# Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and
Mirror Symmetry

Luca Giovenzana  School of Mathematics and Statistics, Hicks Building,
Hounsfield Road, Sheffield, S3 7RH, United Kingdom
<EMAIL_ADDRESS>and  Alan Thompson  Department of Mathematical
Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU,
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Tyurin degenerations of K3 surfaces are degenerations whose central fibre
consists of a pair of rational surfaces glued along a smooth elliptic curve.
We study the lattice theory of such Tyurin degenerations, establishing a
notion of lattice polarisation that is compatible with existing definitions
for the general fibre and the rational surfaces comprising the central fibre.
We separately consider elliptically fibred K3 surfaces, where the base of the
fibration admits a splitting into a pair of discs with specified monodromy
around the boundary. In this setting we establish a notion of lattice
polarisation for the induced elliptic fibrations over discs, which is
compatible with the existing definition for K3 surfaces. Finally, we discuss
the mirror symmetric correspondence between these two settings.

The authors were supported by Engineering and Physical Sciences Research
Council (EPSRC) New Investigator Award EP/V005545/1.
For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of the paper
  2. 2 Pseudolattices
    1. 2.1 Background
    2. 2.2 Further properties of quasi del Pezzo homomorphisms
    3. 2.3 From pseudolattices to lattices
    4. 2.4 Point-like vectors and the Néron-Severi lattice
    5. 2.5 Lattice polarisations
  3. 3 Tyurin Degenerations of K3 Surfaces
    1. 3.1 Hodge theory
    2. 3.2 Clemens-Schmid and pseudolattices over the rational numbers
    3. 3.3 Integrality
    4. 3.4 Point-like vectors and the Néron-Severi lattice
    5. 3.5 Lattice polarisations
    6. 3.6 Primitivity
    7. 3.7 Moduli spaces
  4. 4 Elliptic Fibrations on K3 Surfaces
    1. 4.1 Pseudolattices from elliptic fibrations over discs
    2. 4.2 Elliptically fibred K3 surfaces
    3. 4.3 Point-like vectors and the Néron-Severi lattice
    4. 4.4 Lattice polarisations
    5. 4.5 Primitivity
    6. 4.6 Relationship with lattice polarised rational elliptic surfaces
  5. 5 Mirror Symmetry
    1. 5.1 Mirror Pairs
    2. 5.2 Compatibility with other forms of mirror symmetry
      1. 5.2.1 Lattice polarised mirror symmetry for K3 surfaces
      2. 5.2.2 Homological mirror symmetry for quasi del Pezzo surfaces and quasi LG models
      3. 5.2.3 Mirror symmetry for lattice polarised del Pezzo surfaces
    3. 5.3 The DHT conjecture

##  1\. Introduction

A K3 surface of degree 2 is given by a double cover of the projective plane
branched along a sextic curve $(f_{6}=0)\subset\mathbb{P}^{2}$; such a K3
surface can be described explicitly as a weighted projective hypersurface
$(z^{2}=f_{6})\subset\mathbb{P}(1,1,1,3)$. A simple degeneration of such a K3
surface occurs when the sextic curve degenerates to a double cubic.
Explicitly, consider the family of K3 surfaces over the punctured disc
$\Delta^{*}$ defined by the equation
$(z^{2}={f_{3}}^{2}+tf_{6})\subset\mathbb{P}(1,1,1,3)\times\Delta^{*}$, where
$t\in\Delta^{*}$. This family can be completed to a flat family
$\mathcal{X}\to\Delta$ with smooth total space $\mathcal{X}$ and central fibre
$X_{0}=V_{1}\cup_{C}V_{2}$ , where $V_{1}\cong\mathbb{P}^{2}$ and
$V_{2}\cong\textrm{Bl}_{18}\mathbb{P}^{2}$ are rational surfaces glued along a
smooth anticanonical curve isomorphic to $(f_{3}=0)\subset\mathbb{P}^{2}$.

One may associate a lattice to this degeneration in a canonical way. On
$X_{0}$, the orthogonal complement of the hyperplane section inside
$K_{V_{1}}^{\perp}\oplus
K_{V_{2}}^{\perp}\subset\operatorname{NS}(V_{1})\oplus\operatorname{NS}(V_{2})$
is generated by classes $(e_{i}-e_{i+1})$, where
$e_{1},\ldots,e_{18}\in\operatorname{NS}(V_{2})$ are the classes of the
exceptional divisors in $V_{2}\cong\textrm{Bl}_{18}\mathbb{P}^{2}$. These
classes span the negative definite root lattice $A_{17}$.

Degenerations such as this one, where a K3 surface degenerates into a pair of
smooth rational surfaces $V_{i}$ glued along a smooth curve $C$, are called
_Tyurin degenerations_. By adjunction $C$ is necessarily an elliptic curve and
is anticanonical in each of the surfaces $V_{i}$. We call such smooth rational
surfaces with smooth anticanonical divisor _quasi del Pezzo surfaces_.

Besides the degeneration described above, a K3 surface of degree $2$ admits
three other Tyurin degenerations, which have corresponding lattices
$D_{16}A_{1}$, $E_{8}E_{8}A_{1}$, and $E_{7}D_{10}$ [Fri84, Section 5].
Dolgachev observed that under mirror symmetry such degenerations correspond to
elliptic fibrations on the mirror K3 surfaces, with reducible fibres governed
by the lattices above [Dol96, Remark 7.11]. Indeed, in this case we see that
mirrors to K3 surfaces of degree $2$ (which are polarised by the lattice
$H\oplus E_{8}\oplus E_{8}\oplus A_{1}$) admit four different elliptic
fibrations, with reducible singular fibres of Kodaira types $\mathrm{I}_{18}\
(\widetilde{A}_{17})$, $\mathrm{I}_{12}^{*}\,\mathrm{I}_{2}\
(\widetilde{D}_{16}\widetilde{A}_{1})$,
$\mathrm{I}\mathrm{I}^{*}\,\mathrm{I}\mathrm{I}^{*}\,\mathrm{I}_{2}\
(\widetilde{E}_{8}\widetilde{E}_{8}\widetilde{A}_{1}$), and
$\mathrm{I}\mathrm{I}\mathrm{I}^{*}\,\mathrm{I}_{6}^{*}\
(\widetilde{E}_{7}\widetilde{D}_{10})$ respectively.

The DHT philosophy, introduced in [DHT17], attempts to link this observation
to the theory of mirror symmetry for the surfaces $V_{i}$. It postulates that
the base of each elliptic fibration $Y\to\mathbb{P}^{1}$ on the mirror K3
surface should admit a splitting
$\mathbb{P}^{1}\cong\Delta_{1}\cup_{\gamma}\Delta_{2}$ into a pair of discs
$\Delta_{i}$ glued along a simple loop $\gamma$, so that the restriction
$Y_{i}\to\Delta_{i}$ of the fibration to the disc $\Delta_{i}$ is the Landau-
Ginzburg (LG) model for the quasi del Pezzo surface $V_{i}$.

In our running example of the degeneration of a K3 surface of degree $2$, it
is known that the mirror to $V_{1}\cong\mathbb{P}^{2}$ is a Landau-Ginzburg
model consisting of an elliptic fibration over $\mathbb{A}^{1}$ with three
singular fibres of Kodaira type $\mathrm{I}_{1}$ and monodromy
$\begin{pmatrix}1&-9\\\ 0&1\end{pmatrix}$ around the point at infinity (see,
for instance, [AKO06]). This may be equivalently viewed as a fibration over a
disc $\Delta_{1}$, by simply removing a small neighbourhood of the point at
infinity. In this setting, the DHT philosophy suggests that there should exist
a simple loop $\gamma$ in the base of the mirror fibration
$Y\to\mathbb{P}^{1}$, such that there are three singular fibres of types
$\mathrm{I}_{1}\,\mathrm{I}_{1}\,\mathrm{I}_{1}$ on one side of $\gamma$, four
singular fibres of types
$\mathrm{I}_{1}\,\mathrm{I}_{1}\,\mathrm{I}_{1}\,\mathrm{I}_{18}$ on the other
side of $\gamma$, and such that monodromy around $\gamma$ is given by
$\begin{pmatrix}1&-9\\\ 0&1\end{pmatrix}$. Moreover, there should be some
sense in which an elliptic fibration over a disc with singular fibres of types
$\mathrm{I}_{1}\,\mathrm{I}_{1}\,\mathrm{I}_{1}\,\mathrm{I}_{18}$ may be
thought of as an LG model for $V_{2}\cong\textrm{Bl}_{18}\mathbb{P}^{2}$
(which is not a del Pezzo surface and so does not have an LG model in the
traditional sense).

The aim of this paper is to establish a general framework for this theory.
There are substantial technical obstacles to doing this, related to
establishing compatibility between the various theories of mirror symmetry
involved.

At the highest level, mirror symmetry for $n$-dimensional Calabi-Yau manifolds
predicts an exchange of Hodge numbers $h^{p,q}(X)=h^{p,n-q}(Y)$ between a
Calabi-Yau $X$ and its mirror $Y$. For K3 surfaces, however, this statement is
vacuous, as all K3 surfaces have the same Hodge numbers.

A solution to this problem was given by Dolgachev, Nikulin, and Pinkham
[Pin77, Nik83, Dol96], who realised that mirror symmetry for K3 surfaces could
instead be formulated as an exchange between the algebraic (i.e. Néron-Severi)
and transcendental parts of $H^{1,1}(X)$. Their theory relies upon the notion
of lattice polarisations, which impose constraints on the Néron-Severi lattice
of a K3 surface.

On the other hand, existing theories of mirror symmetry for quasi del Pezzo
surfaces are somewhat less fine-grained. Work of Auroux, Katzarkov, and Orlov
[AKO06] has established a version of homological mirror symmetry between del
Pezzo surfaces and their LG models, and Harder and the second author [HT20]
have extended this to a homological mirror symmetry statement for quasi del
Pezzo surfaces, using Kuznetsov’s theory of pseudolattices [Kuz17]. However,
these theories are not sensitive to much apart from the del Pezzo degree (i.e.
the self-intersection number of the anticanonical divisor); they do not, for
instance, draw any distinction between LG models for different weak del Pezzo
surfaces of the same degree, such as $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
the Hirzebruch surface $\mathbb{F}_{2}$.

To remedy this, one may ask for a notion of lattice polarised mirror symmetry
for quasi del Pezzo surfaces and their LG models, in the style of Dolgachev’s
theory for K3 surfaces. However, it is not obvious how such a concept should
be correctly defined: one has $H^{2}(V;\mathbb{Z})\cong\operatorname{NS}(V)$
for any quasi del Pezzo surface $V$, so there is no algebraic / transcendental
decomposition to work with as there is in the K3 case. An attempt to solve
this problem in the weak del Pezzo case, by defining lattice polarisations in
terms of effective classes instead of algebraic classes, was made by Doran and
the second author in [DT18].

Concretely, the aims of this paper can be summarised as follows.

  1. (1)

Rigorously establish a theory of lattice polarisations for Tyurin
degenerations of K3 surfaces $X\rightsquigarrow X_{0}=V_{1}\cup_{C}V_{2}$,
linking Dolgachev’s [Dol96] theory of lattice polarisations for the K3 surface
$X$ to the (pseudo)lattice theory of the quasi del Pezzo surfaces $V_{i}$
described in [DT18, HT20] (Theorems 3.26 and 3.28).

  2. (2)

Enhance the theory of pseudolattices for elliptic fibrations over discs from
the setting of [HT20], where all singular fibres have Kodaira type
$\mathrm{I}_{1}$, to allow (non-multiple) fibres of arbitrary Kodaira types
(Theorem 4.10).

  3. (3)

Rigorously develop a theory of lattice polarisations for elliptic fibrations
over discs (with arbitrary Kodaira fibres), which is compatible both with
Dolgachev’s [Dol96] theory of lattice polarisations for an elliptically fibred
K3 surface $Y\to\mathbb{P}^{1}$ when there is an appropriate splitting
$\mathbb{P}^{1}\cong\Delta_{1}\cup_{\gamma}\Delta_{2}$ of the base, and with
the theory of lattice polarised rational elliptic surfaces described in [DT18]
(Theorems 4.25 and 4.38).

  4. (4)

Formulate a mirror symmetric correspondence between these two settings.

To elaborate further on (4), we will develop a rigorous definition for when a
lattice polarised Tyurin degeneration $X\rightsquigarrow
X_{0}=V_{1}\cup_{C}V_{2}$ and a lattice polarised elliptically fibred K3
surface $Y\to\mathbb{P}^{1}$ with a splitting
$\mathbb{P}^{1}\cong\Delta_{1}\cup_{\gamma}\Delta_{2}$ form a _mirror pair_
(Definition 5.1). We will show that this definition is compatible with
existing theories of mirror symmetry in the following ways.

  * •

The K3 surfaces $X$ and $Y$ are lattice polarised mirrors in the sense of
[Dol96] (Section 5.2.1);

  * •

$(V_{i},C)$ and the restriction $Y_{i}\to\Delta_{i}$ form a quasi del Pezzo
surface / LG model mirror pair, in the sense of [HT20] (Theorem 5.6);

  * •

We formulate a version of lattice polarised mirror symmetry for weak del Pezzo
surfaces $(V,C)$ and elliptic fibrations over discs $W\to\Delta$ (Conjecture
5.8), which is compatible with the theory above and the formulation of mirror
symmetry for weak del Pezzo surfaces from [DT18] (Theorem 5.10).

Given this, we conclude by rigorously formulating the DHT philosophy for K3
surfaces as a conjecture (Conjecture 5.12), as follows: If $X\rightsquigarrow
X_{0}=V_{1}\cup_{C}V_{2}$ is a Tyurin degeneration of lattice polarised K3
surfaces, then the mirror K3 surface $Y$ to $X$ should admit an elliptic
fibration $Y\to\mathbb{P}^{1}$ with a splitting
$\mathbb{P}^{1}\cong\Delta_{1}\cup_{\gamma}\Delta_{2}$, so that $X$ and $Y$
form a mirror pair.

There remain several obstacles to proving this conjecture, principal amongst
which is appropriate choice of the splitting loop $\gamma$. Evidence from
[DPT23] suggests that this loop cannot be determined by lattice theoretic
considerations alone; additional combinatorial data is likely to be required.

###  1.1. Outline of the paper

This paper is structured as follows. In Section 2 we begin by developing the
abstract theory of lattices and pseudolattices that we will need for the
remainder of the paper. The aim is to develop tools to translate between the
lattice theory of the K3 surfaces $X$ and $Y$ and the theory of pseudolattices
associated with the quasi del Pezzo surfaces $V_{i}$ and the fibrations over
discs $Y_{i}\to\Delta_{i}$ (as described in [HT20]). We then describe how to
define lattice polarisations in each of these settings, in a compatible way.

In Section 3 we apply these tools to the study of Tyurin degenerations of K3
surfaces. Many of the ideas here are extensions of ideas of Friedman [Fri84]
to the lattice polarised setting; there is also a close relationship with
recent work of Alexeev and Engel on lattice polarised degenerations [AE23]. In
particular, in Section 3.5 we define lattice polarisations on Tyurin
degenerations and on their central fibres, and show (Theorems 3.26 and 3.28)
that these notions are compatible with the definition of a lattice
polarisation on a weak del Pezzo surface given in [DT18]. In Section 3.6, we
introduce the concept of a _doubly admissible_ degeneration, which is a
natural extension of Dolgachev’s [Dol96] $1$-admissible condition to the
setting of Tyurin degenerations; in the remainder of Section 3 we show that
this assumption allows some major simplifications of the underlying theory and
we explore its implications for moduli.

In Section 4 we study elliptically fibred K3 surfaces. In this setting there
is much more work to do. The first serious obstacle is that the results of
[HT20] only hold for genus $1$ Lefschetz fibrations over discs, which
necessarily have all singular fibres of Kodaira type $\mathrm{I}_{1}$, whereas
elliptically fibred K3 surfaces may have singular fibres of many different
Kodaira types. Our first main result of this section, Theorem 4.10, shows that
the results of [HT20] may be extended to arbitrary elliptic fibrations over
discs (without multiple fibres).

Using this, in Section 4.2 we proceed to study elliptic fibrations on lattice
polarised K3 surfaces $Y\to\mathbb{P}^{1}$. We describe when a loop
$\gamma\subset\mathbb{P}^{1}$ splits the base of such a fibration into two LG
models and show that the theory of Section 2 applies in this setting. This
allows us, in Section 4.4, to describe a compatibility condition between the
lattice polarisation on $Y$ and the splitting loop
$\gamma\subset\mathbb{P}^{1}$, under which we obtain induced lattice
polarisations on the pieces $Y_{i}\to\Delta_{i}$ (Theorem 4.25). In Section
4.5 we explore the analogue of the doubly admissible condition in this setting
and show that it approximately corresponds to the existence of a section for
the elliptic fibration $Y\to\mathbb{P}^{1}$ along with a $1$-admissible
$0$-cusp in the Baily-Borel compactification of the moduli space of $Y$.

To conclude our treatment of fibrations, in Section 4.6 we explore the
compatibility between the notions of lattice polarisation defined here and in
[DT18]. This is complicated by the fact that our lattice polarisations are
defined for elliptic fibrations over discs, whilst those in [DT18] are defined
for rational elliptic surfaces. Compatibility between the two definitions is
established by Theorem 4.38.

In Section 5 we discuss how mirror symmetry relates the degeneration and
fibration pictures presented in the previous sections. We begin by defining
when a Tyurin degeneration of lattice polarised K3 surfaces $X\rightsquigarrow
X_{0}=V_{1}\cup_{C}V_{2}$ and a lattice polarised elliptically fibred K3
surface $Y\to\mathbb{P}^{1}$ with a splitting
$\mathbb{P}^{1}\cong\Delta_{1}\cup_{\gamma}\Delta_{2}$ form a _mirror pair_
(Definition 5.1). We give the most general form of this definition first,
which is consequently quite complicated to state. In the remainder of Section
5.1 we show that this definition can be made much simpler, at the cost of
certain additional assumptions; in particular, we show that the doubly
admissible assumption allows a dramatic simplification (Proposition 5.4).

In Section 5.2 we prove compatibility between this definition and some
established formulations of mirror symmetry. We demonstrate that our mirror
pairs are compatible with lattice polarised mirror symmetry for the K3
surfaces $X$ and $Y$, in the sense of [Dol96], and homological mirror symmetry
for the quasi del Pezzo surfaces $V_{i}$ and the pieces $Y_{i}\to\Delta_{i}$,
in the sense of [HT20] (Theorem 5.6).

