# Degeneration of Hodge structures on I-surfaces

Stephen Coughlan  Stephen Coughlan
Mathematisches Institut
Lehrstuhl Mathematik VIII
Universitätsstraße 30
95447 Bayreuth
Germany<EMAIL_ADDRESS>,  Marco Franciosi  Marco Franciosi
Dipartimento di Matematica
Università di Pisa
Largo B. Pontecorvo 5
I-56127 Pisa
Italy<EMAIL_ADDRESS>,  Rita Pardini  Rita Pardini
Dipartimento di Matematica
Università di Pisa
Largo B. Pontecorvo 5
I-56127 Pisa
Italy<EMAIL_ADDRESS>and  Sönke Rollenske  Sönke Rollenske
FB 12/Mathematik und Informatik
Philipps-Universität Marburg
Hans-Meerwein-Str. 6
35032 Marburg
Germany<EMAIL_ADDRESS>
###### Abstract.

Work of Green, Griffiths, Laza, and Robles suggests that the moduli space of
(smoothable) stable surfaces should admit a natural stratification defined via
Hodge theoretic data. In the case of stable surfaces with $K_{X}^{2}=1$ and
$\chi(X)=3$ we compute the Hodge type of all examples known to us and show
that all predicted degenerations are geometrically realised.

###### Contents

  1. 1 Introduction
  2. 2 Stable surfaces and Kollár’s glueing
    1. 2.A I-surfaces
  3. 3 Stratification of $\bar{\mathfrak{M}}$ from Hodge structures
    1. 3.A The mixed Hodge structure on $H^{2}(X)$
    2. 3.B The Mayer–Vietoris exact sequence
  4. 4 Examples of degenerations in I-surfaces
    1. 4.A Double covers of the cone
      1. 4.A.1 Normal double covers of the cone
      2. 4.A.2 Non-normal double covers of the cone
    2. 4.B Two K3 surfaces
    3. 4.C Double covers of a different cone
      1. 4.C.1 A K3 surface and a rational surface

##  1\. Introduction

The study of surfaces of general type and their moduli spaces constructed by
Gieseker is a classical topic of investigation. In recent years, additional
interest has arisen from the construction of a modular compactification, the
moduli space of stable surfaces.

Already the Gieseker moduli space ${\mathfrak{M}}_{K^{2},\chi}$ can be
arbitrarily singular and, in general, the compactification
$\overline{{\mathfrak{M}}}_{K^{2},\chi}$ also does not have good geometric
properties. However, in their investigation of compactifications of period
domains, Green, Griffiths, Laza, and Robles suggested the existence of a
stratification of the closure of the Giesecker moduli space determined by
Hodge-theoretic data. In its coarsest form, the Hodge type reflects the
weights with respect to the mixed Hodge structure of the cohomology classes
contributing to $p_{g}$. Several new features (and difficulties) occur for
$p_{g}\geq 2$. We will explain some of this in Section 3, but want to mention
here that the final aim would be to use even more information from the Hodge
structure to study the geometry of the moduli space, compare [GGR21].

One of the easiest test cases for these ideas are minimal surfaces with
$K^{2}_{X}=1$ and $\chi(X)=3$, baptised I-surfaces by Green et al. These had
been studied by the three last named authors in [FPR17] as part of their
investigation of Gorenstein stable surfaces with $K_{X}^{2}=1$. The
interaction of the Hodge theoretic and geometric approaches, also manifest at
the 2021 Meeting “Moduli and Hodge Theory, IMSA Miami” has kept our interest
in this class of surfaces alive and led to the papers [FPRR22] and [CFP+21].

In the present paper, our purpose is modest: we want to exhibit a zoo of
examples of I-surfaces, mostly non-normal, and show how relatively easy
geometric constructions realise all degenerations of the Hodge type predicted
by general theory (cf. [Rob17]). As our understanding of the moduli space of
I-surfaces is still limited, there is no way we can claim to give a complete
discussion in any sense.

### Acknowledgements

We thank Mark Green, Phillip Griffiths, Radu Laza and Colleen Robles for
showing us the role of Hodge theory in analysing the boundary components of
the moduli space of stable surfaces, and, above all, for interesting and
stimulating discussions on these topics.

M.F. and R.P. are partially supported by the project PRIN 2017SSNZAW$\\_$004
“Moduli Theory and Birational Classification” of Italian MIUR and members of
GNSAGA of INDAM.

##  2\. Stable surfaces and Kollár’s glueing

The (semi-log)-canonical reference for the basic definitions and properties of
stable surfaces is [Kol13].

We simply recall that a stable surface $X$ has ample ${\mathbb{Q}}$-Cartier
canonical divisor $K_{X}$ and semi-log-canonical singularities.

If $X$ is a stable normal surface and $(X,x)$ is an isolated surface
singularity, then $(X,x)$ is either a rational or elliptic singularity. In the
second case, considering the minimal resolution
$\sigma\colon(\bar{Y},E)\to(X,x)$, $x$ is said to be simple elliptic if $E$ is
smooth of genus $1$, and is said to be a cusp if $p_{a}(E)=1$ and $E$ is a
cycle of rational curves. The degree of the singularity is $-E^{2}$.

If $X$ is a non-normal stable surface we consider its normalisation
$\pi\colon\bar{X}\to X$ and we recall that the non-normal locus $D\subset X$
and its preimage $\bar{D}\subset\bar{X}$ are pure of codimension $1$, i. e.,
they are curves. Since $X$ has ordinary double points at the generic points of
$D$, the map on normalisations $\bar{D}^{\nu}\to D^{\nu}$ is the quotient by
an involution $\tau$. Kollár’s glueing principle says that $X$ can be uniquely
reconstructed from $(\bar{X},\bar{D},\tau\colon\bar{D}^{\nu}\to\bar{D}^{\nu})$
via the following two push-out squares:

(2.1)
${\bar{X}}$${\bar{D}}$${\bar{D}^{\nu}}$${X}$${D}$${D^{\nu}}$$\scriptstyle{\pi}$$\scriptstyle{\bar{\iota}}$$\scriptstyle{\pi}$$\scriptstyle{\bar{\nu}}$$\scriptstyle{/\tau}$$\scriptstyle{\iota}$$\scriptstyle{\nu}$

More precisely, we have the following.

