# Deformations and moduli of irregular canonical covers with $K^{2}=4p_{g}-8$

Purnaprajna Bangere  Department of Mathematics, University of Kansas,
Lawrence, USA<EMAIL_ADDRESS>,  Francisco Javier Gallego  Departamento de
Álgebra, Geometría y Topología and Instituto de Matemática Interdisciplinar,
Universidad Complutense de Madrid, Spain<EMAIL_ADDRESS>,  Jayan
Mukherjee  The Institute for Computational and Experimental Research in
Mathematics, Providence, USA<EMAIL_ADDRESS>and  Debaditya
Raychaudhury  The Fields Institute for Research in Mathematical Sciences,
Toronto, Canada<EMAIL_ADDRESS>
###### Abstract.

In this article, we study the moduli of irregular surfaces of general type
with at worst canonical singularities satisfying $K_{X}^{2}=4p_{g}(X)-8$, for
any even integer $p_{g}\geq 4$. These surfaces also have unbounded
irregularity $q$. We carry out our study by investigating the deformations of
the canonical morphism $\varphi:X\to\mathbb{P}^{N}$, where $\varphi$ is Galois
of degree 4. These canonical covers are classified in [GP08] into four
distinct families. We show that, when $X$ is general in its family, any
deformation of $\varphi$ has degree greater than or equal to $2$ onto its
image. More interestingly, we prove that, with two exceptions, a general
deformation of $\varphi$ is two–to–one onto its image, whose normalization is
a ruled surface of appropriate irregularity. We also show that with the
exception of one family, the deformations of a general surface $X$ are
unobstructed, and consequently, $X$ belongs to a unique irreducible component
of the Gieseker moduli space, which we prove is uniruled. As a consequence we
show the existence of infinitely many moduli spaces with uniruled components
corresponding to each even $p_{g}\geq 4$, and completely determine the degree
and the image of the canonical morphism of its general element. Remarkably,
this shows the existence of infinitely many moduli components with a proper
”quadruple” subloci where the degree of the canonical morphism jumps up. The
existence of jumping subloci is a contrast with the moduli of surfaces with
$K_{X}^{2}=2p_{g}-4$, (which are double covers of surfaces of minimal degree)
studied by Horikawa. There is a similarity and difference to the moduli of
curves of genus $g\geq 3$, for, like in the case of curves, the degree of the
canonical morphism goes down outside a closed sublocus but, unlike in the case
of curves, it is never birational.

###### Key words and phrases:

Abelian covers, deformations of morphisms, moduli of surfaces of general type.

###### 2020 Mathematics Subject Classification:

14J29, 14J10

Dedicated to N. Mohan Kumar on the occasion of his 70th birthday

##  1\. introduction

Canonical covers of varieties of minimal degree have a ubiquitous presence in
the geometry of algebraic surfaces and higher dimensional varieties. They
appear as extremal cases in a variety of geometric situations. The first,
paradigmatic example of a canonical double cover is the canonical morphism of
a hyperelliptic curve. In the case of surfaces, Horikawa’s celebrated work
(see [Hor76]) shows that minimal surfaces of general type on the Noether line
$K_{X}^{2}=2p_{g}-4$ are all canonical double covers of surfaces of minimal
degree. Horikawa’s results show that the deformations of canonical double
covers of surfaces of minimal degree are again canonical double covers of
surfaces of minimal degree, unlike what happens for canonical double covers in
the case of curves of $g\geq 3$. Therefore, a natural and intriguing question
for an algebraic surface, is:

######  Question 1.1.

If $n>2$, does the degree $n$ of a canonical cover change when we deform the
cover?

Canonical triple covers $X$ of surfaces of minimal degree (these covers
satisfy $K_{X}^{2}=3p_{g}(X)-6$) are very few (their geometric genus
$p_{g}(X)$ is bounded by $5$ and their images are singular surfaces; see
[Hor77], [Hor82] and [Kon91]), and, when $p_{g}(X)\leq 4$, their deformations
are again canonical triple covers of surfaces of minimal degree. In contrast,
the geometry of canonical quadruple covers of minimal degree (these covers
satisfy $K_{X}^{2}=4p_{g}(X)-8$) display a wide range of behaviors. Indeed,
they act like general surfaces of general type from a number of geometric
perspectives (see for example [GP11]). They are the first case among low
degree covers where families of irregularity $q(X)\geq 1$ appear. Moreover, as
Remark 1.2 below indicates, quadruple canonical covers are the only ones,
among canonical covers of smooth surfaces of minimal degree, having both
unbounded geometric genus and irregularity (see Theorem 1.3), with the
possible exception of degree $6$ covers. All this makes canonical quadruple
covers stand out as the most interesting case among canonical covers and are
natural candidates for testing Question 1.1.

######  Remark 1.2.

It follows from a more general result (see [GP04], Theorem 3.2), that there
are no odd degree canonical covers of smooth surfaces of minimal degree other
than $\mathbb{P}^{2}$. This together with [Bea79] implies that, if
$\chi(X)\geq 31$, then the degree of a canonical cover of a smooth surface of
minimal degree could only be $2,4,6$ or $8$ (if $\chi(X)\leq 30$, then $q(X)$
is bounded). Since the irregularity of degree $8$ canonical covers is bounded
above by $3$ when $p_{g}\geq 115$ (see [Xia86]), degree $4$ canonical covers
are the only ones, among covers of smooth surfaces of minimal degree, having
unbounded irregularity (and thus unbounded geometric genus, since
$p_{g}(X)\geq 2q(X)-4$ by [Bea82]) except possibly the degree six canonical
covers. One can show that there are no smooth regular degree $6$ abelian
covers of smooth surfaces of minimal degree (see [BGMR21]). Evidence seems to
indicate that there are no such irregular covers as well.

Thus, in this article we focus on the study of the deformations of irregular
quadruple Galois canonical covers of smooth surfaces of minimal degree. As the
table of Theorem 1.3 shows, Galois covers capture by themselves the complexity
of behavior mentioned above. We completely figure out the behavior of these
under deformations for all but one family. For the one family, we have some
partial results. From this study it follows that the answer to Question 1.1
(see Table 1 and Theorem 3.1) is, in general, positive. Indeed, quadruple
canonical covers of surfaces of minimal degree are the lowest degree covers,
with the possible exception of triple covers of geometric genus $5$, for which
the degree of the cover changes under deformation. Regular quadruple Galois
canonical covers of surfaces of minimal degree also provide positive answers
to Question 1.1, as the authors will show in a forthcoming article. From all
this we derive interesting consequences for the moduli of surfaces of general
type.

Classification of irregular quadruple Galois canonical covers of surface
scrolls. The classification of irregular quadruple Galois canonical covers of
surfaces of minimal degree was done by the first two authors in [GP08]. We
need the technical details of their classification results, for the purpose of
this article, so we will summarize them here. The image of these covers are
smooth rational normal scrolls $Y$. Recall that a smooth rational normal
scroll is a Hirzebruch surface $\mathbb{F}_{e}$ ($e\geq 0$), which is, by
definition, $\mathbb{P}(\mathcal{E})$, where
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-e)$.
Let $p:\mathbb{P}(\mathcal{E})\to\mathbb{P}^{1}$ be the natural projection.
The line bundles on $\mathbb{F}_{e}$ are of the form
$\mathcal{O}_{Y}(aC_{0}+bf)$ where
$\mathcal{O}_{Y}(C_{0})=\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and
$\mathcal{O}_{Y}(f)=p^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$. The line bundle
$\mathcal{O}_{Y}(aC_{0}+bf)$ is very ample if $b\geq ae+1$.

######  Theorem 1.3.

([GP08], Theorem $0.1$) Let $X$ be an irregular canonical surface and let $Y$
be a smooth surface of minimal degree. If the canonical bundle of $X$ is base-
point-free and $\varphi:X\to Y$ is a quadruple Galois canonical cover, then
$Y$ is the Hirzebruch surface $\mathbb{F}_{0}$, embedded by $|C_{0}+mf|$,
$(m\geq 1)$. Let $G$ be the Galois group of $\varphi$.

  * (a)

If $G=\mathbb{Z}_{4}$, then $\varphi$ is the composition of two double covers
$p_{1}:X_{1}\to Y$ branched along a divisor $D_{2}$ and $p_{2}:X\to X_{1}$,
branched along the ramification of $p_{1}$ and $p_{1}^{*}D_{1}$, where $D_{1}$
is a divisor on $Y$ and with trace zero module
$p_{1}^{*}\mathcal{O}_{Y}(-\frac{1}{2}D_{1}-\frac{1}{4}D_{2})$.

  * (b)

If $G=\mathbb{Z}^{\oplus 2}$, then $X$ is the fiber product over $Y$ of two
double covers of $Y$ branched along divisors $D_{1}$ and $D_{2}$, and
$\varphi$ is the natural morphism from the fiber product to $Y$.

More precisely, $\varphi$ has one of the sets of invariants shown in the
following table. Conversely, if $\varphi:X\to Y$ is either

  * (1)

the composition of two double covers $p_{1}:X_{1}\to Y$, branched along a
divisor $D_{2}$, and $p_{2}:X\to X_{1}$, branched along the ramification of
$p_{1}$ and $p_{1}^{*}D_{1}$, and with trace zero module
$p_{1}^{*}\mathcal{O}_{Y}(-\frac{1}{2}D_{1}-\frac{1}{4}D_{2})$, with $D_{1}$
and $D_{2}$ as described in rows $1$ of the table below; or

  * (2)

the fiber product over $Y$ of two double covers $p_{1}:X_{1}\to Y$ and
$p_{2}:X_{2}\to Y$, branched respectively along divisors $D_{2}$ and $D_{1}$,
as described in rows $2$, $3$, and $4$ of the table below,

then $\varphi:X\to Y$ is a Galois canonical cover whose Galois group is
$\mathbb{Z}_{4}$ in case $1$ and $\mathbb{Z}_{2}^{\oplus 2}$ in case $2$.

Type | $p_{g}(X)$ | $Y$ | $G$ | $D_{1}\sim$ | $D_{2}\sim$ | $q(X)$
---|---|---|---|---|---|---
$(1)_{m}$ | $2m+2$ | $\mathbb{F}_{0}$ | $\mathbb{Z}_{4}$ | $(2m+4)f$ | $4C_{0}$ | $1$
$(1^{\prime})_{m}$ | $2m+2$ | $\mathbb{F}_{0}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $2C_{0}+(2m+4)f$ | $4C_{0}$ | $1$
$(2)_{m}$ ($m\geq 2$) | $2m+2$ | $\mathbb{F}_{0}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $(2m+2)f$ | $6C_{0}+2f$ | $m$
$(3)_{m}$ | $2m+2$ | $\mathbb{F}_{0}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $(2m+4)f$ | $6C_{0}$ | $m+3$

General surfaces of type $(1^{\prime})_{m}$, $(2)_{m}$ and $(3)_{m}$ are
smooth (in fact all surfaces of type $(3)_{m}$ are smooth) while surfaces of
type $(1)_{m}$ are necessarily singular, the general ones having only $A_{1}$
singularities. The table shows the existence of families of quadruple Galois
canonical covers with unbounded irregularity.

######  Remark 1.4.

(See also [GP11], Remark 3.4) Let $X$ be as in Theorem 1.3.

  * (1)

If $X$ is of type $(1)_{m}$ or $(1^{\prime})_{m}$, then $X$ contains an
elliptic pencil of genus $m+1$ hyperelliptic curves.

  * (2)

If $X$ is of type $(2)_{m}$, then $X$ contains a genus $m$ pencil of genus 2
curves.

  * (3)

If $X$ is of type $(3)_{m}$, then $X$ contains a genus $m+1$ pencil of genus 2
curves.

In addition, some of the families of Theorem 1.3 are extremal cases for
several inequalities concerning irregular surfaces of general type, such as
$K_{X}^{2}\geq 2\chi(X)$, the slope inequality (see [LP12]), $K_{X}^{2}\geq
2p_{g}(X)$ (see [Deb82]) and $p_{g}(X)\geq 2q(X)-4$ (see [Bea82]). Because of
all of this, quadruple Galois covers are interesting from the perspective of
the geography of irregular surfaces as well.

######  Remark 1.5.

Although the covers of Theorem 1.3 are _simple iterated double covers_ in the
sense of [Man97], they are not _good sequences_. Moreover, the point of view
of our article is to study the deformation of canonical morphisms to
projective spaces, rather than the deformation of finite morphisms between two
surfaces. Therefore our study distinctly differs from the study of simple
iterated double covers carried out by Catanese and Manetti.

Statements of the main results. Let $X$ be a surface as in Theorem 1.3, and
let $\varphi$ be the canonical morphism of $X$. First we present a description
of our results about the algebraic formally semiuniversal deformation space of
$\varphi$ (which exists by Remark 2.17) in the following table (see Theorem
3.1).

Table 1. Deformations of $\varphi$ $X$ is of type ∗ |  Degree of any deformation of ${\varphi}$ |  Description of $\varphi_{t}$ for a general $t$ in the deformation space of $\varphi$ |  Normalization of the image of $\varphi_{t}$ for a general $t$ in the deformation space of $\varphi$
---|---|---|---
$(1)_{m}$ | $\geq 2$ |  Double cover onto a non-normal surface |  Elliptic ruled surface with invariant $0$
$(1^{\prime})_{m}$ | $\geq 2$ |  Double cover onto a non-normal surface |  Elliptic ruled surface with invariant $0$
$(2)_{2}$ | $\geq 2$ |  Double cover onto a smooth surface |  Ruled surface over a curve of genus $2$ with invariant $-2$
$(2)_{m}$ $(m\geq 3)$ | $\geq 2$ | Unknown | Unknown
$(3)_{1}$ | $4$ |  Quadruple cover onto a smooth surface | $\mathbb{P}^{1}\times\mathbb{P}^{1}$
$(3)_{m}$ $(m\geq 2)$ | $\geq 2$ |  Double cover onto a smooth surface |  Product of a curve of genus two with a curve of genus $m+1$

$*$ In Table 1, if $X$ is of type $(1)_{m}$, $(1^{\prime})_{m}$ or $(2)_{m}$,
then we ask $X$ to be smooth or to have $A_{1}$ singularities (these surfaces
are general).

It is illustrative to compare the results summarized in the above table with
the deformations of lower degree canonical covers of surfaces of minimal
degree. As already mentioned, any deformation of a degree 2 canonical morphism
is again of degree 2. More generally, deformations of double or triple
canonical covers of embedded projective bundles over $\mathbb{P}^{1}$ of
arbitrary dimension are, respectively, of degree 2 and 3 (see [GGP13b],
[GGP16a] and [GGP16b]). Thus Theorem 3.1 is in sharp contrast with these
results and, as pointed out before, quadruple covers are the lowest degree
examples of canonical covers for which the degree of the canonical map of a
general deformation drops down, with the possible exception of degree $3$
covers with $p_{g}=5$.

Now we summarize our results on the moduli (see Theorems 4.1 and 4.2 and
Remark 4.3). For a given $m$, there is a unique component of moduli of
surfaces of general type that contains all surfaces of type $(1)_{m}$ and
$(1^{\prime})_{m}$ (see Theorem 4.1). There is a unique moduli component that
contains all surfaces of type $(2)_{2}$ and, for given $m$, there is also a
unique moduli component that contains all surfaces of type $(3)_{m}$. In the
following table we outline the description of these moduli components:

Table 2. Moduli of $X$ $X$ is of type∗∗ |  Obstructions to deformations of $X$ |  Geometry of unique moduli component $\mathcal{M}_{[X]}$ containing $X$ |  Canonical morphism of a general surface in $\mathcal{M}_{[X]}$ |  Normalization of the image of the general canonical morphism
---|---|---|---|---
$(1)_{m}$ or $(1^{\prime})_{m}$ | Unobstructed |  Uniruled of dimension $8m+20$ |  Double cover onto a non-normal surface |  Elliptic ruled surface with invariant $0$
$(2)_{2}$ | Unobstructed |  Uniruled of dimension $28$ |  Double cover onto a smooth surface |  Ruled surface over a curve of genus $2$ with invariant $-2$
$(3)_{1}$ | Unobstructed |  Uniruled of dimension $6$ |  Quadruple cover onto $\mathbb{P}^{1}\times\mathbb{P}^{1}$ | $\mathbb{P}^{1}\times\mathbb{P}^{1}$
$(3)_{m}$ | Unobstructed |  Uniruled of dimension $3m+3$ |  Double cover onto $\mathbb{P}^{1}\times C_{2}$ $g(C_{2})=m+1$ | $\mathbb{P}^{1}\times C_{2}$

$**$ In Table 2, if $X$ is of type $(1)_{m}$, $(1^{\prime})_{m}$ or $(2)_{m}$,
then we ask $X$ to be smooth or to have $A_{1}$ singularities (these surfaces
are general).

The fact that general surfaces $X$ as described in the above table lie on a
unique component of the moduli of surfaces of general type is a consequence of
the unobstructedness of $X$. We prove the unobstructedness of $X$ in these
cases despite $H^{2}(T_{X})$ being non zero.

The results summarized in Table 2 have further consequences for the moduli of
surfaces of general type. They imply that there exist irreducible moduli
components having a proper, locally closed _quadruple sublocus_ parametrizing
those surfaces of general type whose canonical morphism is a quadruple cover
of a surface of minimal degree. In this sense, quadruple canonical covers
(unlike, for instance, Horikawa surfaces), resemble hyperelliptic curves of
genus bigger than $2$ and the existence of a proper hyperelliptic sublocus in
the moduli of these curves. In Remark 4.8 we estimate the dimension of the
quadruple loci and give the dimension of the smaller loci parametrizing
surface of general type whose canonical morphism is a quadruple Galois cover
of a surface of minimal degree.

The classification of quadruple canonical covers shows the existence of
fibrations in all genus, as is illustrated in Remark 1.4. Thus, the results in
this article apply to fibrations of all genus. Deformation and moduli of genus
two fibrations have been studied in [Sei95] and [GGP13a]. Therefore, for the
special case of genus two fibrations, namely, the surfaces of type $(1)_{1}$,
$(1^{\prime})_{1}$ and $(2)_{2}$, their results on unobstructedness and
persistence of genus two fibrations upon deformation does apply. But in this
article we show uniruledness not only of these moduli components but also of
the moduli components of fibrations of all genus. In the course of proving our
main results, we also show that the elliptic pencil of genus $m+1$ found in
Remark 1.4 persists for any deformation of a general surface $X$ of type
$(1)_{m}$ or $(1^{\prime})_{m}$.

The approach followed in this article is new and the techniques employed,
which are a combination of recent and new ones, are used in a novel fashion.
Instead of directly studying the deformation of the canonical morphism of the
quadruple cover to projective space, we study the deformations of the
intermediate double cover of the quadruple cover. This shift of focus is
crucial: on the one hand, looking at the intermediate cover permits a finer
analysis; on the other hand, as previously mentioned, the obstruction space to
Defφ is non-zero, so one cannot prove the smoothness of Defφ directly. Then,
to study the deformations of the intermediate cover, we show the existence of
some embedded, possibly non locally Cohen-Macaulay scheme structures on $Y$,
which are images of special double structures supported on $Y$ called ribbons
(see Remark 2.8). By deforming these scheme structures to reduced schemes, we
prove that the general deformation of the intermediate cover is finite
birational. This is done with the aid of a fundamental theorem of the
deformation theory of finite morphisms, i.e., Theorem 2.10, proved by the
first two authors and González in [GGP10]. Then we prove the existence of a
family, which is generically two-to-one, over the family whose central fiber
is the intermediate cover. This means that a general deformation of $\varphi$
is two–to–one. There is a subtle point, though, that needs to be addressed:
since our surfaces are irregular, such a general deformation need not be a
canonical morphism. We overcome this difficulty and show nevertheless the
existence of a deformation of $\varphi$ which is two-to-one and, indeed,
canonical. We give now a glimpse of the strategy we follow to prove the
unobstructedness of $\varphi$ and $X$ and the uniruledness of the moduli
component of $X$, when $X$ is a surface of type $(1)_{m},(1^{\prime})_{m}$ or
$(2)_{2}$, general as specified in Table 1. First we show that ${\bf
Def}_{\varphi^{\prime}}$ is smooth ($\varphi^{\prime}$ is the composition of
the intermediate cover and the embedding of $Y$ in projective space). Then we
show that, under some assumptions, which are satisfied in our case, the
following chain of implications holds:

${\bf Def}_{\varphi^{\prime}}$ is smooth
$\qquad\stackrel{{\scriptstyle\mathclap{\mbox{Thm.
~{}\ref{Def_1}}}}}{{\implies}}\qquad$ ${\bf Def}_{p/\mathbb{P}^{N}}$ is smooth
$\qquad\stackrel{{\scriptstyle\mathclap{\mbox{Cor.
~{}\ref{Defphi}}}}}{{\implies}}\qquad$ ${\bf Def}_{\varphi}$ is smooth
$\qquad\stackrel{{\scriptstyle\mathclap{\mbox{Cor.
~{}\ref{defcanonicalmorphism }}}}}{{\implies}}\qquad$ ${\bf Def}_{X}$ is
smooth.