Compatibility between our mirror pairs and the notion of lattice polarised
mirror symmetry for lattice polarised del Pezzo surfaces (in the sense of
[DT18]) is more complicated. Firstly, one must address the fact that the
mirror correspondence in [DT18] is stated for rational elliptic surfaces, not
elliptic fibrations over discs; this is dealt with by Conjecture 5.8, which
reformulates the main conjecture of [DT18] in our setting. Secondly, and more
seriously, one must also account for the fact that effective curves on $X$
(resp. $Y$) may not give rise to effective curves supported on just one
component $V_{i}$ (resp. one piece $Y_{i}\to\Delta_{i}$); the failure of this
is controlled by an object that we call the _coupling group_. When this
coupling group is torsion, Theorem 5.10 shows that our mirror pairs are
compatible with the theory from [DT18]. On the other hand, if the coupling
group is not torsion, we do not expect lattice polarised mirror symmetry for
the K3 surfaces $X$ and $Y$ (in the sense of [Dol96]) to induce lattice
polarised mirror symmetry for $V_{i}$ and $Y_{i}\to\Delta_{i}$ (in the sense
of [DT18]).

Finally, Section 5.3 explores the consequences of this work for the DHT
philosophy. In particular, we give a precise conjectural statement (Conjecture
5.12) about the mirror symmetric relationship between Tyurin degenerations of
K3 surfaces and elliptically fibred K3 surfaces.

Data access statement: No datasets were generated or analysed during this
study.

Acknowledgments: LG would like to thank Giulia Gugiatti and Franco Rota for
stimulating discussions. He also wishes to thank Franco Giovenzana for
engaging in fruitful discussions throughout the duration of this project. AT
would like to thank Charles Doran and Andrew Harder for helpful discussions.
The authors would also like to thank Klaus Hulek for helpful comments on an
earlier version. Part of this project was completed whilst AT was attending
the thematic programme on _K-theory, algebraic cycles and motivic homotopy
theory_ at the Isaac Newton Institute in the summer of 2022; he would like to
thank the Isaac Newton Institute for their excellent hospitality over this
period.

##  2\. Pseudolattices

###  2.1. Background

We begin by briefly recapping some relevant background from the theory of
pseudolattices, based on [Kuz17, HT20]; a much more detailed treatment of
these concepts may be found in those two papers. We start with some basic
definitions.

######  Definition 2.1.

[Kuz17, Definition 2.1] A _pseudolattice_
$(\mathrm{G},\langle\cdot,\cdot\rangle_{\mathrm{G}})$ is a finitely generated
free abelian group $\mathrm{G}$ equipped with a (not necessarily symmetric)
nondegenerate bilinear form

$\langle\cdot,\cdot\rangle_{\mathrm{G}}\colon\mathrm{G}\times\mathrm{G}\rightarrow\mathbb{Z}.$

The _rank_ of $\mathrm{G}$ is its rank as a free abelian group and
$\mathrm{G}$ is called _unimodular_ if the form
$\langle\cdot,\cdot\rangle_{\mathrm{G}}$ induces an isomorphism between
$\mathrm{G}$ and $\mathrm{Hom}_{\mathbb{Z}}(\mathrm{G},\mathbb{Z})$. To
simplify notation, we will usually refer to
$(\mathrm{G},\langle\cdot,\cdot\rangle_{\mathrm{G}})$ by its underlying group
$\mathrm{G}$.

A simple example of a pseudolattice, that we will use repeatedly, is the
_elliptic curve pseudolattice_.

######  Definition 2.2.

[HT20, Definition 2.14] The _elliptic curve pseudolattice_ , henceforth
denoted by $\mathrm{E}$, is the unimodular pseudolattice of rank $2$ generated
by primitive elements $a,b$, with bilinear form defined by

$\langle a,b\rangle_{\mathrm{E}}=-1,\quad\quad\langle
b,a\rangle_{\mathrm{E}}=1,\quad\quad\langle a,a\rangle_{\mathrm{E}}=\langle
b,b\rangle_{\mathrm{E}}=0,$

extended to $\mathbb{Z}^{2}$ by linearity.

The pseudolattice $\mathrm{E}$ arises naturally from an elliptic curve $C$ in
two ways (see [HT20, Example 2.15]).

  1. (1)

The numerical Grothendieck group
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))$ associated to the bounded
derived category $\mathbf{D}(C)$ of coherent sheaves on $C$, with its Euler
pairing, is isomorphic to $\mathrm{E}$. The element $a$ may be identified with
the class of the structure sheaf $\mathcal{O}_{p}$ of a point $p\in C$, and
$b$ may be identified with the class of $\mathcal{O}_{C}$.

  2. (2)

The first integral homology $\mathrm{H}_{1}(C;\mathbb{Z})$, with its standard
intersection form, is isomorphic to $\mathrm{E}$. In this case, $(a,-b)$ is a
symplectic basis for $\mathrm{H}_{1}(C;\mathbb{Z})$. This may also be
interpreted as the numerical Grothendieck group associated to a triangulated
category, in this case the _derived Fukaya category_ $\mathbf{F}(C)$
associated to $C$.

These two interpretations can be thought of as mirror to one another [HT20,
Section 5]; the first will be important for us in Section 3 and the second in
Section 4.

We now define some important structures on pseudolattices: _Serre operators_
and _exceptional bases_.

######  Definition 2.3.

[Kuz17, Section 2.1] Let $\mathrm{G}$ be a pseudolattice. A _Serre operator_
$S_{\mathrm{G}}$ is an automorphism of $\mathrm{G}$ so that

$\langle u_{1},u_{2}\rangle_{\mathrm{G}}=\langle
u_{2},S_{\mathrm{G}}(u_{1})\rangle_{\mathrm{G}}$

for all $u_{1},u_{2}\in\mathrm{G}$.

We note that if a Serre operator exists then it is unique, and that every
unimodular pseudolattice has a Serre operator.

######  Definition 2.4.

[Kuz17, Definition 2.2] An element $e\in\mathrm{G}$ is called _exceptional_ if
$\langle e,e\rangle_{\mathrm{G}}=1$. A sequence of elements
$(e_{1},\ldots,e_{n})$ is called an _exceptional sequence_ if $\langle
e_{i},e_{j}\rangle_{\mathrm{G}}=0$ for all $i>j$. An _exceptional basis_ for
$\mathrm{G}$ is an exceptional sequence whose elements form a basis for
$\mathrm{G}$.

A subclass of pseudolattices, that will be important for this paper, is the
_surface-like_ pseudolattices.

######  Definition 2.5.

[Kuz17, Definition 3.1] A pseudolattice $\mathrm{G}$ is _surface-like_ if
there is a primitive element $\mathbf{p}\in\mathrm{G}$ such that

  1. (1)

$\langle\mathbf{p},\mathbf{p}\rangle_{\mathrm{G}}=0$.

  2. (2)

$\langle\mathbf{p},u\rangle_{\mathrm{G}}=\langle
u,\mathbf{p}\rangle_{\mathrm{G}}$ for all $u\in\mathrm{G}$.

  3. (3)

If $u_{1},u_{2}\in\mathrm{G}$ satisfy $\langle
u_{1},\mathbf{p}\rangle_{\mathrm{G}}=\langle
u_{2},\mathbf{p}\rangle_{\mathrm{G}}=0$, then $\langle
u_{1},u_{2}\rangle_{\mathrm{G}}=\langle u_{2},u_{1}\rangle_{\mathrm{G}}$.

An element $\mathbf{p}$ with the above properties is called a _point-like_
element in $\mathrm{G}$.

Using a point-like element, one may define the _Néron-Severi lattice_ of a
surface-like pseudolattice $\rm{G}$.

######  Definition 2.6.

[Kuz17, Section 3.2] Let $\mathrm{G}$ be a surface-like pseudolattice. Define
the _Néron-Severi group_ of $\mathrm{G}$ to be the group

$\operatorname{NS}(\mathrm{G}):={\mathbf{p}^{\perp}}/\mathbf{p}.$

$\operatorname{NS}(\mathrm{G})$ is a finitely generated free abelian group. It
is equipped with a nondegenerate integral symmetric bilinear form
$q(\cdot,\cdot)$, defined as follows: if $u_{1},u_{2}\in{\mathbf{p}}^{\perp}$
and $[u_{1}],[u_{2}]$ are their classes in $\operatorname{NS}(\mathrm{G})$,
then

$q\left([u_{1}],[u_{2}]\right):=-\langle u_{1},u_{2}\rangle_{\mathrm{G}},$

and this definition is independent of the choice of representatives for
$[u_{1}]$, $[u_{2}]$. The pair
$(\operatorname{NS}(\mathrm{G}),q(\cdot,\cdot))$ is called the _Néron-Severi
lattice_ of $\mathrm{G}$. As before, to simplify notation, we will usually
refer to $(\operatorname{NS}(\mathrm{G}),q(\cdot,\cdot))$ by its underlying
group $\operatorname{NS}(\mathrm{G})$.

Following [Kuz17, Section 3.2], one may define a rank function on a surface-
like pseudolattice $\mathrm{G}$ as follows. For $u\in\mathrm{G}$, we have

$\operatorname{rank}(u):=\langle
u,\mathbf{p}\rangle_{\mathrm{G}}=\langle\mathbf{p},u\rangle_{\mathrm{G}}.$

Using this, one may define a map
$\lambda\colon\bigwedge^{2}\mathrm{G}\to\mathrm{NS}(\mathrm{G})$ by

$\lambda(u_{1}\wedge
u_{2}):=[\operatorname{rank}(u_{1})u_{2}-\operatorname{rank}(u_{2})u_{1}].$

This map can be used to define a distinguished class $K_{\mathrm{G}}$ in
$\operatorname{NS}(\mathrm{G})\otimes\mathbb{Q}$ , called the _canonical
class_ of $\mathrm{G}$. Existence and uniqueness of the canonical class is
established in [Kuz17, Lemma 3.12].

######  Definition 2.7.

The _canonical class of $\mathrm{G}$_ is the unique class
$K_{\mathrm{G}}\in\mathrm{NS}(\mathrm{G})\otimes\mathbb{Q}$ such that for
every $u_{1},u_{2}\in\mathrm{G}$,

$\langle u_{1},u_{2}\rangle_{\mathrm{G}}-\langle
u_{2},u_{1}\rangle_{\mathrm{G}}=-q(K_{\mathrm{G}},\lambda(u_{1}\wedge
u_{2})).$

By [Kuz17, Lemma 3.12], if $\mathrm{G}$ is unimodular, then $K_{\mathrm{G}}$
is integral (i.e. $K_{\mathrm{G}}\in\mathrm{NS}(\mathrm{G})$).

We next introduce a special class of maps between pseudolattices.

######  Definition 2.8.

[HT20, Definition 2.16] Let $\mathrm{G}$ and $\mathrm{H}$ be pseudolattices. A
_spherical homomorphism_ from $\mathrm{G}$ to $\mathrm{H}$ is a homomorphism
of abelian groups $f\colon\mathrm{G}\to\mathrm{H}$ with the following
properties:

  1. (1)

The homomorphism $f$ has a right adjoint $r\colon\mathrm{H}\to\mathrm{G}$, in
the sense that

$\langle f(u),v\rangle_{\mathrm{H}}=\langle u,r(v)\rangle_{\mathrm{G}}$

for any $u\in\mathrm{G}$ and $v\in\mathrm{H}$.

  2. (2)

The _twist_ and _cotwist_ endomorphisms, which are defined as

${T}_{f}:=\mathrm{id}_{\mathrm{H}}-fr,\quad\quad{C}_{f}:=\mathrm{id}_{\mathrm{G}}-rf,$

respectively, are invertible. In fact, if ${T}_{f}$ is invertible with inverse
$T_{f}^{-1}$, then ${C}_{f}$ is invertible with inverse
$C_{f}^{-1}=\mathrm{id}_{\mathrm{G}}+rT_{f}^{-1}f$, and vice versa.

Note that, by [HT20, Remark 2.17], if a right adjoint for a homomorphism $f$
exists, then it must be unique, and a right adjoint always exists if
$\mathrm{G}$ is unimodular.

######  Definition 2.9.

[HT20, Definition 2.18] A spherical homomorphism
$f\colon\mathrm{G}\rightarrow\mathrm{H}$ is _relative $(-1)^{n}$-Calabi-Yau_
(usually abbreviated to _relative $(-1)^{n}$-CY_) if $\mathrm{G}$ has a Serre
operator $S_{\mathrm{G}}$ and ${C}_{f}=(-1)^{n}S_{\mathrm{G}}$.

Note that this definition only depends upon the parity of $n$; consequently we
will usually take $n$ to be an element of $\mathbb{Z}/2\mathbb{Z}$. An example
of a relative $(-1)^{0}$-CY spherical homomorphism that will be important for
this paper is as follows.

######  Example 2.10.

[HT20, Example 2.20] Let $\mathrm{E}$ be the elliptic curve pseudolattice and
let $v\in\mathrm{E}$ be a primitive vector. Note that $\langle
v,v\rangle_{\mathrm{E}}=0$, as the bilinear form on $\mathrm{E}$ is
antisymmetric. Let $\mathrm{Z}(v)$ be the unimodular pseudolattice generated
by a single element $z$, with $\langle z,z\rangle_{\mathrm{Z}(v)}=1$. Note
that the Serre operator on $\mathrm{Z}(v)$ is the identity and $(z)$ is an
exceptional basis for $\mathrm{Z}(v)$.

There is a spherical homomorphism
$\zeta\colon\mathrm{Z}(v)\rightarrow\mathrm{E}$ sending $z$ to $v$. The right
adjoint of this homomorphism is

$\displaystyle\rho\colon\mathrm{E}$
$\displaystyle\longrightarrow\mathrm{Z}(v)$ $\displaystyle w$
$\displaystyle\longmapsto\langle v,w\rangle_{\mathrm{E}}z$

The cotwist of $\zeta$ is the identity on $\mathbb{Z}(v)$, so it follows that
$\zeta$ is relative $(-1)^{0}$-CY. The twist of $\zeta$ acts on
$w\in\mathrm{E}$ by

$T_{\zeta}(w)=w-\langle v,w\rangle_{\mathrm{E}}v.$

If we identify $\mathrm{E}$ with $\mathrm{H}_{1}(C;\mathbb{Z})$ for an
elliptic curve $C$, and $v\in\mathrm{H}_{1}(C;\mathbb{Z})$ is the class of a
simple oriented cycle, then $T_{\zeta}$ is the action on
$\mathrm{H}_{1}(C;\mathbb{Z})$ induced by the Dehn twist associated to $v$.

We can use spherical homomorphisms to glue pseudolattices together.

######  Definition 2.11.

[HT20, Definition 2.22] Let $\mathrm{G}_{1},\mathrm{G}_{2}$ and $\mathrm{H}$
be pseudolattices. Assume that

$f_{1}\colon\mathrm{G}_{1}\longrightarrow\mathrm{H},\quad\quad
f_{2}\colon\mathrm{G}_{2}\longrightarrow\mathrm{H}$

are homomorphisms of the underlying groups. Define
$\mathrm{G}_{1}\oright_{\mathrm{H}}\mathrm{G}_{2}$ to be the pseudolattice
with underlying group $\mathrm{G}_{1}\oplus\mathrm{G}_{2}$ and bilinear form
$\langle\cdot,\cdot\rangle_{\mathrm{G}_{1}\oright_{\mathrm{H}}\mathrm{G}_{2}}$
given as follows: for $u_{i}\in\mathrm{G}_{i}$ and $v_{j}\in\mathrm{G}_{j}$,
define

$\langle
u_{i},v_{j}\rangle_{\mathrm{G}_{1}\oright_{\mathrm{H}}\mathrm{G}_{2}}=\begin{cases}\langle
u_{i},v_{j}\rangle_{\mathrm{G}_{i}}&\text{if }i=j\\\ \langle
f_{i}(u_{i}),f_{j}(v_{j})\rangle_{\mathrm{H}}&\text{if }i=1,\ j=2\\\
0&\text{if }i=2,\ j=1\end{cases}$

and extend to $\mathrm{G}_{1}\oplus\mathrm{G}_{2}$ by linearity. By [HT20,
Proposition 2.23], there is a natural spherical homomorphism $f_{1}\oright
f_{2}\colon\mathrm{G}_{1}\oright_{\mathrm{H}}\mathrm{G}_{2}\rightarrow\mathrm{H}$,
sending $(u_{1},u_{2})\in\mathrm{G}_{1}\oright_{\mathrm{H}}\mathrm{G}_{2}$ to
$f_{1}(u_{1})+f_{2}(u_{2})$, which has the property that ${T}_{f_{1}\oright
f_{2}}={T}_{f_{1}}\cdot{T}_{f_{2}}$.

We will mostly be interested in the case of spherical homomorphisms where the
target is the elliptic curve pseudolattice $\mathrm{E}$. In this case we have
the following result, which is a special case of [HT20, Proposition 2.23].

######  Lemma 2.12.

Given two pseudolattices $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ with relative
$(-1)^{0}$-CY spherical homomorphisms
$f_{i}\colon\mathrm{G}_{i}\rightarrow\mathrm{E}$, then $f_{1}\oright f_{2}$ is
also relative $(-1)^{0}$-CY.

As an important example of this gluing process, we can glue the spherical
homomorphisms $\mathrm{Z}(v)\to\mathrm{E}$ from Example 2.10 to build
pseudolattices of higher rank.

######  Example 2.13.

[HT20, Example 2.24] Suppose we have an ordered $n$-tuple of elements
$(v_{1},\ldots,v_{n})$ in $\mathrm{E}$. For each $i\in\\{1,\ldots,n\\}$,
define rank one pseudolattices $\mathrm{Z}(v_{i})$ generated by elements
$z_{i}$ with $\langle z_{i},z_{i}\rangle_{\mathrm{Z}(v_{i})}=1$, as in Example
2.10. By Example 2.10, we have relative $(-1)^{0}$-CY spherical homomorphisms
$\zeta_{i}\colon\mathrm{Z}(v_{i})\to\mathrm{E}$ taking $z_{i}$ to $v_{i}$.

Let $\mathrm{Z}(v_{1},\ldots,v_{n})$ denote the pseudolattice

$\mathrm{Z}(v_{1},\ldots,v_{n}):=\mathrm{Z}(v_{1})\oright_{\mathrm{E}}\mathrm{Z}(v_{2})\oright_{\mathrm{E}}\cdots\oright_{\mathrm{E}}\mathrm{Z}(v_{n}).$

Then $\mathrm{Z}(v_{1},\ldots,v_{n})$ is a unimodular pseudolattice of rank
$n$ generated by $z_{1},\ldots,z_{n}$, with bilinear form given by

$\langle
z_{i},z_{j}\rangle_{\mathrm{Z}(v_{1},\ldots,v_{n})}=\begin{cases}1&\text{ if
}i=j\\\ \langle v_{i},v_{j}\rangle_{\mathrm{E}}&\text{ if }i<j\\\ 0&\text{ if
}i>j.\end{cases}$

$(z_{1},\ldots,z_{n})$ thus form an exceptional basis for $Z(v_{1},v_{2})$.