######  Theorem 2.2 ([Kol13, Thm. 5.13]) —

Associating to a stable surface $X$ the triple
$(\bar{X},\bar{D},\tau\colon\bar{D}^{\nu}\to\bar{D}^{\nu})$ induces a one-to-
one correspondence

$\left\\{\text{\begin{minipage}{52.03227pt} \begin{center} stable surfaces
\end{center}
\end{minipage}}\right\\}\leftrightarrow\left\\{(\bar{X},\bar{D},\tau)\left|\,\text{\begin{minipage}{160.43727pt}
$(\bar{X},\bar{D})$ log-canonical pair with $K_{\bar{X}}+\bar{D}$ ample, \\\
$\tau\colon\bar{D}^{\nu}\to\bar{D}^{\nu}$ involution s.th.\
$\mathrm{Diff}_{\bar{D}^{\nu}}(0)$ is $\tau$-invariant.
\end{minipage}}\right.\right\\}.$

where $\mathrm{Diff}_{\bar{D}^{\nu}}(0)$ is the different (see [Kol13, 5.11]
for the definition).

###  2.A. I-surfaces

An _I-surface_ $X$ is a stable surface with $K_{X}^{2}=1$ and $\chi(X)=3$. If
$X$ is smooth this is equivalent to saying that $X$ is a minimal surface of
general type with $K_{X}^{2}=1$, $p_{g}(X)=2$ and $q(X)=0$.

By Theorem 2.2 a triple $(\bar{X},\bar{D},\tau)$ corresponds to an I-surface
if and only if the following four conditions are satisfied:

Stable pair condition:

$(\bar{X},\bar{D})$ is an lc pair such that $K_{\bar{X}}+\bar{D}$ is an ample
${\mathbb{Q}}$-Cartier divisor.

$K_{X}^{2}$-condition:

$(K_{\bar{X}}+\bar{D})^{2}=1$.

Glueing condition:

$\tau\colon\bar{D}^{\nu}\to\bar{D}^{\nu}$ is an involution such that
$\mathrm{Diff}_{\bar{D}^{\nu}}(0)$ is $\tau$-invariant.

$\chi$-condition:

The holomorphic Euler-characteristic of the non-normal locus $D$ is
$\chi(D)=3-\chi(\bar{X})+\chi(\bar{D})$.

In [FPR17] it was shown that the classical description of smooth surfaces of
general type with $K_{X}^{2}=1$ and $\chi(X)=3$ extends to the Gorenstein
case, i.e.:

  * •

a Gorenstein I-surface $X$ is canonically embedded as a hypersurface of degree
$10$ in (the smooth locus of) ${\mathbb{P}}(1,1,2,5)$;

  * •

the moduli space $\overline{\mathfrak{M}}_{1,3}^{(Gor)}$ of Gorenstein stable
surfaces with $K^{2}=1$ and $\chi=3$ is irreducible and rational of dimension
$28$.

  * •

for a Gorenstein I-surface $X$, the bicanonical map is a degree 2 morphism
$\varphi_{2}\colon X\to{\mathcal{Q}}_{2}\subset{\mathbb{P}}^{3}$, where
${\mathcal{Q}}_{2}$ is the quadric cone, branched on the vertex $o$ and on a
quintic section $B$ of ${\mathcal{Q}}_{2}$ not containing $o$.

  * •

conversely, if $B$ is a quintic section of ${\mathcal{Q}}_{2}$ such that
$({\mathcal{Q}}_{2},\frac{1}{2}B)$ is a log-canonical pair, then the double
cover of ${\mathcal{Q}}_{2}$ branched on $B$ and $o$ is a stable I-surface; it
is Gorenstein if $B$ does not contain $o$, else it is 2-Gorenstein.

##  3\. Stratification of $\bar{\mathfrak{M}}$ from Hodge structures

###  3.A. The mixed Hodge structure on $H^{2}(X)$

Here we follow roughly the exposition of [Ant20].

The stratification we will define and study later is motivated by work
(partially in progress) of Green, Griffiths, Kerr, Laza and Robles [GGR21,
KPR19, Rob16, Rob17] about degenerations of Hodge structures. Roughly, there
should be a stratification of the moduli space of the surfaces under
investigation, according to the type of polarised mixed Hodge structure on
$H^{2}(X)$. It should be noted that the details about this Hodge-theoretic
stratification are not fully understood yet. Therefore, we can only give an
informal description. We refer to Robles’ exposition [Rob17] and the
references therein for more details; for the basic theory of mixed Hodge
structures see Durfee’s short introduction [Dur83] and the comprehensive
account by Peters and Steenbrink [PS08].

Given a flat degeneration ${\mathfrak{X}}\to\Delta$, where
$\Delta\subset{\mathbb{C}}$ is the unit disc and where all fibres
${\mathfrak{X}}_{s}$, $s\in\Delta^{*}=\Delta-0$ are smooth and projective, we
can can associate a _limiting polarised mixed Hodge structure_ (LMHS for
short) with the family of Hodge structures
$H^{2}({\mathfrak{X}}_{s};{\mathbb{C}})$ (see [GGR21]). Moreover, we can
compare this LMHS with Deligne’s natural mixed Hodge structure on the
cohomology of the special fibre $H^{n}({\mathfrak{X}}_{0};{\mathbb{C}})$ via
the specialisation map. In general, the two mixed Hodge structures are related
by the Clemens–Schmid exact sequence (cf. [Cle77]) and its generalizations
(cf. [KLS21, KL20]).

For our application we consider the case where $X={\mathfrak{X}}_{0}$ is a
stable surface. In this case we consider Deligne’s mixed Hodge structure on
$X$, since $X$ is _cohomologically insignificant_. That is, for every $n$, the
Deligne mixed Hodge structure on $H^{n}({\mathfrak{X}}_{0};{\mathbb{C}})$ and
the limiting mixed Hodge structure on $H^{n}({\mathfrak{X}}_{s};{\mathbb{C}})$
agree on $(p,q)$-components where $pq=0$. This follows since a projective
variety with at worst du Bois singularities is cohomologically insignificant
by a result of Steenbrink [Ste81, Theorem 2], and semi-log-canonical surfaces
are Du Bois (cf. Kollár [Kol13, Corollary 6.32] or Kovács, Schwede, Smith
[KSS10, Theorem 4.16]). See also [KLS21, Thm 9.3 and Cor. 9.9] for a
generalisation of the Clemens–Schmid exact sequence and applications to slc
varieties.

