# Tautological families of cyclic covers of projective spaces

Promit Kundu  Department of Mathematics, University of Kansas, Lawrence, USA
<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 existence of tautological families on a Zariski
open set of the coarse moduli space parametrizing certain Galois covers over
projective spaces. More specifically, let ($1$) $\mathscr{H}_{n.r.d}$ (resp.
$M_{n,r,d}$) be the stack (resp. coarse moduli) parametrizing smooth simple
cyclic covers of degree $r$ over the projective space $\mathbb{P}^{n}$
branched along a divisor of degree $rd\geq 4$, and ($2$)
$\mathscr{H}_{1,3,d_{1},d_{2}}$ (resp. $M_{1,3,d_{1},d_{2}}$) be the stack
(resp. coarse moduli) of smooth cyclic triple covers over $\mathbb{P}^{1}$
with $2d_{1}-d_{2}\geq 4$ and $2d_{2}-d_{1}\geq 4$. In the former case, we
show that such a family exists if and only if $\textrm{gcd}(rd,n+1)\mid d$
while in the latter case, we show that it always exists. We further show that
even when such a family exists, often it cannot be extended to the open locus
of objects without extra automorphisms. The existence of tautological families
on a Zariski open set of its coarse moduli can be interpreted in terms of
rationality of the stack if the coarse moduli space is rational. Combining our
results with known results on the rationality of the coarse moduli of points
on $\mathbb{P}^{1}$ and the coarse moduli of plane curves (see [BG10],
[BGK09]), we determine the rationality of $\mathscr{H}_{1,r,d}$ (resp.
$\mathscr{H}_{2,r,d}$) for $rd\geq 4$ (resp. $rd\geq 49$). On the other hand
$\mathscr{H}_{1,3,d_{1},d_{2}}$ is unirational, and we show that its coarse
moduli $M_{1,3,d_{1},d_{2}}$ is unirational and fibred over a rational base by
homogeneous varieties which are rational if $\textrm{char}(\mathbb{k})=0$. Our
study is motivated by [GV09] and [GV08].

###### Key words and phrases:

Tautological families, cyclic covers, rationality, linearization,
Brauer–Severi schemes

###### 2020 Mathematics Subject Classification:

14A20, 14C15, 14D22, 14D23, 14E08, 14F22

##  1\. Introduction

In this article we study families of Galois covers over a projective space.
Fix a finite group $G$. Classically one can consider a functor $F$,
parametrizing Galois covers of the projective space $\mathbb{P}^{n}$ with
Galois group $G$. More precisely we have
$F:(\textrm{Sch}^{\textrm{op}})\to(\textrm{Sets})$, so that for any scheme
$S$, $F(S)=\\{$isomorphism classes of scheme morphisms $\mathscr{X}\to S$,
with an action of $G$ on $\mathscr{X}$ leaving $\mathscr{X}\to S$ invariant,
such that for every geometric point $s\in S$, the action of $G$ on
$\mathscr{X}_{s}$ is faithful and $\mathscr{X}_{s}/G\cong\mathbb{P}^{n}\\}$.
Since in general, $\mathscr{X}_{s}$ will have non-trivial automorphisms, $F$
is not representable and there will not exist a universal family over the
moduli scheme $M$ coarsely representing the functor $F$. One however can
expect to have several tautological families over $M$ or at least over an open
subscheme of $M$. Our article is devoted to investigating when such
tautological families exist.

Since a general smooth curve of genus $g\geq 3$ has no non-trivial
automorphism, there exist tautological families over open sets in $M_{g}$, the
moduli space of smooth curves of genus $g$ with $g\geq 3$ and the open set
could be taken to be $M_{g}^{0}$ which parametrizes smooth curves of genus $g$
without non-trivial automorphisms.

The question is more subtle for $H_{g}$, the moduli of hyperelliptic curves of
genus $g$ since a general hyperelliptic curve always has a hyperelliptic
involution. In [GV09], Gorchinskiy and Viviani show that $H_{g}$ has a
tautological family on an open set if and only if $g$ is odd. They further
show that such a family, even when it exists cannot be extended to the open
subscheme $H_{g}^{0}$, which by definition is the moduli of hyperelliptic
curves with no automorphisms other than the hyperelliptic involution, unlike
the case of smooth curves of genus $g$.

It has been shown by Markman in [Mar21] that tautological families exist over
every irreducible component of the moduli space of marked irreducible
holomorphic symplectic manifolds.

In [BH08], Biswas and Hoffmann prove the existence of tautological families
over the moduli of certain symplectic bundles on a curve.

Existence of a tautological family is key to the concept of rationality of an
algebraic stack introduced in [BH08]. If the algebraic stack generically has
the structure of a gerbe over its coarse moduli space, the stack is rational
if and only if its coarse moduli is rational and there exist a tautological
section over an open subset of its coarse moduli. In [GV09], using the fact
that moduli space $H_{g}$ is rational, the authors conclude that the stack of
hyperelliptic curves of genus $g$ is rational if and only if $g$ is odd.

We generalize the results of [GV09] in two different directions. We study the
existence of tautological section on open subsets of

  * (1)

$M_{n,r,d}:=$ the coarse moduli of smooth simple cyclic covers of degree $r$
and branch degree $rd$ over a projective space of dimension $n$, i.e., the
coarse moduli space of the stack $\mathscr{H}_{n,r,d}$.

  * (2)

$M_{1,3,d_{1},d_{2}}:=$ the coarse moduli of smooth cyclic triple covers on
$\mathbb{P}^{1}$ with branch divisor $D_{1}+D_{2}$, where $D_{1}$ is of degree
$2d_{1}-d_{2}$ and $D_{2}$ is of degree $2d_{2}-d_{1}$, which is the coarse
moduli of a quotient stack $\mathscr{H}_{1,3,d_{1},d_{2}}$.

We explain below why the above two cases are compelling generalizations of the
case $H_{g}$. One can see that $H_{g}\cong M_{1,2,2g+2}$ is a special case of
$M_{n,r,d}$ and hence a natural question is how far the results of [GV09]
generalize for arbitary $n,r,d$ given that by varying the parameters one can
construct a wide range of algebraic varieties in all dimensions. The second
moduli $M_{1,3,d_{1},d_{2}}$ generalizes $H_{g}$, in terms of increased
complexity of the action of the group $\mu_{3}$. Note that hyperelliptic
curves are double covers of $\mathbb{P}^{1}$ and while the action of $\mu_{2}$
on $\mathbb{P}^{1}$ is necessarily simple cyclic, the action of $\mu_{3}$ is
not. It is simple cyclic precisely when $d_{1}=2d_{2}$ or $d_{2}=2d_{1}$.
Since a general trigonal curve has no non-trivial automorphism, it always
admits a tautological family over a Zariski open subscheme of its coarse
moduli. Hence the question of existence of tautological familes is non-trivial
only over the closed subloci of trigonal curves consisting of smooth cyclic
triple covers of $\mathbb{P}^{1}$.

We now state the precise results for the stack of smooth simple cyclic covers.
In [AV04], Arsie and Vistoli give a quotient stack structure to the category
$\mathscr{H}_{n,r,d}$ fibred over $(\textrm{Sch}/\mathbb{k})$ by groupoids
consisting of simple cyclic covers of degree $r$ and branch degree $rd$, over
the projective space of dimension $n$ and isomorphisms between them. More
specifically, they prove the following (see Theorem 2.2 for details)

$\mathscr{H}_{n,r,d}\cong[\mathbb{A}_{sm}(n,rd)/(GL_{n+1}/\mu_{d})].$

where the action of $GL_{n+1}/\mu_{d}$ on $\mathbb{A}_{sm}(n,rd)$ is given by
$[A]\cdot f(x)=f(A^{-1}x)$ ($[A]$ denotes the class of $A\in GL_{n+1}$ in
$GL_{n+1}/\mu_{d}$). Our main result is the following in which we also compute
the Picard group of the coarse moduli $M_{n,r,d}$.

######  Theorem 1.1.

($=$ Theorem 2.16, Corollaries 2.22, 2.23, 2.27, 2.29, Theorem 2.30) Let
$M_{n,r,d}^{0}$ be the open subscheme of $M_{n,r,d}$ parametrizing simple
cyclic covers of $\mathbb{P}^{n}$, whose automorphism group is $\mu_{r}$. Then
the following statements hold.

  * (1)

There exists a tautological family over an open subscheme of $M_{n,r,d}$ if
and only if $\textrm{gcd}(rd,n+1)\mid d$. When such a family exists, the
Brauer–Severi scheme $P\to U$ associated to the tautological family is Zariski
locally trivial if and only if $\textrm{gcd}(rd,n+1)=1$.

  * (2)

Both $\mathscr{H}_{n,r,d}$ and $M_{n,r,d}$ are unirational. Moreover,

    * (i)

if $rd\geq 4$, the stack $\mathscr{H}_{1,r,d}$ is rational if and only if
either $d$ is even or $rd$ is odd,

    * (ii)

if $rd\geq 49$ the stack $\mathscr{H}_{2,r,d}$ is rational if and only if
either $3\mid d$ or $3\nmid rd$.

