let $y$ be the unique vertex not on
any leg. We have $f(c)=\gamma e+\alpha+\beta$ and $f(w)=\gamma e+\beta$ with
$\alpha,\beta>0$ by Lemma 3.18. By ampleness $A.c>0$, so
$f(e_{1})+f(w)+f(y)>2f(c)$ and using that $A.D_{1}=f(e_{1})=e$ we obtain

$f(y)>(\gamma-1)e+2\alpha+\beta.$

Intersecting $A$ with $y$ gives $-2f(y)+f(c)>0$. So
$0>2f(y)-f(c)>2(\gamma-1)e+4\alpha+2\beta-f(c)$, implying
$0>(\gamma-2)e+3\alpha+\beta>0$, a contradiction. Hence a divisor such as $A$
cannot exist.

We note the following variant of Proposition 3.20.

######  Proposition 3.23.

Let $Y$ be a component of a surface $Y_{c}$ in
$\operatorname{Mod}_{2}(\mathscr{P})$, with $\Gamma_{Y}$ non-degenerate. Let
$D=D_{1}+D_{2}$ be the anticanonical divisor. Assume $D^{2}_{1}=1$. Let $k\geq
0$ be an integer. Then there is an integer $\triangle>0$ such that for any
even integer $e>\triangle$ an ample divisor $A$ on $Y$ exists with $A.D_{1}=e$
and $A.D_{2}=\frac{3}{2}e+k$.

###### Proof.

Let $c$ be the fork of $\Gamma_{Y}$ and let $v_{e_{i}}$ denote the exceptional
vertex with $D_{i}.v_{e_{i}}=1$. Let $L(v_{e_{2}})=(f_{0},\dots,f_{n},c)$.
Denote by $y$ the unique vertex not appearing on any of the legs and set
$\triangle=n(n+1)+6n+2k+10$.

Let $e$ be any even integer such that $e>\triangle$. Define a map
$f\colon\Gamma_{Y}\to\mathbb{Z}$ by

$\displaystyle f(v_{e_{1}})=e$ $\displaystyle
f(f_{i})=\frac{3}{2}e+k+\frac{1}{2}i(i+1)$ $\displaystyle f(c)=f(f_{n})+n+1$
$\displaystyle f(y)=\frac{1}{2}e+\frac{1}{2}n(n+1)+2n+3+k.$

Then $-f(v_{e_{1}})+f(c)=f(f_{n})+n+1-e>0.$ As above, we have
$f(f_{1})-f(f_{0})=1$ and

$f(f_{i-1})-2f(f_{i})+f(f_{i+1})=1$

for $1\leq i\leq n-1$. Moreover, $f(f_{i})-f(f_{i-1})=i$ for $1\leq i\leq
n-1$, so

$\displaystyle-2f(f_{n})+f(c)+f(f_{n-1})=1.$

We also have

$\displaystyle-2f(c)+f(f_{n})+f(v_{e_{1}})+f(y)$
$\displaystyle=-2f(f_{n})-2n-2+f(f_{n})+f(v_{e_{1}})+f(y)$
$\displaystyle=-f(f_{n})+f(v_{e_{1}})+f(y)-2n-2$
$\displaystyle=-\frac{3}{2}e-k-\frac{1}{2}n(n+1)-2n-2+e+f(y)$
$\displaystyle=1.$

Finally

$\displaystyle-2f(y)+f(c)$
$\displaystyle=-e-n(n+1)-4n-6-2k+\frac{3}{2}e+k+\frac{1}{2}n(n+1)+n+1$
$\displaystyle=\frac{1}{2}e-\frac{1}{2}n(n+1)-3n-5-k$
$\displaystyle=\frac{1}{2}(e-\triangle)>0.$

Hence $\Gamma_{f}$ is an ample divisor with properties as desired. ∎

We will also need to construct ample divisors on surfaces with degenerate
curve structures. For better readability, we will treat the different cases
separately.

######  Proposition 3.24.

Let $Y$ be a component of a surface
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$. Suppose $\Gamma_{Y}$ is of type
$d_{2}$, regular and degenerate. Suppose there is an exceptional vertex
$v_{0}$ and that $e$ is an integer, $e>2$. Write $D=D_{1}+D_{2}$ with $D_{1}$
denoting the component of $D$ with $D_{1}.v_{0}=1$. Then there is a map
$f\colon\Gamma_{Y}\to\mathbb{Z}$ such that $\Gamma_{f}$ is an ample divisor
with degree $e+m$ on $D_{2}$ and degree $e$ on $D_{1}$ for some integer $m>0$
independent of $e$.