Now we focus on _I-surfaces_ (see §2.A). Assume that $X$ is smoothable. Since
$h^{2,0}(X)=h^{0,2}(X)=p_{g}(X)=2$ for a smooth minimal surface $X$ of general
type satisfying $K_{X}^{2}=1$ and $\chi(X)=3$, our case of interest
corresponds to the Hodge numbers $h=(2,h^{1,1},2)$. That is, the Deligne
splitting $H^{2}(X;{\mathbb{C}})=\bigoplus_{p,q}H^{2}(X;{\mathbb{C}})^{(p,q)}$
of the limiting mixed Hodge structure has a Hodge diamond (indicating $\dim
H^{2}(X;{\mathbb{C}})^{(p,q)}$) of the form

$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{r}$$\scriptstyle{r}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{2-r-s}$$\scriptstyle{2-r-s}$$\scriptstyle{h^{1,1}-r-2s}$

where $r,s\geq 0$, $r+s\leq 2$ and $r+2s\leq h^{1,1}$.

######  Definition 3.1 —

Let $X$ be an I-surface (see §2.A). Then $X$ is said to be of _Hodge type
$\lozenge_{r,s}$_ if $r=\dim H^{2}(X;{\mathbb{C}})^{(0,0)}$ and $s=\dim
H^{2}(X;{\mathbb{C}})^{(1,0)}$, where $H^{2}(X;{\mathbb{C}})^{(p,q)}$ is the
$(p,q)$-component of Deligne’s mixed Hodge structure on
$H^{2}(X;{\mathbb{C}})$.

Among these Hodge types there is the so-called polarised relation, defined by
$\lozenge_{r,s}\leq\lozenge_{t,u}$ if and only if $r\leq t$ and $r+s\leq t+u$.
This relation comes from representation theory and has a geometric meaning
which follows by the analysis of the extension of period domain and the period
map (cf. Robles [Rob17]). Roughly speaking $\lozenge_{r,s}\leq\lozenge_{t,u}$
means that the Hodge type $\lozenge_{t,u}$ is “more degenerate” than
$\lozenge_{r,s}$.

In our case the polarised relations among Hodge types for an I-surface are
illustrated in the Degeneration Diagram in Figure 1 (cf. [Rob17, Example 4.10]
for a detailed explanation).

Smooth varieties have Hodge structures of pure weight, that is, they are of
type $\lozenge_{0,0}$. Our purpose is to show that we can realise in several
different ways all the polarised relations illustrated in Figure 1 by
considering the stratification given by the singularities of I-surfaces. We
will ignore $h^{1,1}$ just as we will ignore canonical surface singularities.

$\lozenge_{0,0}$$\lozenge_{0,1}$$\lozenge_{0,2}$$\lozenge_{1,0}$$\lozenge_{1,1}$$\lozenge_{2,0}$
Figure 1. Degeneration diagram for the Hodge types.

###  3.B. The Mayer–Vietoris exact sequence

Let us provide some information on how to compute Deligne’s mixed Hodge
structure on the second cohomology and more specifically the Hodge type in
practice.

######  Proposition 3.2 —

There are Mayer–Vietoris type exact sequences of mixed Hodge structure in the
following situations:

  1. (i)

Let $X$ be a demi-normal surface, $\pi\colon\bar{X}\to X$ its normalisation,
$D\subset X$ the non-normal locus and $\bar{D}\subset\bar{X}$ its preimage
(see §2). Then the commutative diagram associate to the normalisation map

${\bar{X}}$${\bar{D}}$${X}$${D}$$\scriptstyle{\pi}$$\scriptstyle{\bar{\iota}}$$\scriptstyle{\pi|_{\bar{D}}}$$\scriptstyle{\iota}$

induces an exact sequences of mixed Hodge structures

(3.3) ${H^{1}(X,{\mathbb{C}})}$${H^{1}(\bar{X},{\mathbb{C}})\oplus
H^{1}(D,{\mathbb{C}})}$${H^{1}(\bar{D},{\mathbb{C}})}$${H^{2}(X,{\mathbb{C}})}$${H^{2}(\bar{X},{\mathbb{C}})\oplus
H^{2}(D,{\mathbb{C}})}$${H^{2}(\bar{D},{\mathbb{C}})}$${\dots}$$\scriptstyle{(\pi^{*},\iota^{*})}$$\scriptstyle{\bar{\iota}^{*}-\pi|_{D}^{*}}$$\scriptstyle{(\pi^{*},\iota^{*})}$$\scriptstyle{\bar{\iota}^{*}-\pi|_{D}^{*}}$

  2. (ii)

Let $\bar{X}$ be a normal surface and $\sigma\colon\bar{Y}\to\bar{X}$ a
resolution. Let $S=\\{p_{1},\dots,p_{k}\\}$ be the singular points of $X$ and
$E_{i}$ the exceptional curve over $p_{i}$. Then there is an exact sequence

(3.4)
${H^{1}(\bar{X},{\mathbb{C}})}$${H^{1}(\bar{Y},{\mathbb{C}})}$${\bigoplus_{i}H^{1}(E_{i},{\mathbb{C}})}$${H^{2}(\bar{X},{\mathbb{C}})}$${H^{2}(\bar{Y},{\mathbb{C}})}$${\bigoplus_{i}H^{2}(E_{i},{\mathbb{C}})}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\sigma^{*}}$

  3. (iii)

Let $C$ be a curve with singular points $p_{1},\dots,p_{r}$ and $\nu\colon
C^{\nu}=\bigsqcup_{i}C^{\nu}_{i}\to C$ the normalisation. Then there is an
exact sequence

(3.5)
${0}$${H^{0}(C,{\mathbb{C}})}$${H^{0}(C^{\nu},{\mathbb{C}})\oplus\bigoplus_{i}H^{0}(p_{i},{\mathbb{C}})}$${\bigoplus_{i}H^{0}({\nu}^{-1}(p_{i}),{\mathbb{C}})}$${H^{1}(C,{\mathbb{C}})}$${\bigoplus_{i}H^{1}(C^{\nu}_{i},{\mathbb{C}})}$${0}$$\scriptstyle{\nu^{*}}$

###### Proof.

The Proposition follows from [PS08, Corollary-Definition 5.37]. Concerning
(iii), a detailed exposition in the case where $C$ has only nodes is given in
[Dur83, §4]. $\square$

In particular for a stable surface $X$, if $(X,p)$ is a simple elliptic
singularity with minimal resolution $\sigma\colon(\bar{Y},E)\to(X,p)$, then
$E$ is a smooth elliptic curve, hence $H^{1}(E,{\mathbb{C}})$ carries a pure
Hodge structure of weight $1$. If $p$ is a cusp then $E$ is a cycle of
rational curves, hence $H^{1}(E,{\mathbb{C}})$ carries a mixed Hodge structure
concentrated in weight $0$. The restriction maps in the Mayer–Vietoris exact
sequences described above can be used to compute the mixed Hodge structure on
$X$ (cf. [Ant20, §2]).