  * (3)

    * (i)

If $rd\geq 8$, there does not exist a tautological family over the open subset
$M_{1,r,d}^{0}$.

    * (ii)

If $\textrm{char}(\mathbb{k})=0$ or $\textrm{char}(\mathbb{k})>(rd-1)(rd-2)+1$
and $rd\geq 7$, there do not exist a tautological family over the open subset
$M_{2,r,d}^{0}$.

  * (4)

If $rd\geq 4$, $\textrm{Pic}(M_{n,r,d})=0$.

We now state the precise results for the stack of of smooth cyclic triple
covers over $\mathbb{P}^{1}$. Let as before $\mathscr{H}_{1,3,d_{1},d_{2}}$
denote the category fibred over the category $(\textrm{Sch}/\mathbb{k})$ by
groupoids consisting of cyclic triple covers over $\mathbb{P}^{1}$ and
isomorphisms between them. In [AV04], the authors show that (see Theorem 3.2)

$\mathscr{H}_{1,3,d_{1},d_{2}}\cong[\mathbb{U}/\Gamma(d_{1},d_{2})].\\\ $

We explain the notations: $\Gamma(d_{1},d_{2})\cong\frac{\mathbb{G}_{m}\times
GL_{2}}{\mu_{d_{1}}\times\mu_{d_{2}}}$ where the embedding of
$i:\mu_{d_{1}}\times\mu_{d_{2}}\hookrightarrow\mathbb{G}_{m}\times GL_{2}$ is
given by $(x_{1},x_{2})\mapsto(x_{2}/x_{1},x_{1}I_{2\times 2})$. The space
$\mathbb{U}\subseteq\mathbb{A}_{sm}(1,2d_{1}-d_{2})\times\mathbb{A}_{sm}(1,2d_{2}-d_{1})$
is the open subscheme consisting of pairs of smooth forms with no common root.
The action of $\Gamma(d_{1},d_{2})$ on $\mathbb{U}$ is given by

$[(\alpha,A)]\cdot(f_{1}(x),f_{2}(x))\rightarrow(\alpha^{d_{2}}f_{1}(A^{-1}x),\alpha^{-2d_{2}}f_{2}(A^{-1}x)),$

where $(\alpha,A)\in G_{m}\times GL_{2}$ and
$[(\alpha,A)]\in\Gamma(d_{1},d_{2})$ denotes the corresponding class under the
quotient by $i$.

######  Theorem 1.2.

($=$ Theorem 3.10, Corollary 3.14, Remark 3.8, Proposition 3.15) Let
$M_{1,3,d_{1},d_{2}}^{0}$ be the open subscheme of $M_{1,3,d_{1},d_{2}}$
parametrizing cyclic triple covers of $\mathbb{P}^{1}$, whose automorphism
group is $\mu_{3}$. Assume $2d_{1}-d_{2}\geq 4$ and $2d_{2}-d_{1}\geq 4$. Then
the following statements hold.

  * (1)

There exist a tautological family on an open subset of $M_{1,3,d_{1},d_{2}}$
for any $d_{1}$ and $d_{2}$. The Brauer–Severi scheme associated to the
tautological family is Zariski locally trivial if and only if either $d_{1}$
or $d_{2}$ is odd.

  * (2)

$\mathscr{H}_{1,3,d_{1},d_{2}}$ is unirational. $M_{1,3,d_{1},d_{2}}$ is
unirational and is fibred over a rational base by fibres birational to a
homogeneous space. If $\textrm{char}(\mathbb{k})=0$, $M_{1,3,d_{1},d_{2}}$ is
rationally fibred over a rational base.

  * (3)

There does not exist a tautological family on the open subscheme
$M_{1,3,d_{1},d_{2}}^{0}$ of $M_{1,3,d_{1},d_{2}}$.

######  Remark 1.3.

Note that the group appearing in the denominator of part ($2$) in the above
Theorem, is not solvable (see Remark 3.16). Solvability of the denominator
would guarantee the rationality of the fibres (see [CZ17]) independent of the
characteristic of the base field. $\blacksquare$

The technique introduced in [GV09] is to approach the problem
stack–theoretically. In particular, in [GV09], the authors start from the fact
that $\mathscr{H}_{1,2,2g+2}$ is the moduli stack of hyperelliptic curves and
carry out a sequence of the following steps.

  * (1)

Construct a stack $\mathscr{D}_{1,2,2g+2}$ intermediate between
$\mathscr{H}_{1,2,2g+2}$ and its coarse moduli $H_{g}$, i.e., there exist a
sequence of morphisms $\mathscr{H}_{1,2,2g+2}\to\mathscr{D}_{1,2,2g+2}\to
H_{g}$ and such that $\mathscr{D}_{1,2,2g+2}$ is a quotient stack with generic
trivial stabilizer. This gives a unique section $s$ from an open set $U$ of
$H_{g}$ to $\mathscr{D}_{1,2,2g+2}$.

  * (2)

Use explicit isomorphisms $GL_{2}/\mu_{g+1}\cong\mathbb{G}_{m}\times PGL_{2}$
if $g$ is even and $GL_{2}/\mu_{g+1}\cong GL_{2}$ if $g$ is odd and étale
cohomology associated to the exact sequence

$1\to\mu_{2}\to GL_{2}/\mu_{g+1}\to GL_{2}/\mu_{2g+2}\to 1$

to determine if there exists a section
$t:\mathscr{D}_{1,2,2g+2}\to\mathscr{H}_{1,2,2g+2}$.

We use and build on the techniques introduced in [GV09] in this article. The
first step of the method in [GV09], remains unchanged. We construct and give
quotient structures to two intermediate stacks $\mathscr{D}_{n,r,d}$ and
$\mathscr{D}_{1,3,d_{1},d_{2}}$ with trivial generic stabilizer which is also
known as rigidification of a stack. While the existence of rigidification and
the quotient structure of $\mathscr{D}_{n,r,d}$ is a result of a formal
generalization of the work in [GV09] and [AV04], neither the existence nor the
quotient structure is clear in the second case. In the second step, we study
the conditions imposed on the Picard group of the naturally associated
Brauer–Severi scheme by the algebra structure on the pushforward of the
structure sheaf of a tautological family if such a family exists. The problem
thus reduces to the existence of maps between two torsors $\mathscr{P}$ and
$\mathscr{Q}$ over $PGL_{n+1}$ ($PGL_{2}$ in the second case). Since our
computation involves finite abelian covers $f:X\to\mathbb{P}^{n}$ in this
article, $f_{*}\mathscr{O}_{X}$ splits as a direct sum of line bundles
$\\{\mathscr{L}_{i}\\}_{i=1}^{k}$ and the torsor $\mathscr{Q}$ is usually
$\textrm{Aut}(\mathbb{P}^{n},\\{\mathscr{L}_{i}\\}_{i=1}^{k})$ (which could be
non–special). The torsor $\mathscr{P}$ is obtained from the quotient structure
of the rigidification. We can then explicitly check when such a map exists.
The methods however can be used to handle any finite cover if the cover can be
completely expressed in terms of vector bundles on the base and their sections
(see the general structure theorems for finite covers for example in [CE96],
[Mir85]).

We list further some interesting results for families of maximal variation of
moduli admitting a section. A family of curves over a base $V$ is said to have
maximal variation of moduli if the image of the modular map $V\to M_{g}$ has
dimension $\textrm{min}\left\\{\textrm{dim}(V),3g-3\right\\}$. In [Cap03],
Caporaso showed if a family of smooth curves of genus $g$ over a base $V$ of
dimension $3g-3$, having maximal variation of moduli, admits a rational
section, then the modular map $V\to M_{g}$ is generically finite of degree a
multiple of $2g-2$. This means a tautological family of smooth curves over an
open set of $M_{g}$, never admits a rational section.

In [GV06], it is shown that if a family of smooth hyperelliptic curves of
genus $g$ over a base $V$ of dimension $2g-1$, having maximal variation of
moduli, admits a rational section, then the modular map $V\to H_{g}$ is
generically finite of degree a multiple of $2$. This means the tautological
family of smooth hyperelliptic curves of odd genus over an open set of $H_{g}$
(shown to exist in [GV09]), never admits a rational section.

In the present context and in relation to the results of Caporaso ([Cap03])
and those of Gorchinskiy–Viviani ([GV06]), it will be interesting to know the
divisibility of the modular map of a family of maximum variation of moduli for
either $\mathscr{H}_{n,r,d}$ or $\mathscr{H}_{1,3,d_{1},d_{2}}$ when the the
modular map is generically finite.