###### Proof.

Write $L(v_{0})=(v_{0},\dots,v_{n})$. Let $w$ be the unique element in
$\Gamma_{Y}\backslash L(v_{0})$ which exists by regularity. Let $e\geq 2$.
Define $f\colon\Gamma_{Y}\to\mathbb{Z}$ by setting
$f(v_{i})=e+\frac{1}{2}i(i+1)$ and $f(w)=n+1$. Then $\Gamma_{f}.v_{i}=1$ for
all $i$ and also $\Gamma_{f}.w>0$. It follows from Lemma 3.19 that
$\Gamma_{f}$ is an ample divisor with degree $e$ on $D_{1}$ and degree
$e+(D_{2}.w)(n+1)+\frac{1}{2}n(n+1)$ on $D_{2}$. Since $D_{2}.w>0$ we obtain
the claim by setting $m=(D_{2}.w)(n+1)+\frac{1}{2}n(n+1)$. ∎

Let $\Gamma_{Y}$ be of type $d_{2}$, non-regular and with no exceptional
vertex. Then $\Gamma_{Y}$ is a singleton given by a vertex $v$. Conversely, if
$Y$ is a component of a surface $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$
with $\Gamma_{Y}$ a singleton, then $\Gamma_{Y}$ is of type $d_{2}$, non-
regular and with no exceptional vertex. For later use we note the following
obvious statement.

######  Proposition 3.25.

Let $Y$ be a component of a surface
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$. Suppose $\Gamma_{Y}$ is a
singleton given by a vertex $v$. Then, after relabelling $D$, we have
$D_{1}.v=1$ and $D_{2}.v=2$. Then, for any $e>0$, $v\mapsto e$ defines an
ample divisor with degree $e$ on $D_{1}$ and $2e$ on $D_{2}$ and any ample
divisor is of this form.

######  Proposition 3.26.

Let $Y$ be a component of a surface
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$. Suppose $\Gamma_{Y}$ is not
regular. Suppose there is an exceptional vertex $v_{0}$ and let $e>2$ be an
integer. Write $D=D_{1}+D_{2}$ with $D_{1}$ denoting the component of $D$ with
$D_{1}.v_{0}=1$. Then there is a map $f\colon\Gamma_{Y}\to\mathbb{Z}$ such
that $\Gamma_{f}$ is an ample divisor with degree $2e+m$ on $D_{2}$ and degree
$e$ on $D_{1}$ for some integer $m>0$ independent of $e$.

###### Proof.

As $\Gamma_{Y}$ is not regular and hence degenerate, $L(v_{0})=\Gamma_{Y}$.
Write $L(v_{0})=(v_{0},v_{1},\dots v_{n})$ and setting
$f(v_{i})=e+\frac{1}{2}i(i+1)$ defines an ample divisor (note that
$v^{2}_{n}=0$) with degree $2(e+\frac{1}{2}n(n+1))$ on $D_{2}$ and $e$ on
$D_{1}$. ∎

##  4\. Projective Models

In this section we analyse the elements in the set $\operatorname{PMod}_{2}$.

###  4.1. Models of class $\mathscr{T}$

Recall from Example 1.28 that $Y_{\mathscr{T}}$ has a component with
normalisation $\mathfrak{Y}_{4}$. Denote this component by
$(Y_{\mathscr{T}}){{}_{\omega}}$. Let
$Y_{c}\in\operatorname{Mod}_{2}(\mathscr{T})$. There is a sequence of type I
flops to $Y_{\mathscr{T}}$. Let $Y_{\omega}$ be the image of
$(Y_{\mathscr{T}}){{}_{\omega}}$ under the induced birational map. It is
independent of the chosen sequence. We call $Y_{\omega}$ the _special_
component of $Y_{c}$.

######  Proposition 4.1.

Let $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{T})$, $Y_{\omega}$ the special
component. Let $D_{\omega}$ be the smooth component of the boundary curve of
$Y_{\omega}$. Let $D_{1}$ and $D_{2}$ be the disjoint curves in the preimage
of $D_{\omega}$ under the normalisation $\pi\colon Y^{\nu}_{\omega}\to
Y_{\omega}$. Then $Y_{c}$ is projective if and only if
$D_{1}^{2}=D_{2}^{2}=-1$.