######  Remark 3.6 —

We collect some slogans about what determines the mixed Hodge type on the
second cohomology, that follow from the above exact sequences:

  * •

Smooth varieties have Hodge structures of pure weight.

  * •

Rational singularities do not affect the Hodge type of a stable surface. In
particular, a surface $X$ with only rational singularities has Hodge structure
of pure weight.

  * •

Isolated irrational singularities may introduce the HS of the exceptional
curve.

  * •

Non-normal glueings may introduce the HS of the non-normal locus.

  * •

On a curve, the weight filtration on the first cohomology captures the genus
of the normalisation.

######  Remark 3.7 —

We want to emphasise again that the Hodge type $\lozenge_{r,s}$ is a numerical
datum, which forgets a lot of information contained in the mixed Hodge
structure. In particular, both the integral structure as well as the extension
data of the graded pieces should capture much more of the geometry of the
surface, including information about the rational singularities. We refer to
[GGR21, §5.2] for the period mapping and a detailed description of the
extension data for I-surfaces.

##  4\. Examples of degenerations in I-surfaces

###  4.A. Double covers of the cone

Smooth I-surfaces can be realised as double covers of the quadric cone
${\mathcal{Q}}_{2}\subset{\mathbb{P}}^{3}$, branched over on the vertex $o$
and on a smooth quintic section $B$ of ${\mathcal{Q}}_{2}$ not containing $o.$
Therefore we can obtain degenerations of a smooth I-surface by letting $B$
acquire suitable singularities (see §2.A).

Since the quadric cone can be seen as the image of the second Veronese map of
${\mathbb{P}}(1,1,2)$, we identify ${\mathcal{Q}}_{2}$ with
${\mathbb{P}}(1,1,2)$ and we describe I-surfaces that arise as double covers
of the quadric cone in ${\mathbb{P}}^{3}$ as a hypersurfaces $X$ of degree ten
in ${\mathbb{P}}(1,1,2,5)$ given by an equation

$z^{2}-f_{10}(x_{0},x_{1},y)=0.$

Denote by $B=\\{f_{10}=0\\}\subset{\mathbb{P}}(1,1,2)$ the branch locus of the
double cover. Then the following hold:

  * •

$X$ has slc singularities if and only if $({\mathbb{P}}(1,1,2),\frac{1}{2}B)$
is an lc pair.

  * •

$X$ is non-normal if and only if $f$ has a multiple factor.

  * •

$X$ is Gorenstein if and only if the branch curve $B$ does not pass through
the vertex, that is, the singular point $(0:0:1)$ of ${\mathbb{P}}(1,1,2)$.

Let us quickly review what kind of singularities we can allow on the branch
curve $B$ in a smooth point of ${\mathbb{P}}(1,1,2)$ so that $X$ is slc.

If $p\in B$ is an isolated singularity of $B$, then one of the following
occurs:

  * •

$p$ is a negligible singularity: $p$ is a double point or a triple point such
that every point infinitely near to $p$ is at most double for the strict
transform of $B$. The preimage of $P$ in $X$ is a canonical singularity.

  * •

$p$ is a quadruple point such that for every point $q$ infinitely near to $p$
the local intersection number at $q$ of the strict transform of $B$ and the
exceptional curve is at most 2. The preimage of $p$ in $X$ is an elliptic
Gorenstein singularity of degree $2$. The exceptional curve in the minimal
resolution is a smooth elliptic curve if and only if the quadruple point is
ordinary, else it is a cycle of rational curves.

  * •

$p$ is a $[3,3]$-point, namely $p$ is a triple point with an infinitely near
triple point $p_{1}$ such that for every point $q$ infinitely near to $p_{1}$
the local intersection number at $q$ of the strict transform of $B$ and the
exceptional curve is at most 2. The preimage of $p$ in $X$ is an elliptic
Gorenstein singularity of degree $1$. The exceptional curve in the minimal
resolution is a smooth elliptic curve if and only if the infinitely near
triple point is ordinary, else it is a cycle of rational curves.

####  4.A.1. Normal double covers of the cone

Clearly, a smooth I-surface has Hodge type $\lozenge_{0,0}$ and we want to
show quickly that all possible degenerations of the Hodge type already occur
for normal surfaces. A different stratification by number and degree of
elliptic points was given in [FPR17]. But since cusps were not considered in
loc. cit., we cannot cite all desired degenerations directly from there. For
the reader’s convenience, we reproduce the resulting stratification together
with the corresponding Hodge type in Figure 2. Some information on the general
element in each stratum is given in Table 1, where

${\mathfrak{N}}_{d_{1},\dots,d_{k}}=\left\\{X\in\overline{\mathfrak{M}}_{1,3}^{(Gor)}\left|\,\text{\begin{minipage}{195.12767pt}
$X$ is normal and has exactly $k$ elliptic singularites of degree
$d_{1}\leq\dots\leq d_{k}$ \end{minipage}}\right.\right\\}.$