The spherical homomorphism
$\zeta=\zeta_{1}\oright\cdots\oright\zeta_{n}\colon\mathrm{Z}(v_{1},\ldots,v_{n})\rightarrow\mathrm{E}$
sending $z_{i}$ to $v_{i}$ is relative $(-1)^{0}$-CY and the twist
${T}_{\zeta}={T}_{\zeta_{1}}\cdots{T}_{\zeta_{n}}$.

Spherical homomorphisms to the elliptic curve pseudolattice give a convenient
way to construct surface-like pseudolattices. The following result is a
combination of the results of [HT20, Propositions 3.1, 3.2, 3.3].

######  Proposition 2.14.

Suppose that $(a,b)$ is a basis for $\mathrm{E}$ as in Definition 2.2. Let
$f\colon\mathrm{G}\to\mathrm{E}$ be a relative $(-1)^{0}$-CY spherical
homomorphism with right adjoint $r$. Then $\mathrm{G}$ is surface-like with
point-like vector $\mathbf{p}=r(a)$ if and only if $r(a)$ is primitive and the
twist $T_{f}$ satisfies $T_{f}(a)=a$. Moreover, if this holds, then:

  1. (1)

$r(b)\in\mathbf{p}^{\perp}$ and $[r(b)]=-K_{\mathrm{G}}$, where the square
brackets $[-]$ denote the class in $\operatorname{NS}(\mathrm{G})$.

  2. (2)

In the basis $(a,b)$, the twist $T_{f}$ has matrix $\begin{pmatrix}1&-d\\\
0&1\end{pmatrix}$, where $d:=q(K_{\mathrm{G}},K_{\mathrm{G}})$.

Now we come to the central definition of [HT20].

######  Definition 2.15.

[HT20, Definition 3.7] Let $\mathrm{G}$ be a pseudolattice of rank $n$. A
spherical homomorphism of pseudolattices $f\colon\mathrm{G}\to\mathrm{E}$ is
called a _quasi del Pezzo homomorphism_ if there exists a basis $(a,b)$ for
$\mathrm{E}$, as in Definition 2.2, so that all of the following conditions
hold.

  1. (1)

$\mathrm{G}$ is surface-like with point-like element $\mathbf{p}=r(a)$.

  2. (2)

$f$ is relative $(-1)^{0}$-Calabi-Yau.

  3. (3)

$\mathrm{G}$ admits an exceptional basis $e_{\bullet}=(e_{1},\ldots,e_{n})$
such that $f(e_{i})\in\mathrm{E}$ is primitive for each $1\leq i\leq n$.

  4. (4)

$\operatorname{NS}(\mathrm{G})$ has signature $(1,n-3)$.

Note that conditions (1) and (2) from this definition imply that Proposition
2.14 holds for all quasi del Pezzo homomorphisms. The motivating example
behind this definition is as follows.

######  Example 2.16.

[HT20, Example 3.10] Let $V$ be a nonsingular rational surface and suppose
that the anticanonical linear system $|-K_{V}|$ contains a smooth member $C$;
such surfaces are called _quasi del Pezzo surfaces_. Note that, by adjunction,
$C$ is an elliptic curve. Let
$i^{*}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))\to\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))$
be the map between the numerical Groethendieck groups of the bounded derived
categories of coherent sheaves $\mathbf{D}(V)$ and $\mathbf{D}(C)$ on $V$ and
$C$, respectively, given by the derived pull-back under the inclusion $i\colon
C\hookrightarrow V$. Then $i^{*}$ is a relative $(-1)^{0}$-CY spherical
homomorphism, with right adjoint $i_{*}$ given by derived push-forward.

Identify $\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))\cong\mathrm{E}$ as
above: identify $a$ with the class $[\mathcal{O}_{p}]$ of the structure sheaf
$\mathcal{O}_{p}$ of a point $p\in C$ and $b$ with the class
$[\mathcal{O}_{C}]$ of $\mathcal{O}_{C}$. Then $i^{*}$ is a quasi del Pezzo
homomorphism of pseudolattices.

The point-like element $\mathbf{p}=i_{*}([\mathcal{O}_{p}])$ is the class of
the structure sheaf of a point in
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))$. With respect to this point-
like element we have an isomorphism of lattices
$\mathbf{p}^{\perp}/\mathbf{p}=:\operatorname{NS}(\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V)))\cong\operatorname{NS}(V)\cong
H^{2}(V;\mathbb{Z})$, which takes the canonical class in
$\operatorname{NS}(\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V)))$ (Definition
2.7) to the usual canonical class in $\operatorname{NS}(V)$.

The main result of [HT20] is a classification of quasi del Pezzo
homomorphisms. In order to state it, we first need to say what it means for
two quasi del Pezzo homomorphisms to be isomorphic.

######  Definition 2.17.

[HT20, Definition 3.9] Two quasi del Pezzo homomorphisms
$f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}_{2}\to\mathrm{E}$ are _isomorphic_ if

  * •

there are bases $(a_{1},b_{1})$ and $(a_{2},b_{2})$ for $\mathrm{E}$, so that
$(a_{i},b_{i})$ satisfies the conditions of Definition 2.15 for
$f_{i}\colon\mathrm{G}_{i}\to\mathrm{E}$, and an automorphism
$\varphi\colon\mathrm{E}\to\mathrm{E}$ taking $(a_{1},b_{1})$ to
$(a_{2},b_{2})$,

  * •

there is an isomorphism of pseudolattices
$\psi\colon\mathrm{G}_{1}\to\mathrm{G}_{2}$ (i.e. an isomorphism of the
underlying abelian groups which preserves the bilinear form), and

  * •

we have $\varphi f_{1}=f_{2}\psi$.

It is easy to check that the last part of this definition implies that $\psi
r_{1}=r_{2}\varphi$, where $r_{i}$ is the right adjoint to $f_{i}$.

The next theorem is the main result of [HT20], which classifies quasi del
Pezzo homomorphisms up to isomorphism. The version stated here is a slight
simplification of the original.

######  Theorem 2.18.

[HT20, Theorem 3.25] Let $f\colon\mathrm{G}\to\mathrm{E}$ be a quasi del Pezzo
homomorphism and let $n:=\operatorname{rank}(\mathrm{G})$. Then
$f\colon\mathrm{G}\to\mathrm{E}$ is isomorphic to exactly one of:

  1. (1)

$\mathrm{Z}(b,3a+b,6a+b,a,\ldots,a)\to\mathrm{E}$, for any $n\geq 3$; or

  2. (2)

$\mathrm{Z}(b,2a+b,2a+b,4a+b)\to\mathrm{E}$ for $n=4$.

It is not difficult to show (c.f. [Dem80], [CT88, Proposition 0.4]) that if
$V$ is a quasi del Pezzo surface and $C\subset V$ is a smooth anticanonical
divisor, then $(V,C)$ is either

  * •

a blow-up of a smooth cubic curve in $\mathbb{P}^{2}$ in $k\geq 0$ points in
almost general position (a set of points is in _almost general position_ if no
stage of the blowing-up involves blowing up a point which lies on a rational
$(-2)$-curve; infinitely near points are allowed as long as this condition is
not violated), or

  * •

a smooth curve of bidegree $(2,2)$ in $\mathbb{P}^{1}\times\mathbb{P}^{1}$, or

  * •

a smooth anticanonical curve in the Hirzebruch surface $\mathbb{F}_{2}$.

It then follows from Theorem 2.18, [HT20, Example 3.10], and the analysis in
[HT20, Section 3.3], that every quasi del Pezzo homomorphism is isomorphic to
$i^{*}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))\to\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))$
for $V$ a quasi del Pezzo surface and $i\colon C\hookrightarrow V$ a smooth
anticanonical divisor. Explicitly, case (1) of Theorem 2.18 corresponds to the
case where $V$ is a blow-up of a smooth cubic curve in $\mathbb{P}^{2}$ in
$n-3$ points in almost general position, and case (2) of Theorem 2.18
corresponds to the case where $V\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ or
$V\cong\mathbb{F}_{2}$.

The main observation of [HT20] is that quasi del Pezzo homomorphisms also
arise naturally from genus $1$ Lefschetz fibrations over closed discs
$Y\to\Delta$. In this setting they occur as boundary maps
$H_{2}(Y,F;\mathbb{Z})\to H_{1}(F;\mathbb{Z})$, where $F$ is a smooth fibre
over a point on the boundary of $\Delta$ and we place pseudolattice structures
on $H_{2}(Y,F;\mathbb{Z})$ and $H_{1}(F;\mathbb{Z})$ via the _Seifert pairing_
and the usual intersection form respectively; we refer the reader to Section
4.1 for a more detailed treatment.

The fact that quasi del Pezzo homomorphisms arise in both of these settings
should be thought of as a manifestation of Fano-LG mirror symmetry for quasi
del Pezzo surfaces; see [HT20, Section 5].

###  2.2. Further properties of quasi del Pezzo homomorphisms

In this subsection we will collect some additional properties of quasi del
Pezzo homomorphisms, which do not come from [Kuz17, HT20], that will be useful
to us in the rest of the paper.

######  Lemma 2.19.

Let $f\colon\mathrm{G}\to\mathrm{E}$ be a quasi del Pezzo homomorphism of
pseudolattices and let $r\colon\mathrm{E}\to\mathrm{G}$ denote the right
adjoint of $f$. Let $(a,b)$ be a basis for $\mathrm{E}$ as in Definition 2.15.
Then $r$ is injective and satisfies $fr(a)=0$.

###### Proof.

Injectivity of $r$ is an easy consequence of the explicit classification of
quasi del Pezzo homomorphisms (Theorem 2.18). Moreover, as $f$ is a quasi del
Pezzo homomorphism, Proposition 2.14 gives $T_{f}(a)=a$. But
$T_{f}(a)=a-fr(a)$ by definition, so we must have $fr(a)=0$. ∎

We next collect some information about the Néron-Severi lattice associated to
a quasi del Pezzo homomorphism.

######  Lemma 2.20.

Let $f\colon\mathrm{G}\to\mathrm{E}$ be a quasi del Pezzo homomorphism and let
$n:=\operatorname{rank}(\mathrm{G})$.

  1. (1)

If $f\colon\mathrm{G}\to\mathrm{E}$ is isomorphic to case (1) in Theorem 2.18,
then $\operatorname{NS}(\mathrm{G})$ is the odd unimodular lattice
$\mathrm{I}_{1,n-3}$ of signature $(1,n-3)$. The canonical class
$K_{\mathrm{G}}$ has $q(K_{\mathrm{G}},K_{\mathrm{G}})=12-n$ and its
orthogonal complement $K_{\mathrm{G}}^{\perp}$ in
$\operatorname{NS}(\mathrm{G})$ is given in Table 2.1. This orthogonal
complement is negative definite if $n<12$, degenerate if $n=12$, and
indefinite of signature $(1,n-4)$ if $n>12$. If $n\notin\\{3,12\\}$, the
discriminant group
$\operatorname{disc}(K_{\mathrm{G}}^{\perp})\cong\mathbb{Z}\,/\,|12-n|\mathbb{Z}$.
The quadratic form on $\operatorname{disc}(K_{\mathrm{G}}^{\perp})$ is
generated by an element $e$ with $e^{2}=\frac{1}{n-12}$, with values taken
modulo $\mathbb{Z}$ if $n$ is odd and modulo $2\mathbb{Z}$ if $n$ is even.

  2. (2)

If $f\colon\mathrm{G}\to\mathrm{E}$ is isomorphic to case (2) in Theorem 2.18,
then $\operatorname{NS}(\mathrm{G})$ is the even unimodular lattice
$H=\mathrm{I}\mathrm{I}_{1,1}$ of signature $(1,1)$ (the hyperbolic plane).
The canonical class $K_{\mathrm{G}}$ has $q(K_{\mathrm{G}},K_{\mathrm{G}})=8$
and the orthogonal complement $K_{\mathrm{G}}^{\perp}$ is the root lattice
$A_{1}$, which is negative definite with
$\operatorname{disc}(K_{\mathrm{G}}^{\perp})\cong\mathbb{Z}/2\mathbb{Z}$. The
quadratic form on $\operatorname{disc}(K_{\mathrm{G}}^{\perp})$ is generated
by an element $e$ with $e^{2}=-\frac{1}{2}$, with values taken modulo
$2\mathbb{Z}$.

$n$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $\geq 9$
---|---|---|---|---|---|---|---
$K_{\mathrm{G}}^{\perp}$ | $\\{0\\}$ | $(-8)$ | $\begin{pmatrix}-2&1\\\ 1&-4\end{pmatrix}$ | $A_{2}\oplus A_{1}$ | $A_{4}$ | $D_{5}$ | $E_{n-3}$
Table 2.1. Néron-Severi lattices in case (1) of Lemma 2.20. Here $A_{n}$,
$D_{n}$, $E_{6}$, $E_{7}$, and $E_{8}$ are the usual negative definite root
lattices, and we extend the definition of the $E_{n}$ series of lattices to
$n\geq 9$ in the obvious way.

###### Proof.

This follows from the isomorphism between $\operatorname{NS}(\mathrm{G})$ and
$\operatorname{NS}(V)$ for a quasi del Pezzo surface $V$, along with the well-
known lattice theory for quasi del Pezzo surfaces (see, for example, [Dol12,
Chapter 8]). ∎

Now we come to the main result of this section, which characterises
isomorphisms between quasi del Pezzo homomorphisms in terms of their action on
the Néron-Severi lattices.

######  Proposition 2.21.

Suppose that $f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}_{2}\to\mathrm{E}$ are two quasi del Pezzo
homomorphisms.

If $f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}_{2}\to\mathrm{E}$ are isomorphic, in the sense of
Definition 2.17, then the isomorphism
$\psi\colon\mathrm{G}_{1}\to\mathrm{G}_{2}$ induces an isometry
$\hat{\psi}\colon\operatorname{NS}(\mathrm{G}_{1})\to\operatorname{NS}(\mathrm{G}_{2})$
such that $\hat{\psi}(K_{\mathrm{G}_{1}})=K_{\mathrm{G}_{2}}$.

Conversely, if
$\hat{\psi}\colon\operatorname{NS}(\mathrm{G}_{1})\to\operatorname{NS}(\mathrm{G}_{2})$
is an isometry such that $\hat{\psi}(K_{\mathrm{G}_{1}})=K_{\mathrm{G}_{2}}$,
then there exists an isomorphism between
$f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}_{2}\to\mathrm{E}$ that induces $\hat{\psi}$.

###### Proof.

Let $r_{i}$ denote the right adjoint to $f_{i}$. Suppose first that there
exists an isomorphism between $f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}_{2}\to\mathrm{E}$, in the sense of Definition 2.17.
Then there are bases $(a_{1},b_{1})$ and $(a_{2},b_{2})$ for $\mathrm{E}$ so
that $(a_{i},b_{i})$ satisfies the conditions of Definition 2.15 for
$f_{i}\colon\mathrm{G}_{i}\to\mathrm{E}$, and an automorphism
$\varphi\colon\mathrm{E}\to\mathrm{E}$ taking $(a_{1},b_{1})$ to
$(a_{2},b_{2})$, along with an isomorphism
$\psi\colon\mathrm{G}_{1}\to\mathrm{G}_{2}$ which commutes with $\varphi$.

As $\psi r_{1}=r_{2}\varphi$, we have
$\psi(r_{1}(a_{1}))=r_{2}(\varphi(a_{1}))=r_{2}(a_{2})$, so $\psi$ maps the
point-like vector in $\mathrm{G}_{1}$ to the point-like vector in
$\mathrm{G}_{2}$. It follows that $\psi$ induces an isometry
$\hat{\psi}\colon\operatorname{NS}(\mathrm{G}_{1})\to\operatorname{NS}(\mathrm{G}_{2})$.
Moreover, as $\psi(r_{1}(b_{1}))=r_{2}(\varphi(b_{1}))=r_{2}(b_{2})$ and
$K_{\mathrm{G}_{i}}=-[r_{i}(b_{i})]$ for each $i\in\\{1,2\\}$, we have
$\hat{\psi}(K_{\mathrm{G}_{1}})=K_{\mathrm{G}_{2}}$.

Next, suppose that
$\hat{\psi}\colon\operatorname{NS}(\mathrm{G}_{1})\to\operatorname{NS}(\mathrm{G}_{2})$
is an isometry such that $\hat{\psi}(K_{\mathrm{G}_{1}})=K_{\mathrm{G}_{2}}$.

By the classification of quasi del Pezzo homomorphisms,
$f_{j}\colon\mathrm{G}_{j}\to\mathrm{E}$ is isomorphic to
$i^{*}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))\to\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C_{j}))$
for $V_{j}$ a quasi del Pezzo surface and $i\colon C_{j}\hookrightarrow V_{j}$
a smooth anticanonical divisor. We may identify
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C_{j}))\cong\mathrm{E}$ in the usual
way: identify $a_{j}$ with the class of a point $p\in C_{j}$ and $b_{j}$ with
the class of $\mathcal{O}_{C_{j}}$. Moreover, by [Kuz17, Example 3.5], there
are isomorphisms
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))\cong\mathbb{Z}\oplus\operatorname{NS}(V_{j})\oplus\mathbb{Z}$
given by $v\mapsto(\operatorname{rank}(v),c_{1}(v),\chi(v))$, i.e. the rank,
first Chern class, and Euler characteristic of an element
$v\in\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))$.

Choose bases for $\operatorname{NS}(\mathrm{G}_{j})$ as free
$\mathbb{Z}$-modules for each $j\in\\{1,2\\}$. Let $Q_{j}$ denote the matrix
of the intersection form and let $K_{j}$ denote the column vector representing
$K_{\mathrm{G}_{j}}$ with respect to this basis. Then in the realisation
$\mathrm{G}_{j}\cong\mathbb{Z}\oplus\operatorname{NS}(V_{j})\oplus\mathbb{Z}$
given above, one may compute that the bilinear form on $\mathrm{G}_{j}$ is
given explicitly by

$\begin{pmatrix}-1&0&1\\\ Q_{j}K_{j}&-Q_{j}&0\\\ 1&0&0\end{pmatrix}$

and the map $f_{j}\colon\mathrm{G}_{j}\to\mathrm{E}$ is given explicitly by

$\begin{pmatrix}0&-K_{j}^{T}Q_{j}&0\\\ 1&0&0\end{pmatrix}$

with respect to the basis $a_{j}=\begin{pmatrix}1\\\ 0\end{pmatrix}$ and
$b_{j}=\begin{pmatrix}0\\\ 1\end{pmatrix}$ of $\mathrm{E}$.