For general surfaces of types $(1)_{m}$ and $(1^{\prime})_{m}$, the
obstruction spaces of ${\bf Def}_{\varphi}$ and ${\bf Def}_{X}$ are non-zero,
but still we prove that these functors are smooth (see Remark 4.5). Finally we
show the uniruledness of the algebraic formally semiuniversal deformation
space of ${\bf Def}_{p/\mathbb{P}^{N}}$, which we construct in Theorem 2.21.
From this, we prove the uniruledness of the moduli component of $X$.

In the final section of the article, we show that infinitesimal Torelli holds
for smooth families of type $(1^{\prime})_{m}$. It is known that for $(3)_{m}$
it holds for $m=1$ and does not hold for $m>2$ (see Remark 5.3). We end that
section with an interesting question for families of type $(2)_{m}$.

Organization of this article. In Section 2, we prove the main results we need
to carry out our study. In § 2.1 we recall some basic facts about the normal
sheaf, and in § 2.2 we recall the basics of abelian covers. In § 2.4, we
describe the techniques of deformations of finite morphisms developed by the
first two authors. In § 2.5, we describe how to apply this technique in our
situation, namely to study the covers we are interested in. Finally § 2.6 is
devoted to study the geometry of the moduli components. We prove the results
stated in Table 1 (see Theorem 3.1) in Section 3. The proof of the results
stated in Table 2 (see Theorems 4.1 and 4.2) appear in Section 4. Finally, we
prove that the infinitesimal Torelli theorem holds for smooth surfaces of type
$(1^{\prime})_{m}$ in Section 5. The results show that there are families for
which infinitesimal Torelli hold, and families for which they do not, the
latter being well known for the case of product of curves, which is one of the
families.

Notations and conventions.

  * 1.

We will always work over the field of complex numbers $\mathbb{C}$ and a
variety is an integral separated scheme of finite type over $\mathbb{C}$.

  * 2.

The symbol ‘$\sim$’ denotes linear equivalence and ‘$\equiv$’ denotes
numerical equivalence.

  * 3.

We will use the multiplicative and the additive notation of line bundles
interchangeably. Thus, for line bundles $L_{1},L_{2}$, $L_{1}\otimes L_{2}$
and $L_{1}+L_{2}$ are the same. $L^{-r}$, $L^{\otimes-r}$ (or $-rL$) denotes
$(L^{\vee})^{\otimes r}$.

  * 4.

If $L_{i}$ is a line bundle on the variety $X_{i}$ for $i=1,2$,
$L_{1}\boxtimes L_{2}$ is by definition, the line bundle
$p_{1}^{*}L_{1}\otimes p_{2}^{*}L_{2}$ on $X_{1}\times X_{2}$ where
$p_{i}:X_{1}\times X_{2}\to X_{i}$ is the $i$-th projection for $i=1,2$. When
$X_{i}=\mathbb{P}^{1}$ for $i=1,2$, then
$\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(a,b):=\mathcal{O}_{\mathbb{P}^{1}}(a)\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(b)$.

  * 5.

For a morphism $X\to Y$ between algebraic schemes, $\Omega_{X/Y}$ (or
$\Omega_{X/Y}^{1}$) is the sheaf of relative differentials and
$T_{X/Y}=\mathcal{H}\textrm{om}(\Omega_{X/Y},\mathcal{O}_{X})$ is the relative
tangent sheaf. By convention, $\Omega_{X}$ (or $\Omega_{X}^{1}$) and $T_{X}$
is obtained by taking $Y=\textrm{Spec}(\mathbb{C})$.

  * 6.

For an algebraic scheme $X$, let $T^{i}(X)$ and $\mathcal{T}^{i}(X)$ be the
local and global cohomology of the cotangent complex respectively. It is well
known that when $X$ is a projective variety, the first cotangent sheaf
$\mathcal{T}_{X}^{1}=\mathcal{E}\textrm{xt}^{1}(\Omega_{X},\mathcal{O}_{X})$
and $T^{1}(X)=\textrm{Ext}^{1}(\Omega_{X},\mathcal{O}_{X})$.

  * 7.

For an algebraic scheme, ${\bf Def}_{X}$ (resp. Def${}^{\prime}_{X}$) is the
functor of deformations (resp. locally trivial deformations) of $X$.

  * 8.

For an algebraic scheme $X$ and a line bundle $L$ on it, ${\bf Def}_{(X,L)}$
(resp. ${\bf Def}^{\prime}_{(X,L)}$) is the functor of deformations (resp.
locally trivial deformations) of the scheme and the line bundle i.e. the pair
$(X,L)$.

  * 9.

For a morphism $\varphi:X\to Z$ between algebraic schemes, ${\bf
Def}_{\varphi}$ (resp. ${\bf Def}^{\prime}_{\varphi}$) is the functor of
deformations (resp. locally trivial deformations) of $\varphi$ with fixed
target.

  * 10.

For morphism $X\xrightarrow{\pi}Y\to Z$ of algebraic schemes, ${\bf
Def}_{\pi/Z}$ (resp. ${\bf Def}^{\prime}_{\pi/Z}$) is the functor of
$Z$-deformati-
ons (resp. locally trivial $Z$-deformations) of $\pi$ with varying target.

For a scheme $X$, if ${\bf Def}_{X}={\bf Def}_{X}^{\prime}$, then ${\bf
Def}_{(X,L)}={\bf Def}_{(X,L)}^{\prime}$, ${\bf Def}_{\varphi}={\bf
Def}_{\varphi}^{\prime}$ and ${\bf Def}_{\pi/Z}={\bf Def}^{\prime}_{\pi/Z}$
respectively in the situation of 9, 10 and 11.

##  2\. Deformations of finite morphisms: basic results

In this section, we provide the main technical results regarding the
deformations of morphisms that are essential to carry out our study.

###  2.1. Preliminaries on normal sheaves

Locally trivial deformation theory of a morphism is governed by the normal
sheaf that we define below.

######  Definition 2.1.

([Ser06], 3.4.5) To a morphism $f:X\to Z$ of algebraic schemes there exists an
exact sequence of coherent sheaves which defines the sheaf $\mathcal{N}_{f}$
called the normal sheaf of $f$;

$0\to T_{X/Z}\to T_{X}\to f^{*}T_{Z}\to\mathcal{N}_{f}\to 0.$

The morphism $f$ is called non-degenerate if $T_{X/Z}=0$.

A morphism being non-degenerate is equivalent to being unramified in an open
dense set. Thus, a finite flat morphism between normal Cohen-Macaulay
varieties is non-degenerate and so is the composition of non-degenerate
morphisms between normal Cohen-Macaulay varieties. The following is the
general version of [Gon06], Lemma 3.3 whose proof we omit.

######  Lemma 2.2.

Let $X,Y,Z$ be normal Cohen-Macaulay varieties. Let $\pi:X\to Y$ be a non-
degenerate morphism for which $\pi^{*}$ is an exact functor (this happens if
$\pi$ is finite and flat) and let $\psi:Y\to Z$ be a non-degenerate morphism.
Suppose $\varphi:=\psi\circ\pi$. Then there is an exact sequence;

$0\to\mathcal{N}_{\pi}\to\mathcal{N}_{\varphi}\to\pi^{*}\mathcal{N}_{\psi}\to
0.$

###  2.2. Normal abelian covers of smooth varieties

Our objects of study are canonical morphisms that factor through abelian
covers. We recall some basic facts about these covers, see [Par91] for further
details.

######  Definition 2.3.

Let $Y$ be a variety and let $G$ be a finite abelian group. A Galois cover of
$Y$ with Galois group $G$ is a finite flat morphism $\pi:X\to Y$ together with
a faithful action of $G$ on $X$ that exhibits $Y$ as a quotient of $X$ via
$G$.

Let $\pi:X\to Y$ be a Galois $G$ cover of a smooth variety $Y$ with $X$
normal. Then $\pi_{*}\mathcal{O}_{X}$ splits as a direct sum indexed by the
characters. More precisely,

$\pi_{*}\mathcal{O}_{X}=\bigoplus\limits_{\chi\in G^{*}}L_{\chi}^{-1}.$

Let $D$ be the branch divisor of $\pi$. Let $\mathcal{C}$ be the set of cyclic
subgroups of $G$ and for all $H\in\mathcal{C}$, denote by $S_{H}$ the set of
generators of the group of characters $H^{*}$. Then, we may write

$D=\sum\limits_{H\in\mathcal{C}}\sum\limits_{\psi\in S_{H}}D_{H,\psi}$

where $D_{H,\psi}$ is the sum of all the components of $D$ that have inertia
group $H$ and character $\psi$. The sheaves $L_{\chi}$ and the divisors
$D_{H,\psi}$ are called the building data of the cover. For every pair
$\chi,\chi^{\prime}\in G^{*}$, for every $H\in\mathcal{C}$ and for every
$\psi\in S_{H}$, one may write $\chi|_{H}=\psi^{i_{\chi}}$ and
$\chi^{\prime}|_{H}=\psi^{i_{\chi^{\prime}}}$,
$i_{\chi},i_{\chi^{\prime}}\in\\{0,\dots,m_{H}-1\\}$ where $m_{H}$ is the
order of $H$. Let $S_{\chi}=\\{(H,\psi):\chi|_{H}\neq\psi^{m_{H}-1}\\}$.

Recall that for a variety $Z$, $T_{Z}$ denotes its tangent sheaf. Furthermore,
if $Z$ is smooth and $D$ is a divisor on $Z$,
$\Omega_{Z}^{p}(\textrm{log}\,D)$ denotes the sheaf of logarithmic
differential $p$–forms. The following is a generalization of [Par91],
Corollary $4.1$ to the case $X$ normal.

######  Proposition 2.4.

Let $\pi:X\to Y$ be an abelian cover with Galois group $G$, $X$ normal and $Y$
smooth. Assume the branch divisor $D$ of $\pi$ is normal crossing. Then,

$(\pi_{*}\mathcal{N}_{\pi})^{\chi}\cong\bigoplus_{(H,\psi)\in
S_{\chi}}\mathcal{O}_{D_{H,\psi}}(D_{H,\psi})\otimes L_{\chi}^{-1}.$

Proof. We have an exact sequence

$0\to T_{X}\to\pi^{*}T_{Y}\to\mathcal{N}_{\pi}\to 0.$

Now since $\pi$ is finite and all sheaves in the exact sequence are quasi-
coherent we have that the pushforward is an exact functor and consequently we
get the following exact sequence.

$0\to\pi_{*}T_{X}\to\pi_{*}\pi^{*}T_{Y}\to\pi_{*}\mathcal{N}_{\pi}\to 0$

Now observe that $(\pi_{*}\pi^{*}T_{Y})^{\chi}=T_{Y}\otimes L_{\chi}^{-1}$.
Also note that $\pi_{*}T_{X}$ is a reflexive sheaf and hence it is enough to
determine $\pi_{*}T_{X}$ in an open subset of codimension $\geq 2$. Consider
the set $S$ which is the image of the singular locus of $X$ under $\pi$. Since
$X$ is normal we have that $S$ is of codimension at least two in $Y$. Removing
$S$ and $\pi^{-1}(S)$ (which is also of codimension $2$ in $X$ since $\pi$ is
finite) from $X$ and $Y$ respectively, we can assume that $X$ and $Y$ are
smooth hence by [Par91], Proposition 4.1(b) we have that
$(\pi_{*}T_{X})^{\chi}=T_{Y}(-\textrm{log }D_{H,\psi};(H,\psi)\in
S_{\chi})\otimes L_{\chi}^{-1}$. Since $Y$ is smooth and $D_{H,\psi}$ is a
normal crossing divisor (since $D$ is normal crossing) we have that
$T_{Y}(-\textrm{log }D_{H,\psi};(H,\psi)\in S_{\chi})$ is locally free and the
conclusion follows. $\square$

###  2.3. A generalization of Atiyah exact sequence

For any scheme $X$, there is a natural map $\mathcal{O}_{X}^{*}\to\Omega_{X}$
defined by $u\mapsto du/u$. For a line bundle $L$ on $X$, the natural map
induced between the cohomology groups $H^{1}(\mathcal{O}_{X})\to
H^{1}(\Omega_{X})\cong\textrm{Ext}^{1}(\mathcal{O}_{X},\Omega_{X})$ gives an
extension

$e_{L}:\quad 0\to\Omega_{X}\to\mathcal{Q}_{L}\to\mathcal{O}_{X}\to 0.$

We set
$\mathcal{E}_{L}:=\mathcal{H}\textrm{om}(\mathcal{Q}_{L},\mathcal{O}_{X})$,
and we obtain the following exact sequence which is known as the Atiyah exact
sequence when $X$ is smooth:

(2.1) $0\to\mathcal{O}_{X}\to\mathcal{E}_{L}\to T_{X}\to 0.$

Altmann and Christophersen has generalized [Ser06], Theorem 3.3.11 and showed
that $H^{1}(\mathcal{E}_{L})$ parametr-
izes first order locally trivial deformations of $(X,L)$ when $X$ is reduced.

######  Theorem 2.5.

([AC10], Theorem 3.1 (ii)) If $X$ is a reduced projective scheme, then ${\bf
Def}^{\prime}_{X,L}(\mathbb{C}[\epsilon])=H^{1}(\mathcal{E}_{L})$.

In fact, one can follow the treatment of [Ser06], Section 3.3.4 to define a
map $M:\mathcal{E}_{L}\to H^{0}(L)^{\vee}\otimes L$ that fits into the
following commutative diagram where the left vertical map is the one obtained
in (2.1).

(2.2) ${\mathcal{O}_{X}}$${H^{0}(L)^{\vee}\otimes
L}$${\mathcal{E}_{L}}$${H^{0}(L)^{\vee}\otimes
L}$$\scriptstyle{m}$$\scriptstyle{M}$

The proof of the following follows by repeating the argument of [Ser06],
Proposition 3.3.14 word by word.

######  Proposition 2.6.

Let $X$ be a reduced projective scheme and let $L$ be a line bundle on $X$.
Assume $(\mathcal{X},\mathcal{L})$ is a first order locally trivial
deformation of $(X,L)$ defined by a cohomology class $\eta_{1}\in
H^{1}(\mathcal{E}_{L})$ according to Theorem 2.5. Let $s\in H^{0}(L)$ be a
section of $L$. Then $s$ lifts to a section $\mathcal{L}$ if and only if
$s\in\textrm{ker}(M_{1}(\eta_{1}))$, where
$M_{1}:H^{1}(\mathcal{E}_{L})\to\textrm{Hom}(H^{0}(L),H^{1}(L))$ is induced by
$M$.

When $L$ is base point free, in order to verify the above section-lifting-
criterion, we will make use of [Ser06], (3.39) diagram with exact rows and
columns, which is given for smooth case, and whose existence is a routine
computation when $X$ is reduced and projective. In particular, we have the
exact sequence

(2.3) $0\to\mathcal{E}_{L}\to H^{0}(L)^{\vee}\otimes
L\to\mathcal{N}_{\varphi_{L}}\to 0,$

where $\varphi_{L}:X\to\mathbb{P}(H^{0}(L)^{\vee})$ is the morphism induced by
$|L|$.

###  2.4. Technique to reduce the degree along a deformation

One of the central technique for deforming a finite morphism to a morphism of
smaller degree is to construct a suitable multiple structure on the image of
the morphism which are called ropes.

######  Definition 2.7.

Let $Y$ be a reduced connected scheme and let $\mathcal{E}$ be a vector bundle
of rank $m-1$ on Y. A rope of multiplicity $m$ on $Y$ with conormal bundle
$\mathcal{E}$ is a scheme $\widetilde{Y}$ with
$\widetilde{Y}_{\textrm{red}}=Y$ such that
$\mathcal{I}_{Y/\widetilde{Y}}^{2}=0$, and
$\mathcal{I}_{Y/\widetilde{Y}}=\mathcal{E}$ as $\mathcal{O}_{Y}$ modules. If
$\mathcal{E}$ is a line bundle then $\widetilde{Y}$ is called a ribbon on $Y$.

######  Remark 2.8.

Recall that, for a morphism $\varphi:X\to\mathbb{P}^{N}$ from a smooth,
projective variety which is finite onto a smooth variety
$Y\hookrightarrow\mathbb{P}^{N}$, the space $H^{0}(\mathcal{N}_{\varphi})$
parametrizes the space of infinitesimal deformations of $\varphi$. Suppose
$\mathcal{E}$ is the trace zero module of the induced morphism $\pi:X\to Y$.
It is shown in [Gon06], Proposition 2.1, that the space
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{E})$ parametrizes the
pairs $(\widetilde{Y},\widetilde{i})$ where $\widetilde{Y}$ is a rope on $Y$
with conormal bundle $\mathcal{E}$ and
$\widetilde{i}:\widetilde{Y}\to\mathbb{P}^{N}$ is a morphism that extends $i$.

The relation between these two cohomology groups is given by the following
proposition.

######  Proposition 2.9.

([Gon06], Proposition 3.7) Let $X$ be a normal Cohen-Macaulay projective
variety and let $\varphi:X\to\mathbb{P}^{N}$ be a morphism that factors as
$\varphi=i\circ\pi$, where $\pi$ is a finite cover of a smooth variety $Y$ and
$i:Y\hookrightarrow\mathbb{P}^{N}$ is an embedding. Let $\mathcal{E}$ be the
trace zero module of $\pi$ and let $\mathcal{I}$ be the ideal sheaf of $i(Y)$.
There exists a homomorphism

$H^{0}(\mathcal{N}_{\varphi})\xrightarrow{\psi}\textrm{Hom}(\pi^{*}(\mathcal{I}/\mathcal{I}^{2}),\mathcal{O}_{X})$

that appears when taking cohomology on the commutative diagram [Gon06]
$(3.3.2)$. Since

$\textrm{Hom}(\pi^{*}(\mathcal{I}/\mathcal{I}^{2}),\mathcal{O}_{X})=H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}})\oplus
H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{E}),$

the homomorphism $\psi$ has two components;

$H^{0}(\mathcal{N}_{\varphi})\xrightarrow{\psi_{1}}H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}})\textrm{
and
}H^{0}(\mathcal{N}_{\varphi})\xrightarrow{\psi_{2}}H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{E}).$

We will make use of the following fundamental theorem of the deformation
theory of finite morphisms to reduce the degree of a general deformation of
the canonical morphism.