###### Proof.

Let $Y_{c}=Y_{1}\cup Y_{2}\cup Y_{3}$ and suppose $Y_{\omega}=Y_{2}$. We begin
by showing that $Y_{c}\in\operatorname{PMod}_{2}(\mathscr{T})$ if and only if
$Y_{\omega}$ is projective. Projectivity of $Y_{c}$ clearly implies
projectivity of $Y_{w}$, so we only need to show the opposite implication. So
suppose $Y_{\omega}$ is projective. Because $Y_{c}$ is in
$\operatorname{Mod}_{2}$ and has dual intersection complex $\mathscr{T}$, the
remaining two components of $Y_{c}$ are obtained by blow-ups and blow-downs of
$\mathfrak{Y}_{1}$ in the interior special point, and thus are projective.
Hence there are ample line bundle $A_{i}$ on $Y_{i}$ for $i=1,2,3$. The
components $Y_{1}$ and $Y_{2}$ are glued by identifying a curve $D_{12}\subset
Y_{1}$ with a curve $D_{21}\subset Y_{2}$. Similarly, $Y_{3}$ and $Y_{2}$ are
glued by identifying a curve $D_{32}\subset Y_{3}$ with a curve $D_{23}\subset
Y_{2}$. Replacing the $A_{i}$ by suitable multiples, we can assume that
$A_{1}.D_{12}=A_{2}.D_{21}$ and $A_{2}.D_{23}=A_{3}.D_{32}$. Because $Y_{c}$
has trivial Carlson extension, one obtains an ample bundle on $Y_{c}$ by Lemma
1.10 and [Laz04, Proposition 1.2.16], implying that $Y_{c}$ is projective.

We now show that projectivity of $Y_{w}$ is equivalent to the numerical
conditon stated in the propositon. Suppose first that
$D_{1}^{2}=D_{2}^{2}=-1$. One can, similar as in the proof of Proposition
1.26, find an ample divisor on $Y_{\omega}$. Indeed, take an ample divisor
$L^{\prime}=L^{\prime}_{\omega}$ on the normalisation $Y^{\nu}_{\omega}$.
Suppose without loss of generality $L^{\prime}.D_{1}\geq L^{\prime}.D_{2}$.
Set $L^{\prime\prime}=(L^{\prime}.D_{2})L^{\prime}$. Set
$M=L^{\prime}+(L^{\prime}.D_{1})D_{1}$. Then $M.C\geq 0$ for all curves $C$ on
$Y^{\nu}_{\omega}$ with equality if and only if $C=D_{1}$. Write
$n=L^{\prime}.D_{1}-L^{\prime}.D_{2}$. Note that $D_{1}.D_{2}=0$. Then
$L_{\omega}=L^{\prime\prime}+nM$ is ample by the Nakai-Moisjezon criterion and

$\displaystyle L_{\omega}.D_{2}$
$\displaystyle=L^{\prime\prime}.D_{2}+nM.D_{2}$
$\displaystyle=(L^{\prime}.D_{2})(L^{\prime}.D_{2})+(L^{\prime}.D_{1})(L^{\prime}.D_{2})-(L^{\prime}.D_{2})(L^{\prime}.D_{2})$
$\displaystyle=L^{\prime\prime}.D_{1}+nM.D_{1}$
$\displaystyle=L_{\omega}.D_{1}.$

Hence $L_{\omega}$ induces an ample divisor on $Y_{w}$.

We now show that the condition on the intersection numbers is also necessary.
So suppose that $D_{1}^{2}<-1$. Note that $D_{2}^{2}=-2-D_{1}^{2}$. Suppose
$L^{\prime}$ is an ample line bundel on $Y_{\omega}$. The normalisation
$\nu\colon Y_{\omega}^{\nu}\to Y_{\omega}$ is finite, so $L=\nu^{*}L^{\prime}$
is ample.

By Proposition 3.16, replacing $L$ by a suitable multiple we can write

$L=\sum_{v\in C(Y)}a_{v}C_{v}$

with $a_{v}\in\mathbb{Z}$.