Let $P$ be the matrix of the isometry
$\operatorname{NS}(\mathrm{G}_{1})\to\operatorname{NS}(\mathrm{G}_{2})$ with
respect to the chosen bases of these two lattices. Then $P^{T}Q_{2}P=Q_{1}$
and $PK_{1}=K_{2}$. Define a map $\psi\colon\mathrm{G}_{1}\to\mathrm{G}_{2}$
by the matrix

$\begin{pmatrix}1&0&0\\\ 0&P&0\\\ 0&0&1\end{pmatrix}$

and let $\varphi$ be the automorphism $\varphi\colon\mathrm{E}\to\mathrm{E}$
taking $(a_{1},b_{1})$ to $(a_{2},b_{2})$. Then it is a straightforward
exercise in linear algebra to show that $\psi$ defines an isomorphism of
pseudolattices $\mathrm{G}_{1}\to\mathrm{G}_{2}$ and that $\varphi
f_{1}=f_{2}\psi$, so we obtain an isomorphism of quasi del Pezzo homomorphisms
between $f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$ and
$f_{2}\colon\mathrm{G}\to\mathrm{E}$. ∎

###  2.3. From pseudolattices to lattices

With this background in place, we will explore what happens when one glues
pairs of quasi del Pezzo homomorphisms. In particular, we will show that if
this is done in the correct way, it gives rise to an exact sequence of
lattices.

Let $f_{i}\colon\mathrm{G}_{i}\to\mathrm{E}$, for $i\in\\{1,2\\}$ be two quasi
del Pezzo homomorphisms of pseudolattices, as in Definition 2.15, and suppose
further that the basis $(a,b)$ for $\mathrm{E}$ appearing in this definition
can be taken to be the same for both $i\in\\{1,2\\}$. Let $r_{i}$ denote the
right adjoint of $f_{i}$. Finally, assume that
$q(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})=-q(K_{\mathrm{G}_{2}},K_{\mathrm{G}_{2}})$.

The homomorphism $(-f_{2})\colon\mathrm{G}_{2}\to\mathrm{E}$ is clearly
spherical, with right adjoint $(-r_{2})$ and twist and cotwist
$T_{(-f_{2})}=T_{f_{2}}$ and $C_{(-f_{2})}=C_{f_{2}}$ respectively. From this
it follows immediately that $(-f_{2})$ is also relatively $(-1)^{0}$-CY.

Now define $\mathrm{G}:=\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}$
glued along the homomorphisms $f_{1}$ and $(-f_{2})$, and let
$f\colon\mathrm{G}\to\mathrm{E}$ be the natural homomorphism

$\displaystyle
f\colon\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}\longrightarrow\mathrm{E}.$
$\displaystyle f(u_{1},u_{2})=f_{1}(u_{1})-f_{2}(u_{2}).$

By Lemma 2.12, $f$ is a relative $(-1)^{0}$-CY spherical homomorphism. We
denote the right adjoint of $f$ by $r$. By the proof of [HT20, Proposition
2.23], the map $r$ is given explicitly by

(1)
$r(v)=(r_{1}T_{f_{2}}(v),-r_{2}(v))\in\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}$

for any $v\in\mathrm{E}$. Let $\mathrm{K}\subset\mathrm{G}$ denote the kernel
of $f$.

######  Lemma 2.22.

The bilinear form $\langle\cdot,\cdot\rangle_{\mathrm{G}}$ is symmetric on
$\mathrm{K}$, so $\mathrm{K}$ is a lattice (in the classical sense).

###### Proof.

Let $S_{\mathrm{G}}$ denote the Serre operator of $\mathrm{G}$; as $f$ is
relatively $(-1)^{0}$-CY, we have
$S_{\mathrm{G}}=C_{f}:=\mathrm{id}_{\mathrm{G}}-rf$. Then for any
$u,v\in\mathrm{K}$, we have

$\langle u,v\rangle_{\mathrm{G}}=\langle
v,S_{\mathrm{G}}(u)\rangle_{\mathrm{G}}=\langle
v,u-rf(u)\rangle_{\mathrm{G}}=\langle v,u\rangle_{\mathrm{G}}$

as $f(u)=0$ by assumption. ∎

######  Lemma 2.23.

Let $u\in\mathrm{E}$. Then $r(u)\in\mathrm{K}$ and $\langle
v,r(u)\rangle_{\mathrm{G}}=0$ for all $v\in\mathrm{K}$, so
$\operatorname{im}(r)$ is a totally degenerate sublattice of $\mathrm{K}$.

###### Proof.

By Definition 2.11, the twist $T_{f}=T_{f_{1}}\cdot T_{-f_{2}}=T_{f_{1}}\cdot
T_{f_{2}}$. Moreover, by Proposition 2.14, in the basis $(a,b)$ for
$\mathrm{E}$ we have $T_{f_{i}}=\begin{pmatrix}1&-d_{i}\\\ 0&1\end{pmatrix}$,
where $d_{i}:=q(K_{\mathrm{G}_{i}},K_{\mathrm{G}_{i}})$. Thus, since
$d_{1}=-d_{2}$ by assumption, we have $T_{f}=\mathrm{id}_{\mathrm{E}}$. But
$T_{f}=\mathrm{id}_{\mathrm{E}}-fr$ by definition, so $fr(u)=0$ for all
$u\in\mathrm{E}$ and hence $r(u)\in\mathrm{K}$.

To prove the second part of the statement, note that by adjunction

$\langle v,r(u)\rangle_{\mathrm{G}}=\langle f(v),u\rangle_{\mathrm{E}}=0$

as $f(v)=0$ by assumption. ∎

Let $\overline{\mathrm{E}}\subset\mathrm{K}$ denote the saturation of the
$\mathbb{Z}$-submodule $\operatorname{im}(r)$, equipped with the bilinear form
induced by $\langle\cdot,\cdot\rangle_{\mathrm{G}}$; by the results above we
see that $\overline{\mathrm{E}}$ is isomorphic to $\mathbb{Z}^{2}$ with the
zero form. By Lemma 2.23, we may take the quotient
$\mathrm{K}/\overline{\mathrm{E}}$ to obtain a new lattice, which we denote by
$\mathrm{M}$. We thus have an exact sequence of lattices (in the classical
sense)

(2)
$0\longrightarrow\overline{\mathrm{E}}\longrightarrow\mathrm{K}\longrightarrow\mathrm{M}\longrightarrow
0.$

######  Remark 2.24.

Note that if $(u_{1},u_{2})$ and $(v_{1},v_{2})$ are elements of $\mathrm{K}$,
for $u_{i},v_{i}\in\mathrm{G}_{i}$, then as $\mathrm{K}=\ker(f)$ we have
$f_{1}(u_{1})=f_{2}(u_{2})$ and $f_{1}(v_{1})=f_{2}(v_{2})$. Consequently

$\displaystyle\langle(u_{1},u_{2}),\,(v_{1},v_{2})\rangle_{\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}}$
$\displaystyle=\langle u_{1},v_{1}\rangle_{\mathrm{G}_{1}}-\langle
f_{1}(u_{1}),f_{2}(v_{2})\rangle_{\mathrm{E}}+\langle
u_{2},v_{2}\rangle_{\mathrm{G}_{2}}$ $\displaystyle=\langle
u_{2},v_{2}\rangle_{\mathrm{G}_{2}}-\langle
f_{2}(u_{2}),f_{1}(v_{1})\rangle_{\mathrm{E}}+\langle
u_{1},v_{1}\rangle_{\mathrm{G}_{1}}$
$\displaystyle=\langle(u_{2},u_{1}),\,(v_{2},v_{1})\rangle_{\mathrm{G}_{2}\oright_{\mathrm{E}}\mathrm{G}_{1}}$

It follows that switching the order of the factors in the definition of
$\mathrm{G}$ does not change the lattice $\mathrm{K}$: we obtain the same
$\mathrm{K}$ if we define $\mathrm{G}$ to be
$\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}$ or
$\mathrm{G}_{2}\oright_{\mathrm{E}}\mathrm{G}_{1}$. From this it is easy to
see that the remaining results of this section, particularly the exact
sequence (2), are also independent of the ordering of the factors in
$\mathrm{G}$.

As a consequence of this remark, we may freely switch the labels on
$\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ without affecting the subsequent
development. For concreteness, we will henceforth assume that labels have been
chosen so that $q(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})\geq 0$.

######  Definition 2.25.

Define the _degree_ of $\mathrm{G}$ to be
$\deg(\mathrm{G})=q(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})$, whenever
$q(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})\neq 8$. If
$q(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})=8$ then, by Theorem 2.18 and Lemma
2.20, there are two possibilities for the quasi del Pezzo homomorphism
$f_{1}\colon\mathrm{G}_{1}\to\mathrm{E}$: we say that $\deg(\mathrm{G})=8$ if
$\operatorname{NS}(\mathrm{G}_{1})\cong\mathrm{I}_{1,1}$ (case (1) of Theorem
2.18) and $\deg(\mathrm{G})=8^{\prime}$ if
$\operatorname{NS}(\mathrm{G}_{1})\cong\mathrm{I}\mathrm{I}_{1,1}$ (case (2)
of Theorem 2.18).

As a consequence of the classification of quasi del Pezzo homomorphisms
(Theorem 2.18), we note that $f\colon\mathrm{G}\to\mathrm{E}$ is determined up
to isomorphism by $\deg(\mathrm{G})$, and that $0\leq\deg(\mathrm{G})\leq 9$.

###  2.4. Point-like vectors and the Néron-Severi lattice

Next we consider the vectors $(r_{1}(a),0)$ and $(0,r_{2}(a))$ in
$\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}=\mathrm{G}$. Define $\Psi$
to be the $\mathbb{Z}$-submodule of $\mathrm{G}$ generated by these two
elements.

######  Lemma 2.26.

$\Psi$ has the following properties.

  1. (1)

$\Psi$ is primitive in $\mathrm{G}$.

  2. (2)

If $p\in\Psi$, then $\langle p,v\rangle_{\mathrm{G}}=\langle
v,p\rangle_{\mathrm{G}}$ for any $v\in\mathrm{G}$.

  3. (3)

$\Psi$ is contained in $\mathrm{K}$.

  4. (4)

The restriction of the bilinear form $\langle\cdot,\cdot\rangle_{\mathrm{G}}$
to $\Psi$ is identically zero.

  5. (5)

$\Psi\cap\overline{\mathrm{E}}$ is generated by the primitive element

$r(a)=(r_{1}(a),-r_{2}(a))\in\mathrm{G}_{1}\oright_{\mathrm{E}}\mathrm{G}_{2}=\mathrm{G}.$

###### Proof.

To prove (1), note that $r_{i}(a)$ is primitive in $\mathrm{G}_{i}$ for each
$i\in\\{1,2\\}$, by Proposition 2.14. As $\mathrm{G}$ is isomorphic to
$\mathrm{G}_{1}\oplus\mathrm{G}_{2}$ as a $\mathbb{Z}$-module, it follows that
$\Psi$ is primitive in $\mathrm{G}$.

To prove (2), it suffices to show that $\langle
r_{i}(a),v\rangle_{\mathrm{G}}=\langle v,r_{i}(a)\rangle_{\mathrm{G}}$ for
each $i\in\\{1,2\\}$ and any $v\in\mathrm{G}$. So let $v\in\mathrm{G}$.
Writing $v=(v_{1},v_{2})$ for $v_{i}\in\mathrm{G}_{i}$, by Definition 2.11 we
have

$\displaystyle\langle r_{1}(a),v\rangle_{\mathrm{G}}$ $\displaystyle=\langle
r_{1}(a),v_{1}\rangle_{\mathrm{G}_{1}}+\langle
f_{1}r_{1}(a),-f_{2}(v_{2})\rangle_{\mathrm{E}},$ $\displaystyle\langle
v,r_{1}(a)\rangle_{\mathrm{G}}$ $\displaystyle=\langle
v_{1},r_{1}(a)\rangle_{\mathrm{G}_{1}}.$

But $f_{1}r_{1}(a)=0$ by Lemma 2.19, so

$\langle r_{1}(a),v\rangle_{\mathrm{G}}=\langle
r_{1}(a),v_{1}\rangle_{\mathrm{G}_{1}}.$

To complete the proof, we use the fact that, by Definition 2.15, $r_{1}(a)$ is
point-like in $\mathrm{G}_{1}$, so

$\langle r_{1}(a),v_{1}\rangle_{\mathrm{G}_{1}}=\langle
v_{1},r_{1}(a)\rangle_{\mathrm{G}_{1}};$

we conclude that $\langle r_{1}(a),v\rangle_{\mathrm{G}}=\langle
v,r_{1}(a)\rangle_{\mathrm{G}}$ as required. The proof for $r_{2}$ is
completely analogous.

To prove (3), we have $f(r_{1}(a),0)=f_{1}r_{1}(a)$ and
$f(0,r_{2}(a))=-f_{2}r_{2}(a)$, and both of these are equal to $0$ by Lemma
2.19. So $(r_{1}(a),0),(0,r_{2}(a))\in\ker(f)=\mathrm{K}$, which implies
$\Psi\subset\mathrm{K}$.

To prove (4) we note that, as $r_{i}(a)$ is point-like in $\mathrm{G}_{i}$ for
$i\in\\{1,2\\}$, we must have $\langle
r_{i}(a),r_{i}(a)\rangle_{\mathrm{G}_{i}}=0$. So, by Definition 2.11,

$\langle(r_{1}(a),0),(r_{1}(a),0)\rangle_{\mathrm{G}}=\langle(0,r_{2}(a)),(0,r_{2}(a))\rangle_{\mathrm{G}}=0$

Moreover, $\langle(0,r_{2}(a)),(r_{1}(a),0)\rangle_{\mathrm{G}}=0$ by
definition and

$\langle(r_{1}(a),0),(0,r_{2}(a))\rangle_{\mathrm{G}}=\langle
f_{1}r_{1}(a),-f_{2}r_{2}(a)\rangle_{\mathrm{E}}=0,$

as $f_{i}r_{i}(a)=0$ for each $i\in\\{1,2\\}$ by Lemma 2.19. We conclude that
the restriction of the bilinear form $\langle\cdot,\cdot\rangle_{\mathrm{G}}$
to $\Psi$ is identically zero.

Finally, to prove (5) we observe that, by Equation (1), the map $r$ is given
by

$r(v)=(r_{1}T_{f_{2}}(v),-r_{2}(v))=(r_{1}(v)-r_{1}f_{2}r_{2}(v),-r_{2}(v)).$

Applying this to $a\in\mathrm{E}$, we see that $r(a)=(r_{1}(a),-r_{2}(a))$,
where we have used the fact that $f_{2}r_{2}(a)=0$. Thus
$(r_{1}(a),-r_{2}(a))\in\operatorname{im}(r)\subset\overline{\mathrm{E}}$, and
this element is primitive by the same argument that we used to prove (1).

To complete the proof of (5), as both $\Psi$ and $\overline{\mathrm{E}}$ have
rank $2$, it suffices to find an element of $\Psi$ that is not in
$\overline{\mathrm{E}}$. Consider $(0,-r_{2}(a))\in\Psi$. Assume for a
contradiction that some multiple of this is equal to $r(v)$, for
$v\in\mathrm{E}$. By the formula above, this can only be true if
$r_{2}(v)=kr_{2}(a)$, for $k\in\mathbb{Z}\setminus\\{0\\}$. But $r_{2}$ is
linear and injective by Lemma 2.19, so we must have $v=ka$. It then follows
from our computation above that $r(ka)=(kr_{1}(a),-kr_{2}(a))$; this is a
contradiction. ∎

As a consequence of Lemma 2.26(2), we see that the left and right orthogonal
complements of $\Psi$ in $\mathrm{G}$ agree. We denote this orthogonal
complement by $\Psi^{\perp}_{\mathrm{G}}$. The following lemma describes its
properties.

######  Lemma 2.27.

Equipping $\Psi_{\mathrm{G}}^{\perp}$ with the bilinear form induced from
$-\langle\cdot,\cdot\rangle_{\mathrm{G}}$ (note the negative sign), there is a
natural isomorphism of lattices

$\Psi_{\mathrm{G}}^{\perp}/\Psi\cong\operatorname{NS}(\mathrm{G}_{1})\oplus\operatorname{NS}(\mathrm{G}_{2}),$

where $\operatorname{NS}(\mathrm{G}_{i})$ is equipped with its usual bilinear
form $q(\cdot,\cdot)$.

###### Proof.

Recall that, for each $i\in\\{1,2\\}$, the point-like element in
$\mathrm{G}_{i}$ is given by $r_{i}(a)$, so
$\operatorname{NS}(\mathrm{G}_{i}):=(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}/r_{i}(a)$.

We begin by showing that
$(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$
is a sublattice of $\mathrm{G}$ in the obvious way. To do this we need to show
that the bilinear forms induced on
$(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$
from those on $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$, respectively from
$\mathrm{G}$, agree.

Let
$u,v\in(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$
and write $u=(u_{1},u_{2})$, $v=(v_{1},v_{2})$. Then

$\langle u,v\rangle_{\mathrm{G}}=\langle
u_{1},v_{1}\rangle_{\mathrm{G}_{1}}+\langle
u_{2},v_{2}\rangle_{\mathrm{G}_{2}}+\langle
f_{1}(u_{1}),-f_{2}(v_{2})\rangle_{\mathrm{E}}.$

To show that this agrees with the bilinear form induced from those on
$\mathrm{G}_{1}$ and $\mathrm{G}_{2}$, we need to show that $\langle
f_{1}(u_{1}),-f_{2}(v_{2})\rangle_{\mathrm{E}}=0$. As
$u_{1}\in(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}$, we have

$0=\langle u_{1},r_{1}(a)\rangle_{\mathrm{G}_{1}}=\langle
f_{1}(u_{1}),a\rangle_{\mathrm{E}}.$

By the definition of the bilinear form on $\mathrm{E}$, it follows that
$f_{1}(u_{1})$ must be an integer multiple of $a$. A similar argument shows
that $f_{2}(v_{2})$ must also be an integer multiple of $a$, so $\langle
f_{1}(u_{1}),-f_{2}(v_{2})\rangle_{\mathrm{E}}=c\,\langle
a,a\rangle_{\mathrm{E}}=0$, for some $c\in\mathbb{Z}$, as required.

Next we show that
$\Psi_{\mathrm{G}}^{\perp}=(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$
as subsets of $\mathrm{G}$. For any $v=(v_{1},v_{2})\in\mathrm{G}$, we have

$\langle(v_{1},v_{2}),\,(r_{1}(a),0)\rangle_{\mathrm{G}}=\langle
v_{1},r_{1}(a)\rangle_{\mathrm{G}_{1}}$

and

$\langle(0,r_{2}(a)),\,(v_{1},v_{2})\rangle_{\mathrm{G}}=\langle
r_{2}(a),v_{2}\rangle_{\mathrm{G}_{2}},$

so $v\in\Psi_{\mathrm{G}}^{\perp}$ if and only if
$v_{1}\in(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}$ and
$v_{2}\in(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$.

We thus have an equality of sublattices of $\mathrm{G}$

$\Psi^{\perp}_{\mathrm{G}}=(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}.$

We conclude by quotienting by $\Psi$ and noting that the bilinear form
$q(\cdot,\cdot)$ on $\operatorname{NS}(\mathrm{G}_{i})$ is induced by
$-\langle\cdot,\cdot\rangle_{\mathrm{G}_{i}}$. ∎

As a consequence of Lemma 2.26, we also see that $(r_{1}(a),0)$ and
$(0,r_{2}(a))$ both descend to the same primitive element
$\mathbf{p}\in\mathrm{M}$.