The rationality results in Theorem 1.1, are obtained by known results on the
rationality of moduli of points on $\mathbb{P}^{1}$ and the rationality of
moduli of plane curves. We show that even when a family exists over a Zariski
open subset of the coarse moduli, it cannot be extended to the loci of objects
without extra automorphisms. We use the method introduced in [GV09] involving
computing the Picard group of the rigidification and then show that for both
$\mathscr{H}_{2,r,d}$ and $\mathscr{H}_{1,3,d_{1},d_{2}}$ the objects with
extra automorphisms form a locally closed substack of codimension greater than
or equal to two.

We now provide the structure of this article. Section 2 is devoted to prove
Theorem 1.1 and is divided into four subsections. In subsection 2.1 we
construct the intermediate stack $\mathscr{D}_{n,r,d}$ and give it a quotient
structure. Subsection 2.2 is devoted to prove that $\mathscr{H}_{n,r,d}$ is a
$\mu_{r}$-gerbe over $\mathscr{D}_{n,r,d}$ using which we will identify
$\mathscr{D}_{n,r,d}$ as the rigidification of $\mathscr{H}_{n,r,d}$.
Subsection 2.3 is the computational heart of this section using which we will
prove our main result in Subsection 2.4. We prove Theorem 1.2 in Section 3
which is structured exactly like Section 2 for the convenience of the reader.
Finally we prove a Mestrano-Ramanan type result in the Appendix (Section 4,
Theorem 4.2).

### Notation and conventions

Throughout this article, we work over an algebraically closed base field
$\mathbb{k}$ of characteristic zero or $p>0$ which:

  * (1)

in Section 2 satisfies $\textrm{gcd}(p,2rd)=1$ and $r\geq 2$,

  * (2)

in Section 3 satisfies $\textrm{gcd}(p,2l_{1}l_{2})=1,$ where
$l_{1}=2d_{1}-d_{2}$ and $l_{2}=2d_{2}-d_{1}.$

Further, $\mathbb{A}_{sm}(n,k)$ denotes the quasi-affine scheme consisting of
smooth homogeneous polynomials of degree $k$ in $n+1$ variables.

### Acknowledgements

We thank Professor Purnaprajna Bangere, Professor Brendan Hassett and
Professor Yunfeng Jiang for valuable discussions. It is our pleasure to thank
Professor D.S. Nagaraj for patiently answering our questions on the Picard
group of $PGL_{n}$. The second author was supported by the National Science
Foundation, Grant No. DMS-1929284 while in residence at the Institute for
Computational and Experimental Research in Mathematics in Providence, RI, as
part of the ICERM Bridge program. D.R. thanks the Fields Institute since a
part of the work was done during his stay there for the academic year
$2021-2022$.

##  2\. Stacks of simple cyclic covers of $\mathbb{P}^{n}$

Throughout this section, we work with the stack of simple cyclic covers of
projective spaces with the objective of proving Theorem 1.1.

###  2.1. Rigidification of the stack of cyclic covers over a projective space

We formally introduce the main object of our study, namely the fibered
category $\mathscr{H}_{n,r,d}$.

######  Definition 2.1.

Let $\mathscr{H}_{n,r,d}$ denote the fibred category whose objects over any
$\mathbb{k}$-scheme $S$ consists of a Brauer–Severi scheme $P\rightarrow S$ of
relative dimension $n$, a line bundle $\mathscr{L}$ on $P$ which restricts to
degree $-d$ on a fibre of every geometric point $s$ of $S$ and an injective
homomorphism $i:\mathscr{L}^{\otimes r}\rightarrow\mathscr{O}_{P}$ that
remains injective with smooth associated divisor when restricted to any
geometric fibre and morphisms consist of Cartesian diagrams between
Brauer–Severi schemes preserving the line bundle and the injective
homomorphism.

As has been shown in [AV04], $\mathscr{H}_{n,r,d}$ is isomorphic to the stack
of smooth simple cyclic covers of degree $r$, over $\mathbb{P}^{n}$ branched
along a divisor of degree $rd$ ([AV04], Remark 3.3). The quotient stack
structure of $\mathscr{H}_{n,r,d}$ has been provided by Arsie and Vistoli in
[AV04], we recall the precise result below.

######  Theorem 2.2.

([AV04], Corollary 4.2)
$\mathscr{H}_{n,r,d}\cong[\mathbb{A}_{sm}(n,rd)/(GL_{n+1}/\mu_{d})]$ where the
action of $GL_{n+1}/\mu_{d}$ on $\mathbb{A}_{sm}(n,rd)$ is given by $[A]\cdot
f(x)=f(A^{-1}x)$.

######  Remark 2.3.

Note that given an object $\zeta=(P\to S,\mathscr{L},i:\mathscr{L}^{\otimes
r}\to\mathscr{O}_{P})\rightarrow S\in Ob(\mathscr{H}_{n,r,d})$ we have that
multiplication by an $r-$th root of unity in $\mathscr{O}_{S}^{*}$ is an
automorphism of $\zeta\to S$. Hence

$\mu_{r}(Q\to S)\subseteq\textrm{Aut}(\zeta\to S)(Q\to S)$

for all $S$-scheme $Q\to S$. $\blacksquare$

We now construct a stack $\mathscr{D}_{n,r,d}$, which as we shall see in the
next section, a stack intermediate to $\mathscr{H}_{n,r,d}$ and its coarse
moduli space with generic trivial stabilizer. This key object was used to
study the stack of hyperelliptic curves in [GV09] and [GV08]. We shall see in
Remark 2.11 that $\mathscr{D}_{n,r,d}$ is the $\mu_{r}$-rigidification of
$\mathscr{H}_{n,r,d}$.

######  Definition 2.4.

Let $\mathscr{D}_{n,r,d}$ denote the fibred category whose objects over any
$\mathbb{k}$-scheme $S$ consists of a Brauer severi scheme $P\to S$ of
relative dimension $n$, a line bundle $\mathscr{L}$ which restricts to degree
$-rd$ on a fibre of every geometric point $s\in S$ and an injective
homomorphism $i:\mathscr{L}\to\mathscr{O}_{P}$ that remains injective with
smooth associated divisor when restricted to any geometric fibre and morphisms
consist of Cartesian diagrams between Brauer–Severi schemes preserving the
line bundle and the injective homomorphism.

The techniques of [AV04] gives a quotient structure to $\mathscr{D}_{n,r,d}$
that we record in the following theorem. We omit the proof.

######  Theorem 2.5.

The stack
$\mathscr{D}_{n,r,d}\cong[\mathbb{A}_{sm}(n,rd)/(GL_{n+1}/\mu_{rd})]$ with the
action given by $[A]\cdot f(x)=f(A^{-1}x)$.

###  2.2. Gerbe structure of $\mathscr{H}_{n,r,d}$ over $\mathscr{D}_{n,r,d}$

We aim to realize $\mathscr{H}_{n,r,d}$ as a gerbe over $\mathscr{D}_{n,r,d}$.
We first recall the definition of a gerbe over a site.

######  Definition 2.6.

([StacksProject, Tag 06NY] Definition $8.11.1$) A gerbe over a site
$\mathscr{C}$ is a category $p:\mathscr{S}\to\mathscr{C}$ such that

  * (1)

$p:\mathscr{S}\to\mathscr{C}$ is a stack in groupoids over $\mathscr{C}$

  * (2)

For any $U\in\textrm{Ob}(\mathscr{C})$ there exist a covering $\\{U_{i}\to
U\\}$ such that $\mathscr{S}_{U_{i}}$ is non-empty and

  * (3)

For any $U\in\textrm{Ob}(\mathscr{C})$ and objects $x$ and $y$ in
$\mathscr{S}_{U}$ there exist a covering $\\{U_{i}\to U\\}$ such that
$x|_{U_{i}}\cong y|_{U_{i}}$ in $\mathscr{S}_{U_{i}}$.

The following lemma provides an equivalence of three statements that defines
$G$-gerbes over a stack $\mathscr{Y}$, where $G$ is a group scheme. For
$\mathscr{X}\to X$ a $G$-gerbe where $X$ is the coarse moduli of
$\mathscr{X}$, the question of existence of tautological families on a Zariski
open set of $X$, is the question of Zariski local triviality of the gerbe, or
the vanishing of its gerbe class in $H^{2}(X,G)$.

######  Lemma 2.7.

([StacksProject, Tag 06NY] Lemma $8.11.3$, [StacksProject, Tag 06QH] Lemma
$99.27.7$) Let $\mathscr{C}$ be a site. Let $p:\mathscr{X}\to\mathscr{C}$ and
$q:\mathscr{Y}\to\mathscr{C}$ be stacks in groupoids. Let
$\mathscr{X}\to\mathscr{Y}$ be a one morphism of categories over
$\mathscr{C}$. We say that $F:\mathscr{X}\to\mathscr{Y}$ a gerbe if any of the
following three equivalent conditions are satisfied.