As $L$ is ample, the coefficients of vertices meeting the anticanonical
divisor are strictly positive. This implies $a_{v}>0$ for all $v$ by Lemma
3.18, as either $v$ is contained in the leg defined by some exceptional vertex
or meets a component of the boundary. By the condition on the self-
intersection of $D_{1}$ the curve structure $\Gamma_{Y}$ is degenerate: there
is an exceptional vertex $e$ with $e_{1}.D_{1}=1$ and $L(e)$ ends in a vertex
$w$ with $w.D_{2}=1$. By Lemma 3.18,

$L.D_{2}>a_{w}>a_{e}=L.D_{1},$

so $Y_{\omega}$ cannot be projective. ∎

In particular, the proposition says that $Y_{c}\in\operatorname{Mod}_{2}$ is
in $\operatorname{PMod}_{2}(\mathscr{T})$ if and only if there is a sequence
of type I flops

$Y_{c}\dashrightarrow\dots\dashrightarrow
Y_{i}\dashrightarrow\dots\dashrightarrow Y_{\mathscr{T}},$

such that any flopping curve meets the nodal components of the double locus on
the special component of $Y_{i}$.

###  4.2. Models of class $\mathscr{P}$

######  Proposition 4.2.

Let $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$ and write $Y_{c}=Y_{1}\cup
Y_{2}\cup Y_{3}$.

  * (i)

If $\Gamma_{Y_{i}}$ is degenerate for all $i=1,2,3$, then $Y_{c}$ is not
projective.

  * (ii)

Suppose $\Gamma_{Y_{i}}$ is regular for all $i$. If there is a component
$Y_{i}$ such that $\Gamma_{Y_{i}}$ is non-degenerate, then $Y_{c}$ is
projective.

###### Proof.

Suppose first that the curve structures $\Gamma_{Y_{i}}$ are all degenerate.
We show that this cannot happen if $Y_{c}$ is projective. So let $A$ be an
ample divisor on $Y_{c}$. Denote the restriction to $Y_{i}$ by $A_{i}$. The
set of underlying curves of $\Gamma_{Y_{i}}$ is a basis of
$\operatorname{Pic}(Y_{i})_{\mathbb{Q}}$ and thus after replacing $A$ by a
suitable multiple, we write $A_{i}=\Gamma_{f_{i}}$ for a suitable function
$f_{i}\colon\Gamma_{Y_{i}}\to\mathbb{Z}$. Note that at least one of the
$Y_{i}$, say $Y_{1}$, has an exceptional vertex $v_{e}$. Let the leg of
$v_{e}$ be given by $L(v_{e})=(v_{e},\dots,v_{n})$. Suppose the component of
the anticanonical divisor met by $v_{e}$ is $D_{13}$ while $v_{n}$ meets
$D_{12}$. Necessarily $f(v_{e})>0$, and thus we have $f(v_{n})>f(v_{e})$ by
Lemma 3.18. In particular, $A_{1}.D_{12}>A_{1}.D_{13}$. Now consider $Y_{2}$.
If $\Gamma_{Y_{2}}$ has an exceptional vertex, we find
$A_{2}.D_{23}>A_{2}.D_{21}$ by the same reasoning. If $\Gamma_{Y_{2}}$ has no
exceptional vertex and $|\Gamma_{Y_{2}}|>1$, then $\Gamma_{Y_{2}}$ has two
elements, say $v_{1}$ and $v_{2}$ with indices chosen such that $v_{1}^{2}=0$,
$v_{1}.v_{2}=1$, $D_{21}.v_{2}=0$ and $D_{21}.v_{1}=1$. Write
$A_{2}=a_{1}v_{1}+a_{2}v_{2}$. Then $0<A_{2}.v_{1}=a_{2}$ and
$0<A_{2}.D_{21}=a_{1}$. So $A_{2}.D_{23}\geq a_{1}+a_{2}>a_{1}=A_{2}.D_{21}$.
If $|\Gamma_{Y_{2}}|=1$, we have $A_{2}.D_{23}=2A_{2}.D_{21}$ by virtue of
Proposition 3.25.

The same reasoning applied to $Y_{3}$ yields the chain of inequalites

$A_{2}.D_{23}\geq A_{2}.D_{21}=A_{1}.D_{12}>A_{1}.D_{13}=A_{3}.D_{31}\geq
A_{3}.D_{32}=A_{2}.D_{23},$

where the equalities come from the gluing condition. Hence $Y_{c}$ is not
projective.