######  Theorem 2.10.

([GGP10], Theorem 1.4) Let $X$ be a smooth irreducible projective variety and
let $\varphi:X\to\mathbb{P}^{N}$ be a morphism that factors through an
embedding $Y\hookrightarrow\mathbb{P}^{N}$ with $Y$ smooth and let $\pi:X\to
Y$ be the induced morphism which we assume to be finite of degree $n\geq 2$.
Let $\widetilde{\varphi}:\widetilde{X}\to\mathbb{P}^{N}_{\Delta}$
$(\Delta=\textrm{Spec}\left(\frac{\mathbb{C}[\epsilon]}{\epsilon^{2}}\right))$
be a first order infinitesimal deformation of $\varphi$ and let $\nu\in
H^{0}(\mathcal{N}_{\varphi})$ be the class of $\widetilde{\varphi}$. If

  * (a)

the homomorphism $\psi_{2}(\nu)$ has rank $k>\frac{n}{2}-1$, and

  * (b)

there exists an algebraic formally semiuniversal deformation of $\,\,{\bf
Def}_{\varphi}$ and ${\bf Def}_{\varphi}$ is smooth,

then there exists a flat family of morphisms,
$\Phi:\mathcal{X}\to\mathbb{P}^{N}_{T}$ over $T$, where $T$ is a smooth
irreducible algebraic curve with a distinguished point $0$, such that

  1. (1)

$\mathcal{X}_{t}$ is a smooth irreducible projective variety,

  2. (2)

the restriction of $\Phi$ to the first order infinitesimal neighbourhood of
$0$ is $\widetilde{\varphi}$, and

  3. (3)

for $t\neq 0$, $\Phi_{t}$ is finite and one-to-one onto its image in
$\mathbb{P}^{N}$.

We will use the theorem above to study the deformations of the canonical
morphisms of the varieties we are interested in. However, we are also
interested in the degree of the canonical morphisms of the moduli components
of these varieties. We remark that a general deformation of the canonical
morphism of a regular variety remains canonical by [GGP10], Lemma 2.4 (the
statement requires smoothness, but it holds for varieties with canonical
singularities as well, see [BCG21], proof of Proposition 2.9).

###  2.5. Deformations of iterated double covers of embedded varieties

Throughout this subsection, we will work with the following diagram where $X$,
$Y$ and $Z$ are normal local complete intersection (abbreviated as lci)
projective varieties, $i:Z\hookrightarrow\mathbb{P}^{N}$ is an embedding, and
$g:X\to Z$ is a twice iterated double cover:

${X}$${Y}$${Z\hookrightarrow\mathbb{P}^{N}}$$\scriptstyle{\pi}$$\scriptstyle{g}$$\scriptstyle{p}$

We set $\psi:=i\circ p$, and $\varphi:=\psi\circ\pi$. Notice that
Def${}^{\prime}_{\varphi}$ has a formal semiuniversal deformation space by
[Ser06], Theorem 3.4.8, and our objective is to determine the degree of a
general locally trivial deformation of $\varphi$. We will show that under
suitable hypothesis, $\varphi$ can be locally trivially deformed to a two-to-
one morphism onto its image. We first need the following technical fact that
we will put as a remark for future reference.

######  Remark 2.11.

Let $Y$ be a normal lci projective variety. Let $L$ be a line bundle on $Y$
and $B$ be a divisor in $H^{0}(L)$. Assume $H^{1}(L)=0$. Let $f:\mathcal{Y}\to
T$ be a deformation of $Y$ over a smooth projective pointed variety $(T,0)$
($f$ is assumed to be proper and flat). Assume that $L$ lifts to a line bundle
$\mathcal{L}$ on $\mathcal{Y}$. Then (possibly after shrinking $T$)
$f_{*}(\mathcal{L})$ is locally free of rank $h^{0}(L)$ on $T$. We have a
Cartesian diagram as shown below.

${\mathcal{Y}\times_{T}\mathbb{P}(f_{*}(\mathcal{L}))}$${\mathbb{P}(f_{*}(\mathcal{L}))}$${\mathcal{Y}}$${T}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{g}$$\scriptstyle{f}$

Consequently,
$p:\mathcal{Y}\times_{T}\mathbb{P}(f_{*}(\mathcal{L}))\rightarrow\mathbb{P}(f_{*}(\mathcal{L}))$
is a deformation of $Y$ with $q^{*}(\mathcal{L})$ and the incidence divisor
$\mathcal{B}\in H^{0}(q^{*}(\mathcal{L}))$ giving natural lifts of $L$ and $B$
respectively on $\mathcal{Y}\times_{T}\mathbb{P}(f_{*}(\mathcal{L}))$. Now
since $f_{*}(\mathcal{L})$ is locally free, by shrinking $T$, we can always
construct a section $s:T\to\mathbb{P}(f_{*}(\mathcal{L}))$ and
$\mathcal{B}\times_{\mathbb{P}(f_{*}(\mathcal{L}))}T$ is a lift of the divisor
$B$ to $\mathcal{Y}$. Conversely any lift $\mathcal{B}$ of $B$ on
$\mathcal{Y}$ is obtained by a pullback induced by a section
$s:T\to\mathbb{P}(f_{*}(\mathcal{L}))$. $\blacksquare$

######  Remark 2.12.

Let $X$ and $Y$ be normal lci projective varieties with $Y$ smooth and
consider morphisms $X\xrightarrow{\pi}Y\xrightarrow{\psi}\mathbb{P}^{N}$. Let
$\pi$ be a finite $2:1$ morphism with trace zero module $L^{*}$ on $Y$. Let
$\varphi=\psi\circ\pi$. Let
$L^{*}=\omega_{Y}\otimes\psi^{*}(\mathcal{O}_{\mathbb{P}^{N}}(-1))$. Then
$\varphi^{*}(\mathcal{O}_{\mathbb{P}^{N}}(1))=\omega_{X}$. $\blacksquare$

######  Theorem 2.13.

Let $X$ be a normal lci projective variety and let $Y$ and $Z$ be smooth
projective varieties. Let $\pi:X\to Y$ be a finite, flat morphism of degree
two onto $Y$ with trace zero module $\mathscr{E}_{\pi}=L^{*}$ and branched
along a divisor $B\in H^{0}(L^{\otimes 2})$, and let $p:Y\to Z$ be a finite
(hence flat) morphism of degree two onto $Z$ with trace zero module
$\mathcal{E}_{p}$. Let $i:Z\hookrightarrow\mathbb{P}^{N}$ be an embedding,
$\varphi=i\circ p\circ\pi$ and $\psi=i\circ p$. Suppose

  * (a)

$H^{2}(\mathcal{O}_{Y})=0$,

  * (b)

${\bf Def}_{\psi}$ is smooth,

  * (c)

$\psi_{2}:H^{0}(\mathcal{N}_{\psi})\to
H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{E}_{p})$ is non-zero.

  * (d)

$H^{1}(L^{\otimes 2})=0$

Then there exist a flat family $\mathcal{X}\to T$ of deformations of $X$ over
a smooth pointed affine algebraic curve $(T,0)$ and a $T$-morphism
$\Phi:\mathcal{X}\to\mathbb{P}^{N}_{T}$ satisfying:

  * (1)

$\Phi=\Psi\circ\Pi$, where $\mathcal{Y}\to T$ is a flat family,
$\Psi:\mathcal{Y}\to\mathbb{P}^{N}_{T}$, and $\Pi:\mathcal{X}\to\mathcal{Y}$
are $T$-morphisms with $\mathcal{Y}_{0}=Y$, $\Pi_{0}=\pi$, and
$\Psi_{0}=\psi$,

  * (2)

$\Pi_{t}$ is a finite morphism of degree 2 for all $t$, and $\Psi_{t}$ is
birational onto its image for all $t\in T-\\{0\\}$,

  * (3)

Suppose that $\varphi$ is the canonical morphism of $X$ and that
$\mathcal{E}_{\pi}=L^{*}=\omega_{Y}\otimes\psi^{*}(\mathcal{O}_{\mathbb{P}^{N}}(-1))$,
then $\Phi_{t}$ can be taken to be the canonical morphism of $\mathcal{X}_{t}$

Proof. We will prove the assertions (1) and (2) in two steps.

Step $1$. In this step, we deform $\psi$ into a birational morphism. Notice
that ${\bf Def}_{\psi}$ is unobstructed, and has an algebraic formally
semiuniversal deformation by [BGG20], Proposition 1.5. Moreover, $\psi_{2}$ is
non-zero, hence we apply Theorem 2.10 and one gets that there exists a family
$\mathcal{Y}$ of smooth projective varieties, proper and flat over a smooth
pointed affine algebraic curve $(T,0)$ and a $T$-morphism
$\Psi:\mathcal{Y}\to\mathbb{P}^{N}_{T}$ with:

  * (1)

$\Psi_{0}=\psi$,

  * (2)

$\Psi_{t}$ is birational onto its image for all $t\in T-\\{0\\}$,

Step $2$. We construct a deformation
$\mathcal{X}\to\mathcal{Y}\to\mathbb{P}_{T}^{N}\to T$ of $\varphi$. For this
we need to construct a deformation $\Pi:\mathcal{X}\to\mathcal{Y}\to T$ of the
finite morphism $\pi:X\to Y$. Let $q:\mathcal{Y}\to T$ be the deformation
obtained by applying the forgetful map to $\mathcal{Y}\to\mathbb{P}^{N}_{T}\to
T$. We need to construct lifts $\mathcal{L}^{\otimes 2}$ and $\mathcal{B}$ of
the line bundle $L^{\otimes 2}$ and the divisor $B$ respectively on
$\mathcal{Y}$. Note that since $H^{2}(\mathcal{O}_{Y})=0$, we have that the
map ${\bf Def}_{(Y,L)}\to{\bf Def}_{Y}$ is smooth by ([Ser06], Proposition
$2.3.6$). Hence by [Ser06], Proposition $2.2.5$, (iv), we have a lift
$\mathcal{L}$ of $L$ on $\mathcal{Y}$. The conclusion follows from Remark
2.11. This proves statements $(1)$ and $(2)$.

For part $(3)$ we note that
$\omega_{\mathcal{Y}/T}\otimes\Psi^{*}(\mathcal{O}_{\mathbb{P}^{N}_{T}}(-1))$
is a deformation of $L^{*}$, since
$L^{*}=\omega_{Y}\otimes\psi^{*}(\mathcal{O}_{\mathbb{P}^{N}}(-1))$. Hence
$-2(\omega_{\mathcal{Y}/T}\otimes\Psi^{*}(\mathcal{O}_{\mathbb{P}^{N}_{T}}(-1))$
is a lift of $L^{\otimes 2}$. Now we apply Remark 2.11 to construct a relative
cover using a lift $\mathcal{B}\in
H^{0}(-2(\omega_{\mathcal{Y}/T}\otimes\Psi^{*}(\mathcal{O}_{\mathbb{P}^{N}_{T}}(-1))))$
of the divisor $B\in H^{0}(L^{\otimes 2})$. Since for each $t$, the trace zero
module
$L_{t}^{*}=\omega_{Y_{t}}\otimes\Psi_{t}^{*}(\mathcal{O}_{\mathbb{P}_{t}^{N}}(-1))$,
we have that for each $t$, $\Phi_{t}$ is the canonical morphism of $X_{t}$ by
Remark 2.12. $\square$

######  Remark 2.14.

Let $X$ be a normal lci projective variety with ample and base point free
canonical bundle $K$. Let $T$ be a smooth affine curve and
$\mathcal{Y}\xrightarrow{\Phi}\mathbb{P}^{N}_{T}\to T$ be a deformation of the
canonical morphism of $X$, i.e, for all $t\in T$, $\Phi_{t}$ is given by the
complete linear series $\omega_{\mathcal{Y}_{t}}$. Suppose that the degree of
the finite morphism $\Phi_{t}$ is $d$ for a general $t\in T$. Then there exist
an irreducible component $U_{X}$ of the universal deformation space of $X$
such that for a general closed point $u\in U_{X}$, the canonical morphism of
the fibre $\mathcal{X}_{u}$ of the universal family over $U_{X}$ has degree
less than or equal to $d$.

Proof. Choose an irreducible component $U_{X}$ containing $T$. Since $T$ is
smooth this embedding factors through the reduced induced structure
$U_{X}^{0}$ of $U_{X}$. Consider the pullback
$\mathcal{X}\xrightarrow{p}U_{X}^{0}$ of the universal family to $U_{X}^{0}$.
Since $p_{g}(\mathcal{X}_{s})$ is constant for $s\in U_{X}^{0}$, we have that
$p_{*}(\omega_{\mathcal{X}/U_{X}^{0}})$ is locally free of rank
$h^{0}(\omega_{X})$. Then
$\mathcal{X}\xrightarrow{\Psi}\mathbb{P}(p_{*}(\omega_{\mathcal{X}/U_{X}^{0}}))\to
U_{X}^{0}$ is a deformation of the canonical morphism of $X$ such that for
each $s\in U_{X}^{0}$, $\Psi_{s}$ is the canonical morphism of
$\mathcal{X}_{s}$. Since $U_{X}^{0}$ is integral, we have that degree of
$\Psi_{s}$ is upper semicontinuous. Now
$\mathcal{Y}\xrightarrow{\Phi}\mathbb{P}^{N}_{T}\to T$ is obtained by pulling
back
$\mathcal{X}\xrightarrow{\Psi}\mathbb{P}(p_{*}(\omega_{\mathcal{X}/U_{X}^{0}}))\to
U_{X}^{0}$ to $T$ by the embedding $T\hookrightarrow U_{X}^{0}$. This shows
that degree of $\Psi_{s}$ is less than or equal to $d$ for a general $s\in
U_{X}^{0}$. Since closed points of $U_{X}$ are the same as closed points of
$U_{X}^{0}$, we are done. $\blacksquare$

Now we will find the condition following the proof of [Weh86], Proposition
1.10, under which any deformation of $\varphi$ factors through a deformation
of $\pi$ (with varying target).

######  Proposition 2.15.

Let $\pi:X\to Y$ be a finite, flat morphism with trace zero module
$\mathcal{E}$ between projective varieties with $X$ normal lci, and $Y$
smooth. Let $\psi:Y\to Z$ be a non-degenerate morphism to a smooth projective
variety $Z$. Let $\varphi=\psi\circ\pi$ be the composed morphism. Assume that
any deformation of $X$ is locally trivial (e.g. when $X$ is smooth) and
$H^{0}(\mathcal{N}_{\psi}\otimes\mathcal{E})=0$. Then the natural map between
the functors ${\bf Def}_{\pi/Z}\to{\bf Def}_{\varphi}$ is smooth.

Proof. Note that ${\bf Def}_{\pi/Z}$ and ${\bf Def}_{\varphi}$ has formal
semiuniversal deformations. Because of the following diagram

${\dots}$${H^{0}(\mathcal{N}_{\pi})}$${T^{1}(\pi/Z)}$${H^{0}(\mathcal{N}_{\psi})}$${H^{1}(\mathcal{N}_{\pi})}$${T^{2}(\pi/Z)}$${H^{1}(\mathcal{N}_{\psi})}$${\dots}$${\dots}$${H^{0}(\mathcal{N}_{\pi})}$${H^{0}(\mathcal{N}_{\varphi})}$${H^{0}(\pi^{*}\mathcal{N}_{\psi})}$${H^{1}(\mathcal{N}_{\pi})}$${H^{1}(\mathcal{N}_{\varphi})}$${H^{1}(\pi^{*}\mathcal{N}_{\psi})}$${\dots}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\beta_{2}}$

the map between the functors ${\bf Def}_{\pi/Z}\to{\bf Def}_{\varphi}$ is
smooth if $\alpha_{1}$ is surjective and $\alpha_{2}$ is injective (see
[Ser06], Proposition 2.3.6). But that happens if $\beta_{1}$ is surjective and
$\beta_{2}$ is injective. The assertion follows since $\beta_{2}$ is always
injective and $\beta_{1}$ is surjective if
$H^{0}(\mathcal{N}_{\psi}\otimes\mathcal{E})=0$. $\square$

The following is the main result that we will use to prove Theorem 3.1. The
proof of this result is an immediate consequence of Theorem 2.13 and Corollary
2.15.

######  Corollary 2.16.

Assume the hypotheses (a), (b), (c) and $(d)$ of Theorem 2.13. Furthermore,
assume Defφ has an algebraic formally semiuniversal deformation space, any
deformation of $X$ is locally trivial, and
$H^{0}(\mathcal{N}_{\psi}\otimes\mathcal{E})=0$. Then a general deformation of
$\varphi$ is a composition of a double cover over a deformation $Y^{\prime}$
of $Y$ followed by a morphism of $Y^{\prime}\to\mathbb{P}^{N}$ that is
birational onto its image, consequently, it is a two-to-one morphism onto its
image.

######  Remark 2.17.

Let $X$ be a surface with ample and globally generated canonical bundle
$\omega_{X}$ with at worst canonical singularities. Let $\varphi$ be the
canonical morphism of $X$. Then $H^{0}(T_{X})=0$. Furthermore, DefX, Defφ, and
Def${}_{(X,\omega_{X})}$ have algebraic formally universal deformation spaces.
$\blacksquare$

We have assumed in Proposition 2.15 and Corollary 2.16 that any deformation of
$X$ is locally trivial. This hypothesis is satisfied if $X$ is smooth. We will
show the hypothesis to be true in more generality, namely, if we allow $X$ to
have $A_{1}$ singularities. We will establish this fact using a corollary of
the following proposition.

######  Proposition 2.18.

Let $X$ be a normal lci projective variety. Assume $\pi:X\to Y$ is double
cover of a smooth projective variety $Y$ with ramification divisor $R$ and
branch divisor $B\in H^{0}(L^{\otimes 2})$. Then we have the following
four–term exact sequence

(2.4) $0\to H^{0}(\mathcal{N}_{\pi})\to
H^{0}(\pi^{*}\mathcal{O}_{Y}(B)|_{R})\to H^{0}(\mathcal{T}_{X}^{1})\to
H^{1}(\mathcal{N}_{\pi})$

Proof. Let
$Y^{\prime}:=\mathbb{V}(L^{-1}):=\textrm{Spec}(\textrm{Sym}(L^{-1}))$ denote
the total space of the line bundle $L$. Let $p^{\prime}:Y^{\prime}\to Y$ be
the projection. We have an embedding of $i:X\hookrightarrow Y^{\prime}$ as a
divisor in $Y^{\prime}$ such that $\pi=p^{\prime}\circ i$. The conormal sheaf
$\mathcal{N}^{*}_{X/Y^{\prime}}:=\mathcal{I}/\mathcal{I}^{2}$ of $X$ in
$Y^{\prime}$ is given by $\pi^{*}(-\mathcal{O}_{Y}(B))$ (since $X$ is defined
as the scheme of zeroes of $t^{n}-p^{*}B$ where $t\in p^{*}(L)$). Since $X$ is
a local complete intersection, we have an exact sequence

(2.5)
$0\to\pi^{*}(-\mathcal{O}_{Y}(B))\to\Omega_{Y^{\prime}}\otimes\mathcal{O}_{X}\to\Omega_{X}\to
0.$

Now since $p^{\prime}:Y^{\prime}\to Y$ is a smooth morphism, we have that
$\Omega_{Y^{\prime}/Y}$ is an invertible sheaf isomorphic to $p^{\prime*}(-L)$
and we have an exact sequence

(2.6)
$0\to\pi^{*}(\Omega_{Y})\to\Omega_{Y^{\prime}}\otimes\mathcal{O}_{X}\to\pi^{*}(-L)\to
0.$

We also have another exact sequence as follows

(2.7)
$0\to\pi^{*}(-\mathcal{O}_{Y}(B))\to\pi^{*}(-L)\to\pi^{*}(-L)\otimes\mathcal{O}_{R}\to
0.$

Now apply snake lemma to the following diagram where the first row is (2.5)

${0}$${\pi^{*}(-B)}$${\Omega_{Y^{\prime}}\otimes\mathcal{O}_{X}}$${\Omega_{X}}$${0}$${0}$${\pi^{*}(-L)}$${\pi^{*}(-L)}$${0}$${0}$

and use the previous two short exact sequences (2.6) and (2.7) to get the
following exact sequence.