Figure 2. Hodge type for the normal surfaces from [FPR17] ${{\mathfrak{M}}_{1,3}}$${\lozenge_{0,0}}$${{\mathfrak{N}}_{1}}$${{\mathfrak{N}}_{2}}$${\lozenge_{0,1}}$${{\mathfrak{N}}_{1,1}^{R}}$${{\mathfrak{N}}^{E}_{1,1}}$${{\mathfrak{N}}_{1,2}}$${{\mathfrak{N}}_{2,2}}$${\lozenge_{0,2}}$${{\mathfrak{N}}_{1,1,1}}$${{\mathfrak{N}}_{1,1,2}}$ Table 1. Irreducible strata of normal surfaces in $\overline{\mathfrak{M}}_{1,3}^{(Gor)}$ stratum | dimension | minimal resolution $\widetilde{X}$ | $\kappa(\widetilde{X})$
---|---|---|---
${\mathfrak{N}}_{\varnothing}={\mathfrak{M}}_{1,3}$ | $28$ | general type | $2$
${\mathfrak{N}}_{2}$ | $20$ | blow up of a K3-surface | $0$
${\mathfrak{N}}_{1}$ | $19$ | minimal elliptic surface with $\chi(\widetilde{X})=2$ | $1$
${\mathfrak{N}}_{2,2}$ | $12$ | rational surface | $-\infty$
${\mathfrak{N}}_{1,2}$ | $11$ | rational surface | $-\infty$
${\mathfrak{N}}_{1,1}^{R}$ | $10$ | rational surface | $-\infty$
${\mathfrak{N}}_{1,1}^{E}$ | $10$ | blow up of an Enriques surface | $0$
${\mathfrak{N}}_{1,1,2}$ | $2$ | ruled surface with $\chi(\widetilde{X})=0$ | $-\infty$
${\mathfrak{N}}_{1,1,1}$ | $1$ | ruled surface with $\chi(\widetilde{X})=0$ | $-\infty$

Now we explain one possibility to realise everything in Figure 1: fix two
general points $p_{1},p_{2}$ on quadric the cone and take three general planes
$H_{3},H_{4},H_{5}$ containing both points. Now consider the family of all
pairs of planes $H_{1},H_{2}$ such that if one takes $B=\sum_{i}H_{i}$ as
branch divisor then the resulting surface is a normal stable surface $X$.

Then the following happens:

  1. (i)

if $H_{1},H_{2}$ are general, then $X$ has canonical singularities;

  2. (ii)

if $H_{1}$ is general and $H_{2}$ passes through $p_{2}$ in a general way,
then $p_{2}$ is an ordinary quadruple point for $B$ and $X$ has one simple
elliptic singularity of degree $2$;

  3. (iii)

if $H_{1}$ is general and $H_{2}$ is tangent to $H_{3}$ in $p_{2}$, then
$p_{2}$ is a degenerate quadruple point and $X$ has one cusp singularity of
degree $2$;

  4. (iv)

if $H_{i}$ passes through $p_{i}$ in a general direction, then then $p_{1}$
and $p_{2}$ are ordinary quadruple points and $X$ has two elliptic
singularities of degree $2$;

  5. (v)

if $H_{1}$ passes through $p_{1}$ in a general direction and $H_{2}$ is
tangent to $H_{3}$ in $p_{2}$, then $X$ has one elliptic and one cusp
singularity of degree $2$;

  6. (vi)

if $H_{i}$ is tangent to $H_{3}$ at $p_{i}$, then $X$ has two cusp
singularities of degree $2$.

According to general principles ((3.4)), this should realise all degenerations
in Figure 1, if we check that the exceptional curves in the resolution give
rise to independent classes in $H^{2}(X)$. This can be easily checked using
the classification of cases given in [FPR17], that we have summarized in Table
1.

In case (ii) the minimal resolution $\bar{Y}$ of $X$ is the blow-up of a K3
surface. Denoting by $E_{2}$ the elliptic curve coming from the resolution of
the singular point over $P_{2}$, equation (3.4) reads as follows

${0=H^{1}(\bar{Y},{\mathbb{C}})}$${H^{1}(E_{2},{\mathbb{C}}){}}$${H^{2}(X,{\mathbb{C}})}$${H^{2}(\bar{Y},{\mathbb{C}})}$$\scriptstyle{\sigma^{*}}$

In this case $H^{1}(E_{2},{\mathbb{C}})$ carries a pure Hodge structure of
weight 1 with $h^{1,0}(E_{2})=1.$ Therefore the mixed Hodge structure of $X$
is of type $\lozenge_{0,1}$, that is, we have realised the degeneration of
Hodge types $\lozenge_{0,0}\to\lozenge_{0,1}.$

If we are in case (iv) then our surface $X$ has two simple elliptic
singularities and the minimal resolution $\bar{Y}$ of $X$ is a rational
surface. Arguing as above, equation (3.4) shows that the classes of the
exceptional divisors $E_{1}$ and $E_{2}$ give independent classes in
$H^{2}(X)$, hence we obtain a further degeneration of Hodge types
$\lozenge_{0,1}\to\lozenge_{0,2}$.

If we are in case (iii) then $X$ has one cusp singularity of degree $2$ over
$P_{2}$ and its minimal resolution is again the blow-up of a K3 surface. This
is in fact a degeneration of case (ii). Denoting by
$E_{2}=\sum_{i=1}^{r}E_{2,i}$ the cycle of rational curve coming from the
resolution of the singular point over $p_{2}$, and by
$\\{q_{1},\ldots,q_{r}\\}$ the set of singular points of $E_{2}$, equation
(3.4) and equation (3.5) show that we get a mixed Hodge structure of type
$\lozenge_{1,0}$. Indeed we get

${0=H^{1}(\bar{Y},{\mathbb{C}})}$${H^{1}(E_{2},{\mathbb{C}}){}}$${H^{2}(X,{\mathbb{C}})}$${H^{2}(\bar{Y},{\mathbb{C}})}$${0}$$\scriptstyle{\sigma^{*}}$

To study $H^{1}(E_{2},{\mathbb{C}})$ we consider the exact sequence (3.5)

${0}$${{\mathbb{C}}=H^{0}(E_{2},{\mathbb{C}})}$${\bigoplus_{i}H^{0}(E_{2,i},{\mathbb{C}})\oplus\bigoplus_{i}H^{0}(q_{i},{\mathbb{C}})}$${\bigoplus_{i}H^{0}({\nu}^{-1}(q_{i}),{\mathbb{C}})}$${H^{1}(E_{2},{\mathbb{C}})}$${\bigoplus_{i}H^{1}(E_{2,i},{\mathbb{C}})}$$\scriptstyle{\nu^{*}}$

Now it is
$\bigoplus_{i}H^{0}(E_{2,i},{\mathbb{C}})\cong{\mathbb{C}}^{r}\cong\bigoplus_{i}H^{0}(q_{i},{\mathbb{C}})$,
$\bigoplus_{i}H^{0}({\nu}^{-1}(q_{i}),{\mathbb{C}})\cong{\mathbb{C}}^{2r}$ and
$\bigoplus_{i}H^{1}(E_{2,i},{\mathbb{C}})=0$. Moreover
$(H^{1}(\bar{Y}))^{0,0}=(H^{2}(\bar{Y}))^{0,0}=0$, hence $(H^{2}(X))^{0,0}$ is
isomorphic to the $(0,0)$ components of $H^{1}(E_{2})$. Summing up, we have
obtained the degeneration $\lozenge_{0,0}\to\lozenge_{0,1}\to\lozenge_{1,0}$.