  * (1)

For some (equivalently any) factorization $F=F^{\prime}\circ a$, where
$a:\mathscr{X}\to\mathscr{X}^{\prime}$ is an equivalence of categories over
$\mathscr{C}$ and $\mathscr{X^{\prime}}$ is fibred in groupoids over
$\mathscr{C}$, the map $F^{\prime}:\mathscr{X^{\prime}}\to\mathscr{Y}$ is a
gerbe with the topology on $\mathscr{Y}$ inherited by $\mathscr{C}$.

  * (2)

    * (a)

Given an object $y\in\textrm{Ob}(\mathscr{Y})$ over
$U\in\textrm{Ob}(\mathscr{C})$, there exist an étale cover $\\{U_{i}\to U\\}$
and objects $x_{i}$ of $\mathscr{X}$ over $U_{i}$ such that $F(x_{i})\cong
y|_{U_{i}}$ in $\mathscr{Y}_{U_{i}}$.

    * (b)

For $U\in\textrm{Ob}(\mathscr{C})$,
$x,x^{\prime}\in\textrm{Ob}(\mathscr{X}_{U})$ and a morphism $b:F(x)\to
F(x^{\prime})$ in $\mathscr{Y}_{U}$, there exist an étale cover ${U_{i}\to U}$
and morphisms $a_{i}:x|_{U_{i}}\to x^{\prime}|_{U_{i}}$ in
$\mathscr{X}_{U_{i}}$ such that $F|_{U_{i}}=a_{i}$.

  * (3)

There exist an algebraic space and a group algebraic space $G$ flat and
locally of finite presentation over $U$ and a surjective flat and locally
finitely presented morphism $U\to\mathscr{Y}$ such that
$\mathscr{X}\times_{\mathscr{Y}}U\cong[U/G]$ over $U$.

We take this opportunity to recall two lemmas.

######  Lemma 2.8.

([StacksProject, Tag 06NY] Lemma $8.11.8$) Let $\mathscr{S}\to\mathscr{C}$ is
a gerbe over a site $\mathscr{C}$. Assume that for all
$U\in\textrm{Ob}({\mathscr{C}})$ and $x\in\mathscr{Ob}(\mathscr{S}_{U})$, the
sheaf of groups $\textrm{Aut}(x)=\textrm{Isom}(x,x)$ on $\mathscr{C}/U$ is
abelian. Then there exist

  * (1)

a sheaf $\mathscr{G}$ of abelian groups over $\mathscr{C}$,

  * (2)

for every object $U\in\textrm{Ob}(\mathscr{C})$ and an object
$x\in\textrm{Ob}(\mathscr{S}|_{U})$ an isomorphism
$\mathscr{G}|_{U}\to\textrm{Aut}(x)$ such that for every
$U\in\textrm{Ob}(\mathscr{C})$ and every morphism $\phi:x\to y$ in
$\mathscr{S}_{U}$ the following diagram is commutative.

${\mathscr{G}|_{U}}$${\mathscr{G}|_{U}}$${\textrm{Aut}(x)}$${\textrm{Aut}(y)}$$\scriptstyle{\alpha\mapsto\phi\circ\alpha\circ\phi^{-1}}$

######  Lemma 2.9.

([StacksProject, Tag 06QB] Lemma $99.27.6)$ Let $\pi:\mathscr{X}\to U$ is a
morphism from an algebraic stack $\mathscr{X}$ to an algebraic space. Let
$s:U\to\mathscr{X}$ be a section to $\pi$. If $\mathscr{X}$ is a gerbe over
$U$ then there is an isomorphism of stacks in groupoids
$\mathscr{X}\cong[U/G]$ where $[U/G]$ is the quotient stack for the trivial
action of $G$ on $U$.

The following is the main theorem of this section, it shows that the stack of
simple cyclic covers over a projective space is a $\mu_{r}$-gerbe over the
stack $\mathscr{D}_{n,r,d}$.

######  Theorem 2.10.

The morphism $F:\mathscr{H}_{n,r,d}\to\mathscr{D}_{n,r,d}$ of algebraic stacks
over an algebraically closed field $\mathbb{k}$, realizes
$\mathscr{H}_{n,r,d}$ as a gerbe with relative automorphism group sheaf
$\mu_{r}$ (which exists by Lemma 2.8).

Proof. We use Lemma 2.7, ($2$). We note the explicit description of the map
$F$. Given an object $\zeta=(P\to S,\mathscr{L},i:\mathscr{L}^{\otimes
r}\to\mathscr{O}_{P})$ of $\mathscr{H}_{n,r,d}$ as in Definition 2.1,
$F(\zeta)=(P\to S,\mathscr{L}^{\otimes r},i:\mathscr{L}^{\otimes
r}\to\mathscr{O}_{P})$ which is an element of $\mathscr{D}_{n,r,d}$ over $S$.
Given two objects $\zeta=(P\to S,\mathscr{L},i:\mathscr{L}^{\otimes
r}\to\mathscr{O}_{P})$ and $\zeta^{\prime}=(P^{\prime}\to
S,\mathscr{L^{\prime}},j:\mathscr{L^{\prime}}^{\otimes r}\to\mathscr{O}_{P})$,
a morphism $\Phi$ between $\zeta$ and $\zeta^{\prime}$ is equivalent to giving
a morphism $\psi:P\to P^{\prime}$ as $S$-schemes and a morphism
$\phi:\mathscr{L}\to\psi^{*}\mathscr{L^{\prime}}$ as $\mathscr{O}_{S}$ modules
such that $(\psi^{*}j)\circ\phi=i$. Then $F(\Phi)=(\psi,\phi^{\otimes r})$.

Now suppose that $\alpha=(P\to S,\mathscr{M},i:\mathscr{M}\to\mathscr{O}_{P})$
be an object of $\mathscr{D}_{n,r,d}$ over $S$. After an étale pullback
$\\{S_{i}\to S\\}$ we have
$\alpha|_{S_{i}}=(\mathbb{P}_{S}^{n},\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd),i:\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd))\to\mathscr{O}_{\mathbb{P}_{S}^{n}})$.
Then
$\zeta_{i}=(\mathbb{P}_{S}^{n},\mathscr{O}_{\mathbb{P}_{S}^{n}}(-d),i:\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd))\to\mathscr{O}_{\mathbb{P}_{S}^{n}})$
is an object of $\mathscr{H}_{n,r,d}$ over $S_{i}$ such that
$F(\zeta_{i})=\alpha|_{S_{i}}$.

Suppose $\zeta=(P\to S,\mathscr{L},i:\mathscr{L}^{\otimes
r}\to\mathscr{O}_{P})$ and $\zeta^{\prime}=(P^{\prime}\to
S,\mathscr{L^{\prime}},i:\mathscr{L^{\prime}}^{\otimes
r}\to\mathscr{O}_{P^{\prime}})$ have images $\alpha=(P\to
S,\mathscr{L}^{\otimes r},i:\mathscr{L}^{\otimes r}\to\mathscr{O}_{P})$ and
$\alpha^{\prime}=(P^{\prime}\to S,\mathscr{L^{\prime}}^{\otimes
r},i:\mathscr{L^{\prime}}^{\otimes r}\to\mathscr{O}_{P^{\prime}})$ and let
$(\psi,\beta)$ be a morphism between $\alpha$ and $\alpha^{\prime}$. Then
there exist an étale cover $\\{S_{i}\to S\\}$ such that $\mathscr{L}^{\otimes
r}|_{S_{i}}\cong\mathscr{L^{\prime}}^{\otimes
r}|_{S_{i}}\cong\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd)$ and hence
$\beta|_{S_{i}}:\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd)\to\mathscr{O}_{\mathbb{P}_{S}^{n}}(-rd)$.
Since this automorphism induces identity on $\mathscr{O}_{S}$,
$\beta|_{S_{i}}\in\mathbb{G}_{m}(\mathbb{P}_{k}^{n})=k^{*}$. Choosing a $r$-th
root of $\gamma_{i}$ of $\beta|_{S_{i}}$ we have that
$(\phi|_{S_{i}},\gamma_{i})$ is a morphism between $\zeta|_{S_{i}}$ and
$\zeta^{\prime}|_{S_{i}}$ such that
$F(\psi|_{S_{i}},\gamma_{i})=(\psi|_{S_{i}},\beta|_{S_{i}})$.