$0\to\pi^{*}(\Omega_{Y})\to\Omega_{X}\to\pi^{*}(-L)\otimes\mathcal{O}_{R}\to
0$

Since $\pi$ is non-degenerate, we get that $T_{X/Y}=0$ and hence by dualizing
the above sequence we have the following exact sequence

(2.8) $0\to
T_{X}\to\pi^{*}T_{Y}\to\mathcal{E}\textrm{xt}^{1}(\pi^{*}(-L)\otimes\mathcal{O}_{R},\mathcal{O}_{X})\to\mathcal{T}_{X}^{1}\to
0.$

Notice that
$\mathcal{E}\textrm{xt}^{1}(\pi^{*}(-L)\otimes\mathcal{O}_{R},\mathcal{O}_{X})=\pi^{*}(L\otimes\mathcal{O}_{R}(R))=\pi^{*}(\mathcal{O}_{Y}(B))\otimes\pi^{*}\mathcal{O}_{R}=\pi^{*}(\mathcal{O}_{Y}(B))\otimes\mathcal{O}_{R}$,
and consequently (2.8) becomes

(2.9) $0\to
T_{X}\to\pi^{*}T_{Y}\to\pi^{*}(\mathcal{O}_{Y}(B))\otimes\mathcal{O}_{R}\to\mathcal{T}_{X}^{1}\to
0.$

The exact sequence of the proposition follows from the fact that
$\mathcal{N}_{\pi}=\textrm{ker}\left(\pi^{*}(\mathcal{O}_{Y}(B))\otimes\mathcal{O}_{R}\to\mathcal{T}_{X}\right)$.
$\square$

######  Corollary 2.19.

In the set-up of Proposition 2.18, assume that $B$ has normal crossing and
$H^{1}(\mathcal{N}_{\pi})=0$. Then ${\bf Def}^{\prime}_{X}={\bf Def}_{X}$.

Proof. Since $B$ has normal crossing, it follows from Proposition 2.4 that
$H^{0}(\mathcal{N}_{\pi})\cong H^{0}(\pi^{*}\mathcal{O}_{Y}(B)|_{R})$.
Consequently, $H^{0}(\mathcal{T}_{X}^{1})=0$ by the exact sequence of
Proposition 2.18. The assertion then follows from the five-term exact sequence

$0\to H^{1}(T_{X})\to\textrm{Ext}^{1}(\Omega_{X},\mathcal{O}_{X})\to
H^{0}(\mathcal{T}_{X}^{1})\to H^{2}(T_{X})\to T^{2}(X)$

associated to the spectral sequence
$E^{p,q}_{2}=H^{p}(\mathcal{T}_{X}^{q})\implies T^{p+q}(X)$ and general
deformation theory. $\square$

######  Remark 2.20.

Assume all the hypotheses of Corollary 2.19. Further assume $X$ is a surface
with at worst canonical singularities and ${\bf Def}_{X}$ is smooth. Then it
follows from the work of Burns and Wahl (see [BW74]) that if $X^{\prime}$ is
the minimal resolution of $X$, then ${\bf Def}_{X^{\prime}}$ is smooth.

###  2.6. Geometry of the deformation spaces

One of our objectives is to describe the moduli components of surfaces of type
$(1)_{m}$ and $(1^{\prime})_{m}$. We will see that for a fixed $m$, there is a
unique component of the moduli space that contains all surfaces of both types,
and that this component is uniruled. The proof of this fact is based on the
following result.

######  Theorem 2.21.

Assume the hypothesis (a), (b) and (d) of Theorem 2.13. Further assume $B$ is
normal crossing (in particular any deformation of $X$ is locally trivial by
Proposition 2.4 and Corollary 2.19). Then ${\bf Def}_{\pi/\mathbb{P}^{N}}$ has
a smooth uniruled algebraic formally semiuniversal deformation space
$V_{\pi/\mathbb{P}^{N}}$.

Proof. We construct an algebraic formally semiuniversal family of deformations
of the functor ${\bf Def}_{\pi/\mathbb{P}^{N}}$

$\mathcal{X}_{\pi/\mathbb{P}^{N}}\to\mathbb{P}^{N}_{V_{\pi/\mathbb{P}^{N}}}\to
V_{\pi/\mathbb{P}^{N}}$

over a smooth pointed irreducible base $(V_{\pi/\mathbb{P}^{N}},0)$.

Let $\mathcal{Y}_{\psi}\to\mathbb{P}_{U_{\psi}}^{N}\to U_{\psi}$ be the
algebraic formally semiuniversal family of deformations of the functor ${\bf
Def}_{\psi}$ (this space exists, see for example the proof of Theorem 2.13).

Let $(\mathscr{Y}_{L},\mathscr{L})\to U_{L}$ be the algebraic formally
semiuniversal deformation space of the functor ${\bf Def}_{(Y,L)}$. Let
$\mathscr{Y}\to U$ be the algebraic formally semiuniversal deformation space
of the functor ${\bf Def}_{Y}$. Forgetful maps between functors induce a
cartesian diagram, which in turn induces a cartesian diagram of algebraic
formally semiuniversal deformation spaces as shown below.

${{\bf Def}_{\psi}\times_{{\bf Def}_{Y}}{\bf Def}_{(Y,L)}}$${{\bf
Def}_{(Y,L)}}$${{\bf Def}_{\psi}}$${{\bf Def}_{Y}}$

${U_{\psi}\times_{U}U_{L}}$${U_{L}}$${U_{\psi}}$${U}$

Since $H^{2}(\mathscr{O}_{Y})=0$, we have that the forgetful map ${\bf
Def}_{(Y,L)}\to{\bf Def}_{Y}$ is smooth (see [Ser06], Proposition $2.3.6$) and
hence the map ${\bf Def}_{\psi}\times_{{\bf Def}_{Y}}{\bf Def}_{(Y,L)}\to{\bf
Def}_{\psi}$ is smooth. Now using the fact that ${\bf Def}_{\psi}$ is smooth,
we have that ${\bf Def}_{\psi}\times_{{\bf Def}_{Y}}{\bf Def}_{(Y,L)}$ is
smooth and hence $U_{\psi}\times_{U}U_{L}$ is smooth. We set
$U_{(\psi,L)}:=U_{\psi}\times_{U}U_{L}$. The semiuniversal families form the
following cartesian diagram

${(\mathcal{Y}_{\psi}\times_{\mathcal{Y}}\mathcal{Y}_{L}\to\mathbb{P}_{U_{(\psi,L)}}^{N},\mathcal{L}_{\psi})}$${(\mathcal{Y}_{L},\mathscr{L})}$${(\mathcal{Y}_{\psi}\to\mathbb{P}_{U_{\psi}}^{N})}$${\mathcal{Y}}$

Hence
$(\mathcal{Y}_{\psi}\times_{\mathcal{Y}}\mathcal{Y}_{L}\to\mathbb{P}_{U_{(\psi,L)}}^{N}\to
U_{(\psi,L)},\mathcal{L}_{\psi})$ is a smooth algebraic formally semiuniversal
deformation of ${\bf Def}_{\psi}\times_{{\bf Def}_{Y}}{\bf Def}_{(Y,L)}$ where
$\mathcal{L}_{\psi}$ is the pullback of $\mathcal{L}$ under the morphism
$\mathcal{Y}_{\psi}\times_{\mathcal{Y}}\mathcal{Y}_{L}\to\mathcal{Y}_{L}$. Let
the map
$\mathcal{Y}_{\psi,L}:=\mathcal{Y}_{\psi}\times_{\mathcal{Y}}\mathcal{Y}_{L}\to
U_{(\psi,L)}$ be denoted by $p_{(\psi,L)}$. Since $H^{1}(L^{\otimes 2})=0$, we
have that $p_{(\psi,L)*}(\mathcal{L_{\psi}}^{\otimes 2})$ is free after
possibly shrinking $U_{(\psi,L)}$. Let
$V_{\pi/\mathbb{P}^{N}}:=\mathbb{P}(p_{(\psi,L)*}(\mathcal{L_{\psi}}^{\otimes
2}))$ and consider the Cartesian diagram

${\mathcal{Y}_{\psi,L}\times_{U_{\psi,L}}V_{\pi/\mathbb{P}^{N}}}$${V_{\pi/\mathbb{P}^{N}}}$${\mathcal{Y}_{\psi,L}}$${U_{\psi,L}}$

Choose a basis $\bigoplus\limits_{i=0}^{M}\mathcal{O}_{U_{(\psi,L)}}s_{i}$ of
$p_{(\psi,L)*}(\mathcal{L_{\psi}}^{\otimes 2})$. Let $X_{i}\in
H^{0}(\mathcal{O}_{V_{\pi/\mathbb{P}^{N}}}(1))=H^{0}(p_{(\psi,L)*}(\mathcal{L_{\psi}}^{\otimes
2})^{*})$, with $0\leq i\leq M$ be the dual basis. Now on
$\mathcal{Y}_{\psi,L}\times_{U_{(\psi,L)}}V_{\pi/\mathbb{P}^{N}}$, consider
the divisor $\mathcal{B}=\sum\limits_{i=0}^{M}X_{i}s_{i}$. One can construct a
relative Galois double cover
$\mathcal{X}_{\pi/\mathbb{P}^{N}}\to\mathcal{Y}_{\psi,L}\times_{U_{(\psi,L)}}V_{\pi/\mathbb{P}^{N}}$
given by the equation $t^{2}-\mathcal{B}$ in the total space of
$q^{*}(\mathcal{L}_{\psi})$ where $t$ is the tautological section of
$q^{*}(\mathcal{L}_{\psi})$ and
$q:\mathcal{Y}_{\psi,L}\times_{U_{(\psi,L)}}V_{\pi/\mathbb{P}^{N}}\to\mathcal{Y}_{\psi,L}$.
The fibre of this relative double cover over a point $(u,[r])\in
V_{\pi/\mathbb{P}^{N}}$ with $u\in U_{(\psi,L)}$ and $r\in
H^{0}(\mathcal{L}_{{\psi},u}^{\otimes 2})$ is the double cover
$\mathcal{X}_{\pi/\mathbb{P}^{N},u}\to\mathcal{Y}_{\psi,L,u}$ given by the
line bundle $L_{\psi,u}$ and the divisor $r\in
H^{0}(\mathcal{L}_{{\psi},u}^{\otimes 2})$. This is therefore a smooth
algebraic deformation of the functor ${\bf Def}_{\pi/\mathbb{P}^{N}}$.

Now note that given a flat family of polarized schemes
$f:(\mathcal{C},\mathcal{M})\to S$ over an affine scheme $S$ with
$f_{*}(\mathcal{M})$ free, giving a divisor $\mathcal{D}\in
H^{0}(\mathcal{M})$ flat over $S$ is equivalent to giving a unique $S$ \-
valued point in $\mathbb{P}(f_{*}(\mathcal{M}))$ and hence a section
$S\to\mathbb{P}(f_{*}(\mathcal{M}))$. This along with the fact that
$U_{(\psi,L)}$ is formally semiuniversal implies that $V_{\pi/\mathbb{P}^{N}}$
is a smooth algebraic formally semiuniversal deformation of the functor ${\bf
Def}_{\pi/\mathbb{P}^{N}}$. Also since it is a projective bundle over a smooth
affine scheme, it is uniruled. $\square$

######  Remark 2.22.

Under the assumptions of Theorem 2.21, it is easy to prove the smoothness of
${\bf Def}_{\pi/\mathbb{P}^{N}}$ only using the existence of a formally
semiuniversal deformation, without the explicit construction. Indeed, since
$H^{1}(\mathcal{N}_{\pi})=0$ by Proposition 2.4, and the assumptions
$H^{1}(L^{\otimes 2})=H^{2}(\mathcal{O}_{Y})=0$, it follows from the following
exact sequence

$\cdots\to H^{0}(\mathcal{N}_{\pi})\to T^{1}(\pi/\mathbb{P}^{N})\to
H^{0}(\mathcal{N}_{\psi})\to H^{1}(\mathcal{N}_{\pi})\to
T^{2}(\pi/\mathbb{P}^{N})\to H^{1}(\mathcal{N}_{\psi})\to\cdots$

Since $H^{1}(\mathcal{N}_{\pi})=0$, and [Ser06], Proposition 2.3.6 that the
forgetful map ${\bf Def}_{\pi/\mathbb{P}^{N}}\to{\bf Def}_{\psi}$ is smooth.
Consequently ${\bf Def}_{\pi/\mathbb{P}^{N}}$ is smooth as ${\bf Def}_{\psi}$
is smooth by hypothesis. $\blacksquare$

The following corollary shows that if ${\bf Def}_{\varphi}$ has an algebraic
formally universal deformation space, then that space is also smooth and
uniruled under suitable assumptions. In fact, one can also expect to determine
the degree of a general deformation of $\varphi$.

######  Corollary 2.23.

Assume the hypotheses (a), (b), (c), and (d) of Theorem 2.13. Further assume
$B$ is normal crossing, and

  * (a)

Defφ has an algebraic formally universal deformation space,

  * (b)

$H^{0}(N_{\psi}\otimes\mathscr{E}_{\pi})=0$.

Then the the following happens:

  * (1)

the natural forgetful map ${\bf Def}_{\pi/\mathbb{P}^{N}}\to{\bf
Def}_{\varphi}$ is smooth,

  * (2)

the algebraic formally universal deformation space of ${\bf Def}_{\varphi}$ is
smooth and uniruled, and

  * (3)

a general deformation of ${\varphi}$ is a composition of a double cover over a
deformation $Y^{\prime}$ of $Y$ followed by a morphism of
$Y^{\prime}\to\mathbb{P}^{N}$ that is birational onto its image.

Proof. The smoothness of ${\bf Def}_{\pi/\mathbb{P}^{N}}\to{\bf
Def}_{\varphi}$ is a consequence of Proposition 2.15, thanks to assumption
(b). Moreover, ${\bf Def}_{\varphi}$ is smooth since ${\bf
Def}_{\pi/\mathbb{P}^{N}}$ is smooth (see [Ser06], Proposition 2.2.5 (iii)),
thanks to Theorem 2.21.

Now we show that the algebraic formally semiuniversal deformations space of
${\bf Def}_{\varphi}$, which we denote by $U_{\varphi}$, is uniruled. In the
notation of Theorem 2.21, after possibly shrinking $V_{\pi/\mathbb{P}^{N}}$ we
can assume that $V_{\pi/\mathbb{P}^{N}}=U_{(\psi,L)}\times\mathbb{P}^{m}$
where over a point $u\in U_{(\psi,L)}$, the fibre which is a projective space
that parametrizes the divisors in the linear system of $\mathcal{L}_{\psi,u}$
which are branch divisors of the finite morphism $X_{u}\to Y_{u}$. The
conclusion follows since a branch divisor is uniquely determined by the finite
morphism.

Finally, part (3) follows from Corollary 2.16. $\square$

Now we provide the consequences of the above results on the deformations of
$X$.

######  Corollary 2.24.

Assume all the hypotheses of Corollary 2.23. Furthermore assume ${\bf
Def}_{X}$ has an algebraic formally semiuniversal deformation space. If the
natural forgetful map ${\bf Def}_{\varphi}\to{\bf Def}_{X}$ has surjective
differential map then ${\bf Def}_{X}$ is smooth and the algebraic formally
semiuniversal deformation space of $X$ is uniruled.

Proof. Since ${\bf Def}_{\varphi}$ is smooth by Corollary 2.23, the smoothness
of ${\bf Def}_{X}$ follows from [Ser06], Proposition $2.3.7$. Composing by the
smooth surjection $V_{\pi/\mathbb{P}^{N}}\to U_{\varphi}$, we have a smooth
surjection $V_{\pi/\mathbb{P}^{N}}\to U$ where $U$ is the algebraic formally
universal deformation space of ${\bf Def}_{X}$. Lastly, $U$ is uniruled since
$X$ is normal and for a normal abelian cover, the branch divisors are uniquely
determined by $X$. $\square$

##  3\. Deformations of irregular covers of surface scrolls

The objective of this section is to study the deformations of the canonical
morphisms of surfaces of each of the four types $(1)_{m}$ ,
$(1^{\prime})_{m}$, $(2)_{m}$ and $(3)_{m}$ (general in the first three cases
and arbitrary in the fourth) described in Theorem 1.3. In particular, we aim
to prove the following

######  Theorem 3.1.

Let $X$ be an irregular surface with at worst canonical singularities. Assume
the canonical bundle $\omega_{X}$ is ample and globally generated, and the
canonical morphism $\varphi$ is a quadruple Galois cover onto a smooth surface
of minimal degree, i.e, $X$ belongs to one of the four families described in
Theorem 1.3. If $X$ is of type $(1)_{m}$ , $(1^{\prime})_{m}$ or $(2)_{m}$,
assume furthermore that $X$ is smooth or has $A_{1}$ singularities. Then we
have the following description of the algebraic formally semiuniversal
deformation space of $\varphi$ (that exists by Remark 2.17).

  * (1)

If $X$ belongs to the family of type $(1)_{m}$ $(m\geq 1)$, $(1^{\prime})_{m}$
$(m\geq 1)$, $(2)_{2}$ or $(3)_{m}$ $(m\geq 2)$, then there exist an
irreducible component in the algebraic formally semiuniversal deformation
space of $\varphi$, such that its general element is a two-to-one morphism
onto its image

    * (I)

which is a non-normal variety whose normalization is an elliptic ruled surface
with invariant $e=0$, if $X$ is a surface of type $(1)_{m}$ or
$(1^{\prime})_{m}$ (we will show in Theorem 4.1 that in this case $\varphi$ is
unobstructed and hence this is the only component);

    * (II)

which is a smooth surface ruled over a smooth curve of genus $2$ with
invariant $e=-2$, if $X$ is a surface of type $(2)_{2}$ (we will show in
Theorem 4.1 that $\varphi$ is unobstructed and hence this is the only
component);

    * (III)

which is a product of a smooth curve of genus $2$ with a smooth non-
hyperelliptic curve of genus $m+1$ if $X$ is a surface of type $(3)_{m}$
$(m\geq 2)$;

and is induced by the complete linear series of a line bundle numerically
equivalent to $\omega_{X}$.

  * (2)

If $X$ is of type $(3)_{1}$, any deformation of $\varphi$ is a morphism of
degree four onto its image which is $\mathbb{F}_{0}$ and is induced by the
complete linear series of a line bundle numerically equivalent to
$\omega_{X}$.

  * (3)

Any deformation of $\varphi$ is of degree $\geq 2$. In particular, there do
not exist any irreducible component in the algebraic formally semiuniversal
deformation space of $\varphi$, such that its general element is birational
onto its image.

First we fix the notations that we are going to use throughout this section.
It follows from Theorem 1.3 that if $\pi:X\to Y$ is an irregular quadruple
Galois canonical cover of a smooth variety of minimal degree $Y$ with trace
zero module $\mathcal{E}$, then
$i:Y=\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{N}$ and the
embedding is given by the complete linear series
$|\mathcal{O}_{Y}(C_{0}+mf)|$. We have $N=2m+1$ and we identify
$\mathcal{O}_{Y}(C_{0})$ with $\mathcal{O}_{Y}(1,0)$ and $\mathcal{O}_{Y}(f)$
with $\mathcal{O}_{Y}(0,1)$. Notice that
$T_{Y}=\mathcal{O}_{Y}(2,0)\oplus\mathcal{O}_{Y}(0,2)$, whence
$h^{0}(T_{Y})=6$. One has the following two exact sequences;

(3.1) $0\to\mathcal{O}_{Y}(2,0)\oplus\mathcal{O}_{Y}(0,2)\to
T_{\mathbb{P}^{N}}|_{Y}\to\mathcal{N}_{Y/\mathbb{P}^{N}}\to 0,$ (3.2)
$0\to\mathcal{O}_{Y}\to\mathcal{O}_{Y}(1,m)^{\oplus N+1}\to
T_{\mathbb{P}^{N}}|_{Y}\to 0.$

######  Lemma 3.2.

Let $Y=\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{N}$ be the
embedding is given by the complete linear series
$|\mathcal{O}_{Y}(C_{0}+mf)|$.

  * (1)

$H^{0}(T_{\mathbb{P}^{N}}|_{Y})=(N+1)^{2}-1$,
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}})=(N+1)^{2}-7$.

  * (2)

$H^{1}(T_{\mathbb{P}^{N}}|_{Y})=0$, $H^{1}(\mathcal{N}_{Y/\mathbb{P}^{N}})=0$.