In case (v) and case (vi) we repeat the above arguments for the two singular
points, obtaining the degeneration
$\lozenge_{0,0}\to\lozenge_{0,1}\to\lozenge_{1,0}\to\lozenge_{1,1}$ and
$\lozenge_{0,0}\to\lozenge_{0,1}\to\lozenge_{1,0}\to\lozenge_{1,1}\to\lozenge_{2,0}$,
respectively.

######  Remark 4.1 —

In Table 1 we find degenerations $X$ with three elliptic points and minimal
resolution a (possibly non-minimal) surface $\widetilde{X}$ ruled over an
elliptic curve. Therefore, the only possible exceptional elliptic cycles are
smooth elliptic cuves embedded as multi-sections in $\widetilde{X}$. Thus the
relevant part of (3.4) becomes

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because each multisection carries, via its map to the base curve of the ruled
surface the first rational cohomology. Thus the Hodge type turns out to be
$\lozenge_{0,2}$. Note that none of the elliptic points can degenerate to a
cusp because of the geometry of the minimal resolution. If we want to
degenerate the Hodge type further we need to degenerate $X$ to a non-normal
surface.

####  4.A.2. Non-normal double covers of the cone

If we want to consider non-normal double covers of the cone then we have a
non-reduced branch divisor, that is,

$f=gd^{2}\text{ and }B=B_{0}+2D,$

and the normalisation diagram (2.1) becomes

${\bar{X}}$${\bar{D}}$${X}$${D}$${{\mathbb{P}}(1,1,2)}$$\scriptstyle{\pi}$$\scriptstyle{\bar{\iota}}$branched
over $\scriptstyle B_{0}$ and vertexbranched over $\scriptstyle B_{0}\cap D$
and vertex$\scriptstyle{\pi}$$\scriptstyle{\iota}$

If both $B_{0}$ and $D$ are general then the normalisation is smooth and so
are the conductor curves. The possibilities are listed in Table 2. The first
two cases yield Gorenstein non-normal I-surfaces and are described in [FPR17].
The other three cases are new and yield 2-Gorenstein I-surfaces, which have
singular normalisation.

The list is obtained by considering the possible degrees of $B_{0}$ and $D$.
The genus of $D$ follows by using the adjunction formula for a complete
intersection curve in $\mathbb{P}^{3}$. $K_{X}$ and $K_{\bar{D}}$ can be
computed via the Riemann-Hurwitz formula looking at the double covers
$\bar{X}\to{\mathbb{P}}(1,1,2)$ branched over $B_{0}$ and the vertex,
respectively, $\bar{D}\to D$ branched over $B_{0}\cap D$ and the vertex.

Note that a smooth curve in ${\mathcal{Q}}_{2}$ passes through the vertex if
its degree is odd. If such a curve is a component of $B$ then $X$ is not
Gorenstein.

The curve $D$ can have at most nodes as singularities. If $p$ is a node of $D$
then $p\not\in B_{0}$ and in the double cover we get a degenerate cusp locally
analytically given by the equation $z^{2}-u^{2}v^{2}=0$. If $p$ is a smooth
point of $D$, then one of the following occurs:

  * •

$p\not\in B_{0}$ and the preimage in $X$ is a normal crossing point.

  * •

$D$ and $B_{0}$ intersect transversely at $p$ and the preimage in $X$ is a
pinch point.

  * •

The local intersection multiplicity $(D,B_{0})_{p}=2$ and $B_{0}$ is either
smooth or has an $A_{n}$ singularity at $p$. The preimage is a degenerate
cusp, usually denoted with $T_{2,q,\infty}$ for $q\geq 3$ and locally
described by the equation $z^{2}-u^{2}(v^{2}-u^{q-2})$.

One could view $T_{2,3,\infty}$ as a non-normal version of the elliptic or
cusp singularity coming from a $[3,3]$ point, while the others are non-normal
versions of the quadruple point.

This can be checked by blowing up and analysing the cases similarly to what is
done in [LR12] or from the point of view of log-canonical threshold in [KSC04,
6.5].

Table 2. Non-normal double covers of ${\mathbb{P}}(1,1,2)$ for general $D$ and $B_{0}$ Hodge-Type | $\deg d$ | $\deg g$ | $\bar{X}$ | $g(\bar{D})$ | $g(D)$
---|---|---|---|---|---
$\lozenge_{0,2}$ | 2 | 6 | del Pezzo of degree 1 | 2 | 0
$\lozenge_{0,2}$ | 4 | 2 | ${\mathbb{P}}^{2}$ | 3 | 1
$\lozenge_{0,1}$ | 1 | 8 | K3 | 1 | 0
$\lozenge_{0,2}$ | 3 | 4 | del Pezzo of degree 4 | 2 | 0
$\lozenge_{0,2}$ | 5 | 0 | two copies of ${\mathbb{P}}(1,1,2)$ | 2 + 2 | 2

Equipped with this knowledge of possible singularities we will use the ideas
from Remark 3.6 to show that in each of the cases in Table 2, an appropriately
chosen degeneration of the branch data will allow for all possible
degenerations of the Hodge type depicted in Figure 1.

$(\deg d,\deg g)=(2,6)$:

In this case (3.3) gives an exact sequence

$0\to H^{1}(\bar{D},{\mathbb{C}})\to H^{2}(X,{\mathbb{C}})\to
H^{2}(\bar{X},{\mathbb{C}})\to 0$

where $\bar{X}$ is a del Pezzo surface of degree one, $D$ a rational curve and
$\bar{D}$ is a genus 2 curve (see [FPR17, Proposition 4.4]); the Hodge type is
determined by the curve only, because the del Pezzo surface $\bar{X}$ has
$p_{g}(\bar{X})=0$.