Conversely, suppose the curve structure $\Gamma_{Y_{1}}$ is non-degenerate.
For large enough $e$, there is an ample divisor $A_{2}$ on $Y_{2}$ with $\deg
A_{2}{{}_{|D_{21}}}=e$ and $\deg A_{2}{{}_{|D_{23}}}=e+k_{1}$, with $k_{1}\geq
0$ and independent of $e$, by Proposition 3.20 if $\Gamma_{Y_{2}}$ is non-
degenerate or Proposition 3.24 in the degenerate case. Similarly, maybe
increasing $e$, one finds an ample divisor $A_{3}$ on $Y_{3}$ with
$A_{3}.D_{32}=e+k_{1}$ and $A_{3}.D_{31}=e+k_{2}$, $k_{2}\geq 0$. Then, maybe
again increasing $e$, there is an ample divisor $A_{1}$ on $Y_{1}$ with
$A_{1}.D_{13}=e+k_{2}$ and $A_{1}.D_{12}=e$, as $\Gamma_{Y_{1}}$ is non-
degenerate and hence Proposition 3.20 applies. Hence $Y$ carries an ample
bundle and thus is projective. ∎

######  Remark 4.3.

The non-degeneracy condition of Propositon 4.2 implies that
$Y_{c}\in\operatorname{PMod}_{2}(\mathscr{P})$ if and only if there is an
index $i$ such that $D^{2}_{ij}\leq 1$ for all $j\neq i$, with self
intersections considered on $Y_{i}$.

Recall that if $\Gamma_{Y_{i}}$ is a curve structure of type $d_{2}$ that is
not regular and $D_{ij}$ is a component of the anticanonical cycle, then there
is a unique vertex $v_{D_{ij}}\in\Gamma_{Y_{i}}$ such that
$v_{D_{ij}}.D_{ij}\neq 0$. Note that if $Y=Y_{1}\cup Y_{2}\cup Y_{3}$ is of
class $\mathscr{P}$ such that $\Gamma_{Y_{1}}$ is non-degenerate and
$\Gamma_{Y_{i}}$ is not regular for $i=2,3$, then there exists an $i$ such
that $v_{D_{i1}}.D_{i1}=2$.

######  Proposition 4.4.

Let $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$ and write $Y_{c}=Y_{1}\cup
Y_{2}\cup Y_{3}$. Suppose $\Gamma_{Y_{1}}$ is non-degenerate and
$\Gamma_{Y_{i}}$ is not regular for $i=2,3$.

  * (i)

If $v_{D_{i1}}.D_{i1}=2$ for $i=2,3$, then $Y_{c}$ is projective.

  * (ii)

If $v_{D_{i1}}.D_{i1}=1$ for exactly one $i\in\\{2,3\\}$, then $Y_{c}$ is
projective if and only if $D^{2}_{1i}\leq 0$.

###### Proof.

We begin by proving the first claim. By Propositions 3.25 and 3.26, given any
sufficiently large $e$, there are ample divisors $A_{2}$ and $A_{3}$ on
$Y_{2}$ and $Y_{3}$ with $A_{2}.D_{23}=A_{3}.D_{32}=e$ and
$A_{i}.D_{i1}=2e+n_{i}$, $i=2,3$, with $n_{i}\geq 0$ and $n_{i}$ independent
of $e$. As $Y_{1}$ is non-degenerate, after maybe increasing $e$, by
Proposition 3.20, there is an ample divisor $A_{1}$ on $Y_{1}$ with
$A.D_{1i}=2e+n_{i}$, hence the $A_{i}$’s glue to an ample divisor on $Y_{c}$.

In the second case, we can assume $v_{D_{21}}.D_{21}=1$. If $D^{2}_{12}>0$,
then in fact $D^{2}_{12}=1$, as $Y_{1}$ is non-degenerate. We want to show
that there is no ample bundle on $Y_{c}$. To get a contradiction, suppose $A$
is ample on $Y_{c}$. Write $A_{i}$ for the restriction to $Y_{i}$. Write
$A_{2}.D_{21}=e>0$. Note that $|\Gamma_{Y_{2}}|>1$. So by Proposition 3.26
$A_{2}.D_{23}=2e+n_{1}$ for some number $n_{1}>0$.