######  Proposition 2.28.

$\mathrm{M}$ is a surface-like pseudolattice with point-like element
$\mathbf{p}$. The Néron-Severi group of $\mathrm{M}$, defined by
$\operatorname{NS}(\mathrm{M}):=\mathbf{p}^{\perp}/\mathbf{p}$, fits into the
following exact sequence

$0\longrightarrow\mathbb{Z}\zeta\longrightarrow\Psi^{\perp}_{\mathrm{K}}/\Psi\longrightarrow\operatorname{NS}(\mathrm{M})\longrightarrow
0,$

where $\Psi^{\perp}_{\mathrm{K}}$ denotes the orthogonal complement of $\Psi$
in $\mathrm{K}$ and $\zeta$ is a primitive generator of
$\overline{\mathrm{E}}/r(a)$.

If $\Psi^{\perp}_{\mathrm{K}}/\Psi$ is equipped with the bilinear form induced
by the natural inclusion

$\Psi^{\perp}_{\mathrm{K}}/\Psi\subset\Psi_{\mathrm{G}}^{\perp}/\Psi\cong\operatorname{NS}(\mathrm{G}_{1})\oplus\operatorname{NS}(\mathrm{G}_{2}),$

$\operatorname{NS}(\mathrm{M})$ is equipped with its usual bilinear form
$q(\cdot,\cdot)$, and $\zeta$ is isotropic ($\zeta.\zeta=0$), then the above
sequence becomes an exact sequence of lattices.

###### Proof.

Note first that $\zeta$ is well-defined as $r(a)$ is primitive in
$\overline{\mathrm{E}}$, so $\overline{\mathrm{E}}/r(a)$ is a torsion-free
$\mathbb{Z}$-module of rank $1$.

The element $\mathbf{p}$ is isotropic in $\mathrm{M}$, as the bilinear form on
$\mathrm{M}$ is induced from that on $\mathrm{K}$ and $\Psi$ is isotropic in
$\mathrm{K}$. So, by [Kuz17, Lemma 3.3(a)], $\mathrm{M}$ is surface-like with
point-like vector $\mathbf{p}$.

To prove that the given sequence is exact, we first observe that, by Lemma
2.26, we have an exact sequence

$0\longrightarrow\mathbb{Z}r(a)\longrightarrow\Psi\longrightarrow\mathbb{Z}\mathbf{p}\longrightarrow
0,$

and the restriction of the bilinear forms to the terms of this sequence are
all identically zero. Moreover, as $\overline{\mathrm{E}}$ is orthogonal to
$\Psi$ (by Lemma 2.23), we have
$\overline{\mathrm{E}}\subset\Psi^{\perp}_{\mathrm{K}}$, so we obtain a second
exact sequence

$0\longrightarrow\overline{\mathrm{E}}\longrightarrow\Psi^{\perp}_{\mathrm{K}}\longrightarrow\mathbf{p}^{\perp}\longrightarrow
0,$

and this is an exact sequence of lattices if we equip its terms with the
induced bilinear forms from $\overline{\mathrm{E}}$, $\mathrm{G}$, and
$\mathrm{M}$. Taking the quotient of the second sequence by the first, using
the snake lemma, and flipping the signs on the bilinear forms gives the
result. ∎

To conclude this section, the following lemma describes the image of
$\mathbb{Z}\zeta$ inside $\Psi^{\perp}_{\mathrm{K}}/\Psi$.

######  Lemma 2.29.

The inclusion $\mathbb{Z}\zeta\hookrightarrow\Psi^{\perp}_{\mathrm{K}}/\Psi$
realises $\mathbb{Z}\zeta$ as a totally degenerate rank $1$ sublattice of
$\Psi^{\perp}_{\mathrm{K}}/\Psi\subset\operatorname{NS}(\mathrm{G}_{1})\oplus\operatorname{NS}(\mathrm{G}_{2})$.
This sublattice contains the nonzero element
$(-K_{\mathrm{G}_{1}},K_{\mathrm{G}_{2}})$.

###### Proof.

The first statement follows immediately from Lemma 2.23. For the second
statement, note that
$r(b)\in\overline{\mathrm{E}}\subset\Psi_{\mathrm{K}}^{\perp}$ and that, by
Equation (1), we have

$r(b)=(r_{1}T_{f_{2}}(b),-r_{2}(b)).$

By Proposition 2.14, we have $T_{f_{2}}(b)=-d_{2}a+b$, where
$d_{2}:=q(K_{\mathrm{G}_{2}},K_{\mathrm{G}_{2}})\in\mathbb{Z}$, so

$r(b)=(-d_{2}r_{1}(a)+r_{1}(b),-r_{2}(b)),$

which is congruent to $(r_{1}(b),-r_{2}(b))$ modulo $\Psi$. The result follows
by descending to the quotient and noting that, by Proposition 2.14, we have
$K_{\mathrm{G}_{i}}=-[r_{i}(b)]$, where the square brackets $[-]$ denote the
class in $\operatorname{NS}(\mathrm{G}_{i})$. ∎

###  2.5. Lattice polarisations

We define various notions of lattice polarisation on our Néron-Severi
lattices.

######  Definition 2.30.

  1. (1)

A _lattice polarisation_ on $\operatorname{NS}(\mathrm{M})$ is defined to be a
primitive sublattice $L\subset\operatorname{NS}(\mathrm{M})$.

  2. (2)

A _lifted polarisation_ on $\Psi^{\perp}_{\mathrm{K}}/\Psi$ is a primitive
sublattice $\hat{L}\subset\Psi^{\perp}_{\mathrm{K}}/\Psi$ such that
$\zeta\in\hat{L}$.

  3. (3)

Given a lifted polarisation $\hat{L}\subset\Psi^{\perp}_{\mathrm{K}}/\Psi$,
the _intersection polarisation_ on $\operatorname{NS}(\mathrm{G}_{i})$ is
defined to be $L_{i}:=\hat{L}\cap\operatorname{NS}(\mathrm{G}_{i})$. Note that
$L_{i}$ is primitive in $\operatorname{NS}(\mathrm{G}_{i})$.

Our first result shows that lattice polarisations and lifted polarisations are
two sides of the same coin.

######  Lemma 2.31.

If $L\subset\operatorname{NS}(\mathrm{M})$ is a lattice polarisation, then the
preimage of $L$ under the map
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$ is a lifted
polarisation. Conversely, if $\hat{L}\subset\Psi^{\perp}_{\mathrm{K}}/\Psi$ is
a lifted polarisation, then the image of $\hat{L}$ under the map
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$ is a lattice
polarisation.

###### Proof.

The first statement is obvious from the definitions. For the converse, we need
to check that the image of $\hat{L}$ is primitive in
$\operatorname{NS}(\mathrm{M})$. Let $\varphi$ denote the map
$\varphi\colon\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$.
Suppose that $v\in\operatorname{NS}(\mathrm{M})$ is a primitive element such
that $mv\in\varphi(\hat{L})$, for some integer $m>1$. As $\varphi$ is
surjective, we may find $w\in\Psi^{\perp}_{\mathrm{K}}/\Psi$ such that
$\varphi(w)=v$. Then $mw+n\zeta\in\hat{L}$ for some $n\in\mathbb{Z}$, as
$\varphi(mw)=mv\in\varphi(\hat{L})$ and $\zeta$ generates the kernel of
$\varphi$. Now, as $\zeta\in\hat{L}$, it follows that $mw\in\hat{L}$. But
$\hat{L}$ is primitive, so we must also have $w\in\hat{L}$, and hence
$\varphi(w)=v\in\varphi(\hat{L})$ and $\varphi(\hat{L})$ is primitive in
$\operatorname{NS}(\mathrm{M})$. ∎

Next we look at the properties of the intersection polarisation.

######  Lemma 2.32.

We have

$\Psi^{\perp}_{\mathrm{K}}/\Psi\cap\operatorname{NS}(\mathrm{G}_{i})=K_{\mathrm{G}_{i}}^{\perp}.$

In particular, if $\hat{L}\subset\Psi^{\perp}_{\mathrm{K}}/\Psi$ is a lifted
polarisation, then the corresponding intersection polarisation on
$\operatorname{NS}(\mathrm{G}_{i})$ satisfies ${L}_{i}\subset
K_{\mathrm{G}_{i}}^{\perp}$

###### Proof.

Begin by noting that, by Proposition 2.14, we have
$K_{\mathrm{G}_{i}}=-[r_{i}(b)]$, where the square brackets $[-]$ denote the
class in $\operatorname{NS}(\mathrm{G}_{i})$.

Now, if
$v\in\operatorname{NS}(\mathrm{G}_{i})=(r_{i}(a))^{\perp}_{\mathrm{G}_{i}}/r_{i}(a)$
is any class, let $\hat{v}\in(r_{i}(a))^{\perp}_{\mathrm{G}_{i}}$ denote a
representative of $v$. From the proof of Lemma 2.27 we have
$\Psi^{\perp}_{\mathrm{G}}=(r_{1}(a))^{\perp}_{\mathrm{G}_{1}}\oplus(r_{2}(a))^{\perp}_{\mathrm{G}_{2}}$,
so $\hat{v}\in\Psi^{\perp}_{\mathrm{G}}\subset\mathrm{G}$. Therefore, by
definition of $f$, we have $f(\hat{v})=(-1)^{i-1}f_{i}(\hat{v})$. Thus, in
particular, we find

$q(v,K_{\mathrm{G}_{i}})=\langle\hat{v},r_{i}(b)\rangle_{\mathrm{G}_{i}}=\langle
f_{i}(\hat{v}),b\rangle_{\mathrm{E}}=(-1)^{i-1}\langle
f(\hat{v}),b\rangle_{\mathrm{E}}.$

Now suppose that
$v\in\Psi^{\perp}_{\mathrm{K}}/\Psi\cap\operatorname{NS}(\mathrm{G}_{i})$.
Then $\hat{v}\in\Psi^{\perp}_{\mathrm{K}}\subset\mathrm{K}$. So we have

$q(v,K_{\mathrm{G}_{i}})=(-1)^{i-1}\langle
f(\hat{v}),b\rangle_{\mathrm{E}}=0,$

as $\mathrm{K}=\ker(f)$. So
$\Psi^{\perp}_{\mathrm{K}}/\Psi\cap\operatorname{NS}(\mathrm{G}_{i})\subset
K_{\mathrm{G}_{i}}^{\perp}$.

Conversely, suppose that $v\in K_{\mathrm{G}_{i}}^{\perp}$. Then

$\langle f(\hat{v}),b\rangle_{\mathrm{E}}=(-1)^{i-1}q(v,K_{\mathrm{G}_{i}})=0$

and

$\langle f(\hat{v}),a\rangle_{\mathrm{E}}=(-1)^{i-1}\langle
f_{i}(\hat{v}),a\rangle_{\mathrm{E}}=(-1)^{i-1}\langle\hat{v},r_{i}(a)\rangle_{\mathrm{G}_{i}}=0$

as $\hat{v}\in(r_{i}(a))^{\perp}_{\mathrm{G}_{i}}$. Therefore we must have
$f(\hat{v})=0$, so $\hat{v}\in\mathrm{K}$. As we also have
$\hat{v}\in\Psi^{\perp}_{\mathrm{G}}$, it follows that
$\hat{v}\in\Psi^{\perp}_{\mathrm{K}}$, and so
$v\in\Psi^{\perp}_{\mathrm{K}}/\Psi$. Thus
$K_{\mathrm{G}_{i}}^{\perp}\subset\Psi^{\perp}_{\mathrm{K}}/\Psi\cap\operatorname{NS}(\mathrm{G}_{i})$
and the proof is complete. ∎

Unfortunately, in general the intersection polarisations $L_{1}$ and $L_{2}$
are not sufficient to determine the corresponding lattice/lifted polarisation.
However, we can recover some information in many cases.

######  Lemma 2.33.

Let $\varphi_{i}\colon
K_{\mathrm{G}_{i}}^{\perp}\to\operatorname{NS}(\mathrm{M})$ denote the
morphism induced by the morphism
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$ from
Proposition 2.28, for $i\in\\{1,2\\}$, and let
$\varphi:=\varphi_{1}\oplus\varphi_{2}\colon K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp}\to\operatorname{NS}(\mathrm{M})$. Then the
$\varphi_{i}$ are injective and moreover:

  * •

If $\deg(\mathrm{G})\neq 0$, then $\varphi$ is injective. Its image is a
finite index sublattice of $\operatorname{NS}(\mathrm{M})$.

  * •

If $\deg(\mathrm{G})=0$, then $\zeta\in K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp}$ and $\ker(\varphi)=\mathbb{Z}\zeta$.

In particular, $\varphi_{i}$ realises $L_{i}$ as a sublattice of $L$ for each
$i\in\\{1,2\\}$ and if $\deg(\mathrm{G})\neq 0$, then $\varphi$ realises
$L_{1}\oplus L_{2}$ as a sublattice of $L$.

###### Proof.

Injectivity of $\varphi_{i}$ follows from the fact that the kernel
$\mathbb{Z}\zeta$ of the morphism
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$ has trivial
intersection with $K_{\mathrm{G}_{i}}^{\perp}$ for each $i$.

So suppose that $\deg(\mathrm{G})\neq 0$. Let $(v_{1},v_{2})$ and
$(w_{1},w_{2})$ be any two distinct elements of
$K_{\mathrm{G}_{1}}^{\perp}\oplus K_{\mathrm{G}_{2}}^{\perp}$ and assume for a
contradiction that $\varphi(v_{1},v_{2})=\varphi(w_{1},w_{2})$. Then
$(v_{1},v_{2})$ and $(w_{1},w_{2})$ differ by an integer multiple of $\zeta$
so, by Lemma 2.29, we have

$(v_{1},v_{2})=(w_{1}-cK_{\mathrm{G}_{1}},w_{2}+cK_{\mathrm{G}_{2}}),$

for some nonzero $c\in\mathbb{Q}$. Now, as $v_{1},w_{1}\in
K_{\mathrm{G}_{1}}^{\perp}$, we have

$0=q(v_{1},K_{\mathrm{G}_{1}})=q(w_{1},K_{\mathrm{G}_{1}})-cq(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}})=-cq(K_{\mathrm{G}_{1}},K_{\mathrm{G}_{1}}).$

But this contradicts the assumption that $\deg(\mathrm{G})\neq 0$. Thus
$\varphi$ must be injective. The fact that the image is a finite index
sublattice then follows if we note that both $K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp}$ and $\operatorname{NS}(\mathrm{M})$ have rank
$18$.

If $\deg(\mathrm{G})=0$, then $q(K_{\mathrm{G}_{i}},K_{\mathrm{G}_{i}})=0$ for
$i\in\\{1,2\\}$, so $K_{\mathrm{G}_{i}}\in K_{\mathrm{G}_{i}}^{\perp}$. It
follows from Lemma 2.29 that $\mathbb{Z}\zeta\subset
K_{\mathrm{G}_{1}}^{\perp}\oplus K_{\mathrm{G}_{2}}^{\perp}$, which is the
kernel of the morphism
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$. ∎

######  Remark 2.34.

Using the fact that $\operatorname{NS}(\mathrm{M})\cong H\oplus E_{8}\oplus
E_{8}$ (Proposition 3.17), which is unimodular, and the discriminant groups of
$K_{\mathrm{G}_{i}}^{\perp}$ computed in Lemma 2.20, one may show that the
index of $\operatorname{im}(\varphi)$ in $\operatorname{NS}(\mathrm{M})$ is
equal to $3$ if $\deg(\mathrm{G})=9$, equal to $4$ if
$\deg(\mathrm{G})=8^{\prime}$, and equal to $\deg(\mathrm{G})$ if
$1\leq\deg(\mathrm{G})\leq 8$. If $\deg(\mathrm{G})=0$, then the quotient
$\operatorname{NS}(M)/\operatorname{im}(\varphi)$ is a free
$\mathbb{Z}$-module of rank $1$.

######  Definition 2.35.

Assume that $\deg(\mathrm{G})\neq 0$, so that $\varphi(L_{1}\oplus L_{2})$ is
a sublattice of $L$. The quotient group $Q(L):=L\,/\,\varphi(L_{1}\oplus
L_{2})$ is called the _coupling group_ of $L$.

The coupling group measures how far the lattice polarisation
$L\subset\operatorname{NS}(\mathrm{M})$ is from being induced by the two
intersection polarisations $L_{i}\subset\operatorname{NS}(\mathrm{G}_{i})$. It
consists of elements of $L$ that “cross between” or “couple” the part of $L$
contained in $\operatorname{NS}(\mathrm{G}_{1})$ and the part of $L$ contained
in $\operatorname{NS}(\mathrm{G}_{2})$. It fits into an exact sequence of
abelian groups

(3) $0\longrightarrow L_{1}\oplus
L_{2}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}L\longrightarrow
Q(L)\longrightarrow 0.$

Our next result shows that the coupling group can be used to measure the
failure of orthogonal complements to descend to the intersection polarisation.

######  Proposition 2.36.

Assume that $\deg(\mathrm{G})\neq 0$. Then the coupling group $Q(L)$ is
torsion if and only if
$(L_{i})^{\perp}_{K_{\mathrm{G}_{i}}^{\perp}}=\hat{L}^{\perp}\cap\operatorname{NS}(\mathrm{G}_{i})$
for each $i=\\{1,2\\}$, where $(L_{i})^{\perp}_{K_{\mathrm{G}_{i}}^{\perp}}$
denotes the orthogonal complement of $L_{i}$ taken in
$K_{\mathrm{G}_{i}}^{\perp}$.

###### Proof.

Note first that, as the class $\zeta$ is totally isotropic, without ambiguity
we can take $\hat{L}^{\perp}$ to be either the orthogonal complement to
$\hat{L}$ in $\Psi^{\perp}_{\mathrm{K}}/\Psi$ or the lift of $L^{\perp}$ under
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$, as these
both give the same sublattice of $\Psi^{\perp}_{\mathrm{K}}/\Psi$. Note also
that $L_{i}\subset\hat{L}$, so
$\hat{L}^{\perp}\cap\operatorname{NS}(\mathrm{G}_{i})\subset(L_{i})^{\perp}_{K_{\mathrm{G}_{i}}^{\perp}}$.