Thus we can degenerate further by choosing $B_{0}$ and $D$ such that they are
tangent in one or two points. Each tangency will introduce a node in $\bar{D}$
and by (3.5) introduce a one-dimensional piece of weight zero in the Hodge
structure. This gives the degeneration of Hodge types
$\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$.

To obtain the entire string
$\lozenge_{0,0}\to\lozenge_{0,1}\to\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$,
we consider a degeneration of the branch divisor $B$. Take
$B=B_{0}+D_{1}+D_{2}$, with $D_{1}$ and $D_{2}$ linearly equivalent but
distinct. For $D_{1}$ and $D_{2}$ general we have negligible singularities,
i.e. Hodge type $\lozenge_{0,0}$. Letting $D_{1}$ and $D_{2}$ pass through a
point $p_{0}\in B_{0}$, with suitable tangent conditions, we obtain a
$[3,3]$-point, hence a degeneration of type $\lozenge_{0,1}$. Letting
$D_{1}=D_{2}$, we get our surface $X$.

$(\deg d,\deg g)=(4,2)$:

In this case $\bar{X}={\mathbb{P}}^{2}$ and (3.3) gives an exact sequence

$0\to H^{1}(D,{\mathbb{C}})\to H^{1}(\bar{D},{\mathbb{C}})\to
H^{2}(X,{\mathbb{C}})\to H^{2}({\mathbb{P}}^{2},{\mathbb{C}})\to 0.$

where $D$ is an elliptic curve and $\bar{D}$ is a genus 3 curve (see [FPR17,
Proposition 4.4]); as in the previous case we can choose $D$ and $B_{0}$ to be
smooth but tangent in one or two points to achieve the degeneration of Hodge
types $\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$.

Note how in the case of two tangencies the Hodge structure on
$H^{1}(\bar{D},{\mathbb{C}})$ still has a part of weight one, which comes from
the elliptic curve and thus does not show up in the Hodge structure on
$H^{2}(X,{\mathbb{Z}})$.

As above, to obtain the entire string of Hodge type degenerations, we consider
$B=B_{0}+D_{1}+D_{2}$, with $D_{1}$ and $D_{2}$ linearly equivalent passing
through a point $p_{0}\in B_{0}$, so that we obtain a $[3,3]$-point, and then
letting $D_{1}=D_{2}$, we get our surface $X$.

$(\deg d,\deg g)=(1,8)$:

$\bar{X}$ is a singular K3 surface (it has two singularities of type $A_{1}$
over the vertex of the cone), $D$ is rational curve and $\bar{D}$ is a genus
one curve. In this case it is not enough to make $B_{0}$ and $D$ tangent, as
we also need to degenerate the K3 surface to get rid of the $(2,0)$ part of
its Hodge structure.

This can be achieved by first letting $B_{0}$ acquire an ordinary quadruple
point and then a degenerate quadruple point where two local branches are
tangent. Identifying ${\mathbb{P}}(1,1,2)$ with the quadric cone
${\mathcal{Q}}_{2}\subset{\mathbb{P}}^{3}$, we can obtain such degeneration
choosing four suitable plane sections through a point.

This gives the degeneration $\lozenge_{0,1}\to\lozenge_{0,2}\to\lozenge_{1,1}$
if $D$ remains general. If in addition we allow $D$ to be tangent to $B_{0}$
then $\bar{D}$ will be a nodal curve of genus one, giving the missing
degenerations from Figure 1.

$(\deg d,\deg g)=(3,4)$:

In this case $\bar{X}$ is a singular del Pezzo surface of degree four, $D$ is
rational curve and $\bar{D}$ is a genus two curve. Again the Hodge type is
carried by $\bar{D}$ and we can degenerate
$\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$ by letting $D$ and $B_{0}$
acquire one or two tangencies.

As in the previous cases we obtain the entire string of Hodge type
degenerations, considering $B=B_{0}+D_{1}+D_{2}$, with $D_{1}$ and $D_{2}$
linearly equivalent passing through a point $p_{0}\in B_{0}$, and then letting
$D_{1}=D_{2}$.

$(\deg d,\deg g)=(5,0)$:

This is the first case we encounter where $X$ is reducible, with
$\bar{D}=D\sqcup D$ and (3.3) becomes

$0\to H^{1}(D,{\mathbb{C}})\to H^{1}(D,{\mathbb{C}})^{\oplus 2}\to
H^{2}(X,{\mathbb{C}})\to\\\ \to H^{2}({\mathcal{Q}}_{2},{\mathbb{C}})^{\oplus
2}\oplus H^{2}(D,{\mathbb{C}})\to H^{2}(D,{\mathbb{C}})^{\oplus 2}\to 0.$

We see that the Hodge type is purely determined by $D$ and can produce the
degeneration $\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$ by letting $D$
acquire one or two nodes.

###  4.B. Two K3 surfaces

It was observed in [FPR17, Example 4.7] that one can obtain examples of non-
normal I-surfaces as follows:

Consider in ${\mathbb{P}}^{3}$ two planes $P_{1}$ and $P_{2}$ and let
$B=B_{1}+B_{2}$ be a quintic section of $P_{1}\cup P_{2}$. Let $L$ be the
intersection line and $L_{i}\subset P_{i}$ its preimages. Then the double
cover

${\bar{X}=X_{1}\sqcup X_{2}}$${X}$${P_{1}\sqcup P_{2}}$${P_{1}\cup
P_{2}}$$\scriptstyle{\pi}$branched over $\scriptstyle B_{i}+L_{i}$

defines, for a general choice of quintic section a 2-Gorenstein I-surface.

For a general quintic sections the curves $B_{i}$ are smooth and intersect the
lines $L_{i}$ transversely, so $X_{i}$ is a singular K3 surface with five
nodes over $L_{i}\cap B_{i}$. It is at these points that $X$ is not
Gorenstein.

To compute the mixed Hodge structure we use (3.3) again and get

$0=(H^{1}(L_{1},{\mathbb{C}})\oplus H^{1}(L_{2},{\mathbb{C}}))\to
H^{2}(X,{\mathbb{C}})\to H^{2}(X_{1},{\mathbb{C}})\oplus
H^{2}(X_{2},{\mathbb{C}})\oplus H^{2}(L,{\mathbb{C}})$

so that $X$ has Hodge type $\lozenge_{0,0}$ if the branch curve is general.
111One might want to compare it with the fact that if we take two elliptic
curves meeting in a node, then the Hodge structure on the first cohomology is
still pure, despite the curve being singular.