We treat $\mathscr{H}_{n,r,d}$ as a stack over the site $\mathscr{D}_{n,r,d}$
(with topology induced from étale topology on the site
$(\textrm{Sch/}\mathbb{k})$) and show that the automorphism group of an object
$x\to F(x)$ is $\mu_{r}$. Note that since $\textrm{Aut}(x/F(x))$ consists of
automorphisms $\mathscr{L}$ that induces identity on $\mathscr{L}^{\otimes r}$
and $\mathscr{O}_{S}$, we have that $\textrm{Aut}(x/F(x))=\mu_{r}.$ $\square$

######  Remark 2.11.

A general smooth polynomial of degree $k\geq 4$ has automorphism group
$\mu_{k}$. Hence the generic automorphism group of $\mathscr{H}_{n,r,d}$ is
$\mu_{r}$ and by Theorem 2.5, $\mathscr{D}_{n,r,d}$ has generic trivial
stabilizer. Since by Theorem 2.10 $\mathscr{H}_{n,r,d}$ is a gerbe over
$\mathscr{D}_{n,r,d}$, the latter is the $\mu_{r}$-rigidification of
$\mathscr{H}_{n,r,d}$. Since smooth hypersurfaces of degree atleast four are
stable under the action of $GL_{n+1}/\mu_{rd}$ or $GL_{n+1}/\mu_{d}$, we have
that the quotient stacks have finite stabilizers and hence by [Edi13], Theorem
$4.7$ and Theorem $4.18$, we have that both are separated DM stacks (A DM
stack is separated if and only if the diagonal is finite by Remark $4.16$ of
[Edi13]). The stability of the hypersurfaces also show that there exist a
geometric quotient of $\mathbb{A}_{sm}(n,rd)$ by either $GL_{n+1}/\mu_{d}$ or
$GL_{n+1}/\mu_{rd}$. Since $\mathscr{D}_{n,r,d}$ is a $\mu_{r}$-rigidification
of $\mathscr{H}_{n,r,d}$ they both have the same coarse moduli space. But by
[Edi13], Proposition $4.26$ the respective coarse moduli are the respective
geometric quotients and hence the geometric quotients are isomorphic

$\mathbb{A}_{sm}(n,rd)//(GL_{n+1}/\mu_{d})\cong\mathbb{A}_{sm}(n,rd)//(GL_{n+1}/\mu_{rd}).$

We denote by $M_{n,r,d}$ the common coarse moduli of the stacks. Since
$\mathscr{D}_{n,r,d}$ has generic trivial stabilizer, there exist an open
substack $\mathscr{D}_{n,r,d}^{0}\subset\mathscr{D}_{n,r,d}$ and an open
subscheme $M_{n,r,d}^{0}\subset M_{n,r,d}$ such that
$\mathscr{D}_{n,r,d}^{0}\cong M_{n,r,d}^{0}$. $\blacksquare$

Theorem 2.10 immediately shows the following

######  Corollary 2.12.

Let $\mathscr{H}_{n,r,d}^{0}$ be the open substack of $\mathscr{H}_{n,r,d}$
which is the inverse image of $M_{n,r,d}^{0}$ under the coarse moduli map
$p:\mathscr{H}_{n,r,d}\to M_{n,r,d}$. Then $p:\mathscr{H}_{n,r,d}^{0}\to
M_{n,r,d}^{0}$ realizes $\mathscr{H}_{n,r,d}^{0}$ as a $\mu_{r}$-gerbe over
the scheme $M_{n,r,d}^{0}$.

###  2.3. Degree of line bundles of on the universal Brauer–Severi scheme
$(P\to M_{n,r,d}^{0})$

In this section, we aim to study the least possible fibrewise degree of a line
bundle on the universal Brauer–Severi scheme $(P\to M_{n,r,d}^{0})$.

######  Lemma 2.13.

There exist a map of $PGL_{n+1}$ torsors $GL_{n+1}/\mu_{d}\to
GL_{n+1}/\mu_{d^{\prime}}$ if and only if there exist an integer $k$ such that
$d\mid k(n+1)+d^{\prime}$, i.e, $\textrm{gcd}(d,n+1)\mid d^{\prime}$.

Proof. Suppose $d\mid k(n+1)+d^{\prime}$. Then consider the map $\varphi$ of
$PGL_{n+1}$ torsors $GL_{n+1}/\mu_{d}\to GL_{n+1}/\mu_{d^{\prime}}$ given by
$\varphi([A])=[\textrm{det}(A)^{\frac{k}{d^{\prime}}}A]$. We show that
$\varphi$ is well defined. Suppose that $\zeta_{d}\in\mu_{d}$. Then

$\varphi([\zeta_{d}])=[\textrm{det}([\zeta_{d}])^{\frac{k}{d^{\prime}}}[\zeta_{d}]]=\left[\zeta_{d}^{\frac{k(n+1)}{d^{\prime}}}[\zeta_{d}]\right]=\left[\zeta_{d}^{\frac{k(n+1)+d^{\prime}}{d^{\prime}}}\right]=\left[\zeta_{d}^{\frac{d}{d^{\prime}}}\right].$

Now, since $\zeta_{d}^{\frac{d}{d^{\prime}}}$ is a $d^{\prime}$-th root of
unity we have that $\varphi$ is well-defined. The fact that $\varphi$ is a
homomorphism can easily be seen from the following

$\varphi([AB])=[\textrm{det}(AB)^{\frac{k}{d^{\prime}}}AB]=[\textrm{det}(A)^{\frac{k}{d^{\prime}}}\textrm{det}(B)^{\frac{k}{d^{\prime}}}AB]=[\textrm{det}(A)^{\frac{k}{d^{\prime}}}A][\textrm{det}(B)^{\frac{k}{d^{\prime}}}B]=\varphi([A])([B]).$

Conversely, suppose that $\varphi:GL_{n+1}/\mu_{d}\to
GL_{n+1}/\mu_{d^{\prime}}$ be a homomorphism of $PGL_{n+1}$ torsor. Then
$\varphi([A])=[f(A)A]$ where $f(A)\in\mathbb{G}_{m}$. Note that, since
$\varphi$ is a homomorphism of groups, we have that

$\varphi([AB])=[f(AB)AB]=\varphi([A])\varphi([B])=[f(A)A][f(B)B]=[f(A)f(B)AB].$

Hence $f(AB)=f(A)f(B)\zeta_{d^{\prime}}$ where $\zeta_{d^{\prime}}$ is a
$d^{\prime}$-th root of unity and therefore
$\overline{f}:GL_{n+1}\to\mathbb{G}_{m}/\mu_{d^{\prime}}$ defined by
$\overline{f}(A)=[f(A)]$ is a well-defined homomorphism. Now consider the
isomorphism $\psi:\mathbb{G}_{m}/\mu_{d^{\prime}}\to\mathbb{G}_{m}$ given by
$x\to x^{d^{\prime}}$. Composing with $\psi$ we have a homomorphism
$g=\overline{f}^{d^{\prime}}:GL_{n+1}/\mu_{d}\to\mathbb{G}_{m}$. Hence
$[f(A)]^{d^{\prime}}=\textrm{det}(A)^{k}$ for some $k\in\mathbb{Z}$ and hence
$\overline{f}(A)=[\textrm{det}(A)^{\frac{k}{d^{\prime}}}]$. Hence

$\varphi([A])=[f(A)A]=[f(A)][A]=\overline{f}(A)[A]=[\textrm{det}(A)^{\frac{k}{d^{\prime}}}A].$

We now find out the condition imposed by $\varphi$ being well-defined. Let
$\zeta_{d}$ is a $d$-th root of unity and notice

$\varphi([\zeta_{d}])=[\textrm{det}([\zeta_{d}])^{\frac{k}{d^{\prime}}}[\zeta_{d}]]=\left[\zeta_{d}^{\frac{k(n+1)}{d^{\prime}}}[\zeta_{d}]\right]=\left[\zeta_{d}^{\frac{k(n+1)+d^{\prime}}{d^{\prime}}}\right].$

Thus, since $\varphi$ is well-defined, we must have that
$\zeta_{d}^{k(n+1)+d^{\prime}}=1$ which implies $d|k(n+1)+d^{\prime}$.
$\square$

The proof of the following lemma has been outlined for $n=1$ in [GV09] and
follows along the same lines for $n\geq 1$. We give the proof since it is
essential to the argument.

######  Lemma 2.14.

Let the unique section $p$ to $\mathscr{D}_{n,r,d}^{0}\to M_{n,r,d}^{0}$ be
denoted by the object $(P\to M_{n,r,d}^{0},D)$ where $P\to S$ is Brauer–Severi
and $D$ is a divisor of degree $rd$ on every geometric fibre. Let $U\subseteq
M_{n,r,d}^{0}$ be open. Then

$\textrm{Pic}(P|_{U}/U)\cong\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n}).$

Proof. We write the proof taking $U=M_{n,r,d}^{0}$ and note that same proof
goes for any $U\subseteq M_{n,r,d}^{0}$. Let
$\mathbb{A}_{sm}^{0}(n,rd)\subset\mathbb{A}_{sm}(n,rd)$ be the open set
consisting of smooth forms with trivial stabilizer inside $GL_{n+1}/\mu_{rd}$.
Further, let $\mathbb{D}$ be the incidence divisor inside
$\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}$ defined by

$\mathbb{D}:=\left\\{(x,y)\in\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}|y\in
x\right\\}.$

There is a natural free action of $GL_{n+1}/\mu_{rd}$ on
$\mathbb{A}^{0}_{sm}(n,rd)$ such that $M_{n,r,d}^{0}$ is its geometric
quotient. Let $GL_{n+1}/\mu_{rd}$ act diagonally on
$\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}$ so that the action is free and
hence admits a geometric quotient $P^{\prime}$. Now clearly $\mathbb{D}$ is
invariant under the action of $GL_{n+1}/\mu_{rd}$ and the induced action of
$GL_{n+1}/\mu_{rd}$ is free. Hence there exist a geometric quotient
$D^{\prime}$. Consider the commutative diagram with the left vertical chain
consisting $GL_{n+1}/\mu_{rd}$ equivariant maps and the horizontal maps being
geometric quotients under the free action of $GL_{n+1}/\mu_{rd}$.