Now assume that $Q(L)$ is torsion. For simplicity of notation, in this part of
the proof we will assume $i=1$; the proof for $i=2$ is identical. Let
$v\in(L_{1})^{\perp}_{K_{\mathrm{G}_{1}}^{\perp}}\subset\operatorname{NS}(\mathrm{G}_{1})$.
Then $\varphi(v,0)\in\operatorname{NS}(\mathrm{M})$. Let $w\in L$. Then as
$Q(L)$ is torsion, there exists $k\in\mathbb{Z}$ such that
$w=k\varphi(w_{1},w_{2})$ for some $w_{i}\in L_{i}$. Then
$q(\varphi(v,0),w)=q_{1}(v,w_{1})+q_{2}(0,w_{2})=0$ as $v\in L_{i}^{\perp}$;
here $q$ denotes the bilinear form on $\operatorname{NS}(\mathrm{M})$ and
$q_{i}$ denotes the bilinear form on $\operatorname{NS}(\mathrm{G}_{i})$. So
$\varphi(v,0)\subset L^{\perp}$ and consequently $(v,0)\in\hat{L}^{\perp}$.
This proves that
$\hat{L}^{\perp}\cap\operatorname{NS}(\mathrm{G}_{1})=(L_{1})^{\perp}_{K_{\mathrm{G}_{1}}^{\perp}}$

To prove the converse, assume that $Q(L)$ is not torsion. Then
$\operatorname{rank}(L_{1})+\operatorname{rank}(L_{2})<\operatorname{rank}(L)$.
Taking orthogonal complements, we find that

$\operatorname{rank}\left((L_{1})^{\perp}_{K_{\mathrm{G}_{1}}^{\perp}}\right)+\operatorname{rank}\left((L_{2})^{\perp}_{K_{\mathrm{G}_{2}}^{\perp}}\right)>\operatorname{rank}\left(L^{\perp})\right.$

But then we cannot have
$(L_{i})^{\perp}_{K_{\mathrm{G}_{i}}^{\perp}}=\hat{L}^{\perp}\cap\operatorname{NS}(\mathrm{G}_{i})$
for both $i=\\{1,2\\}$. ∎

We conclude this section with a result about automorphisms of Néron-Severi
lattices, which will be useful in Section 5.

######  Proposition 2.37.

For each $i\in\\{1,2\\}$, suppose that $\psi_{i}$ is an isometry of
$\operatorname{NS}(\mathrm{G}_{i})$ which fixes $K_{\mathrm{G}_{i}}$. Then
$\psi_{1}\oplus\psi_{2}$ induces an isometry $\psi$ of
$\operatorname{NS}(\mathrm{M})$ which satisfies
$\psi(K_{\mathrm{G}_{i}}^{\perp})=K_{\mathrm{G}_{i}}^{\perp}$ for each
$i\in\\{1,2\\}$ (where we identify $K_{\mathrm{G}_{i}}^{\perp}$ with its image
under the injective map $\varphi_{i}$ from Lemma 2.33).

Conversely, suppose that $\deg(\mathrm{G})\neq 0$ and that $\psi$ is an
isometry of $\operatorname{NS}(\mathrm{M})$ satisfying
$\psi(K_{\mathrm{G}_{i}}^{\perp})=K_{\mathrm{G}_{i}}^{\perp}$ for each
$i\in\\{1,2\\}$ . Then, after possibly multiplying $\psi$ by $(-1)$, for each
$i\in\\{1,2\\}$ there are isometries $\psi_{i}$ of
$\operatorname{NS}(\mathrm{G}_{i})$ which fix $K_{\mathrm{G}_{i}}$ and induce
$\psi$.

###### Proof.

To prove the first part we use the matrix representation from the proof of
Proposition 2.21. For each $i\in\\{1,2\\}$, choose bases for
$\operatorname{NS}(\mathrm{G}_{i})$, let $Q_{i}$ denote the matrix of the
intersection form and let $K_{i}$ denotes the column vector representing
$K_{\mathrm{G}_{i}}$ with respect to this basis. Let $P_{i}$ denote the matrix
of $\psi_{i}$; note that $P_{i}^{T}Q_{i}P_{i}=Q_{i}$ and $P_{i}K_{i}=P_{i}$ by
assumption. Let $P$ denote the matrix

$P=\begin{pmatrix}P_{1}&0\\\ 0&P_{2}\end{pmatrix}$

of $\psi_{1}\oplus\psi_{2}$.

From the proof of Proposition 2.21, the map $f$ induces the map
$F\colon\operatorname{NS}(\mathrm{G}_{1})\oplus\operatorname{NS}(\mathrm{G}_{2})\to\mathrm{E}$
given by $F(v_{1},v_{2})=(-K_{1}^{T}Q_{1}v_{1}+K_{2}^{T}Q_{2}v_{2})a$, where
$a$ is one of the standard basis elements $(a,b)$ for $\mathrm{E}$. An easy
matrix calculation then shows that $FP=F$. It follows that $F$ induces an
isometry on $\ker(F)=\Psi^{\perp}_{\mathrm{K}}/\Psi$. This isometry fixes
$\zeta=c(-K_{\mathrm{G}_{1}},K_{\mathrm{G}_{2}})$ (for some $c\in\mathbb{Q}$),
so further descends to an isometry of $\operatorname{NS}(\mathrm{M})$.

To prove the converse, assume that $\deg(\mathrm{G})\neq 0$. Then
$K_{\mathrm{G}_{1}}^{\perp}$ and $K_{\mathrm{G}_{2}}^{\perp}$ are non-
degenerate by Lemma 2.20. The map $\psi$ induces isometries of the lattices
$K_{\mathrm{G}_{i}}^{\perp}$, which descend to automorphisms of the
discriminant groups $\operatorname{disc}(K_{\mathrm{G}_{i}}^{\perp})$ that
preserve the quadratic forms. These in turn induce an automorphism
$\overline{\psi}$ of $\operatorname{disc}(K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp})\cong\operatorname{disc}(K_{\mathrm{G}_{1}}^{\perp})\oplus\operatorname{disc}(K_{\mathrm{G}_{2}}^{\perp})$.

Let $A:=\operatorname{NS}(\mathrm{M})\,/\,(K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp})$. Then $A$ is a subgroup of
$\operatorname{disc}(K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp})$ which is isotropic with respect to the quadratic
form, and the order of $A$ is the index of $K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp}$ in $\operatorname{NS}(\mathrm{M})$. By [Nik80,
Proposition 1.4.2], the automorphism $\overline{\psi}$ must preserve $A$.

Now, using the explicit descriptions of the discriminant groups from Lemma
2.20, one may compute that the only automorphisms of
$\operatorname{disc}(K_{\mathrm{G}_{1}}^{\perp}\oplus
K_{\mathrm{G}_{2}}^{\perp})$ preserving $A$ and the quadratic forms are
$\pm\mathrm{Id}$. After multiplying $\psi$ by $(-1)$ if necessary, we may
therefore assume that $\overline{\psi}=\mathrm{Id}$.

In particular, $\psi$ acts trivially on
$\operatorname{disc}(K_{\mathrm{G}_{i}}^{\perp})$ for each $i\in\\{1,2\\}$. By
[Dol12, Proposition 8.2.5], it follows that $\psi$ extends to isometries
$\psi_{i}$ of $\operatorname{NS}(\mathrm{G}_{i})$ which fix
$K_{\mathrm{G}_{i}}$, as required. ∎

##  3\. Tyurin Degenerations of K3 Surfaces

Throughout this section we use the following notation. Let
$\pi\colon\mathcal{X}\to\Delta$ denote a semistable Type II degeneration of K3
surfaces over the open unit disc in $\mathbb{C}$; note in particular that as
part of the Type II condition we assume that $K_{\mathcal{X}}\sim 0$. Let
$X\subset\mathcal{X}$ denote a general fibre and let $X_{p}\subset\mathcal{X}$
denote the fibre over a point $p\in\Delta$. We assume further that
$\pi\colon\mathcal{X}\to\Delta$ is a _Tyurin degeneration_ , meaning that the
central fibre $X_{0}=V_{1}\cup_{C}V_{2}$ is a _stable K3 surface of Type II_
in the sense of the following definition, originally due to Friedman. We let
$i_{j}\colon C\to V_{j}$ denote the inclusion.

######  Definition 3.1.

[Fri84, Definition 3.1] A _stable K3 surface of Type II_ is a surface with
normal crossings $X_{0}=V_{1}\cup_{C}V_{2}$ consisting of two smooth rational
surfaces $V_{1}$ and $V_{2}$ glued along a smooth anticanonical elliptic curve
$C$, satisfying $N_{C/V_{1}}\otimes N_{C/V_{2}}\cong\mathcal{O}_{C}$.

######  Remark 3.2.

Note that the normal bundle condition implies that if
$X_{0}=V_{1}\cup_{C}V_{2}$ is a stable K3 surface of Type II and $q$ denotes
the bilinear form on $\operatorname{NS}(V_{i})$, then
$q(K_{V_{1}},K_{V_{1}})=-q(K_{V_{2}},K_{V_{2}})$. In line with our convention
in Section 2.3, we will henceforth assume that labels have been chosen so that
$q(K_{V_{1}},K_{V_{1}})\geq 0$.

######  Remark 3.3.

Tyurin degenerations are in some sense primitive in the class of Type II
degenerations: given a Type II degeneration of polarised K3 surfaces, after
twisting the polarisation and performing some birational modifications on the
central fibre, one may obtain a Tyurin degeneration. We refer the interested
reader to [Fri84, Theorems 2.2 and 2.3] for details.

###  3.1. Hodge theory

We begin by studying the Hodge theory of such degenerations. The material in
this section largely follows from work of Friedman [Fri84] and Scattone
[Sca87].

In the limiting mixed Hodge structure $H^{2}_{\lim}(X)$, the weight filtration
is defined over $\mathbb{Z}$ and is given by

$0\subset I\subset I^{\perp}\subset H^{2}(V;\mathbb{Z}),$

where $I\subset H^{2}(X;\mathbb{Z})$ is a primitive rank two isotropic
sublattice, the choice of which is unique up to isometries of
$H^{2}(X;\mathbb{Z})$ (see [Sca87, Section 2.2]).

######  Lemma 3.4.

The quotient $I^{\perp}/I$ is isomorphic to the lattice $H\oplus E_{8}\oplus
E_{8}$, where $H$ denotes the even unimodular lattice of signature $(1,1)$
(the hyperbolic plane) and $E_{8}$ denotes the usual negative definite root
lattice.

###### Proof.

$H^{2}(X;\mathbb{Z})$ is isomorphic to the K3 lattice
$\Lambda_{\mathrm{K3}}:=H\oplus H\oplus H\oplus E_{8}\oplus E_{8}$. Note that
each copy of $H$ contains a primitive isotropic vector. As $I\subset
H^{2}(X;\mathbb{Z})$ is unique up to isometry, without loss of generality we
may take $I$ to be generated by taking one primitive isotropic vector from
each of the first two factors of $H$. It follows that the quotient
$I^{\perp}/I$ is isomorphic to $H\oplus E_{8}\oplus E_{8}$. ∎

With this notation, we have the following proposition.

######  Proposition 3.5.

The Clemens-Schmid exact sequence induces exact sequences over $\mathbb{Z}$ as
follows:

$\displaystyle 0\longrightarrow H^{0}(X_{0};\mathbb{Z})\longrightarrow
H^{0}_{\lim}(X)\longrightarrow 0,$ $\displaystyle
0\longrightarrow\mathbb{Z}\xi\longrightarrow\xi^{\perp}\longrightarrow
I^{\perp}/I\longrightarrow 0,$ $\displaystyle
0\longrightarrow\mathbb{Z}\eta\longrightarrow
H^{4}(X_{0};\mathbb{Z})\longrightarrow H^{4}_{\lim}(X)\longrightarrow 0,$

where

  * •

$\xi\in H^{2}(V_{1};\mathbb{Z})\oplus H^{2}(V_{2};\mathbb{Z})$ is the class of
$(C,-C)$,

  * •

$\eta\in H^{4}(V_{1};\mathbb{Z})\oplus H^{4}(V_{2};\mathbb{Z})$ is the class
of $(p,-p)$, for $p$ a point.

###### Proof.

The difficult part of this proposition, namely the second sequence, follows
immediately from a result of Friedman [Fri84, Lemma 3.5]. We follow Friedman’s
approach to prove exactness of the remaining two sequences.

Exactness of these sequences follows from the fact that the Clemens-Schmid
sequence is obtained by splicing together the Wang sequence

$\ldots\to H^{i-1}(X;\mathbb{Z})\longrightarrow
H^{i}(\mathcal{X}^{*};\mathbb{Z})\longrightarrow
H^{i}(X;\mathbb{Z})\stackrel{{\scriptstyle
N}}{{\longrightarrow}}H^{i}(X;\mathbb{Z})\to\cdots$

and the exact sequence of a pair

$\ldots\to H^{i}_{V_{0}}(\mathcal{X};\mathbb{Z})\longrightarrow
H^{i}(X_{0};\mathbb{Z})\longrightarrow
H^{i}(\mathcal{X}^{*};\mathbb{Z})\longrightarrow
H^{i+1}_{V_{0}}(\mathcal{X},\mathbb{Z})\to\cdots,$

where $\mathcal{X}^{*}=\mathcal{X}\setminus X_{0}$. When $i=0$ or $i=4$ we
have $H^{i-1}(X,\mathbb{Z})=0$ and $N=0$, so the Wang sequence gives an
isomorphism $H^{i}(\mathcal{X}^{*},\mathbb{Z})\cong H^{i}(X,\mathbb{Z})$.
Furthermore, we have $H^{i}_{X_{0}}(\mathcal{X};\mathbb{Z})=0$ for $i=0,1,5$,
which completes the proof of the first sequence. To conclude the proof for the
third sequence, note that the image of the map
$H^{4}_{X_{0}}(\mathcal{X};\mathbb{Z})\to H^{4}(X_{0};\mathbb{Z})$ is
generated by the class $\eta$. ∎

######  Remark 3.6.

Note that, by [Fri84, Lemma 3.5], the second sequence here arises from the
second weight graded piece of the Clemens-Schmid sequence

$\operatorname{Gr}_{2}^{W}H^{2}_{X_{0}}(\mathcal{X})\longrightarrow\operatorname{Gr}_{2}^{W}H^{2}(X_{0})\longrightarrow\operatorname{Gr}_{2}^{W}H^{2}_{\lim}(X)\longrightarrow
0,$

where the class $\xi$ generates the image of the first map and
$\xi^{\perp}=\operatorname{Gr}_{2}^{W}H^{2}(X_{0})$.

Using the viewpoint of perverse and weight filtrations $P_{\bullet}$ and
$W_{\bullet}$ described in [DT21, Sections 4.1 and 5.2.1], the classes $\xi$
and $\eta$ are generators of the graded pieces
$\operatorname{Gr}_{2}^{W}\operatorname{Gr}_{2}^{P}H^{2}(X_{0})$ and
$\operatorname{Gr}_{4}^{W}\operatorname{Gr}_{4}^{P}H^{4}(X_{0})$,
respectively.

We may take the direct sum of the exact sequences from Proposition 3.5 to
obtain an exact sequence over $\mathbb{Z}$

(4) $0\to\mathbb{Z}\xi\oplus\mathbb{Z}\eta\to
H^{0}(X_{0})\oplus\xi^{\perp}\oplus H^{4}(X_{0})\to H^{0}_{\lim}(X)\oplus
I^{\perp}/I\oplus H^{4}_{\lim}(X)\to 0.$

###  3.2. Clemens-Schmid and pseudolattices over the rational numbers

Our next aim is to show how the results of Section 3.1 are naturally recovered
from the abstract considerations in Section 2.3, for quasi del Pezzo
homomorphisms $f_{i}\colon\mathrm{G}_{i}\to\mathrm{E}$ arising from the
anticanonical pairs $(V_{i},C)$ making up the central fibre. In this
subsection we begin by working over the rational numbers $\mathbb{Q}$; in the
subsequent subsections we will discuss to what extent this continues to work
over $\mathbb{Z}$.

For any surface $V$, it is well-known that the Chern character map

$\mathrm{ch}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))\longrightarrow
H^{\mathrm{even}}(V;\mathbb{Q}):=\bigoplus_{i=0}^{2}H^{2i}(V;\mathbb{Q})$

is a ring homomorphism. Moreover, it follows easily from [Kuz17, Example 3.5]
that this map is injective and its $\mathbb{Q}$-linear extension induces an
isomorphism

$\mathrm{ch_{\mathbb{Q}}}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))\otimes\mathbb{Q}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(V;\mathbb{Q})\oplus(\operatorname{NS}(V)\otimes\mathbb{Q})\oplus
H^{4}(V;\mathbb{Q}).$

The Chern character maps the class of $\mathcal{O}_{V}$ to $(1,0,0)$, the
class of the structure sheaf $\mathcal{O}_{p}$ of a point to $(0,0,1)$, and
the class of $\mathcal{O}_{V}(D)$, for any divisor $D$ on $V$, to
$(1,D,\chi(\mathcal{O}_{V}(D))+\frac{1}{2}q(K_{V},D)-\chi(\mathcal{O}_{V}))$,
where $q(\cdot,\cdot)$ denotes the usual bilinear form on
$H^{2}(V;\mathbb{Q})$ induced by the cup product; it follows from [Kuz17,
Example 3.5] that these classes span the image of $\mathrm{ch}$.

######  Remark 3.7.

Note that the Chern character map is _not_ the same as the isomorphism
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))\cong\mathbb{Z}\oplus\operatorname{NS}(V)\oplus\mathbb{Z}$
from [Kuz17, Example 3.5] that we used in the proof of Proposition 2.21

Using this, it is easy to show that the Euler form on
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V))$ is induced by the bilinear form
on $H^{\mathrm{even}}(V;\mathbb{Q})$ with matrix

(5) $\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
H^{0}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle H^{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}\\\H^{0}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\chi(\mathcal{O}_{V})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-\frac{1}{2}K^{T}Q$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\H^{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\frac{1}{2}QK$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{4}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt\crcr}}}}\right]$}},$

where $Q$ denotes the matrix of the bilinear form $q$ with respect to an
appropriate choice of basis for $H^{2}(V;\mathbb{Q})$ and $K$ denotes the
column vector representing the canonical divisor $K_{V}$ in this basis.

Suppose that $X_{0}=V_{1}\cup_{C}V_{2}$ is a stable K3 surface of Type II.
Each of the surfaces $V_{j}$ is a quasi del Pezzo surface. So, by Example
2.16, the derived pull-back map
$i_{j}^{*}\colon\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))\to\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))$
is a quasi del Pezzo homomorphism of pseudolattices, with right adjoint given
by derived push-forward $i_{j*}$. The class of the structure sheaf of a point
defines a point-like element in
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))$ and with respect to this
point-like element we have
$\operatorname{NS}(\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j})))\cong\operatorname{NS}(V_{j})\cong
H^{2}(V_{j};\mathbb{Z})$. This isomorphism identifies the canonical classes,
so by Remark 3.2 we have $q(K_{V_{1}},K_{V_{1}})=-q(K_{V_{2}},K_{V_{2}})$. It
follows that we are in the setting of Section 2.3, with
$\mathrm{G}_{j}=\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j})))$ and
$f_{j}=i_{j}^{*}$.