We can get all possible degenerations from Figure 1 by letting $B_{1}$ and
then $B_{2}$ independently acquire first an ordinary and then a degenerate
quadruple point.

Explicitly, pick a point $p_{i}\in P_{i}$ and consider three general planes
through both points. The resulting ordinary triple point does not affect the
Hodge type, since it gives rise to a canonical singularity. Now it is easy to
choose different quadrics, which have the required intersection or tangency at
the $p_{i}$.

###  4.C. Double covers of a different cone

As proved in [FPRR22] an I-surface with a unique T-singularity of type
$\frac{1}{18}(1,5)$ is a complete intersection in ${\mathbb{P}}(1,1,2,3,5)$
given by equations

$x_{1}^{3}-yx_{2}=z^{2}-f_{10}(x_{1},x_{2},y,u)=0$

(where $\deg x_{i}=1,\deg y=2,\deg u=3,\deg z=5$) for a sufficiently general
polynomial $f_{10}$ and we can thus consider $X$ as a double cover over a
rational surface

$X\to
S=\operatorname{\mathrm{Proj}}{\mathbb{C}}[x_{1},x_{2},y,u]/(x_{1}^{3}-x_{2}y)$

branched over $B=\\{f_{10}=0\\}$.

For a general branch divisor $B$ the surface $X$ has a unique singularity,
which is rational, and thus has Hodge type $\lozenge_{0,0}$. We want to
explore some degenerations to non-normal surfaces by degenerating the branch
locus but have to be more careful than in Section 4.A to remain slc.

Thus for simplicity we will only consider the case where the non reduced part
of the branch divisor avoids the point $(0:0:0:1)\in{\mathbb{P}}(1,1,2,3)$,
that is, we consider equations of the form

$f_{10}=g_{4}(x_{1},x_{2},y,u)\cdot(u-h_{3}(x_{1},x_{2},y))^{2}.$

Then the homogenous coordinate ring of the non-reduced part of the branch
divisor $D$ is

${\mathbb{C}}[x_{1},x_{2},y,u]/(x_{1}^{3}-x_{2}y,u-h_{3})\cong{\mathbb{C}}[x_{1},x_{2},y]/(x_{1}^{3}-x_{2}y),$

which describes a smooth rational curve and does not depend on $h_{3}$ at all.

The normalisation $\bar{X}$ is given by adjoining the ratio $w:=z/(u-h_{3})$
of weight $2$ to the coordinate ring of $X$. Thus we may eliminate $z$ using
$w$ and $\bar{X}$ is

$\\{x_{1}^{3}-x_{2}y=w^{2}-g_{4}=0\\}\subset{\mathbb{P}}(1,1,2,3,2).$

This is a rational surface with ample anti-canonical sheaf and for general
$g_{4}$, $\bar{X}$ has two $A_{1}$ singularities occurring at the intersection
points of $\bar{X}$ and ${\mathbb{P}}(2,2)\subset{\mathbb{P}}(1,1,2,3,2)$ and
one $\frac{1}{18}(1,5)$ singularity, occurring at $(0:0:0:1:0)$.

In total, by (3.3) the Hodge type is completely controlled by $\bar{D}$, which
is the induced double cover of $D$ described by using $h_{3}$ to eliminate $u$
from the equations for $\bar{X}$:

$\\{x_{1}^{3}-x_{2}y=w^{2}-g_{4}(x_{1},x_{2},y,h_{3})=0\\}\subset{\mathbb{P}}(1,1,2,2)$

For suitable choices of $g_{4}$ and $h_{3}$ we can arrange for $\bar{D}$ to be
either smooth of genus 2 or to have one or two nodes. These give rise to
degenerate cusps on $X$ thus giving the degenerations
$\lozenge_{0,2}\to\lozenge_{1,1}\to\lozenge_{2,0}$.

####  4.C.1. A K3 surface and a rational surface

We consider now a non-normal degeneration of Example 4.C. Let $X$ be the
complete intersection in ${\mathbb{P}}(1,1,2,3,5)$ given by equations

$x_{2}y=z^{2}-f_{10}(x_{1},x_{2},y,u)=0$

for sufficiently general $f_{10}$. As in section 4.C, $X$ is a double cover of
$S\colon(x_{2}y=0)$ in ${\mathbb{P}}(1,1,2,3)$. This time, $S$ has two
components $S_{1}\colon(y=0)$ and $S_{2}\colon(x_{2}=0)$ which are the
weighted projective planes $S_{1}\cong{\mathbb{P}}(1,1,3)$ and
$S_{2}\cong{\mathbb{P}}(1,2,3)$. They are glued along a common weighted
projective line $L:={\mathbb{P}}(1,3)\cong{\mathbb{P}}^{1}$. The double cover
$X_{1}$ of $S_{1}$ is a singular del Pezzo surface with one $\frac{1}{3}(1,1)$
singularity over the point $(0:0:1)$ in ${\mathbb{P}}(1,1,3)$ (see [RS03] for
many more details). The double cover $X_{2}$ of $S_{2}$ is a K3 surface with
an $A_{2}$ singularity over the point $(0:0:1)$ in ${\mathbb{P}}(1,2,3)$. The
double curve of $X$ is an elliptic curve $D$ which is the preimage of $L$. The
two singular points are identified to give a singularity on $X$ which is
analytically isomorphic to the quotient of
$\\{x_{2}y=0\\}\subset{\mathbb{A}}^{3}$ by the order three action
$(x_{2},y,z)\mapsto(\omega x_{2},\omega^{2}y,\omega^{2}z)$ where $\omega$ is a
primitive third root of unity.

Using (3.3) again, we compute the mixed Hodge structure:

$0\to H^{1}(D,{\mathbb{C}})\to H^{1}(D,{\mathbb{C}})^{\oplus 2}\to
H^{2}(X,{\mathbb{C}})\to H^{2}(X_{1},{\mathbb{C}})\oplus
H^{2}(X_{2},{\mathbb{C}})\oplus H^{2}(D,{\mathbb{C}}),$

so $X$ has Hodge type $\lozenge_{0,1}$.

If we allow $D$ to get a node (i.e. $f_{10}$ to become tangent to $L$) then we
get Hodge type $\lozenge_{1,0}$ and so we have recovered all the types in
Figure 1.

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