Proof. Since $Y$ is regular with $H^{2}(\mathcal{O}_{Y})=0$, the assertions
about $H^{j}(T_{\mathbb{P}^{N}}|_{Y})$ for $j=0,1$ follows from (3.2).
Consequently, it is easy to compute $H^{j}(\mathcal{N}_{Y/\mathbb{P}^{N}})$
for $j=0,1$ using (3.1). $\square$

Now we fix our notations for nonrational ruled surfaces. A nonrational ruled
surface over a nonrational smooth curve $C$ of genus $g\neq 0$ is by
definition a projective bundle $W=\mathbb{P}(\mathcal{E^{\prime}})$ where
$\mathcal{E^{\prime}}$ is a rank 2 vector bundle on $C$. We will always assume
that $\mathcal{E^{\prime}}$ is normalized, i.e., $\mathcal{E^{\prime}}$ has
sections, but any twist of $\mathcal{E^{\prime}}$ by any line bundle of
negative degree has no section. By definition
$e:=-\textrm{deg}(\textrm{det}(\mathcal{E^{\prime}}))$ is the invariant of
$W$. A section of $p^{\prime}:W\to C$ determines a sectional curve
$C_{0}^{\prime}$ with self intersection $-e$, and let $f^{\prime}$ be the
numerical class of a fiber of $p^{\prime}$. It is known that
$\textrm{Pic}(W)=\mathbb{Z}C_{0}^{\prime}\oplus p^{\prime*}\textrm{Pic}(C)$.
In particular, the Néron-Severi group
$\textrm{NS}(W)=\mathbb{Z}C_{0}^{\prime}\oplus\mathbb{Z}f^{\prime}$ satisfying

$C_{0}^{\prime 2}=-e,\quad C_{0}^{\prime}f^{\prime}=1,\quad f^{\prime 2}=0.$

If ${\bf a}$ is a divisor on $E^{\prime}$, ${\bf a}f^{\prime}$ denotes the
pull-back $p^{\prime*}{\bf a}$. The canonical bundle
$\omega_{W}=-2C_{0}^{\prime}+({\bf e}+\omega_{C})f^{\prime}$, where ${\bf
e}:=\textrm{det}(\mathcal{E^{\prime}})$, consequently

$\omega_{W}\equiv-2C_{0}^{\prime}-(e+2-2g)f^{\prime}.$

###  3.1. Deformations of canonical morphisms for types $(1)_{m}$ and
$(1^{\prime})_{m}$

For these surfaces, we have the following diagram.

${X}$${X_{1}}$${Y\overset{{i}}{\hookrightarrow}\mathbb{P}^{N}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi}$$\scriptstyle{p_{1}}$

We also know that
${p_{1}}_{*}\mathcal{O}_{X_{1}}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-2C_{0})$.
Since we have identified $\mathcal{O}_{Y}(C_{0})$ with $\mathcal{O}_{Y}(1,0)$,
we can write
${p_{1}}_{*}\mathcal{O}_{X_{1}}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-2,0)$.
It is easy to see that $X_{1}=E\times\mathbb{P}^{1}$ where
$\psi:E\to\mathbb{P}^{1}$ is a smooth double cover, with
$\psi_{*}\mathcal{O}_{E}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-2)$,
i.e. $E$ is a smooth elliptic curve. We set $\varphi_{1}:=i\circ p_{1}$ and
call $B$ the branch divisor of $\pi_{1}$.

######  Proposition 3.3.

Let $X$ be a surface of type $(1)_{m}$ or $(1^{\prime})_{m}$. Then the
following happens:

  * (1)

$h^{1}(\mathcal{O}_{X_{1}})=1$ and $h^{2}(\mathcal{O}_{X_{1}})=0$,

  * (2)

$h^{0}(\mathcal{N}_{p_{1}})=4$ and $h^{1}(\mathcal{N}_{p_{1}})=0$,

  * (3)

$h^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0))=1$ and
$h^{1}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0))=0$,

  * (4)

$h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2C_{0}))=3$ and
$h^{1}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2C_{0}))=0$,

  * (5)

$h^{0}(\mathcal{N}_{\varphi_{1}})=(N+1)^{2}$ and
$h^{1}(\mathcal{N}_{\varphi_{1}})=0$; consequently $\varphi_{1}$ is
unobstructed.

Proof. (1) Follows from
$h^{j}(\mathcal{O}_{X_{1}})=h^{j}(\mathcal{O}_{Y})\oplus
h^{j}(\mathcal{O}_{Y}(-2,0)$ and Künneth formula.

(2) We apply Proposition 2.4. Since $Y$ is regular, it follows that
$h^{0}(\mathcal{N}_{p_{1}})=h^{0}(\mathcal{O}_{Y}(4,0))-1=4$. Furthermore,
since $H^{2}(\mathcal{O}_{Y})=0$, we obtain
$h^{1}(\mathcal{N}_{p_{1}})=h^{1}(\mathcal{O}_{Y}(4,0))=0$.

(3) Tensor (3.2) by $\mathcal{O}_{Y}(-2,0)$ to obtain the following exact
sequence

$0\to\mathcal{O}_{Y}(-2,0)\to\mathcal{O}_{Y}(-1,m)^{\oplus N+1}\to
T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0)\to 0.$

It follows that
$h^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0))=1$,
$h^{1}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0))=0$.

(4) This is a consequence of the long exact sequence associated to the exact
sequence (3.1) tensored by $\mathcal{O}_{Y}(-2,0)$ and part (3).

(5) We have the following short exact sequence by lemma 2.2

(3.3) $0\to\mathcal{N}_{p_{1}}\to\mathcal{N}_{\varphi_{1}}\to
p_{1}^{*}\mathcal{N}_{Y/\mathbb{P}^{N}}\to 0.$

It follows that
$h^{0}(\mathcal{N}_{\varphi_{1}})=h^{0}(\mathcal{N}_{p_{1}})+h^{0}(p_{1}^{*}\mathcal{N}_{Y/\mathbb{P}^{N}})=h^{0}(\mathcal{N}_{p_{1}})+h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}})+h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2C_{0}))$,
since $h^{1}(\mathcal{N}_{p_{1}})=0$ by part (2). We obtain
$h^{0}(\mathcal{N}_{\varphi_{1}})=(N+1)^{2}$ thanks to part (2) and Lemma 3.2.

The fact $h^{1}(\mathcal{N}_{\varphi_{1}})=0$ follows from the vanishings of
$h^{1}(\mathcal{N}_{p_{1}})$ (proven in part (2)),
$h^{1}(\mathcal{N}_{Y/\mathbb{P}^{N}})$ (proven in Lemma 3.2), and
$h^{1}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2C_{0}))$ (proven
in part (4)). $\square$

######  Corollary 3.4.

Let $X$ be a surface of type $(1)_{m}$ or $(1^{\prime})_{m}$. If $X$ is smooth
or has $A_{1}$ singularities, then there exists a smooth, affine irreducible
algebraic curve $T$ and a flat family of morphisms
$\Phi:\mathcal{X}\to\mathbb{P}^{N}_{T}$ over $T$ for which the following
happens;

  * (a)

$\Phi_{t}:\mathcal{X}_{t}\to\mathbb{P}^{2m+1}$ is a morphism of degree two
from a normal projective surface with at worst canonical singularities for all
$t\in T-\\{0\\}$. Further for any $t\in T-\\{0\\}$, the normalization of
Im($\varphi_{t}$) is an elliptic ruled surface which is the projectivization
of a rank two split vector bundle on the elliptic curve and has invariant
$e=0$. Further one can take $\Phi_{t}$ to be the canonical morphism of
$\mathscr{X}_{t}$.

  * (b)

$\Phi_{0}:\mathcal{X}_{0}\to\mathbb{P}^{2m+1}$ is the canonical morphism
$\varphi:X\to\mathbb{P}^{N}$.

Moreover the forgetful map from ${\bf Def_{\pi_{1}/\mathbb{P}^{N}}}\to{\bf
Def_{\varphi}}$ is smooth and hence any deformation of $\varphi$ is a morphism
of degree $\geq 2$, and a general deformation of $\varphi$ is a morphism of
degree $2$ onto its image. Hence in particular $\varphi$ cannot be deformed to
a birational morphism.

Proof. We check the hypotheses of Theorem 2.13. Hypothesis (a) has been
checked in Proposition 3.3 (1). To check hypothesis (b), we need to prove that
${\bf Def}_{\varphi_{1}}$ is smooth, which we have showed in Proposition 3.3
(5). To check hypothesis (c), we need to check
$H^{0}(\mathcal{N}_{\varphi_{1}})\to
H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2,0))$ is non-
zero. This is a consequence of the long exact sequence associated to (3.3),
and the facts that $h^{1}(\mathcal{N}_{p_{1}})=0$ (proven in Proposition 3.3
(2)), and
$h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2,0))\neq 0$
(proven in Proposition 3.3 (4)). The fact that $\mathcal{X}_{t}$ is a normal
projective surface with at worst canonical singularities follows thanks to
[Kaw99]. Now note that since
$h^{1}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-2,0))=h^{1}(T_{\mathbb{P}^{N}}|_{Y})=0$
(by Proposition 3.3 (3) and Lemma 3.2 (2)), we have that
$h^{1}(\varphi_{1}^{*}(T_{\mathbb{P}^{N}}))=0$ and ${\bf
Def_{\varphi_{1}}}\to{\bf Def}_{X_{1}}$ is smooth. Hence the map
$H^{0}(\mathcal{N}_{\varphi_{1}})\to H^{1}(T_{X_{1}})$ is surjective. By
[Sei92], Lemma $12$, there exist an open set in $H^{1}(T_{X_{1}})$ such that
for a smooth curve along a first order deformation belonging to the open set a
general deformation of $X_{1}$ along the curve is an elliptic ruled surface
which is the projectivization of a split rank two vector bundle with invariant
$e=0$. Also $H^{0}(\mathcal{N}_{\varphi_{1}})\to
H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2,0))$ is
surjective and there exist an open set of non-zero elements in
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2,0))$. Hence one
can choose an element (in fact an open set of elements) from
$H^{0}(\mathcal{N}_{\varphi_{1}})$ such that it maps to a non-zero element in
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(-2,0))$ and the
general induced deformation $\mathcal{X}_{1t}$ of $X_{1}$ is an elliptic ruled
surface which is the projectivization of a split rank two vector bundle with
invariant $e=0$. Finally to check hypothesis ($d$), note that
$H^{1}(L^{\otimes 2})=H^{1}(B)=0$ by Proposition 4.4, ($1$).

Note that $\mathcal{E}_{\pi_{1}}=p_{1}^{*}(\mathcal{O}_{Y}(-C_{0}-(m+2)f))$.
Let $\mathcal{E}_{p_{1}}=\mathcal{O}_{Y}(-2C_{0})$. Then

$\omega_{X_{1}}=p_{1}^{*}(\omega_{Y}\otimes\mathcal{O}_{Y}(2C_{0}))=p_{1}^{*}(\mathcal{O}_{Y}(-2C_{0}-2f)\otimes\mathcal{O}_{Y}(2C_{0}))=p_{1}^{*}(-2f).$

Thus, we obtain
$\omega_{X_{1}}\otimes\varphi_{1}^{*}(\mathcal{O}_{\mathbb{P}^{N}}(-1))=p_{1}^{*}(-2f)\otimes
p_{1}^{*}(-C_{0}-mf)=p_{1}^{*}(\mathcal{O}_{Y}(-C_{0}-(m+2)f))=\mathcal{E}_{\pi_{1}}$.
Hence by Theorem 2.13, we can take $\Phi_{t}$ to be the canonical morphism of
$X_{t}$.

The second assertion follows from Corollary 2.16. The existence of an
algebraic formally semiuniversal deformation space of $\varphi$ follows from
Remark 2.17. Since $X$ has at worst $A_{1}$ singularities, $B$ has normal
crossings, so it is easy to check that $H^{1}(\mathcal{N}_{\pi_{1}})=0$ (which
we have verified Proposition 4.4 (1)). Consequently, Corollary 2.19 shows that
any deformation of $X$ is locally trivial. To finish the proof, we need to
show that $H^{0}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}})=0$
where $\mathcal{E}_{\pi_{1}}$ is the trace zero module of $\pi_{1}$. We make
use of the fact that $X_{1}=E\times\mathbb{P}^{1}$ for an elliptic curve $E$.
Recall that $\psi:E\to\mathbb{P}^{1}$ is the morphism induced by the
restriction of $p_{1}$, satisfies
$\psi_{*}\mathcal{O}_{E}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-2)$.
We have

${\pi_{1}}_{*}\mathcal{O}_{X}=\mathcal{O}_{X_{1}}\oplus(\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(-m-2)).$

Also recall that
$T_{X_{1}}=(\mathcal{O}_{E}\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(2))\oplus\mathcal{O}_{X_{1}}$
and
$\mathcal{E}_{\pi_{1}}=\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(-m-2)$.
It is easy to check that $H^{1}(T_{X_{1}}\otimes\mathcal{E}_{\pi_{1}})=0$. One
has the following pullback of the Euler sequence;

(3.4) $0\to\mathcal{O}_{X_{1}}\to p_{1}^{*}\mathcal{O}_{Y}(1,m)^{\oplus
N+1}\to\varphi_{1}^{*}T_{\mathbb{P}^{N}}\to 0.$

We tensor (3.4) by $\mathcal{E}_{\pi_{1}}$ and take the long exact sequence of
cohomology. Notice that $H^{1}(\mathcal{E}_{\pi_{1}})=0$, and
$H^{0}(p_{1}^{*}\mathcal{O}_{Y}(1,m)\otimes\mathcal{E}_{\pi_{1}})=0$,
consequently
$H^{0}(\varphi_{1}^{*}T_{\mathbb{P}^{N}}\otimes\mathcal{E}_{\pi_{1}})=0$. Now
consider the exact sequence;

(3.5) $0\to
T_{X_{1}}\to\varphi_{1}^{*}T_{\mathbb{P}^{N}}\to\mathcal{N}_{\varphi_{1}}\to
0.$

It follows from the long exact sequence of cohomology that
$H^{0}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}})=0$. $\square$

Before moving on to the next case, we make a remark that will help us to see
that for these surfaces $H^{1}(\mathcal{N}_{\varphi})\neq 0$.

######  Remark 3.5.

It follows from the vanishing of
$H^{1}(T_{X_{1}}\otimes\mathcal{E}_{\pi_{1}})$ and the long exact sequence
associated to (3.5) that
$h^{1}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}})\geq
h^{1}(\varphi_{1}^{*}T_{\mathbb{P}^{N}}\otimes\mathcal{E}_{\pi_{1}})\geq
h^{1}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-1,-m-2))=N+1$, where the
last equality follows from (3.2). $\blacksquare$

######  Corollary 3.6.

Let $X$ be a surface of type $(1)_{m}$ or $(1^{\prime})_{m}$. If $X$ is smooth
or has $A_{1}$ singularities, then there exists an irreducible component
$U_{\varphi}$ of the of the algebraic formally semiuniversal deformation space
of ${\varphi}$ (that exists by Remark 2.17) whose general elements are two-to-
one morphisms onto their image whose normalization is an elliptic ruled
surface with invariant $e=0$. Further, there does not exist any component of
the algebraic formally semiuniversal deformation space of ${\varphi}$ whose
general elements are morphisms that are birational onto their image.

Proof. Since the curve constructed in Corollary 3.4 is irreducible, it is
contained in an irreducible component. Now the assertion follows by applying
semicontinuity to the reduced induced structure of the irreducible component
(note that a general closed point of an irreducible scheme is the same as a
general closed point of its reduced induced structure). $\square$

The corollary of the following proposition shows that the image of a general
morphism in the irreducible component $U_{\varphi}$ constructed above is
necessarily non-normal. We remark that what we prove in the following
proposition is a slightly stronger statement than what we need in order to
prove Corollary 3.8; to prove Corollary 3.8, we only need the conclusion of
the following proposition for $e=0$.

######  Proposition 3.7.

There does not exist a surface of general type $X^{\prime}$ with at worst
canonical singularities and $K_{X^{\prime}}^{2}=4p_{g}(X^{\prime})-8$ that
satisfies both of the following properties.

  * (1)

There exist an ample and base point free line bundle
$K\equiv\omega_{X^{\prime}}$ with $h^{0}(K)=p_{g}(X^{\prime})$.

  * (2)

The morphism $\varphi^{\prime}$ induced by the complete linear series $|K|$ is
two-to-one onto its image which is a smooth elliptic ruled surface with
invariant $e\geq 0$.

Proof. Suppose there exists such a surface $X^{\prime}$ with a numerically
canonical bundle $K$ satisfying the properties in the proposition. Let the
image of the morphism $\varphi$ given by $|K|$ be $W$ so that the morphism
$\varphi$ factors as

$X^{\prime}\xrightarrow{\pi^{\prime}}W\hookrightarrow\mathbb{P}^{N^{\prime}}$

where $N^{\prime}+1=p_{g}(X^{\prime})$. Let the very ample line bundle on $W$
be denoted by $aC_{0}^{\prime}+bf^{\prime}$. Note that we have
$\varphi^{\prime*}(\mathcal{O}_{\mathbb{P}^{N^{\prime}}}(1))=K$. The morphism
is induced by the complete linear series and hence

$h^{0}(K)=h^{0}(aC_{0}^{\prime}+bf^{\prime})+h^{0}((aC_{0}^{\prime}+bf^{\prime})\otimes\mathcal{E}_{\pi^{\prime}})=N^{\prime}+1,$

where $\mathcal{E}_{\pi^{\prime}}$ is the trace zero module of $\pi^{\prime}$.
But now $\mathcal{E}_{\pi^{\prime}}\equiv
K_{W}\otimes(-aC_{0}^{\prime}-bf^{\prime})$. Hence
$h^{0}((aC_{0}^{\prime}+bf^{\prime})\otimes\mathcal{E}_{\pi^{\prime}})=0$
which gives $h^{0}(aC_{0}^{\prime}+bf^{\prime})=p_{g}(X^{\prime})$. Since
$h^{i}(aC_{0}^{\prime}+bf^{\prime})=0$ for $i=1,2$ (see for example [GP96],
Proposition 3.1), we obtain by Riemann-Roch

$\frac{1}{2}(-a^{2}e-ae+2ab+2b)=p_{g}(X^{\prime}).$

Now note that
$K=\varphi^{*}(\mathcal{O}_{\mathbb{P}^{N}}(1))=\pi^{*}(aC_{0}^{\prime}+bf^{\prime})$.
Hence $K^{2}=2(aC_{0}^{\prime}+bf^{\prime})^{2}=2(-a^{2}e+2ab)$. Using the
relation $K^{2}=4p_{g}(X^{\prime})-8$, we obtain

$2(-a^{2}e+2ab)=2(-a^{2}e-ae+2ab+2b)-8.$

This gives $2b-4=ae$. But very ampleness of $aC_{0}^{\prime}+bf^{\prime}$
implies $a\geq 1$, $b\geq ae+3$ which implies $-ae\geq 2$ which is a
contradiction if $e\geq 0$. $\square$

######  Corollary 3.8.

Consider the irreducible component $U_{\varphi}$ obtained in Corollary 3.6.
There exist an open set $U_{\varphi}^{0}\subseteq U_{\varphi}$ such that for a
closed point $t\in U_{\varphi}^{0}$, Im($\varphi_{t}$) is non-normal, whose
normalization is an elliptic ruled surface with $e=0$ which is the
projectivization of a rank two split vector bundle on the elliptic curve.