${\mathbb{D}}$${\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}}$${P^{\prime}}$${\mathbb{A}_{sm}^{0}(n,rd)}$${M_{n,r,d}^{0}}$

Now the incidence divisor $\mathbb{D}$ is in the linear system of the
$GL_{n+1}/\mu_{rd}$ linearized line bundle
$\mathscr{O}_{\mathbb{P}_{\mathbb{A}_{sm}^{0}(n,rd)}^{n}}(rd)$. Since the
action of $GL_{n+1}/\mu_{rd}$ on
$\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}$ is free, $P^{\prime}$ is
actually the quotient stack

$[\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}/GL_{n+1}/\mu_{rd}]$

and hence $D^{\prime}$ determines a divisor on $P^{\prime}$ of relative degree
$rd$ on each fibre over a geometric point of $M_{n,r,d}^{0}$. Hence we have
the following diagram where horizontal rows are geometric quotients of free
actions of $GL_{n+1}/\mu_{rd}$.

${\mathbb{D}}$${D}$${\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n}}$${P^{\prime}}$${\mathbb{A}_{sm}^{0}(n,rd)}$${M_{n,r,d}^{0}}$

Hence the object $(P^{\prime}\to M_{n,r,d}^{0},D)$ induces a section
$M_{n,r,d}^{0}\to\mathscr{D}_{n,r,d}^{0}$. But by uniqueness of the section,
we have that $(P^{\prime}\to M_{n,r,d}^{0},D^{\prime})=(P\to
M_{n,r,d}^{0},D)$. Hence

$\displaystyle\textrm{Pic}(P)$ $\displaystyle=\textrm{Pic}(P^{\prime})$
$\displaystyle=\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{A}_{sm}^{0}(n,rd)\times\mathbb{P}^{n})$
$\displaystyle=\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{A}_{sm}^{0}(n,rd))\times\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n})$
$\displaystyle=\textrm{Pic}(M_{n,r,d}^{0})\times\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n}).$

Thus,
$\textrm{Pic}(P/M_{n,r,d}^{0})=\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n})$.
$\square$

The following lemma computes the least possible degree of a line bundle on the
Brauer–Severi scheme associated to the universal rational section
$M_{n,r,d}^{0}\to\mathscr{D}_{n,r,d}^{0}$. This in particular determines
whether the universal Brauer–Severi scheme $P\to M_{n,r,d}^{0}$ is a
projective bundle or not.

######  Lemma 2.15.

Let $s:=$ min{$|t||\exists$ a line bundle $\mathscr{L}$ on the universal
Brauer–Severi scheme $P\to M_{n,r,d}^{0}$ of degree $t$ on every geometric
fibre}. Then $s=\textrm{gcd}(rd,n+1)$

Proof. By Lemma 2.14,

$s=$ min{$|t||\exists$ a line bundle $\mathscr{L}$ on $\mathbb{P}^{n}$ of
degree $t$ such that $\mathscr{L}$ admits a $GL_{n+1}/\mu_{rd}$
linearization}.

Since a $GL_{n+1}/\mu_{rd}$ linearization of $\mathscr{O}(t)$ implies a
$GL_{n+1}/\mu_{rd}$ linearization of $\mathscr{O}(-t)$, we can assume $t>0$.
Since Aut ($\mathbb{P}^{n},\mathscr{O}(t))=GL_{n+1}/\mu_{t}$, we have that by
definition $s\leq rd$. Now if $t=n+1$, then by Lemma 2.13, we have that there
is a morphism of $PGL_{n+1}$ torsors from $GL_{n+1}/\mu_{rd}$ to
$GL_{n+1}/\mu_{n+1}$ (since $\textrm{gcd}(rd,n+1)\mid n+1$) and hence there
exists a $GL_{n+1}/\mu_{rd}$ linearization of $\mathscr{O}(n+1)$. Therefore
there exists a $GL_{n+1}/\mu_{rd}$ linearization of
$\mathscr{O}(\textrm{gcd}(rd,n+1))$. Hence $s\leq\textrm{gcd}(rd,n+1)$. Now
suppose that there exists a $GL_{n+1}/\mu_{rd}$ linearization of
$\mathscr{O}(t)$. Then there exists a map of $PGL_{n+1}$ torsors from
$GL_{n+1}/\mu_{rd}\to GL_{n+1}/\mu_{t}$. Hence $rd\mid k(n+1)+t$ for some
integer $k$. Then $\textrm{gcd}(rd,n+1)\mid t$. Since both are positive we
conclude

$t\geq\textrm{gcd}(rd,n+1).$

Hence $s=\textrm{gcd}(rd,n+1)$. $\square$

###  2.4.  Tautological family on zariski open sets of coarse moduli and
rationality of $\mathscr{H}_{n,r,d}$

This section is devoted to the proof of Theorem 1.1.

####  2.4.1. Rationality of $\mathscr{H}_{n,r,d}$

The following is the main theorem on the existence of tautological sections
over an open subset of $M_{n,r,d}$. This together with Lemma 2.9 determines
whether the gerbe (see Corollary 2.12)

$\mathscr{H}_{n,r,d}^{0}\to M_{n,r,d}^{0}$

is trivial over a Zariski open set of $M_{n,r,d}$ or not and measures
rationality of $\mathscr{H}_{n,r,d}$ given the rationality of $M_{n,r,d}$.

######  Theorem 2.16.

There exist a tautological family of cyclic covers over an open set $U$ of the
coarse moduli $M_{n,r,d}$ if and only if $\textrm{gcd}(rd,n+1)\mid d$. When
such a family exists, the Brauer–Severi scheme $P\to U$ associated to the
tautological family is trivial if and only if $\textrm{gcd}(rd,n+1)=1$.

Proof. Let the unique section to the natural map $p:\mathscr{D}_{n,r,d}^{0}\to
M_{n,r,d}^{0}$ be denoted by $(\pi:P\to M_{n,r,d}^{0},D)$ where $P\to
M_{n,r,d}^{0}$ is a Brauer–Severi scheme of relative dimension $n$ and $D$ is
a divisor smooth over $S$ with relative degree $rd$.

Now suppose that $\textrm{gcd}(rd,n+1)\mid d$. By Lemma 2.13 there exist a
morphism from $GL_{n+1}/\mu_{rd}\to GL_{n+1}/\mu_{d}$. Hence
$\mathscr{O}_{\mathbb{P}^{n}}(d)$ has a $GL_{n+1}/\mu_{rd}$ linearization and
consequently
$\mathscr{O}_{\mathbb{P}^{n}}(d)\in\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n})=\textrm{Pic}(P/M_{n,r,d}^{0})$
and we have a line bundle $\mathscr{L}$ on $P$ with branch degree $d$. Then
the divisor $D$ belongs to the linear system of the line bundle
$\mathscr{L}^{\otimes r}\otimes\pi^{*}(M)$ for some line bundle $M$ on
$M_{n,r,d}^{0}$. Then, after possibly shrinking $M_{n,r,d}^{0}$, we can assume
that $D$ is in the linear system of an $r$ divisible line bundle on $P$. So
one can construct a $r$ cyclic cover over $P\to S$ branched along $D$ to get a
tautological family of cyclic covers over $\mathbb{P}^{n}$.

Now suppose that there exist a tautological family of cyclic covers over an
open set $U$ of $M_{n,r,d}$. Intersecting with $M_{n,r,d}^{0}$ we can assume
that $U$ is contained inside $M_{n,r,d}^{0}$. This implies that the divisor
$D|_{U}$ on $\pi:P|_{U}\to U$ is contained in the linear system of an $r$
divisible line bundle $\mathscr{L}^{\otimes r}$. Now by Lemma 2.14 we have
that
$\textrm{Pic}(P|_{U}/U)=\textrm{Pic}^{GL_{n+1}/\mu_{rd}}(\mathbb{P}^{n})$.
Since $\mathscr{O}(d)$ has a linearization under the action of
$GL_{n+1}/\mu_{rd}$ we have that there exist a homomorphism of $PGL_{n+1}$
torsors between $GL_{n+1}/\mu_{rd}\to GL_{n+1}/\mu_{d}$ and our result follows
from Lemma 2.13.

Note that a Brauer–Severi scheme $P$ is Zariski locally trivial if and only if
there exists a line bundle of degree one when restricted to a fibre. Now our
second assertion follows from Lemma 2.15. $\square$

We now recall the definition of rationality of stack.