As $v_{D_{31}}.D_{31}=2$, by the same results and the gluing condition, we
have $A_{3}.D_{32}=2e+n_{1}$ and $A_{3}.D_{31}=4e+n_{2}$ for some $n_{2}>0$.
It follows that $A_{1}.D_{12}=e$ and $A_{1}.D_{13}=4e+n_{2}$ by the gluing
condition. But then $A_{1}$ cannot be ample by Example 3.22.

If $D^{2}_{12}\leq 0$, we can, by Lemma 3.18 and Proposition 3.25, find an
ample divisor on $A_{2}$ on $Y_{2}$ with $A_{2}.D_{21}=e$ and
$A_{2}.D_{23}=2e+n_{1}$ with $n_{1}$ independent of $e$ for any $e$
sufficiently large. Similarly, we find an ample $A_{3}$ on $Y_{3}$ with
$A_{3}.D_{32}=2e+n_{1}$ and $A_{3}.D_{31}=4e+n_{2}$, $n_{2}$ independent of
$e$. By Proposition 3.21, there is a ample $A_{1}$ on $Y_{1}$ with
$A_{1}.D_{12}=e$ and $A_{1}.D_{13}=4e+n_{2}$. By construction, the $A_{i}$
glue to an ample bundle $A$ on $Y_{c}$, so $Y_{c}$ is projective. ∎

The following proposition is proven by the same reasoning as Proposition 4.4.

######  Proposition 4.5.

Let $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$ and write $Y_{c}=Y_{1}\cup
Y_{2}\cup Y_{3}$. Suppose $\Gamma_{Y_{1}}$ is non-degenerate, $\Gamma_{Y_{2}}$
is not regular and $\Gamma_{Y_{3}}$ is degenerate and regular.

  * (i)

If $v_{D_{21}}.D_{21}=1$ then $Y_{c}$ is projective if and only if
$D^{2}_{12}\leq 0$.

  * (ii)

If $v_{D_{21}}.D_{21}=2$ then $Y_{c}$ is projective if and only if
$D^{2}_{13}\leq 0$.

######  Proposition 4.6.

Let $Y_{c}\in\operatorname{Mod}_{2}(\mathscr{P})$ and write $Y_{c}=Y_{1}\cup
Y_{2}\cup Y_{3}$. Suppose $\Gamma_{Y_{1}}$ is not regular and $\Gamma_{Y_{i}}$
is non-degenerate for $i=2,3$. Also, assume $v_{D_{12}}.D_{12}=2$. Then
$Y_{c}$ is projective.

###### Proof.

Note that by non-degeneracy, $D^{2}_{31}\leq 1$ and $D^{2}_{23}\leq 1$. If
$D^{2}_{31}\leq 0$ or $D^{2}_{23}\leq 0$ then $Y_{c}$ is projective: this
follows as in the proof above. If $D^{2}_{31}=1$ and $D^{2}_{23}=1$, the
intersection numbers and the conditions on degeneracy and regularity fix the
components, so there is only one such surface. It follows from Propositon 3.23
that for sufficiently large $e$ there is an ample divisor $A_{3}$ on $Y_{3}$
with $A_{3}.D_{31}=e$ and $A_{3}.D_{32}=\frac{3}{2}e$.

We can take $e$ to be divisible by $4$. Since $D^{2}_{31}=1$, $\Gamma_{Y_{1}}$
has an exceptional vertex $v_{0}$. Let $L(v_{0})=(v_{0},v_{1},\dots,v_{n})$ be
the associated leg. Then $v_{n}^{2}=0$ as $\Gamma_{Y_{3}}$ is not regular. So
$C_{v_{n}}$ is a nef divisor on $Y_{1}$. By Proposition 3.26, there is an
ample divisor $A^{\prime}_{1}$ on $Y_{1}$ with $A^{\prime}_{1}.D_{13}=e$ and
$A^{\prime}_{1}.D_{12}=2e+k$, with $k$ independent of $e$. Set
$A_{1}=A^{\prime}_{1}+\frac{1}{4}eC_{v_{n}}$. This is an ample divisor with
$A_{1}.D_{13}=e$ and $A_{1}.D_{12}=\frac{9}{4}e+k$. By Proposition 3.23, there
is an ample divisor $A_{2}$ on $Y_{2}$ with $A_{2}.D_{23}=\frac{3}{2}e$ and
$A_{2}.D_{21}=\frac{9}{4}e+k$. Hence the $A_{i}$ glue to an ample divisor on
$Y$. ∎