Proof. Since we are concerned with closed points $t\in U_{\varphi}$, we can
take the reduced induced structure of $U_{\varphi}$ and consider the pullback
of the formally semiuniversal family. Thus, without loss of generality, one
can assume that $U_{\varphi}$ is integral. Let
$\mathcal{X}\xrightarrow{\Phi}\mathbb{P}_{U_{\varphi}}^{N}\to U_{\varphi}$ be
the algebraic formally semiuniversal family of $\varphi$. Since the forgetful
map ${\bf Def_{\pi_{1}/\mathbb{P}^{N}}}\to{\bf Def_{\varphi}}$ is smooth we
have that the above deformation factors as
$\mathcal{X}\xrightarrow{\Pi_{1}}\mathcal{X}_{1}\xrightarrow{\Phi_{1}}\mathbb{P}_{U_{\varphi}}^{N}\to
U_{\varphi}$ ($\Phi=\Pi_{1}\circ\Phi_{1}$). Let
$\mathcal{Y}=\textrm{Im}(\Phi)=\textrm{Im}(\Phi_{1})$. Since $\mathcal{X}$ is
integral, $\mathcal{Y}$ is integral. Since $U_{\varphi}$ is integral, we have
by generic flatness that $\mathcal{Y}\to U_{\varphi}$ is flat (after possibly
shrinking $U_{\varphi}$). Consider the induced deformation $\mathcal{X}_{1}\to
U_{\varphi}$. By our choice of $U_{\varphi}$, we have that there exist $t\in
U_{\varphi}$ such that $\mathcal{X}_{1t}$ is an elliptic ruled surface which
is the projectivization of a rank two split vector bundle on the elliptic
curve with invariant $e=0$. Then by [Sei92], Lemma $12$, we have that for a
general $t\in U_{\varphi}$, $\mathcal{X}_{1t}$ has the same property. Now for
a general $t\in U_{\varphi}$, Im$(\Phi_{t})$ =
Im$(\Phi_{1t})=\mathcal{Y}_{t}$. Also for a general $t\in U_{\varphi}$, we
have that $\mathcal{X}_{1t}\to\mathcal{Y}_{t}$ is the normalization map.
Assume for a general $t$, that $\mathcal{Y}_{t}$ is smooth. Then
$\mathcal{Y}_{t}\cong\mathcal{X}_{1t}=\textrm{Im}(\varphi_{t})$. But this is a
contradiction to Proposition 3.7. $\square$

###  3.2. Deformation of canonical morphism for type $(2)_{m}$ ($m\geq 2$)

Now we do analogous calculations for surfaces type $(2)_{m}$. In this case, We
have the following diagram where
${p_{1}}_{*}\mathcal{O}_{X_{1}}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-3,-1)$
and
${p_{2}}_{*}\mathcal{O}_{X_{2}}=\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(0,-m-1)$.

(3.6)
${X}$${X_{1}}$${X_{2}}$${Y}$${\mathbb{P}^{N}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{i}$

Notice $X_{2}=\mathbb{P}^{1}\times C$ for a smooth curve $C$ that is a double
cover $\psi:C\to\mathbb{P}^{1}$ with
$\psi_{*}\mathcal{O}_{C}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-m-1)$.
Set $\varphi_{j}=p_{j}\circ i$ for $j=1,2$ and call $B$ the branch divisor of
$\pi_{2}$.

######  Proposition 3.9.

Let $X$ be a surface of type $(2)_{m}$ ($m\geq 2$). Then the following
happens;

  * (1)

$h^{1}(\mathcal{O}_{X_{2}})=m$ and $h^{2}(\mathcal{O}_{X_{2}})=0$,

  * (2)

$h^{0}(\mathcal{N}_{p_{2}})=2m+2$ and $h^{1}(\mathcal{N}_{p_{2}})=0$,

  * (3)

$h^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(0,-m-1))=h^{1}(\mathcal{O}_{Y}(0,-m-1))=m$
and $h^{1}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(0,-m-1))=0$,

  * (4)

$h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(0,-m-1))=5m-2$ and
$h^{1}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(0,-m-1))=0$,

  * (5)

$h^{0}(\mathcal{N}_{\varphi_{2}})=(N+1)^{2}+7m-7$ and
$h^{1}(\mathcal{N}_{\varphi_{2}})=0$ where $\varphi_{2}=p_{2}\circ i$;
consequently, $\varphi_{2}$ is unobstructed.

Proof. ($1$) Follows from
$h^{j}(\mathcal{O}_{X_{2}})=h^{j}(\mathcal{O}_{Y})\oplus
h^{j}(\mathcal{O}_{Y}(0,-m-1)$ and Künneth formula.

($2$) We apply Proposition 2.4. Since $Y$ is regular, it follows that
$h^{0}(\mathcal{N}_{p_{0}})=h^{0}(\mathcal{O}_{Y}(0,2m+2))-1=2m+2$.
Furthermore, since $H^{2}(\mathcal{O}_{Y})=0$, we obtain
$h^{1}(\mathcal{N}_{p_{2}})=h^{1}(\mathcal{O}_{Y}(0,2m+2))=0$.

($3$) The assertion follows by tensoring the exact sequence (3.2) by
$\mathcal{O}_{Y}(0,-m-1)$ and taking the long exact sequence of cohomology.

($4$) This is obtained by tensoring (3.1) by $\mathcal{O}_{Y}(0,-m-1)$ and
taking cohomology.

($5$) As before, the assertion follows from the following exact sequence (see
Lemma 2.2)

$0\to\mathcal{N}_{p_{2}}\to\mathcal{N}_{\varphi_{2}}\to
p_{2}^{*}\mathcal{N}_{Y/\mathbb{P}^{N}}\to 0.$

Since $h^{1}(\mathcal{N}_{p_{2}})=0$, we obtain
$h^{0}(\mathcal{N}_{\varphi_{2}})=h^{0}(\mathcal{N}_{p_{2}})+h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}})+h^{0}(\mathcal{N}_{Y/\mathbb{P}^{N}}\otimes\mathcal{O}_{Y}(0,-m-1))$.
We get the value of $h^{0}(\mathcal{N}_{\varphi_{2}})$ from part (2), (4) and
Lemma 3.2. Finally, $h^{1}(\mathcal{N}_{\varphi_{2}})=0$ by part (2), (4) and
Lemma 3.2. $\square$

######  Corollary 3.10.

Let $X$ be a surface of type $(2)_{m}$ ($m\geq 2$) which is smooth or has
$A_{1}$ singularities.

  * (1)

Suppose $m=2$, then there exists a smooth, affine irreducible algebraic curve
$T$ and a flat family of morphisms $\Phi:\mathcal{X}\to\mathbb{P}^{N}_{T}$
over $T$ for which the following happens;

    * (a)

$\Phi_{t}:\mathcal{X}_{t}\to\mathbb{P}^{5}$ is a morphism of degree two from a
normal projective surface with at worst canonical singularities for all $t\in
T-\\{0\\}$. Further for any $t\in T-\\{0\\}$, the normalization of
Im($\varphi_{t}$) is a ruled surface over a smooth curve of genus $2$ and has
invariant $e=-2$. Further one can take $\Phi_{t}$ to be the canonical morphism
of $\mathscr{X}_{t}$.

    * (b)

$\Phi_{0}:\mathcal{X}_{0}\to\mathbb{P}^{5}$ is the canonical morphism
$\varphi:X\to\mathbb{P}^{N}$.

  * (2)

The forgetful map from ${\bf Def_{\pi_{2}/\mathbb{P}^{N}}}\to{\bf
Def_{\varphi}}$ is smooth and hence any deformation of $\varphi$ will be a
morphism of degree $\geq 2$ onto its image. Hence in particular

    * (a)

$\varphi$ cannot be deformed to a birational morphism;

    * (b)

a general deformation of $\varphi$ is a morphism of degree $2$ onto its image
if $m=2$.

Proof. (1) The existence of the curve $T$ so that for a general $t\in T$,
$\Phi_{t}$ has degree two follows from Theorem 2.13, Proposition 3.9 and
[Kaw99]. Note that since
$h^{1}(T_{\mathbb{P}^{5}}|_{Y}\otimes\mathcal{O}_{Y}(0,-3))=h^{1}(T_{\mathbb{P}^{5}}|_{Y})=0$
(by Proposition 3.9 (3), Lemma 3.2 (2)), we have that
$h^{1}(\varphi_{2}^{*}(T_{\mathbb{P}^{5}}))=0$ and ${\bf
Def_{\varphi_{2}}}\to{\bf Def}_{X_{2}}$ is smooth. Hence the map
$H^{0}(\mathcal{N}_{\varphi_{2}})\to H^{1}(T_{X_{2}})$ is surjective. By
[Sei92], Lemma $12$, there exist an open set in $H^{1}(T_{X_{2}})$ such that
for a smooth curve along a first order deformation belonging to the open set a
general deformation of $X_{2}$ along the curve is a ruled surface over a
smooth curve of genus $2$ with invariant $e=-2$. Also
$H^{0}(\mathcal{N}_{\varphi_{2}})\to
H^{0}(\mathcal{N}_{Y/\mathbb{P}^{5}}\otimes\mathcal{O}_{Y}(0,-3))$ is
surjective and there exist an open set of non-zero elements in
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{5}}\otimes\mathcal{O}_{Y}(0,-3))$. Hence one
can choose an element (in fact an open set of elements) from
$H^{0}(\mathcal{N}_{\varphi_{2}})$ such that it maps to a non-zero element in
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{5}}\otimes\mathcal{O}_{Y}(0,-3))$ and the
general induced deformation $\mathcal{X}_{2t}$ of $X_{2}$ is a ruled surface
over a smooth curve of genus $2$ with invariant $e=-2$. The trace zero module
of the finite map $\pi_{2}$ is
$\mathcal{E}_{\pi_{2}}=\omega_{X_{2}}\otimes\varphi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{N}}(-1))$;
the proof of this statement follows exactly as in Proposition 3.3, ($6$). The
fact that $\Phi_{t}$ can be taken to be the canonical morphism of $X_{t}$
follows from Theorem 2.13.

(2) Recall that $X_{2}=\mathbb{P}^{1}\times C$ for a smooth curve $C$ that is
a double cover $\psi:C\to\mathbb{P}^{1}$ with
$\psi_{*}\mathcal{O}_{C}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-m-1)$.
We have the following splitting of ${\pi_{2}}_{*}\mathcal{O}_{X}$:

${\pi_{2}}_{*}\mathcal{O}_{X}=\mathcal{O}_{X_{2}}\oplus(\mathcal{O}_{\mathbb{P}^{1}}(-3)\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)).$

Also,
$T_{X_{2}}=(\mathcal{O}_{\mathbb{P}^{1}}\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(1-m))\oplus\mathcal{O}_{\mathbb{P}^{1}}(2)\boxtimes\mathcal{O}_{C}$,
and
$\mathcal{E}_{\pi_{2}}=\mathcal{O}_{\mathbb{P}^{1}}(-3)\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)$
is the trace zero module of $\pi_{2}$. It is easy to check that
$H^{1}(T_{X_{2}}\otimes\mathcal{E}_{\pi_{2}})=0$. One has the following
pullback of the Euler sequence;

$0\to\mathcal{O}_{X_{2}}\to p_{2}^{*}\mathcal{O}_{Y}(1,m)^{\oplus
N+1}\to\varphi_{2}^{*}T_{\mathbb{P}^{N}}\to 0.$

By tensoring the above exact sequence by $\mathcal{E}_{\pi_{1}}$ and taking
cohomology, we obtain
$H^{0}(\varphi_{2}^{*}T_{\mathbb{P}^{N}}\otimes\mathcal{E}_{\pi_{2}})=0$.
Consequently, the exact sequence;

$0\to
T_{X_{2}}\to\varphi_{2}^{*}T_{\mathbb{P}^{N}}\to\mathcal{N}_{\varphi_{2}}\to
0.$

shows that $H^{0}(\mathcal{N}_{\varphi_{2}}\otimes\mathcal{E}_{\pi_{2}})=0$,
and the assertion follows from Proposition 2.15 (and Corollary 2.16).
$\square$

The following corollary is analogous to Corollary 3.6 and follows immediately
from Corollary 3.10.

######  Corollary 3.11.

Let $X$ be a surface of type $(2)_{m}$ $(m\geq 2)$ which is smooth or has
$A_{1}$ singularities.

  * (1)

If $m=2$, there exists an irreducible component $U_{\varphi}$ of the algebraic
formally semiuniversal deformation space of ${\varphi}$ (that exists by Remark
2.17) whose general elements are a two-to-one morphisms onto their image,
whose normalization is a ruled surface over a smooth curve of genus $2$ and
has invariant $e=-2$.

  * (2)

There does not exist any component of the algebraic formally semiuniversal
deformation space of ${\varphi}$ whose general elements are morphisms that are
birational onto their image.

The following propositions and corollary show that the image of a general
morphism in the irreducible component constructed above is smooth. It also
shows that this open set intersects the locally closed subloci where the
deformed morphism is again the canonical morphism.

Recall that for a ruled surface $X\to C$ of invariant $e$ over a smooth curve
of genus $g\geq 1$, we denote $C_{0}$ and $f$ denote the numerical classes of
a section and a fibre respectively satisfying $C_{0}^{\prime 2}=-e,f^{\prime
2}=0$ and $C_{0}^{\prime}\cdot f^{\prime}=1$.

######  Proposition 3.12.

Suppose that there exist a flat family $(\mathcal{X}_{2}\to T,\mathcal{L})$ of
polarized surfaces ruled over a curve $C$ of genus $m$ over a smooth one
dimensional base $T$ with

  * (1)

$\mathcal{X}_{20}$ has invariant $0$ and $\mathcal{L}_{0}\equiv
C_{0}^{\prime}+2mf^{\prime}$

  * (2)

$\mathcal{X}_{2t}$ has invariant $-m$ or $-(m-1)$ accordingly as $m$ is even
or odd.

Then after possibly shrinking $T$, $\mathcal{L}_{t}\equiv
C_{0}^{\prime}+\frac{3m}{2}f^{\prime}$ if $m$ is even and
$\mathcal{L}_{t}\equiv C_{0}^{\prime}+\frac{3(m-1)}{2}f^{\prime}$ if $m$ is
odd.

Proof. We prove the statement for $m$ even. The proof is identical for $m$
odd. Let $\mathcal{X}_{20}=X_{2}$. Let $X_{2}\xrightarrow{q}C$ be the smooth
morphism. Consider the following commutative diagram.

${\dots}$${T^{1}(X_{2}/C)=H^{1}(T_{X_{2}/C})}$${T^{1}(q)}$${H^{1}(T_{C})}$${T^{2}(X_{2}/C)=H^{2}(T_{X_{2}/C})}$${T^{2}(q)}$${H^{2}(T_{C})}$${\dots}$${\dots}$${T^{1}(X_{2}/C)=H^{1}(T_{X_{2}/C})}$${H^{1}(T_{X_{2}})}$${H^{1}(q^{*}(T_{C}))}$${T^{2}(X_{2}/C)=H^{2}(T_{X_{2}/C})}$${H^{2}(T_{X_{2}})}$${H^{2}(q^{*}(T_{C}))}$${\dots}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\beta_{2}}$

Note that since $q_{*}(\mathcal{O}_{X_{2}})=\mathcal{O}_{C}$,
$H^{i}(q^{*}(T_{C}))=H^{i}(T_{C})$ for $i=1,2$. Hence the maps $\beta_{1}$ is
surjective and the map $\beta_{2}$ is injective and therefore $\alpha_{1}$ is
surjective and $\alpha_{2}$ is injective. This implies that the forgetful map
${\bf Def}_{q}\to{\bf Def}_{X_{2}}$ is smooth. Hence there exist a deformation
of $\mathcal{C}\to T$ of $C$ so that $\mathcal{X}_{2}\to T$ factors as
$\mathcal{X}_{2}\xrightarrow{Q}\mathcal{C}\to T$. Fix a line bundle
$\mathcal{O}_{C}(1)$ of degree one on $C$. Since $H^{2}(\mathcal{O}_{C})=0$,
we have that the line bundle $\mathcal{O}_{C}(1)$ lifts to a line bundle
$\mathcal{O}_{\mathcal{C}}(1)$. Now the numerical class of
$Q^{*}(\mathcal{O}_{\mathcal{C}}(1))$ restricts to the numerical class of
$f^{\prime}$ on the central fibre and it is the pullback of a degree one line
bundle on $\mathcal{C}_{t}$ on a general fibre. Hence the numerical class of
$f^{\prime}$ on $X_{2}$ deforms to the numerical class of $f^{\prime}$ on
$\mathcal{X}_{2t}$. Now suppose that a line bundle of numerical class
$C_{0}^{\prime}$ in $X_{2}$ deforms to a line bundle of numerical class
$aC_{0}^{\prime}+bf^{\prime}$ on $\mathcal{X}_{2t}$ for $t\neq 0$. Using the
fact that their self intersections are the same and noting that $C_{0}^{\prime
2}=0$ on $X_{2}$ while $C_{0}^{\prime 2}=m$ on $\mathcal{X}_{2t}$, we have
that $a^{2}m+2ab=0$. Suppose that $a=0$. Then for sufficiently large $k$,
$C_{0}^{\prime}+kf^{\prime}$ which is very ample on $X_{20}$ deforms to
$(k+b)f^{\prime}$ which is not ample. Hence $a\neq 0$ and $b=\frac{-am}{2}$.
Then $C_{0}^{\prime}$ on $X_{20}$ deforms to
$a(C_{0}^{\prime}-\frac{m}{2}f^{\prime})$ on $\mathcal{X}_{2t}$. Considering
that on $\mathcal{X}_{20}$, $C_{0}^{\prime}\cdot f^{\prime}=1$, we have that
$a=1$. Hence our statement is proven. $\square$

We prove a slightly stronger version of a result we need to prove Corollary
3.14. More precisely we will use the result proven below for $m=2$.

######  Proposition 3.13.

Suppose $X$ is a ruled surface over a curve $C$ of genus $m$ with invariant
$-m$ or $-(m-1)$ accordingly as $m$ is even or odd. Then a line bundle
$L\equiv C_{0}+\frac{3m}{2}f$ is very ample if $m$ is even and a line bundle
$L\equiv C_{0}+\frac{3(m-1)}{2}f$ is very ample if $m$ is odd.

Proof. We use the following criterion for very ampleness (see [LM05] Corollary
$2.13$): let $|H|$ be the complete linear series of a line bundle $H\equiv
C_{0}+bf$. Then $|H|$ is very ample if and only for any two points $P$ and $Q$
on $C$, $h^{0}(H-(P+Q)f)=h^{0}(H)-4$.

Let $m$ be even and consider $L\equiv C_{0}+\frac{3m}{2}f$ on a ruled surface
with invariant $-m$ over a curve of genus $m$ with $m\geq 2$. We need to show,
for any two points $P$ and $Q$ on $C$,

$h^{0}(L-(P+Q)f)=h^{0}(L)-4.$

Let $X=\mathbb{P}(\mathcal{E})$ where $\mathcal{E}$ is normalized and since
$e<0$, we have that $\mathcal{E}$ is stable. Since higher pushforward of any
bundle of numerical equivalence class $C_{0}+bf$ is zero we can compute the
above cohomology by pushing forward to the base curve of genus $m$. Notice
that $L-(P+Q)f\equiv C_{0}+(\frac{3m}{2}-2)f$. Hence it is enough to show that
for any two line bundles $L_{1}$ and $L_{2}$ on $C$ with degree $3m/2-2$ and
$3m/2$ respectively, we have

$h^{0}(L_{1}\otimes\mathcal{E})=h^{0}(L_{2}\otimes\mathcal{E})-4.$

For simplicity, let us denote $L_{1}$ by $\mathcal{O}_{C}(\frac{3m}{2}-2)$,
and $L_{2}$ by $\mathcal{O}_{C}(\frac{3m}{2})$. Applying Riemann-Roch to the
vector bundles $\mathcal{O}_{C}(\frac{3m}{2}-2)\otimes\mathcal{E}$ and
$\mathcal{O}_{C}(\frac{3m}{2})\otimes\mathcal{E}$ and taking subtracting we
get

$h^{0}\left(\mathcal{E}\left(\frac{3m}{2}\right)\right)-h^{0}\left(\mathcal{E}\left(\frac{3m}{2}-2\right)\right)=h^{1}\left(\mathcal{E}\left(\frac{3m}{2}\right)\right)-h^{1}\left(\mathcal{E}\left(\frac{3m}{2}-2\right)\right)+c_{1}\left(\mathcal{E}\left(\frac{3m}{2}\right)\right)-c_{1}\left(\mathcal{E}\left(\frac{3m}{2}-2\right)\right).$

Since $\mathcal{E}$ is of rank two we have that
$c_{1}(\mathcal{E}(\frac{3m}{2})-c_{1}(\mathcal{E}(\frac{3m}{2}-2))=4$. Hence
we are done if we show

$h^{1}\left(\mathcal{E}\left(\frac{3m}{2}\right)\right)=h^{1}\left(\mathcal{E}\left(\frac{3m}{2}-2\right)\right)=0.$

Note that
$h^{1}(\mathcal{E}(\frac{3m}{2}))=h^{0}(\mathcal{E}^{*}(\frac{m}{2}-2)$. Now
the slope of the vector bundle $\mu(\mathcal{E}^{*}(\frac{m}{2}-2))=-2$. Also
since $\mathcal{E}$ is stable we have that $\mathcal{E}^{*}(\frac{m}{2}-2)$ is
stable. Since its slope is negative we have that
$h^{0}(\mathcal{E}^{*}(\frac{m}{2}-2))=0$.