######  Definition 2.17.

(see [BH08], Section $4$)

  * (i)

A stack $\mathscr{X}$ over an algebraically closed field $\mathbb{k}$ is
called rational if it is birational ($1$\- isomorphism between non-empty open
substacks) to $\mathbb{A}^{n}\times BG$ for some group scheme $G$ over
$\mathbb{k}$.

  * (ii)

A stack $\mathscr{X}$ over an algebraically closed field $\mathbb{k}$ is
called unirational if it admits a dominant $1$-morphism from a dense open
subscheme of $\mathbb{A}^{n}$ for some $n$.

We briefly discuss some examples of rational stacks below.

######  Example 2.18.

Toric Deligne-Mumford stacks: a Deligne-Mumford Torus $\mathscr{T}$ is
isomorphic as a Picard stack to $T\times BG$ where $T$ is a torus and $G$ is a
finite abelian group ([FMN10] Proposition $2.6$). A smooth toric Deligne-
Mumford stack is a smooth separated Deligne-Mumford stack $X$ together with an
open immersion of a Deligne-Mumford torus $\mathscr{T}\xrightarrow[]{i}X$ with
dense image such that the action of $\mathscr{T}$ on itself extends to an
action $a:\mathscr{T}\times X\rightarrow X.$ ([FMN10], Definition $3.1$).
Hence it is clear that a smooth Toric Deligne-Mumford stack $X$ is a rational
stack. An important class of Toric Deligne-Mumford stacks given by the
Weighted Projective Stack ([FMN10] Example 7.27) that we describe below.
Weighted Projective Stack. Let

$w=(w_{1},w_{2}..,w_{n})\in\mathbb{N}^{n}_{>0},\quad\textrm{gcd}(w_{1},w_{2},,w_{n})=d.$

The weighted projective stack is given by

$\mathbb{P}(w)=[\mathbb{k}^{n}-(0)/\mathbb{k}^{*}],$

where the action of $\mathbb{k}^{*}$ on $\mathbb{k}^{n}-(0)$ is as follows

$(\lambda,(x_{1},x_{2},..,x_{n}))=(\lambda^{w_{1}}x_{1},\lambda^{w_{2}}x_{2},..,\lambda^{w_{n}}x_{n}).$

Here we denote the $(x_{1},x_{2},..,x_{n})$ as choice of co-ordinates of
$\mathbb{k}^{n}$.

######  Example 2.19.

Root of a Line Bundle: ([FMN10] Subsection 1.3) Let $X$ be a rational scheme
over $\textrm{Spec}(\mathbb{k}).$ Consider $L$ to be a line bundle over $X$.
Denote the root stack corresponding to the $r\geq 0,r-$th root of the line
bundle $L$ over $X,$ to be $\sqrt[r]{L|X}$ which is a Deligne-Mumford stack
over $X$, where $\mu_{r}$ is the $r$-th root of unity, whose objects on
$(U\xrightarrow{\psi}X)$ are given as follows:
$\sqrt[r]{L|X}(U\xrightarrow{\psi}X)=\\{(U\xrightarrow{\psi}X,N\in\textrm{
Pic}(U),\phi:N^{\otimes r}\cong\psi^{*}L)\\}$. It is easy to see that there
exist an open substack isomorphic to $V\times B\mu_{r}$ where
$V\xhookrightarrow{i}X$ is such that $L|_{V}\cong\mathscr{O}_{V}.$

We resume our study the rationality of $\mathcal{H}_{n,r,d}$. Denote by
$C(n,k)$ the coarse moduli parametrizing isomorphism classes of
$\textrm{deg}(k)$ hypersurfaces in $\mathbb{P}^{n}$. The following lemma
relates the coarse moduli of $\mathscr{H}_{n,r,d}$ and $\mathscr{D}_{n,r,d}$
to the moduli of hypersurfaces in projective space.

######  Lemma 2.20.

The coarse moduli space of $\mathscr{H}_{n,r,d}$ is isomorphic to the coarse
moduli of degree $rd$ hypersurfaces in $\mathbb{P}^{n}$, i.e,
$M_{n,r,d}=\mathbb{P}(n+1,rd)//PGL_{n+1}=C(n,rd)$ .

Proof. By Remark 2.11, the coarse moduli of $\mathscr{H}_{n,r,d}$ is
isomorphic to the coarse moduli of $\mathscr{D}_{n,r,d}$. Hence it is enough
to show that
$\mathbb{A}_{sm}(n,rd)//(GL_{n+1}/\mu_{rd})\cong\mathbb{P}(n+1,rd)//PGL_{n+1}$.
The natural map $f:\mathbb{A}(n+1,rd)\to\mathbb{P}(n+1,rd)$ is a
$GL_{n+1}/\mu_{rd}$ equivariant morphism. Consider the normal subgroup
$\mathbb{G}_{m}/\mu_{rd}$ embedded diagonally in $GL_{n+1}/\mu_{rd}$. Note
that $(GL_{n+1}/\mu_{rd})/(\mathbb{G}_{m}/\mu_{rd})\cong PGL_{n+1}$ and that
$f:\mathbb{A}(n+1,rd)\to\mathbb{P}(n+1,rd)$ is a $\mathbb{G}_{m}/\mu_{rd}$
torsor. Then the conclusion follows from [KS19], Lemma $2.3$. $\square$

Following is our main result on the rationality of $\mathscr{H}_{n,r,d}$.

######  Theorem 2.21.

Both $\mathscr{H}_{n,r,d}$ and $M_{n,r,d}$ are unirational. Moreover,

  * (1)

if $\textrm{gcd}(rd,n+1)\nmid d$, the stack $\mathscr{H}_{n,r,d}$ is not
rational,

  * (2)

if $\textrm{gcd}(rd,n+1)\mid d$, then $\mathscr{H}_{n,r,d}$ is rational if and
only if $M_{n,r,d}$ is rational.

Proof. Unirationality of both $\mathscr{H}_{n,r,d}$ and $M_{n,r,d}$ are clear
from the quotient structure.

Note that if a stack $\mathscr{X}$ is rational then its coarse moduli space
$M$ is rational and the stack admits a section from a Zariski open subset of
the coarse moduli. Conversely, if $\mathscr{X}$ is a gerbe over a Zariski open
subset of $M$, then the rationality of $M$ and the existence of a section from
a Zariski open subset of $M$ implies the rationality of the stack
$\mathscr{X}$. Hence $\mathscr{H}_{n,r,d}$ is rational if and only if
$M_{n,r,d}$ is rational and $\mathscr{H}_{n,r,d}$ admits a section from a
Zariski open subset of $M_{n,r,d}$. The conclusion follows from Theorem 2.16.
$\square$

######  Corollary 2.22.

Assume $rd\geq 4$.

  * (i)

If $rd$ is odd, then $\mathscr{H}_{1,r,d}$ is rational.

  * (ii)

If $rd$ is even then $\mathscr{H}_{1,r,d}$ is rational if and only if $d$ is
even.

Proof. The proof follows from Theorem 2.21 and the fact that the moduli of
points on $\mathbb{P}^{1}$ denoted by $M_{0,n}$ is rational for $n\geq 4$.
$\square$

######  Corollary 2.23.

Assume $rd\geq 4$.

  * (i)

If $rd\geq 49$ the stack $\mathscr{H}_{2,r,d}$ is rational if and only if
either $3\mid d$ or $3\nmid rd$.

  * (ii)

If $(r,rd)\in
S:=\left\\{(3,6),(3,12),(3,15),(9,18),(3,24),(6,24),(12,24),(3,48),(6,48)\right\\}$,
then $\mathscr{H}_{2,r,d}$ is not rational.

  * (iii)

If $rd\leq 48$ and $(r,rd)\notin S$, $\mathscr{H}_{2,r,d}$ is rational if and
only if $C(2,rd)$ is rational.

Proof. Follows from Theorem 2.16, Theorem 2.21 and results on rationality of
plane curves in [BG10] and [BGK09] . $\square$

####  2.4.2. Picard groups and obstructions to existence of tautological
family over $M_{n,r,d}^{0}$

It was noted in [GV09], that although hyperelliptic curves of odd genus admits
a tautological family over an open subset of its coarse moduli, the
tautological family cannot be extended to the open set of hyperelliptic curves
without extra automorphisms. The key points of their argument rely on showing
that hyperelliptic curves with extra automorphisms have codimension at least
two in their moduli and an analysis of the class group of the coarse moduli.
We show that even when $\mathscr{H}_{1,r,d}$ or $\mathscr{H}_{2,r,d}$ admits a
section from an open set of the coarse moduli the section cannot be extended
to either $M_{1,r,d}^{0}$ or $M_{2,r,d}^{0}$. While the analysis of the class
group follows from general constructions of [AV04], the major point of this
section is to show that plane curves with non-trivial linear automorphisms
have codimension at least two in $M_{n,r,d}$ using results of [BB16].

######  Lemma 2.24.

Suppose that $k\mid r$.