Let $f\colon\mathrm{G}\to\mathrm{E}$ denote the spherical homomorphism
obtained by gluing the pseudolattices
$\mathrm{G}_{1}=\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{1}))$ and
$\mathrm{G}_{2}=\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{2}))$ along
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))\cong\mathrm{E}$ via the
homomorphisms $i_{1}^{*}$ and $(-i_{2}^{*})$, as in Section 2.3, and let $r$
be its right adjoint. The results of Section 2.3 give rise to lattices
$\overline{\mathrm{E}}$, $\mathrm{K}$, and $\mathrm{M}$, related by the short
exact sequence (2).

######  Lemma 3.8.

The Chern character map induces a natural isomorphism of $\mathbb{Q}$-modules

$\kappa\colon\mathrm{G}_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{\mathrm{even}}(V_{1};\mathbb{Q})\oplus
H^{\mathrm{even}}(V_{2};\mathbb{Q}),$

where $\mathrm{G}_{\mathbb{Q}}:=\mathrm{G}\otimes\mathbb{Q}$. The restriction
of $\kappa$ to $\mathrm{K}_{\mathbb{Q}}:=\mathrm{K}\otimes\mathbb{Q}$ induces
an isomorphism

$\mathrm{K}_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(X_{0};\mathbb{Q})\oplus\operatorname{Gr}_{2}^{W}H^{2}(X_{0};\mathbb{Q})\oplus
H^{4}(X_{0};\mathbb{Q}),$

where $W_{\bullet}$ is the weight filtration arising from Deligne’s mixed
Hodge structure on a normal crossing variety.

###### Proof.

As $H^{2}(V_{j};\mathbb{Z})\cong\operatorname{NS}(V_{j})$, the Chern character
induces an isomorphism between
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))\otimes\mathbb{Q}$ and
$H^{\mathrm{even}}(V_{j};\mathbb{Q})$. Under this identification, the bilinear
form
$\langle\cdot,\cdot\rangle_{\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{j}))}$
is given by Equation (5) and the maps $i_{j}^{*}$ and $i_{j*}$ are given by

$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
H^{0}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle H^{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}\\\\{[\mathcal{O}_{p}]}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-
K_{j}^{T}Q_{j}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\\{[\mathcal{O}_{C}]}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt\crcr}}}}\right]$ and $\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle{[\mathcal{O}_{p}]}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle{[\mathcal{O}_{C}]}\\\H^{0}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-K_{j}\\\H^{4}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-\frac{1}{2}q(K_{V_{j}},K_{V_{j}})$\hfil\kern
5.0pt\crcr}}}}\right]$

respectively, where $Q_{j}$ denotes the matrix of the usual bilinear form on
$H^{2}(V_{j};\mathbb{Q})$ with respect to an appropriate choice of basis, and
$K_{j}$ denotes the column vector representing the canonical divisor
$K_{V_{j}}$ in this basis.

We may thus identify $\mathrm{G}\otimes\mathbb{Q}$ with
$H^{\mathrm{even}}(V_{1};\mathbb{Q})\oplus
H^{\mathrm{even}}(V_{2};\mathbb{Q})$ as $\mathbb{Q}$-modules, proving the
first statement. The bilinear form on $\mathrm{G}$ induces a bilinear form on
$H^{\mathrm{even}}(V_{1};\mathbb{Q})\oplus
H^{\mathrm{even}}(V_{2};\mathbb{Q})$ with matrix

$\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
H^{0}(V_{1})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle H^{2}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{0}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{2}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}(V_{2})\\\H^{0}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-\frac{1}{2}K_{1}^{T}Q_{1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle K_{2}^{T}Q_{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\H^{2}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\frac{1}{2}Q_{1}K_{1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-Q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-Q_{1}K_{1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{4}(V_{1})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{0}(V_{2})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-\frac{1}{2}K_{2}^{T}Q_{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\H^{2}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\frac{1}{2}Q_{2}K_{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-Q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{4}(V_{2})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt\crcr}}}}\right]$}}.$

Now recall that $f(u_{1},u_{2})=i_{1}^{*}(u_{1})-i_{2}^{*}(u_{2})$ for
$u_{i}\in\mathrm{G}_{i}$ and, by the proof of [HT20, Proposition 2.23],
$r(v)=(i_{1*}(v)-i_{1*}i_{2}^{*}i_{2*}(v),-i_{2*}(v))$ for $v\in\mathrm{E}$.
The maps $f$ and $r$ from Section 2.3 are therefore given respectively by

$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
H^{0}(V_{1})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle H^{2}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{0}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{2}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}(V_{2})\\\\{[\mathcal{O}_{p}]}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-
K_{1}^{T}Q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
K_{2}^{T}Q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\\{[\mathcal{O}_{C}]}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt\crcr}}}}\right]$

and

$\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle{[\mathcal{O}_{p}]}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle{[\mathcal{O}_{C}]}\\\H^{0}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{2}(V_{1})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-K_{1}\\\H^{4}(V_{1})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\frac{1}{2}q(K_{V_{1}},K_{V_{1}})\\\H^{0}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{2}(V_{2})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle K_{2}\\\H^{4}(V_{2})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\frac{1}{2}q(K_{V_{2}},K_{V_{2}})$\hfil\kern
5.0pt\crcr}}}}\right]$}}.$

If we write a general vector in $H^{\mathrm{even}}(V_{j};\mathbb{Q})$ as
$(r_{j},D_{j},s_{j})$, it follows that the kernel $\mathrm{K}_{\mathbb{Q}}$ of
$f$ is identified with the subspace of
$H^{\mathrm{even}}(V_{1};\mathbb{Q})\oplus
H^{\mathrm{even}}(V_{2};\mathbb{Q})$ defined by

$\\{(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})\mid r_{1}=r_{2}\text{ and
}q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})\\}.$

Now, Deligne’s mixed Hodge structure on a normal crossing variety gives that

$\displaystyle H^{0}(X_{0})$ $\displaystyle\cong\\{(r_{1},r_{2})\in
H^{0}(V_{1})\oplus H^{0}(V_{2})\mid r_{1}=r_{2}\\},$
$\displaystyle\operatorname{Gr}_{2}^{W}H^{2}(X_{0})$
$\displaystyle\cong\\{(D_{1},D_{2})\in H^{2}(V_{1})\oplus H^{2}(V_{2})\mid
q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})\\},$ $\displaystyle H^{4}(X_{0})$
$\displaystyle\cong H^{4}(V_{1})\oplus H^{4}(V_{2}),$

so we see that $H^{0}(X_{0})\oplus\operatorname{Gr}_{2}^{W}H^{2}(X_{0})\oplus
H^{4}(X_{0})\cong\mathrm{K}_{\mathbb{Q}}$, as required. ∎

######  Notation 3.9.

In light of the isomorphisms from this lemma, we will write elements of
$\mathrm{G}_{\mathbb{Q}}$ as $6$-tuples

$(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2}),$

where $r_{i}\in H^{0}(V_{i};\mathbb{Q})$, $D_{i}\in H^{2}(V_{i},\mathbb{Q})$,
and $s_{i}\in H^{4}(V_{i};\mathbb{Q})$. By the proof of this lemma, we can
express elements of the submodule $\mathrm{K}_{\mathbb{Q}}$ as $6$-tuples as
above satisfying the two conditions $r_{1}=r_{2}$ and
$q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})$.

Now suppose that $X_{0}=V_{1}\cup_{C}V_{2}$ occurs as the central fibre of a
Tyurin degeneration $\pi\colon\mathcal{X}\to\Delta$. Then the following result
gives an interpretation of the lattice $\mathrm{M}$.

######  Lemma 3.10.

There is a natural isomorphism of $\mathbb{Q}$-modules

$\mu\colon\mathrm{M}_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}_{\lim}(X)\oplus\operatorname{Gr}_{2}^{W}H^{2}_{\lim}(X)\oplus
H^{4}_{\lim}(X),$

where $\mathrm{M}_{\mathbb{Q}}:=\mathrm{M}\otimes\mathbb{Q}$ and $W_{\bullet}$
is the weight filtration arising from the limiting mixed Hodge structure.
Moreover, under the isomorphisms $\kappa$, $\mu$ the map
$\mathrm{K}_{\mathbb{Q}}\to\mathrm{M}_{\mathbb{Q}}$ from the short exact
sequence (2) agrees with the one induced by the Clemens-Schmid exact sequence
(4).

###### Proof.

By Proposition 3.5 and Remark 3.6, the Clemens-Schmid exact sequence gives
rise to an isomorphism $\varphi_{0}\colon
H^{0}(X_{0})\stackrel{{\scriptstyle\sim}}{{\to}}H^{0}_{\lim}(X)$, along with
surjective maps
$\varphi_{2}\colon\operatorname{Gr}_{2}^{W}H^{2}(X_{0})\to\operatorname{Gr}_{2}^{W}H^{2}_{\lim}(X)$
and $\varphi_{4}\colon H^{4}(X_{0})\to H^{4}_{\lim}(X)$. Composing
$\varphi:=\varphi_{0}\oplus\varphi_{2}\oplus\varphi_{4}$ with the isomorphism
$\kappa$ from Lemma 3.8, we thus obtain a surjective map of
$\mathbb{Q}$-modules

$\varphi\circ\kappa\colon\mathrm{K}_{\mathbb{Q}}\longrightarrow
H^{0}_{\lim}(X)\oplus\operatorname{Gr}_{2}^{W}H^{2}_{\lim}(X)\oplus
H^{4}_{\lim}(X).$

To prove the proposition, it suffices to show that the kernel of
$\varphi\circ\kappa$ is isomorphic to
$\overline{\mathrm{E}}\otimes\mathbb{Q}=\operatorname{im}(r)\otimes\mathbb{Q}\subset\mathrm{K}_{\mathbb{Q}}$.

From the proof of Lemma 3.8, we see that
$\overline{\mathrm{E}}\otimes\mathbb{Q}\subset\mathrm{K}_{\mathbb{Q}}$ is
generated by the columns of the matrix of the map $r$. Writing a general
element of $\mathrm{K}_{\mathbb{Q}}$ as
$(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})$, and recalling that
$q(K_{V_{1}},K_{V_{1}})=-q(K_{V_{2}},K_{V_{2}})$, we may simplify this set of
generators to $\\{(0,0,1,0,0,-1),(0,-K_{V_{1}},0,0,K_{V_{2}},0)\\}$.

By Proposition 3.5, we see that $\varphi_{0}$ is an isomorphism and the
kernels of $\varphi_{2}$ and $\varphi_{4}$ are generated by the classes
$\xi=(-K_{V_{1}},K_{V_{2}})\in\operatorname{Gr}_{2}^{W}H^{2}(X_{0})$ and
$\eta=(1,-1)\in H^{4}(X_{0})$ respectively. It thus follows immediately that
$\ker(\varphi)=\overline{\mathrm{E}}\otimes\mathbb{Q}$; this completes the
proof.∎

Putting everything in this subsection together, we have the following result,
which gives a relationship between the sequences (2) and (4) over
$\mathbb{Q}$.

######  Proposition 3.11.

Let $\pi\colon\mathcal{X}\to\Delta$ be a Tyurin degeneration with central
fibre $X_{0}$ a stable K3 surface of Type II. Then there is a commutative
diagram of $\mathbb{Q}$-modules

${0}$${\overline{\mathrm{E}}_{\mathbb{Q}}}$${\mathrm{K}_{\mathbb{Q}}}$${\mathrm{M}_{\mathbb{Q}}}$${0}$${0}$${\mathbb{Q}\xi\oplus\mathbb{Q}\eta}$${H^{0}(X_{0})\oplus\xi^{\perp}\oplus
H^{4}(X_{0})}$${H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus
H^{4}_{\lim}(X)}$${0}$$\scriptstyle{\kappa}$$\scriptstyle{\mu}$$\scriptstyle{\varphi}$

where the vertical arrows are isomorphisms.

###  3.3. Integrality

Next we investigate to what extent Proposition 3.11 holds over the integers.
We note first that the map $\kappa$ is not integral: indeed, let
$D_{1}\in\operatorname{NS}(V_{1})\cong H^{2}(V_{1};\mathbb{Z})$ and
$D_{2}\in\operatorname{NS}(V_{2})\cong H^{2}(V_{2};\mathbb{Z})$. Then
$(\mathcal{O}_{V_{1}}(D_{1}),\mathcal{O}_{V_{2}}(D_{2}))$ is an integral
element of $\mathrm{G}$. Applying the map $\kappa$ gives

$\kappa(\mathcal{O}_{V_{1}}(D_{1}),\mathcal{O}_{V_{2}}(D_{2}))=(1,D_{1},s_{1},1,D_{2},s_{2})$

with

$\displaystyle s_{1}$
$\displaystyle=\chi(\mathcal{O}_{V_{1}}(D_{1}))+\tfrac{1}{2}q(K_{V_{1}},D_{1})-\chi(\mathcal{O}_{V_{1}}),$
$\displaystyle s_{2}$
$\displaystyle=\chi(\mathcal{O}_{V_{2}}(D_{2}))+\tfrac{1}{2}q(K_{V_{2}},D_{2})-\chi(\mathcal{O}_{V_{2}}).$

From these expressions we see that
$\kappa(\mathcal{O}_{V_{1}}(D_{1}),\mathcal{O}_{V_{2}}(D_{2}))$ is not
integral unless $q(K_{V_{1}},D_{1})$ and $q(K_{V_{2}},D_{2})$ are even. The
following lemma shows that this is, essentially, the only way in which
integrality can fail.

######  Lemma 3.12.

The map $\kappa$ induces an isomorphism between $\mathrm{G}$ and the subset of
$H^{\mathrm{even}}(V_{1};\mathbb{Q})\oplus
H^{\mathrm{even}}(V_{2};\mathbb{Q})$ defined by those $6$-tuples
$(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})$ satisfying

  * •

$r_{i}\in H^{0}(V_{i};\mathbb{Z})$ for each $i=1,2$;

  * •

$D_{i}\in H^{2}(V_{i};\mathbb{Z})$ for each $i=1,2$;

  * •

$2s_{i}\in H^{4}(V_{i};\mathbb{Z})$ for each $i=1,2$, and $s_{i}\in
H^{4}(V_{i};\mathbb{Z})$ if and only if $q(K_{V_{i}},D_{i})$ is even.

This isomorphism takes $\mathrm{K}$ to the set of $6$-tuples as above
satisfying $r_{1}=r_{2}$ and $q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})$.

###### Proof.

This is a straightforward consequence of the explicit description of the Chern
character map given at the start of Section 3.2. ∎

Whilst the isomorphism $\kappa$ is not integral, somewhat surprisingly we find
that the isomorphism $\mu$ is.

######  Lemma 3.13.

The map $\mu\colon\mathrm{M}\to H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus
H^{4}_{\lim}(X)$ is an isomorphism over $\mathbb{Z}$.

###### Proof.

We first note that the top and bottom rows of the commutative diagram in the
statement of Proposition 3.11 are defined over $\mathbb{Z}$, by the results of
Sections 2.3 and 3.1. By commutativity of this diagram, it suffices to show
that the composition $\varphi\circ\kappa\colon\mathrm{K}\to
H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X)$ is well-defined and
surjective over $\mathbb{Z}$.

The map $\varphi$ is given explicitly by

$\varphi(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})=(r_{1},D_{1}+D_{2},s_{1}+s_{2})\in
H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X).$

Combining this with the description of the image of $\kappa$ from Lemma 3.12,
we find that the image of $\varphi\circ\kappa$ is integral in
$H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X)$, so
$\varphi\circ\kappa$ is defined over $\mathbb{Z}$ .

It remains to show that $\varphi\circ\kappa$ is surjective. Indeed, as the map
$\varphi$ is surjective over $\mathbb{Z}$, for any $x\in H^{0}_{\lim}(X)\oplus
I^{\perp}/I\oplus H^{4}_{\lim}(X)$ there exists

$y=(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})\in
H^{0}(X_{0};\mathbb{Z})\oplus\xi^{\perp}\oplus H^{4}(X_{0};\mathbb{Z})$

with $\varphi(y)=x$. To complete the proof, note that if
$y\notin\kappa(\mathrm{K})$, we may use the description $\kappa(\mathrm{K})$
from Lemma 3.12 and the fact that

$\varphi(r_{1},D_{1},s_{1},r_{2},D_{2},s_{2})=\varphi(r_{1},D_{1},s_{1}+\tfrac{1}{2},r_{2},D_{2},s_{2}-\tfrac{1}{2}),$

to obtain an element of $\kappa(\mathrm{K})$ mapping to $x$. ∎

As noted in Section 2.3, the bilinear form on $\mathrm{K}$ descends under the
map $\mathrm{K}\to\mathrm{M}$ to give a bilinear form on $\mathrm{M}$.
However, $H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X)$ also comes
equipped with a bilinear form, induced by the bilinear form on
$H^{\mathrm{even}}_{\lim}(X)\cong H^{\mathrm{even}}(X;\mathbb{Z})$ defined by
Equation (5) (which is integral as $K_{X}=0$). The following result shows that
these bilinear forms agree.

######  Lemma 3.14.

Equipping $\mathrm{M}$ and $H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus
H^{4}_{\lim}(X)$ with bilinear forms as above, the map $\mu$ is an isomorphism
of lattices.

###### Proof.

Using Equation (5) and the explicit computations in the proofs of Lemmas 3.8
and 3.10, one may show that both bilinear forms are given explicitly by the
matrix

$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
H^{0}_{\lim}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\operatorname{Gr}_{2}^{W}H^{2}_{\lim}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle H^{4}_{\lim}\\\H^{0}_{\lim}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1\\\\\operatorname{Gr}_{2}^{W}H^{2}_{\lim}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\H^{4}_{\lim}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt\crcr}}}}\right]$

where $Q$ denotes the matrix induced by the cup-product form on
$\operatorname{Gr}_{2}^{W}H^{2}_{\lim}(X)$ with respect to an appropriate
basis. ∎

Putting this lemma together with the exact sequence (2), we thus obtain the
following result.

######  Proposition 3.15.

In the context of a Tyurin degeneration of K3 surfaces, the exact sequence (2)
can be realised as

$0\longrightarrow\overline{\mathrm{E}}\longrightarrow\mathrm{K}\longrightarrow
H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X)\longrightarrow 0,$

where

  * •

$\overline{\mathrm{E}}$ is isomorphic to $\mathbb{Z}^{2}$ with the zero form,

  * •

$\mathrm{K}$ is the kernel of the spherical homomorphism
$f\colon\mathrm{G}\to\mathrm{E}$ obtained by gluing
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{1}))$ and
$\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(V_{2}))$ along
$\mathrm{E}\cong\mathrm{K}_{0}^{\mathrm{num}}(\mathbf{D}(C))$, as in Section
2.3, and

  * •

the bilinear form on $H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus H^{4}_{\lim}(X)$
is induced by the bilinear form on $H^{\mathrm{even}}_{\lim}(X)\cong
H^{\mathrm{even}}(X;\mathbb{Z})$ defined by Equation (5).