Now note that
$h^{1}(\mathcal{E}(\frac{3m}{2}-2))=h^{0}(\mathcal{E}^{*}(\frac{m}{2}))$. Note
that $\textrm{deg}(\mathcal{E}^{*}(\frac{m}{2}))=0$ and
$\mathcal{E}^{*}(\frac{m}{2})$ is stable since $\mathcal{E}$ is stable. Then
$h^{0}(\mathcal{E}^{*}(\frac{m}{2}))=0$ since for degree $0$ vector bundles
the existence of a section contradicts stability. The proof for the case $m$
odd follows exactly along the same lines. $\square$

######  Corollary 3.14.

In Corollary 3.10 (1), we can choose the curve $T$ so that after possibly
shrinking $T$, for $t\in T$, $t\neq 0$, $\textrm{Im}(\Phi_{t})$ is smooth and
$\Phi_{t}$ can be taken to be the canonical morphism of $\mathcal{X}_{t}$.

Proof. We resume notations of Corollary 3.10. Consider the factorization
$X\xrightarrow{\pi_{2}}X_{2}\xrightarrow{p_{2}}Y\xrightarrow{i}\mathbb{P}^{N}$.
Let $\varphi_{2}=i\circ p_{2}$. Note that since $X_{2}=C\times\mathbb{P}^{1}$,
where $C$ is a smooth curve of genus $2$, we have that it is a ruled surface
over $C$ and has invariant $e=0$. Consider
$L=\varphi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{N}}(1))=p_{2}^{*}(\mathcal{O}_{Y}(C_{0}+2f))$
(recall $C_{0}$ and $f$ are the classes of a section and fibre of $Y$). Since
$X_{2}$ is a double cover branched along $8f$, we have that
$p_{2}^{*}(\mathcal{O}_{Y}(C_{0}))\equiv aC_{0}^{\prime}$ and
$p_{2}^{*}(\mathcal{O}_{Y}(f))\equiv bf^{\prime}$. We have $ab=2$. Then
setting $h^{0}(p_{2}^{*}\mathcal{O}_{Y}(C_{0}))=2$ we have that $a=1$ and
hence $b=2$. Hence $p_{2}^{*}(\mathcal{O}_{Y}(C_{0}+2f))\equiv
C_{0}^{\prime}+4f^{\prime}$.

Note that the pair $(X_{2},L)$ is unobstructed since
$h^{2}(\mathcal{E}_{L})=0$ (since
$h^{2}(\mathcal{O}_{X_{2}})=h^{2}(T_{X_{2}})=0$). Also since
$h^{2}(\mathcal{O}_{X_{2}})=0$, ${\bf Def}_{(X_{2},L)}\to{\bf Def}_{X_{2}}$ is
smooth. Choose a smooth curve $T$ from the smooth versal deformation space of
$(X_{2},L)$. Let $(\mathcal{X}_{2}\xrightarrow{\sigma}T,\mathcal{L})$ be the
family obtained. Then for a general such curve, for $t\neq 0$,
$\mathcal{X}_{2t}$ has invariant $-2$. By Proposition 3.12,
$\mathcal{L}_{t}\equiv C_{0}^{\prime}+3f^{\prime}$ which is very ample by
Proposition 3.13. Since $H^{1}(L)=0$, (easy to check by projection formula) we
have that (after shrinking $T$), $\sigma_{*}(\mathcal{L})$ is locally free of
rank $h^{0}(L)$ and we get a morphism
$\mathcal{X}_{2}\xrightarrow{\Phi_{2}}\mathbb{P}(\sigma_{*}(\mathcal{L}))\to
T$ which is an embedding for $t\neq 0$ since it is given by the complete
linear series of a line bundle numerically equivalent to
$C_{0}^{\prime}+3f^{\prime}$, which is very ample. Note that
$\mathcal{E}_{\pi_{2}}=\omega_{X_{2}}\otimes\varphi_{2}^{*}(L^{*})$ and let
$B\in H^{0}(-2(\omega_{X_{2}}\otimes\varphi_{2}^{*}(L^{*})))$ be the divisor
giving $\pi_{2}$. Note that
$-2(\omega_{\mathcal{X}_{2}/T}\otimes\Phi_{2}^{*}(\mathcal{O}_{\mathbb{P}_{T}^{N}}(-1)))$
is a lift of $-2(\omega_{X_{2}}\otimes\varphi_{2}^{*}(L^{*}))$. Then by Remark
2.11 one can construct a lift $\mathcal{B}$ of $B$ and hence a relative double
cover $\Pi_{2}$ since $H^{1}(\mathcal{O}_{X_{2}}(B))=0$ (easy to check, see
the proof of Proposition 4.6 (1)). Consider $\Phi=\Pi_{2}\circ\Phi_{2}$. For
$t\neq 0$, $\Phi_{t}$ is the composition of a double cover $\Pi_{2t}$ followed
by an embedding $\Phi_{2t}$ of a smooth surface given by the complete linear
series of a very ample line bundle $\mathcal{L}_{t}$. Moreover $\Pi_{2t}$ is
branched along $-2(\omega_{\mathcal{X}_{2t}}\otimes\mathcal{L}_{t}^{*})$.
Hence $\Phi_{t}$ is the canonical morphism of $\mathcal{X}_{t}$ (by Remark
2.12) and its image is smooth. $\square$

###  3.3. Deformation of canonical morphism for type $(3)_{m}$

Note that the surfaces of type $(3)_{m}$ are of the form $C_{1}\times C_{2}$
where $C_{1}$ is a smooth hyperelliptic curve of genus $2$ and $C_{2}$ is a
smooth hyperelliptic curve of genus $m+1$. Let $\psi_{1}$ and $\psi_{2}$ be
the canonical morphisms of $C_{1}$ and $C_{2}$ respectively. Notice that the
morphism $\varphi$ is the morphism $\psi:C_{1}\times
C_{2}\xrightarrow{}\mathbb{P}^{1}\times\mathbb{P}^{m}$, where
$\psi=\psi_{1}\times\psi_{2}$, composed by the Segre embedding
$\mathbb{P}^{1}\times\mathbb{P}^{m}\hookrightarrow\mathbb{P}^{2m+1}$ given by
the complete linear series
$|\mathcal{O}_{\mathbb{P}^{1}}(1)\boxtimes\mathcal{O}_{\mathbb{P}^{m}}(1)|$.
Thus, we have the following diagram.

(3.7) ${X=C_{1}\times
C_{2}}$${Y=\mathbb{P}^{1}\times\mathbb{P}^{1}}$${Z=\mathbb{P}^{1}\times\mathbb{P}^{m}}$${\mathbb{P}^{N}}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$

We have
${\psi_{1}}_{*}\mathcal{O}_{C_{1}}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-3)$.
Further, $\psi_{2}=i^{\prime}\circ\psi_{2}^{\prime}$ where
$\psi_{2}^{\prime}:C_{2}\to\mathbb{P}^{1}$ is a double cover satisfying
${\psi_{2}^{\prime}}_{*}\mathcal{O}_{C_{2}}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-m-2)$,
and $i^{\prime}:\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{m}$ is embedding. We
also have the following diagram

(3.8)
$X\xrightarrow{\pi_{2}}X_{2}\xrightarrow{p_{2}}Y\hookrightarrow\mathbb{P}^{N}.$

Notice that $X_{2}=\mathbb{P}^{1}\times C_{2}$, and set $\varphi_{2}:=i\circ
p_{2}$.

######  Corollary 3.15.

Let $X$ be a (smooth) surface of type $(3)_{m}$ and let $U_{\varphi}$ is the
algebraic formally semiuniversal deformation space of ${\bf Def}_{\varphi}$.

  * (1)

Assume $m=1$. Then any deformation of $\varphi$ is a morphism of degree 4 onto
its image which is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

  * (2)

If $m\geq 2$ then there exists a smooth, affine irreducible algebraic curve
$T$ and a flat family of morphisms $\Phi:\mathcal{X}\to\mathbb{P}^{N}_{T}$
over $T$ for which;

    * (a)

$\Phi_{t}:\mathcal{X}_{t}\to\mathbb{P}^{2m+1}$ is a morphism of degree 2 onto
its image which is isomorphic to $\mathbb{P}^{1}\times C$ for a smooth non-
hyperelliptic curve $C$ of genus $m+1$ for all $t\in T-\\{0\\}$,

    * (b)

$\Phi_{0}:\mathcal{X}_{0}\to\mathbb{P}^{2m+1}$ is the canonical morphism
$\varphi:X\to\mathbb{P}^{N}$.

Moreover, the forgetful map from ${\bf Def_{\pi_{2}/\mathbb{P}^{N}}}\to{\bf
Def_{\varphi}}$ is smooth and hence any deformation of $\varphi$ will be a
morphism of degree $\geq 2$ onto its image.

Proof. (1) Notice that when $m=1$, $Y$ is a quadric hypersurface in
$\mathbb{P}^{3}$ (i.e., $N=3$), and
$\mathcal{N}_{Y/\mathbb{P}^{3}}=\mathcal{O}_{Y}(2,2)$. Notice that Defφ has an
algebraic formally semiuniversal deformation space. It is enough to prove that
Def${}_{\pi/\mathbb{P}^{3}}\to$Defφ is smooth. By Proposition 2.15, we just
need the vanishing of
$H^{0}(\mathcal{N}_{Y/\mathbb{P}^{3}}\otimes\mathcal{E})$. It is easy to see
this vanishing since
$\mathcal{E}=\mathcal{O}_{Y}(-3,0)\oplus\mathcal{O}_{Y}(0,-3)\oplus\mathcal{O}_{Y}(-3,-3)$.

(2) Choose a smooth algebraic curve $T$ for which there is a deformation
$\Psi:\mathcal{C}\to\mathbb{P}_{T}^{m}$ satisfying the following two
conditions (it exists because $m\geq 2$);

  * (a)

$\Psi_{t}:\mathcal{C}_{t}\to\mathbb{P}^{m}$ is the canonical embedding of a
smooth non-hyperelliptic curve of genus $m+1$ for $t\neq 0$.

  * (b)

$\Psi_{0}:\mathcal{C}_{0}\to\mathbb{P}^{m}$ is the canonical morphism
$\psi:C_{2}\to\mathbb{P}^{m}$.

We define $\Phi$ to be the composition
$C_{1}\times\mathcal{C}\xrightarrow{\psi_{1}\times\Psi}\mathbb{P}^{1}\times\mathbb{P}_{T}^{m}\to\mathbb{P}_{T}^{2m+1}$
where the last morphism is given by the relatively very ample line bundle
$\mathcal{O}_{\mathbb{P}^{1}}(1)\boxtimes\mathcal{O}_{\mathbb{P}_{T}^{m}}(1)$,
and the first assertion follows.

To see the second assertion, we need to show that the map
Def${}_{\pi_{2}/\mathbb{P}^{N}}\to$Defφ is smooth, which follows (thanks to
Proposition 2.15) if
$H^{0}(\mathcal{N}_{\varphi_{2}}\otimes\mathcal{E}_{\pi_{2}})=0$, where
$\mathcal{E}_{\pi_{2}}=\mathcal{O}_{X_{2}}(-3,0)$ is the trace zero module of
$\pi_{2}$. Notice that we have the following exact sequence

$0\to
T_{X_{2}}\to\varphi_{2}^{*}T_{\mathbb{P}^{N}}\to\mathcal{N}_{\varphi_{2}}\to
0.$

Identify $T_{X_{2}}$ with
$(\mathcal{O}_{\mathbb{P}^{1}}(2)\boxtimes\mathcal{O}_{C_{2}})\oplus(\mathcal{O}_{\mathbb{P}^{1}}\boxtimes(\psi_{2}^{\prime*}\mathcal{O}_{\mathbb{P}^{1}}(-m)))$.
One checks that $H^{1}(T_{X_{2}}\otimes\mathcal{E}_{\pi_{2}})=0$ using Künneth
formula.

Now we aim to show that
$H^{0}(\varphi_{2}^{*}T_{\mathbb{P}^{N}}\otimes\mathcal{E}_{\pi_{2}})=0$. By
projection formula, we need to check the vanishings of
$H^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,0))$ and
$H^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,-m-2))$. Let $C$ be a
general hyperplane section of $Y$; it is enough to show that
$H^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,0)|_{C})$ and
$H^{0}(T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,-m-2)|_{C})$. We have
the following diagram with exact rows.

${0}$${\mathcal{O}_{\mathbb{P}^{N-1}}}$${\mathcal{O}_{\mathbb{P}^{N-1}}(1)^{\oplus
N}}$${T_{\mathbb{P}^{N-1}}}$${0}$${0}$${\mathcal{O}_{\mathbb{P}^{N-1}}}$${\mathcal{O}_{\mathbb{P}^{N-1}}(1)^{\oplus
N+1}}$${T_{\mathbb{P}^{N}}|_{\mathbb{P}^{N-1}}}$${0}$

By snake lemma and the splitting of the middle vertical map, we obtain
$T_{\mathbb{P}^{N}}|_{\mathbb{P}^{N-1}}=T_{\mathbb{P}^{N-1}}\oplus\mathcal{O}_{\mathbb{P}^{N-1}}(1)$.
Since hyperplane section of $Y$ is a rational normal curve of degree $N-1$, we
get

$T_{\mathbb{P}^{N}}|_{C}=\mathcal{O}_{\mathbb{P}^{1}}(N)^{\oplus
N-1}\oplus\mathcal{O}_{\mathbb{P}^{1}}(N-1).$

Thus, we have the following to equalities:

  * (I)

$T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,0)|_{C}=\mathcal{O}_{\mathbb{P}^{1}}(N-3m)^{\oplus
N-1}\oplus\mathcal{O}_{\mathbb{P}^{1}}(N-3m-1)$, and

  * (II)

$T_{\mathbb{P}^{N}}|_{Y}\otimes\mathcal{O}_{Y}(-3,-m-2)|_{C}=\mathcal{O}_{\mathbb{P}^{1}}(N-4m-2)^{\oplus
N-1}\oplus\mathcal{O}_{\mathbb{P}^{1}}(N-4m-3)$.

The conclusion follows since $N=2m+1$. The proof is now complete. $\square$

######  Corollary 3.16.

Let $X$ be a (smooth) surface of type $(3)_{m}$.

  * (1)

If $m=1$, a general element of any irreducible component of $U_{\varphi}$ is a
four to one morphism onto its image which is
$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

  * (2)

If $m\geq 2$, there exist a component in the algebraic formally semiuniversal
deformation space of $\varphi$ whose general element is a two-to-one morphism
onto its image which is a product of a smooth curve of genus $2$ with a smooth
non-hyperelliptic curve of genus $m+1$. Further, there does not exist any
component in the algebraic formally semiuniversal deformation space of
$\varphi$ whose general element is a morphism birational onto its image.

Proof. Since the curve constructed in Corollary 3.15 is irreducible, it is
contained in an irreducible component. Now the assertion follows by applying
semicontinuity to the reduced induced structure of the irreducible component.
$\square$

Proof of Theorem 3.1. It follows immediately from Corollary 3.6, Corollary
3.8, Corollary 3.11, Corollary 3.14, and Corollary 3.16. $\square$

##  4\. Moduli components of irregular covers of surface scrolls

In this section we will study the moduli components of irregular quadruple
covers of minimal degree. Furthermore, if the cover is unobstructed, we know
that there is a unique component of the moduli of surfaces of general type; in
that case we would like to understand the geometry of this moduli component.
Regarding surfaces $X$ of type $(1)_{m}$ and $(1^{\prime})_{m}$, our result is
as follows.

######  Theorem 4.1.

Let $X$ be a surface of type $(1)_{m}$ or $(1^{\prime})_{m}$. If $X$ is smooth
or has $A_{1}$ singularities, then ${\bf Def}_{\varphi}$ and ${\bf Def}_{X}$
are smooth, where $\varphi$ is the canonical morphism of $X$, in particular
$X$ is contained in a unique irreducible component of the moduli of surfaces
of general type. Furthermore, for a given $m$,

  * (1)

There exists a unique irreducible component of the moduli of surfaces of
general type $\mathcal{M}_{8m,1,2m+2}$ containing all surfaces of both types.
This component is uniruled of dimension $8m+20$, and

  * (2)

the canonical morphism of a general element of this component is a two-to-one
morphism onto its image which is a non-normal variety whose normalization is
an elliptic ruled surface which is the projectivization of a rank two split
vector bundle over an elliptic curve and has invariant $e=0$.

The situation is not as clean as the previous theorem for general surfaces of
type $(2)_{m}$ for $m\geq 3$, even though the result is neat for $m=2$. In
particular, we show the following.

######  Theorem 4.2.

Let $X$ be a surface of type $(2)_{m}$. If $X$ is smooth or has $A_{1}$
singularities, then the following happens.

  * (1)

If $m=2$, then ${\bf Def}_{\varphi}$ and ${\bf Def}_{X}$ are smooth, in
particular $X$ is contained in a unique irreducible component of the moduli of
surfaces of general type. Furthermore,

    * (a)

there exists a unique irreducible component of the moduli space of surfaces of
general type $\mathcal{M}_{16,2,6}$ containing $X$ (and all other surfaces of
type $(2)_{2}$). This component is uniruled of dimension 28, and

    * (b)

the canonical morphism of a general element in that component is double cover
onto its image whose normalization is a ruled surface over a smooth curve of
genus $2$ and has invariant $e=-2$.

  * (2)

If $m\geq 3$, there do not exist an irreducible component of the moduli of
surfaces of general type $\mathcal{M}_{8m,m,2m+2}$ containing $X$, such that
the canonical morphism of a general element in that component is birational
onto its image.

The proofs of Theorems 4.1 and 4.2 are quite involved and we devote the next
two subsections to them. In contrast, surfaces of type $(3)_{m}$, since they
are product of two curves, are much easier to handle:

######  Remark 4.3.

If $X$ is a surface of type $(3)_{m}$, then $X$ is a product of smooth
hyperelliptic curves of genus $2$ and $m+1$. Consequently, $X$ is unobstructed
(see for example [Ops05], Corollary 2.3), and for a fixed $m$, the surfaces of
type $(3)_{m}$ lie on a unique irreducible component of the moduli space of
surfaces of general type. Now we have (see for example from [Ops05], Theorem
$4.1$. $4.2$) that if $g(C_{1}),g(C_{2})\geq 2$, $\mathcal{M}_{[C_{1}\times
C_{2}]}=\mathcal{M}_{[C_{1}]}\times\mathcal{M}_{[C_{2}]}$ if $g(C_{1})\neq
g(C_{2})$ and $\mathcal{M}_{[C_{1}\times
C_{2}]}=\textrm{Sym}^{2}(\mathcal{M}_{[C_{1}]})$ if $g(C_{1})=g(C_{2})$. It
follows that $\mathcal{M}_{[X]}$ is of dimension $3m+3$; this component is
uniruled since in our case $g(C_{1})=2$ and $\mathcal{M}_{2}$ is uniruled (in
fact rational, see [Bog86], [Kat86]). If $m\neq 1$, then the canonical
morphism of a surface $C_{1}\times C_{2}$, general in the moduli component, is
of degree 2, and its image is isomorphic to $\mathbb{P}^{1}\times C_{2}$. If
$m=1$, the canonical morphism is a quadruple cover of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ that remains quadruple upon deformation.

In contrast to Theorems 4.1 $(2)$, 4.2 $(2)$ and Remark 4.3, it is shown in
[GGP10] (see also [AK90] and [GGP13c]) that there are irreducible components
of moduli spaces of surfaces of general type such that the canonical morphism
of a general element is birational onto its image. At the same time, in these
components there is a proper sublocus in which the canonical morphism has
degree $2$.

###  4.1. Description of moduli components of surfaces of types $(1)$ and
$(1^{\prime})$

First we aim to prove Theorem 4.1. Throughout this subsection, we work with
the notations of § 3.1. Let
$B\in|\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(2)\otimes\mathcal{O}_{\mathbb{P}^{1}}(2m+4)|$
be the branch divisor of $\pi_{1}$. In order to do that, we need the following
cohomology computations.

######  Proposition 4.4.

Let $X$ be a surface of type $(1)_{m}$ or $(1^{\prime})_{m}$. If $X$ is smooth
or has $A_{1}$ singularities, then the following happens:

  * (1)

$h^{0}(\mathcal{O}_{X_{1}}(B))=h^{0}(\mathcal{N}_{\pi_{1}})=8m+20$,
$h^{1}(\mathcal{O}_{X_{1}}(B))=h^{1}(\mathcal{N}_{\pi_{1}})=0$ and
$h^{2}(\mathcal{O}_{X_{1}}(B))=h^{2}(\mathcal{N}_{\pi_{1}})=0$,

  * (2)

$h^{0}(\pi_{1}^{*}T_{X_{1}})=4$, $h^{1}(\pi_{1}^{*}T_{X_{1}})=4$ and
$h^{2}(\pi_{1}^{*}T_{X_{1}})=4m$,

  * (3)

$h^{1}(T_{X})=8m+20$, $h^{2}(T_{X})=4m$,

  * (4)

$h^{0}(\mathcal{N}_{\varphi})=4m^{2}+16m+24$,
$h^{1}(\mathcal{N}_{\varphi})\geq 2m+2$.

Proof. (1) It is easy to see from Künneth formula, and projection formula that

$h^{0}(\mathcal{O}_{X_{1}}(B))=4m+20,\textrm{
}h^{1}(\mathcal{O}_{X_{1}}(B))=0,\textrm{ and
}h^{2}(\mathcal{O}_{X_{1}}(B))=0.$

The remaining assertions follow from the following exact sequence

$0\to\mathcal{O}_{X_{1}}\to\mathcal{O}_{X_{1}}(B)\to\mathcal{O}_{B}(B)\to 0$

Proposition 2.4 and Proposition 3.3 (1).

(2) One checks this readily by Proposition 3.3 (1), Künneth formula, and
projection formula since $h^{j}(\pi_{1}^{*}T_{X_{1}})$ is nothing but the
following sum

$h^{j}(\mathcal{O}_{X_{1}})+h^{j}(\mathcal{O}_{E}\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(2))+h^{j}(\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(-m-2))+h^{j}(\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^{1}}(-m)).$

(3) We use the following two exact sequences

(4.1) $0\to T_{X}\to\pi_{1}^{*}T_{X_{1}}\to\mathcal{N}_{\pi_{1}}\to 0.$

Since $h^{0}(T_{X})=0$, and
$h^{1}(\mathcal{N}_{\pi_{1}})=h^{2}(\mathcal{N}_{\pi_{1}})=0$ by part (1), we
get the following two exact sequences:

$0\to H^{0}(\pi_{1}^{*}T_{X_{1}})\to H^{0}(\mathcal{N}_{\pi_{1}})\to
H^{1}(T_{X})\to H^{1}(\pi_{1}^{*}T_{X_{1}})\to 0,$ $0\to H^{2}(T_{X})\to
H^{2}(\pi_{1}^{*}T_{X_{1}})\to 0.$

The conclusion now follows from part (2).

(4) We get the following exact sequence from Lemma 2.2:

(4.2)
$0\to\mathcal{N}_{\pi_{1}}\to\mathcal{N}_{\varphi}\to\pi_{1}^{*}\mathcal{N}_{\varphi_{1}}\to
0.$

We first compute $h^{0}(\mathcal{N}_{\varphi}$. From part (1), we get
$h^{1}(\mathcal{N}_{\pi_{1}})=0$. It follows from projection formula that

$h^{0}(\mathcal{N}_{\varphi})=h^{0}(\mathcal{N}_{\pi_{1}})+h^{0}(\mathcal{N}_{\varphi_{1}})+h^{0}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}}).$

Recall that we have checked the vanishing of
$h^{0}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}})$ in the proof of
Corollary 3.4. Thus,

$h^{0}(\mathcal{N}_{\varphi})=8m+20+(N+1)^{2}$

by part (1) and Proposition 3.3 (5). The conclusion follows since $N+1=2m+2$.

Notice that $h^{2}(\mathcal{N}_{\pi_{1}})=0$ by part (1). From (4.2), we
obtain that
$h^{1}(\mathcal{N}_{\varphi})=h^{1}(\mathcal{N}_{\varphi_{1}})+h^{1}(\mathcal{N}_{\varphi_{1}}\otimes\mathcal{E}_{\pi_{1}})$.
The conclusion follows from Remark 3.5. $\square$

Proof of Theorem 4.1. We resume the notations of § 3.1. Fix a surface $X$ of
type $(1)_{m}$ or $(1^{\prime})_{m}$ smooth or with $A_{1}$ singularities;
then $B$ has normal crossing. First note that ${\bf Def}_{\varphi}$ and ${\bf
Def}_{X}$ has algebraic formally semiuniversal deformation space by Remark
2.17.

We apply Corollary 2.23. All the hypotheses have been verified in the proofs
of Proposition 4.4 and Corollary 3.4. It follows that ${\bf Def}_{\varphi}$ is
unobstructed.

To show ${\bf Def}_{X}$ is unobstructed, we use Corollary 2.24. It remains to
verify that the differential of the map ${\bf Def}_{\varphi}\to{\bf Def}_{X}$
is surjective. Since the canonical bundle $\omega_{X}$ lifts to any first
order deformation of $X$, it is enough to show that any section of
$\omega_{X}$ lifts to any first order deformation of $(X,\omega_{X})$. We aim
to use the section lifting criterion (Proposition 2.6). Consider the
generalized Atiyah extension (2.1)

(4.3) $0\to\mathcal{O}_{X}\to\mathcal{E}_{\omega_{X}}\to T_{X}\to 0.$

Since the canonical bundle $\omega_{X}$ lifts to any first order deformation
of $X$, $H^{1}(\mathcal{E}_{\omega_{X}})\to H^{1}(T_{X})$ is surjective. This,
together with $H^{0}(T_{X})=0$ implies that

$h^{0}(\mathcal{E}_{\omega_{X}})=1\textrm{ and
}h^{1}(\mathcal{E}_{\omega_{X}})=h^{1}(\mathcal{O}_{X})+h^{1}(T_{X})=8m+21$

where the last equality follows from Proposition 4.4 (3). Now consider the
following exact sequence (2.3)

(4.4) $0\to\mathcal{E}_{\omega_{X}}\to
H^{0}(\omega_{X})^{\vee}\otimes\omega_{X}\to\mathcal{N}_{\varphi}\to 0.$

Recall that a section in $H^{0}(\omega_{X})$ lifts to a first order
deformation $\eta\in H^{1}(\mathcal{E}_{\omega_{X}})$ of the pair
$(X,\omega_{X})$ if and only if its image under the map
$H^{1}(\mathcal{E}_{\omega_{X}})\to\textrm{Hom}(H^{0}(\omega_{X}),H^{1}(\omega_{X}))$
induced from (4.4) is zero (see Proposition 2.6). Thus, it is enough to show
that the map $H^{1}(\mathcal{E}_{\omega_{X}})\to
H^{0}(\omega_{X})^{\vee}\otimes H^{1}(\omega_{X}))$ induced from (4.4) is
zero. Now, (4.4) gives rise to the following exact sequence

(4.5) $0\to H^{0}(\mathcal{E}_{\omega_{X}})\to H^{0}(\omega_{X})^{\vee}\otimes
H^{0}(\omega_{X})\to H^{0}(\mathcal{N}_{\varphi})\to
H^{1}(\mathcal{E}_{\omega_{X}}).$

Thus, the dimension of the image of $H^{0}(\mathcal{N}_{\varphi})\to
H^{1}(\mathcal{E}_{\omega_{X}})$ is

$h^{0}(\mathcal{E}_{\omega_{X}})-(h^{0}(\omega_{X}))^{2}+h^{0}(\mathcal{N}_{\varphi})=1-(2m+2)^{2}+4m^{2}+16m+24=8m+21$

where the last equality follows from Proposition 4.4. But this dimension is
same as $h^{1}(\mathcal{E}_{\omega_{X}})$. This shows that (4.5) is surjective
on the right, and consequently any section of $\omega_{X}$ lifts to any first
order deformation of $(X,\omega_{X})$. Thus the differential of ${\bf
Def}_{\varphi}\to{\bf Def}_{X}$ is surjective. Thus ${\bf Def}_{X}$ is
unobstructed, and the algebraic formally semiuniversal (in fact universal)
deformation space $U_{X}$ of this functor is smooth, irreducible and uniruled.

(1) We show that there exist a unique component of moduli of surfaces of
general type containing all surfaces of both types $(1)_{m}$ and
$(1^{\prime})_{m}$. We do this by the following few steps.

Step 1. We claim that all bidouble covers i.e, the family $(1^{\prime})_{m}$
is contained in an irreducible component of the moduli. We will show this by
showing that the bidouble covers in $(1^{\prime})_{m}$ are parametrized by an
open set of $\mathbb{P}(H^{0}(2C_{0}+(2m+4)f))\times\mathbb{P}(H^{0}(4C_{0}))$
which is irreducible.

Let $L_{1}=-C_{0}-(m+2)f$ and $L_{2}=-2C_{0}$ on
$Y=\mathbb{P}^{1}\times\mathbb{P}^{1}$. Note that $Y$ is rigid and consider
the following Cartesian square.

${Y\times\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))}$${\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))}$${Y}$${\textrm{Spec}(\mathbb{C})}$$\scriptstyle{q}$$\scriptstyle{p}$

Furthermore, consider the divisor $\mathscr{B}_{1}=\\{(x,r,s)\in
Y\times\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))|r(x)=0\\}=q^{*}(L_{1}^{\otimes-2})$
and $\mathscr{B}_{2}=\\{(x,r,s)\in
Y\times\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))|s(y)=0\\}=q^{*}(L_{2}^{\otimes-2})$.
Let

$T=Y\times\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2})).$

Let $\mathscr{L}_{1}^{\otimes-1}\xrightarrow{f_{1}}T$ and
$\mathscr{L}_{2}^{\otimes-1}\xrightarrow{f_{2}}T$ denote the total space of
the line bundles $q^{*}(L_{1}^{\otimes-1})$ and $q^{*}(L_{2}^{\otimes-1})$ on
$T$. Moreover, let $t_{1}\in H^{0}(f_{1}^{*}q^{*}(L_{1}^{\otimes-1}))$ and
$t_{2}\in H^{0}(f_{2}^{*}q^{*}(L_{2}^{\otimes-1}))$ be the corresponding
tautological sections. Then one can consider relative double covers on $T$
given as the zero locus of $t_{i}^{2}-f_{i}^{*}\mathscr{B}_{i}$ inside
$\mathscr{L}_{i}^{\otimes-1}$.

${T_{1}=(t_{i}^{2}-f_{i}^{*}\mathscr{B}_{i})_{0}}$${\mathscr{L}_{i}^{\otimes-1}}$${T}$

Now consider the fibre product of the relative double covers $T_{1}$ and
$T_{2}$ over $T$. Consider the flat family $T_{1}\times_{T}T_{2}\to
T\to\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))$.
Pulling back the composed morphism at a point
$(r,s)\in\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))$
we have the following Cartesian square

${Y_{r}\times_{Y}Y_{s}}$${T_{1}\times_{T}T_{2}}$${Y}$${T}$${(r,s)}$${\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))}$

where $Y_{r}$ and $Y_{s}$ denote the double covers constructed by $r\in
H^{0}(L_{1}^{\otimes-2})$ and $s\in H^{0}(L_{2}^{\otimes-2})$. We now know by
classification of bidouble canonical covers that smooth bidouble covers are
parametrized an open set of
$\mathbb{P}(p_{*}(L_{1}^{\otimes-2}))\times_{k}\mathbb{P}(p_{*}(L_{2}^{\otimes-2}))$.

Step 2. In this step we note that there is an unique irreducible component,
say $D$, of the moduli containtaing all surfaces of type $(1^{\prime})_{m}$.
This comes from Step 1 and the unobstructedness of the smooth surfaces of type
$(1^{\prime})_{m}$.

Step 3. We claim that any cyclic cover in $(1)_{m}$ can be deformed to a
smooth bidouble cover along an irreducible curve whose general fibre
parametrizes smooth surfaces of general type. In particular, given any cyclic
cover, there exists an irreducible component $D^{\prime}$ containing the
cyclic cover and a smooth bidouble cover. Indeed, take the cyclic cover $X$.
Its intermediate cover $X_{1}$ is smooth. The cyclic cover is obtained by a
(special) choice of branch divisor from $p_{1}^{*}(2C_{0}+2(m+2)f)$. Since
$X_{1}$ is smooth, by a different (special) choice of branch divisor, one can
construct a smooth bidouble cover over $X_{1}$. Since
$\mathbb{P}(H^{0}(p_{1}^{*}(2C_{0}+2(m+2)f)))$ is irreducible and a general
member is smooth by Bertini, we have that one can deform the cyclic cover to a
smooth bidouble cover along a curve whose general member is a smooth surface
of general type (in this case a smooth double cover over an elliptic ruled
surface).

Step 4. We claim that $D^{\prime}=D$. Indeed, if $D^{\prime}\neq D$, the
smooth bidouble cover lies in both $D$ and $D^{\prime}$ contradicting its
unobstructedness.

The dimension of the moduli component containing surfaces of type $(1)_{m}$
and $(1^{\prime})_{m}$ follows from Proposition 4.4 (3) and the
unobstructedness of ${\bf Def}_{X}$. That completes the proof of part (1).

(2) Since there is a unique component of the moduli space containing all
surfaces of type $(1)_{m}$ and $(1^{\prime})_{m}$, therefore to describe the
canonical morphism of a general surface in this component, it is enough to
start with a general surface $X$ of either type. It follows from Corollary
3.4, Remark 2.14, and the fact that $X$ is unobstructed, that for a general
surface of the algebraic formally universal deformation space of $X$, the
canonical morphism is of degree two onto its image which is non-normal and
whose normalization is an elliptic ruled surface which is the projectivization
of a split vector bundle over an elliptic curve with invariant $e=0$. Since
$X$ is a smooth surface with ample canonical bundle we have that the same
holds for its unique irreducible moduli component. That completes the proof.
$\square$

######  Remark 4.5.

It is interesting to note that $H^{1}(\mathcal{N}_{\varphi})\neq 0$ by
Proposition 4.4, but ${\bf Def}_{\varphi}$ is still unobstructed by the above
proof. Another example of such an instance is when $\phi:H\to\mathbb{P}^{L}$
is a morphism that is finite onto its image where $H$ is a hyperkähler
variety. It has been proven in [MR20], Lemma 3.1, that in this case ${\bf
Def}_{\phi}$ is unobstructed but $H^{1}(\mathcal{N}_{\phi})\cong H^{2}(T_{H})$
which is non-zero in general. $\blacksquare$

###  4.2. Description of moduli components of surfaces of type $(2)_{m}$

Now we aim to prove Theorem 4.2. Throughout this subsection, we work with the
notations of § 3.2. Recall that
$B\in|\mathcal{O}_{\mathbb{P}^{1}}(6)\otimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(2)|$
is the branch divisor of $\pi_{2}$. In order to do that, we need the following
cohomology computations.

######  Proposition 4.6.

Let $X$ be a surface of type $(2)_{2}$. If $X$ is smooth or has $A_{1}$
singularities, then the following happens:

  * (1)

$h^{0}(\mathcal{N}_{\pi_{2}})=22$ and $h^{1}(\mathcal{N}_{\pi_{2}})=0$,

  * (2)

$h^{0}(\pi_{2}^{*}T_{X_{2}})=3$ and $h^{1}(\pi_{2}^{*}T_{X_{2}})=9$,

  * (3)

$h^{1}(T_{X})=28$,

  * (4)

$h^{0}(\mathcal{N}_{\varphi})=65$.

Proof. (1) It is easy to see from Künneth formula, and projection formula that

$h^{0}(\mathcal{O}_{X_{2}}(B))=21,\textrm{ and
}h^{1}(\mathcal{O}_{X_{2}}(B))=0.$

The remaining assertions follow from the following exact sequence

$0\to\mathcal{O}_{X_{2}}\to\mathcal{O}_{X_{2}}(B)\to\mathcal{O}_{B}(B)\to 0$

Proposition 2.4 and the fact that $h^{1}(\mathcal{O}_{X_{2}})=2$ (see
Proposition 3.9 (1)).

(2) This one follows from Proposition 3.9 (1), Künneth formula, and projection
formula since $h^{j}(\pi_{2}^{*}T_{X_{2}})$ is is the following sum

$h^{j}(\mathcal{O}_{\mathbb{P}^{1}}(2)\boxtimes\mathcal{O}_{C})+h^{j}(\mathcal{O}_{\mathbb{P}^{1}}\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))+h^{j}(\mathcal{O}_{\mathbb{P}^{1}}(-1)\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))+h^{j}(\mathcal{O}_{\mathbb{P}^{1}}(-3)\boxtimes\psi^{*}\mathcal{O}_{\mathbb{P}^{1}}(-2)).$

(3) We use the following two exact sequence

(4.6) $0\to T_{X}\to\pi_{2}^{*}T_{X_{2}}\to\mathcal{N}_{\pi_{2}}\to 0.$

Since $h^{0}(T_{X})=0$, and $h^{1}(\mathcal{N}_{\pi_{2}})=0$ by part (1), we
get the following exact sequence:

$0\to H^{0}(\pi_{2}^{*}T_{X_{2}})\to H^{0}(\mathcal{N}_{\pi_{2}})\to
H^{1}(T_{X})\to H^{1}(\pi_{2}^{*}T_{X_{2}})\to 0.$

The conclusion now follows from part (2).

(4) We get the following exact sequence from Lemma 2.2:

(4.7)
$0\to\mathcal{N}_{\pi_{2}}\to\mathcal{N}_{\varphi}\to\pi_{2}^{*}\mathcal{N}_{\varphi_{2}}\to
0.$

We first compute $h^{0}(\mathcal{N}_{\varphi})$. From part (1), we get
$h^{1}(\mathcal{N}_{\pi_{2}})=0$. It follows from projection formula that

$h^{0}(\mathcal{N}_{\varphi})=h^{0}(\mathcal{N}_{\pi_{2}})+h^{0}(\mathcal{N}_{\varphi_{2}})+h^{0}(\mathcal{N}_{\varphi_{2}}\otimes\mathcal{E}_{\pi_{2}}).$

The vanishing of
$h^{0}(\mathcal{N}_{\varphi_{2}}\otimes\mathcal{E}_{\pi_{2}})$ has been shown
in the proof of Corollary 3.10. Thus,

$h^{0}(\mathcal{N}_{\varphi})=22+36+7=65$

by part (1) and Proposition 3.9 (5). $\square$

Proof of Theorem 4.2. It is enough to show that show that, for a surface $X$