###  3.4. Point-like vectors and the Néron-Severi lattice

Next we ask how the ideas of Section 2.4 arise in this setting.

In this setting, by [HT20, Example 3.10], $\Psi$ is the sublattice of
$\mathrm{K}$ generated by the classes of
$(r_{1}(a),0)=([\mathcal{O}_{p_{1}}],0)$ and
$(0,r_{2}(a))=(0,[\mathcal{O}_{p_{2}}])$, for $p_{i}$ a point on $X_{i}$.
Using Notation 3.9, $\Psi$ is generated by $(0,0,1,0,0,0)$ and
$(0,0,0,0,0,1)$. By Lemma 2.27, the lattice isomorphisms
$\operatorname{NS}(\mathrm{G}_{i})\cong\operatorname{NS}(V_{i})\cong
H^{2}(V_{i};\mathbb{Z})$ induce an isomorphism of lattices

$\Psi^{\perp}_{\mathrm{G}}/\Psi\cong H^{2}(V_{1};\mathbb{Z})\oplus
H^{2}(V_{2};\mathbb{Z}).$

Using the description of the bilinear form on $\mathrm{G}$ in the proof of
Lemma 3.8, we see that this induces an isomorphism of lattices

$\Psi^{\perp}_{\mathrm{K}}/\Psi\cong\\{(D_{1},D_{2})\in
H^{2}(V_{1};\mathbb{Z})\oplus H^{2}(V_{2};\mathbb{Z})\mid
q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})\\}=\xi^{\perp},$

where $\Psi^{\perp}_{\mathrm{G}}/\Psi$ and $\Psi^{\perp}_{\mathrm{K}}/\Psi$
are equipped with the bilinear forms from Lemma 2.27 and Proposition 2.28.

######  Remark 3.16.

Note that these isomorphisms are _not_ the ones induced by the map $\kappa$
coming from the Chern character (see Lemma 3.8). Indeed, by Lemma 3.12, the
image of $\kappa$ may even contain some non-integral classes. Instead, they
may be thought of as a composition of $\kappa$ with a projection to the
subspace $H^{2}(V_{1};\mathbb{Z})\oplus H^{2}(V_{2};\mathbb{Z})$.

The map to $\mathrm{M}\cong H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus
H^{4}_{\lim}(X)$ takes the generators $(0,0,1,0,0,0)$ and $(0,0,0,0,0,1)$ of
$\Psi$ to a generator $\mathbf{p}\in H^{4}_{\lim}(X)$. By the description of
the bilinear form on $\mathrm{M}$ from Lemma 3.14, we see that the natural
isomorphism $\mu\colon\mathrm{M}\to H^{0}_{\lim}(X)\oplus I^{\perp}/I\oplus
H^{4}_{\lim}(X)$ induces an isomorphism
$\operatorname{NS}(\mathrm{M})=\mathbf{p}^{\perp}/\mathbf{p}\cong
I^{\perp}/I$.

Finally, by Lemma 2.29, we see that the kernel of the map
$\Psi^{\perp}_{\mathrm{K}}/\Psi\to\operatorname{NS}(\mathrm{M})$ is generated
by $\zeta=(-K_{V_{1}},K_{V_{2}})=(C,-C)=\xi$. Putting things together, we thus
obtain the following result.

######  Proposition 3.17.

If $X_{0}=V_{1}\cup_{C}V_{2}$ is a stable K3 surface of type II, then the
lattice isomorphisms
$\operatorname{NS}(\mathrm{G}_{i})\cong\operatorname{NS}(V_{i})$ induce an
isomorphism of lattices

$\Psi^{\perp}_{\mathrm{G}}/\Psi\cong H^{2}(V_{1};\mathbb{Z})\oplus
H^{2}(V_{2};\mathbb{Z}).$

If $X_{0}$ is the central fibre of a Tyurin degeneration
$\pi\colon\mathcal{X}\to\Delta$, then the isomorphism above induces an
isomorphism between the exact sequence

$0\longrightarrow\mathbb{Z}\zeta\longrightarrow\Psi^{\perp}_{\mathrm{K}}/\Psi\longrightarrow\operatorname{NS}(\mathrm{M})\longrightarrow
0$

from Proposition 2.28 and the the exact sequence

$0\longrightarrow\mathbb{Z}\xi\longrightarrow\xi^{\perp}\longrightarrow
I^{\perp}/I\longrightarrow 0$

from Proposition 3.5. From Lemma 3.4, it follows that
$\operatorname{NS}(\mathrm{M})\cong H\oplus E_{8}\oplus E_{8}$.

###  3.5. Lattice polarisations

We now consider how the constructions we have discussed so far work in the
presence of a lattice polarisation. We will make use of the following standard
definition.

######  Definition 3.18.

Let $L$ be a non-degenerate lattice. A class $\delta\in L$ with
$\langle\delta,\delta\rangle_{L}=-2$ is called a _root_. A subset
$\Delta(L)^{+}$ of _positive roots_ is a set of roots with the following
properties:

  * •

for every root $\delta\in L$, we either have $\delta\in\Delta(L)^{+}$ or
$-\delta\in\Delta(L)^{+}$, but not both;

  * •

if $\delta$ is a root that can be written as a non-negative integer
combination of classes from $\Delta(L)^{+}$, then $\delta\in\Delta(L)^{+}$.

Now suppose that $L$ is a non-degenerate primitive sublattice of the K3
lattice $\Lambda_{\text{K3}}\cong H\oplus H\oplus H\oplus E_{8}\oplus E_{8}$
of signature $(1,\operatorname{rank}(L)-1)$. We next define the notion of a
lattice polarisation on a K3 surface, following Alexeev’s and Engel’s [AE23]
modification of Dolgachev’s [Dol96] original definition.

Begin by fixing a vector $h\in L\otimes\mathbb{R}$, such that $\langle
h,h\rangle_{L}>0$ and $h\notin L^{\prime}\otimes\mathbb{R}$ for any proper
sublattice $L^{\prime}\subsetneq L$; Alexeev and Engel call such vectors _very
irrational_.

######  Definition 3.19.

[AE23, Definition 2.6] An _$L$ -quasipolarised K3 surface_ is a K3 surface $X$
along with a primitive lattice embedding $j\colon
L\hookrightarrow\operatorname{NS}(X)$ for which $j(h)$ is nef and big.

Let $\Delta(L)^{+}$ denote the set of classes

$\Delta(L)^{+}:=\\{\delta\in L\mid\langle\delta,\delta\rangle_{L}=-2,\
\langle\delta,h\rangle_{L}>0\\}.$

The fact that $h$ is very irrational implies that $\Delta(L)^{+}$ is a set of
positive roots for $L$. Moreover, it follows from [Dol96, Remark 1.1] that if
$X$ is an $L$-quasipolarized K3 surface, then
$j(\Delta(L)^{+})\subset\operatorname{NS}(X)$ is precisely the set of classes
of effective $(-2)$-curves contained in $j(L)$.

The following definition is based on [DHNT15, Definition 2.1], appropriately
modified for compatibility with Definition 3.19. Compare also Alexeev’s and
Engel’s definition of a _nef model_ [AE23, Definition 3.10] in the presence of
a lattice polarisation.

######  Definition 3.20.

A _$L$ -polarised Tyurin degeneration of K3 surfaces_ is a Tyurin degeneration
$\pi\colon\mathcal{X}\to\Delta$ such that

  1. (1)

(monodromy invariance) over the punctured disc $\Delta^{*}$ there is a trivial
local subsystem $\mathcal{L}$ of $R^{2}\pi_{*}\mathbb{Z}$ with fibre $L$, so
that for each $p\in\Delta^{*}$ the fibre $\mathcal{L}_{p}\subset
H^{2}(X_{p};\mathbb{Z})$ defines an $L$-quasipolarisation of $X_{p}$,

  2. (2)

(positivity) there exists an effective, flat, and nef Cartier divisor $A$ on
$\mathcal{X}$ such that the restriction $A_{p}$ of $A$ to any fibre $X_{p}$
over $p\in\Delta^{*}$ is nef and big.

######  Remark 3.21.

We note that, given a Tyurin degeneration $\pi\colon\mathcal{X}\to\Delta$
along with a Cartier divisor $A$ on $\mathcal{X}$ such that the restriction
$A_{p}$ of $A$ to any fibre $X_{p}$ over $p\in\Delta^{*}$ is nef and big, by
[SB83, Theorem 1] one may twist $A$ by a Cartier divisor $Z$ supported on
components of $X_{0}$ and perform a series of flops to arrange that $A$ is
effective, flat, and nef on $\mathcal{X}$.

The following lemma relates Definition 3.20 to the description in terms of
pseudolattices.

######  Lemma 3.22.

Let $\pi\colon\mathcal{X}\to\Delta$ be an $L$-polarised Tyurin degeneration of
K3 surfaces and let

$0\subset I\subset I^{\perp}\subset H^{2}(X;\mathbb{Z})$

denote the associated limiting mixed Hodge structure. Then $L$ is orthogonal
to $I$ and there is an embedding $L\hookrightarrow I^{\perp}/I$.

###### Proof.

From the definition of the limiting mixed Hodge structure, we have
$I^{\perp}=\ker(N)\subset H^{2}(X;\mathbb{Z})$, where $N$ is the logarithm of
the monodromy. As $L\subset H^{2}(X;\mathbb{Z})$ is primitive and monodromy
invariant, we have that $L$ is a primitive sublattice of $\ker(N)=I^{\perp}$;
in particular we see that $L$ is orthogonal to $I$. Moreover, $I\subset
I^{\perp}$ is a totally degenerate sublattice and $L$ is nondegenerate, so
$L\cap I=0$. It follows that $L$ embeds into $I^{\perp}/I$. ∎

######  Definition 3.23.

A Tyurin degeneration of K3 surfaces is called _primitively $L$-polarised_ if
it is $L$-polarised and the embedding $L\hookrightarrow I^{\perp}/I$ from
Lemma 3.22 is primitive.

######  Remark 3.24.

It is not the case that all $L$-polarised Tyurin degenerations of K3 surfaces
are primitively $L$-polarised. Indeed, from Shimada’s list of elliptically
fibred K3 surfaces [Shi00] there exists an elliptically fibred K3 surface $X$
with Néron-Severi lattice $H\oplus D_{16}$ which embeds primitively into
$H^{2}(X;\mathbb{Z})$ (see [Shi05] for the full list, where this case is
number 1903 with trivial Mordell-Weil group). However, $H\oplus D_{16}$ does
not embed primitively into $I^{\perp}/I\cong H\oplus E_{8}\oplus E_{8}$;
instead it is a sublattice of index $2$. Some conditions that are sufficient
to ensure primitivity are explored in Section 3.6.

Following the discussion of the previous section, composing the embedding
$L\hookrightarrow\operatorname{NS}(\mathrm{M})$ from Lemma 3.22 with the
isomorphism $\operatorname{NS}(\mathrm{M})\cong I^{\perp}/I$ gives an
embedding $L\hookrightarrow\operatorname{NS}(\mathrm{M})$. If the Tyurin
degeneration is primitively $L$-polarised, then we obtain a lattice
polarisation on $\operatorname{NS}(\mathrm{M})$ in the sense of Definition
2.30(1).

We next define the notion of a lattice polarisation on a stable K3 surface of
type II $X_{0}=V_{1}\cup_{C}V_{2}$; this definition generalises the notion of
polarisation by a divisor from [Fri84, Definition 3.11]. Recall from Section
3.1 that we have an isomorphism

$\operatorname{Gr}^{W}_{2}H^{2}(X_{0})\cong\xi^{\perp}=\\{(D_{1},D_{2})\in
H^{2}(V_{1})\oplus H^{2}(V_{2})\mid q(K_{V_{1}},D_{1})=q(K_{V_{2}},D_{2})\\},$

where $\xi$ denotes the isotropic class $(-K_{V_{1}},K_{V_{2}})=(C,-C)$.

######  Definition 3.25.

An _$\hat{L}$ -polarisation_ on a stable K3 surface of Type II is a primitive
sublattice $\hat{L}\subset\xi^{\perp}\subset H^{2}(V_{1};\mathbb{Z})\oplus
H^{2}(V_{2};\mathbb{Z})$ satisfying the following conditions:

  1. (1)

$\xi\in\hat{L}$;

  2. (2)

$\hat{L}$ contains the class of a nef Cartier divisor $A$ on $X_{0}$ with
strictly positive self-intersection;

  3. (3)

For every $(-2)$-class $\delta\in\hat{L}$, either $\delta$ or $-\delta$ is
congruent modulo $\xi$ to the class of an effective Cartier divisor of self-
intersection $(-2)$.

Following the discussion of the previous section, an $\hat{L}$-polarisation on
a stable K3 surface of Type II defines a primitive embedding
$\hat{L}\hookrightarrow\xi^{\perp}\cong\Psi^{\perp}_{\mathrm{K}}/\Psi$ and the
condition that $\xi\in\hat{L}$ corresponds precisely to the condition that the
image of this embedding contains the class $\zeta$. We therefore obtain a
lifted polarisation on $\Psi^{\perp}_{\mathrm{K}}/\Psi$ in the sense of
Definition 2.30(2).

We can relate this to lattice polarised Tyurin degenerations as follows. Let
$\pi\colon\mathcal{X}\to\Delta$ be a primitively $L$-polarised Tyurin
degeneration of K3 surfaces. Then $L$ is a primitive sublattice of
$I^{\perp}/I\cong\operatorname{NS}(\mathrm{M})$. Let $\hat{L}$ denote the
preimage of $L$ under the map from
$\xi^{\perp}\cong\Psi_{\mathrm{K}}^{\perp}/\Psi$. Then $\hat{L}$ is the lifted
polarisation determined by $L$, in the sense of Lemma 2.31, and we have the
following result.

######  Theorem 3.26.

With notation as above, if $\pi\colon\mathcal{X}\to\Delta$ is a primitively
$L$-polarised Tyurin degeneration of K3 surfaces, then the central fibre
$X_{0}$ of $\pi\colon\mathcal{X}\to\Delta$ is an $\hat{L}$-polarised stable K3
surface of Type II.

###### Proof.

The discussion above shows that there is a primitive embedding of $\hat{L}$
into $\xi^{\perp}$. The image of this embedding contains $\xi$, as $\xi$
generates the kernel of the map $\xi^{\perp}\to I^{\perp}/I$. Denote this map
by $\varphi$.

Let $A$ denote the Cartier divisor from Definition 3.20(2). Then the
restriction of $A$ to the central fibre $X_{0}$ is a nef Cartier divisor with
positive self-intersection that lies in $\hat{L}$.

Finally, Definition 3.20(1) implies that effective $(-2)$-curves in
$\Delta(L)^{+}\subset L$ are monodromy invariant, so we may take flat limits
to obtain an embedding of $\Delta(L)^{+}$ into $\hat{L}\subset\xi^{\perp}$
which commutes with $\varphi$. The image of this embedding consists of classes
of effective Cartier divisors of self-intersection $(-2)$ in $\hat{L}$. If
$\delta$ is any $(-2)$-class in $\hat{L}$, then either $\varphi(\delta)$ or
$-\varphi(\delta)$ lies in $\Delta(L)^{+}\subset L$ (as $\Delta(L)^{+}$ is a
system of positive roots in $L$) so, by the argument above, either $\delta$ or
$-\delta$ is congruent modulo $\xi$ to the class of an effective Cartier
divisor of self-intersection $(-2)$. ∎

We conclude this section by showing that the definitions of lattice
polarisation above are compatible with the definition of a lattice
polarisation on a weak del Pezzo surface from [DT18].

######  Definition 3.27.

[DT18, Definition 4.4] Let $V$ be a weak del Pezzo surface and let $C\subset
V$ be a smooth anticanonical divisor. Let $N$ be a negative definite lattice.
An _$N$ -polarisation_ of $(V,C)$ is a primitive embedding $j\colon
N\hookrightarrow\operatorname{NS}(V)$ such that

  * •

$j(v).C=0$ for all $v\in N$, and

  * •

there exists a set of positive roots $\Delta(N)^{+}\subset N$ such that
$j(\Delta(N)^{+})$ is contained in the effective cone in
$\operatorname{NS}(V)$.

The following result shows that this is compatible with our definitions.

######  Theorem 3.28.

Let $X_{0}=V_{1}\cup_{C}V_{2}$ be an $\hat{L}$-polarised stable K3 surface of
Type II and suppose that $V_{1}$ is weak del Pezzo. Then the intersection
polarisation $L_{1}:=\hat{L}\cap\operatorname{NS}(V_{1})$ defines an
$L_{1}$-polarisation of $(V_{1},C)$, in the sense of Definition 3.27.

###### Proof.

The condition that $j(v).C=0$ for all $v\in L_{1}$ follows from Lemma 2.32,
and negative definiteness follows from the fact that
$C^{\perp}=K_{V_{i}}^{\perp}\subset\operatorname{NS}(V_{1})$ is negative
definite for any weak del Pezzo $V_{1}$. To show the second condition from
Definition 3.27, it suffices to show that if $\delta\in L_{1}$ is a class of
self-intersection $(-2)$, then either $\delta$ or $(-\delta)$ is effective.

So let $\delta\in L_{1}$ be a class of self-intersection $(-2)$. Write
$\delta=(v,0)\in\operatorname{NS}(V_{1})\oplus\operatorname{NS}(V_{2})$. After
possibly exchanging $\delta$ for $(-\delta)$, by Definition 3.25, there exists
$k\in\mathbb{Z}$ such that $\delta-k\xi=(v-kC,kC)$ is effective. As $C$ is an
effective curve in $\operatorname{NS}(V_{2})$, this can only occur if $k\geq
0$. But then $v=(v-kC)+kC\in\operatorname{NS}(V_{1})$ is a sum of effective
classes, so is effective. ∎

######  Remark 3.29.

The condition that $V_{1}$ is weak del Pezzo holds if and only if
$\deg(\mathrm{G})=q(K_{V_{1}},K_{V_{1}})>0$. As our labelling convention
ensures that $q(K_{V_{1}},K_{V_{1}})\geq 0$ and we have
$q(K_{V_{1}},K_{V_{1}})=-q(K_{V_{2}},K_{V_{2}})$ by Remark 3.2, $V_{2}$ can
never be weak del Pezzo, as the canonical divisor on any weak del Pezzo
surface always has positive self-intersection.

Putting the results of this section together we obtain the following. Let
$\pi\colon\mathcal{X}\to\Delta$ be a primitively $L$-polarised Tyurin
degeneration of K3 surfaces. Then $L$ is a primitive sublattice of
$I^{\perp}/I\cong\operatorname{NS}(\mathrm{M})$. If
$\hat{L}\subset\xi^{\perp}\cong\Psi_{\mathrm{K}}^{\perp}/\Psi$ is the lifted
polarisation induced by $L$, in the sense of Definition 2.30(2), then by
Theorem 3.26 the central fibre $X_{0}=V_{1}\cup_{C}V_{2}$ of