  * (1)

The natural map
$\textrm{Pic}([\mathbb{A}_{sm}(n,rd)/GL_{n+1}/\mu_{kd}])\to\textrm{Pic}(\mathscr{H}_{n,r,d})$
is injective of index
$\frac{k\cdot\textrm{gcd}(d,n+1)}{\textrm{gcd}(kd,n+1)}$.

  * (2)

$\textrm{Pic}(\mathscr{H}_{n,r,d})$ is generated by
$\textrm{det}^{\frac{d}{\textrm{gcd}(d,n+1)}}$ satisfying
$\textrm{det}^{rd(rd-1)^{n}}=\textrm{id}$

  * (3)

$\textrm{Pic}([\mathbb{A}_{sm}(n,rd)/GL_{n+1}/\mu_{kd}])$ is generated by
$\textrm{det}^{\frac{kd}{\textrm{gcd}(kd,n+1)}}$ satisfying
$\textrm{det}^{rd(rd-1)^{n}}=\textrm{id}$

Proof. We first note the following chain of maps of algebraic groups

$\mathbb{G}_{m}\hookrightarrow GL_{n+1}\rightarrow GL_{n+1}/\mu_{d}\rightarrow
GL_{n+1}/\mu_{kd}.$

This induces the following chain of inclusion of character groups

$\widehat{GL_{n+1}/\mu_{kd}}\hookrightarrow\widehat{GL_{n+1}/\mu_{d}}\hookrightarrow\widehat{GL_{n+1}}\hookrightarrow\widehat{\mathbb{G}_{m}}.$

The fact that the first two maps of character groups is injective follows from
the fact that

$GL_{n+1}\to GL_{n+1}/\mu_{d}\to GL_{n+1}/\mu_{kd}$

is a surjective sequence of maps. The fact that the last map injective is due
to the fact that the map is induced by the restriction of the character
$\textrm{det}\in\widehat{GL_{n+1}}$ to $\mathbb{G}_{m}\hookrightarrow
GL_{n+1}$ embedded diagonally. Let us note the index of the injective group
homomorphisms. The index of
$\widehat{GL_{n+1}}\hookrightarrow\widehat{\mathbb{G}_{m}}$ is $n+1$. The
index of $\widehat{GL_{n+1}/\mu_{d}}\hookrightarrow\widehat{GL_{n+1}}$ is
$\frac{d}{\textrm{gcd}(d,n+1)}$. Similarly the index of
$\widehat{GL_{n+1}/\mu_{kd}}\hookrightarrow\widehat{GL_{n+1}}$ is
$\frac{kd}{\textrm{gcd}(kd,n+1)}$. Hence index of
$\widehat{GL_{n+1}/\mu_{kd}}\hookrightarrow\widehat{GL_{n+1}/\mu_{d}}$ is
$\frac{k\cdot\textrm{gcd}(d,n+1)}{\textrm{gcd}(kd,n+1)}$.

Now let $V=\mathbb{A}(n,rd)$, $X=\mathbb{A}_{sm}(n,rd)$ so that
$\Delta=V\setminus X$ is the irreducible locus of singular forms. We have the
following two exact sequences from [AV04]

$A_{n-1}^{GL_{n+1}/\mu_{d}}(V\setminus
X)\xrightarrow{f}\widehat{GL_{n+1}/\mu_{d}}\to\textrm{Pic}(\mathscr{H}_{n,r,d})\to
0,$ $A_{n-1}^{GL_{n+1}/\mu_{kd}}(V\setminus
X)\xrightarrow{g}\widehat{GL_{n+1}/\mu_{kd}}\to\textrm{Pic}([\mathbb{A}_{sm}(n,rd)/\mu_{kd}])\to
0.$

By [AV04] we have that the image of the map $f$ or $g$ is the subgroup
generated by $\Delta$ which fits in the following sequence of injective group
homomorphisms

$\Delta\hookrightarrow\widehat{GL_{n+1}/\mu_{kd}}\hookrightarrow\widehat{GL_{n+1}/\mu_{d}}\hookrightarrow\widehat{GL_{n+1}}\hookrightarrow\widehat{\mathbb{G}_{m}}.$

Hence we have the following commutative diagram where the rows are exact.

${0}$${\Delta}$${\widehat{GL_{n+1}/\mu_{d}}}$${\textrm{Pic}(\mathscr{H}_{n,r,d})}$${0}$${0}$${\Delta}$${\widehat{GL_{n+1}/\mu_{kd}}}$${\textrm{Pic}([\mathbb{A}_{sm}(n,rd)/\mu_{kd}])}$${0}$

This implies that the index of
$\textrm{Pic}([\mathbb{A}_{sm}(n,rd)/\mu_{kd}])$ in
$\textrm{Pic}(\mathscr{H}_{n,r,d})$ is equal to the index of
$\widehat{GL_{n+1}/\mu_{kd}}$ in $\widehat{GL_{n+1}/\mu_{d}}$ which is equal
to $\frac{k\cdot\textrm{gcd}(d,n+1)}{\textrm{gcd}(kd,n+1)}$. This proves
$(1)$.

The index of $\Delta$ in $\mathbb{G}_{m}$ for the given action is
$rd(n+1)(rd-1)^{n}$. This together with the index of
$\widehat{GL_{n+1}/\mu_{kd}}$ in $\widehat{\mathbb{G}_{m}}$ gives $(2)$ and
$(3)$. $\square$

######  Corollary 2.25.

Suppose The complement of $M_{n,r,d}^{0}$ in $M_{n,r,d}$ has codimension
greater than or equal to $2$ and that
$\frac{r\cdot\textrm{gcd}(d,n+1)}{\textrm{gcd}(rd,n+1)}>1$. Then there do not
exist a global section $s:M_{n,r,d}^{0}\to\mathscr{H}_{n,r,d}$ of the coarse
moduli map $p:\mathscr{H}_{n,r,d}\to M_{n,r,d}$.

Proof. Suppose there exist a section $s:M_{n,r,d}^{0}\to\mathscr{H}_{n,r,d}$.
Then this gives a map
$\textrm{Pic}(\mathscr{H}_{n,r,d})\to\textrm{Pic}(M_{n,r,d}^{0})$. Note that
since $M_{n,r,d}^{0}\cong\mathscr{D}_{n,r,d}^{0}$. Hence we have a section
$s:\textrm{Pic}(\mathscr{H}_{n,r,d})\to\textrm{Pic}(\mathscr{D}_{n,r,d}^{0})\cong\textrm{Pic}(\mathscr{D}_{n,r,d})$
which is a section to the natural map
$\textrm{Pic}(\mathscr{D}_{n,r,d})\to\textrm{Pic}(\mathscr{H}_{n,r,d})$. But
this is a contradiction to the fact that the index of
$\textrm{Pic}(\mathscr{D}_{n,r,d})\to\textrm{Pic}(\mathscr{H}_{n,r,d})$ is
greater than $1$ (see Lemma 2.24). $\square$

The proof of the following theorem is a consequence of methods used in [GV08],
Proposition $2.1$. We include the proof in Appendix (Section 4).

######  Proposition 2.26.

The complement of $M_{1,r,d}^{0}$ in $M_{1,r,d}$ has codimension $\geq 2$ for
$rd\geq 8$.

The above proposition combined with Corollary 2.25 immediately gives the
following result.

######  Corollary 2.27.

There does not exist a global section $M_{1,r,d}^{0}\to\mathscr{H}_{1,r,d}$
for $rd\geq 8$.

We aim to prove an analogue of the corollary above in dimension two. Using a
stratification of the locus plane curves of degree $k$ with non-trivial linear
automorphisms, we can show that analogous conclusion holds for $M_{2,r,d}$.

######  Proposition 2.28.

If $\textrm{char}(\mathbb{k})=0$ or
$\textrm{char}(\mathbb{k})>(rd-1)(rd-2)+1$, the complement of $M_{2,r,d}^{0}$
in $M_{2,r,d}$ has codimension greater than or equal to $2$ for $rd\geq 7$

Proof. We use the stratification given in [BB16], Theorem $7$ of the locus of
plane curves with non-trivial cyclic automorphisms into six stratas, to give
upper bounds to $\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))$ which is
the loci of smooth plane curves of degree $d$ containing the subgroup of
$PGL_{3}$, abstractly isomorphic to $\mathbb{Z}/m\mathbb{Z}$ and which injects
into $PGL_{3}$ by the map $\zeta\to\textrm{diag}\langle
1,\zeta^{a},\zeta^{b}\rangle$ and $m$ is a prime. Let $L_{j,*}$ denote a
homogeneous polynomial of degree $j$ such that the variable $*\in\\{X,Y,Z\\}$
does not appear. In what follows, we follow the notations of [BB16], Theorem
7.

(1) $C\in\rho_{m,0,1}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))$ with $m\mid d-1$
and any such polynomial is of the form

$Z^{d-1}Y+\sum_{j\in S(2)_{m}}Z^{d-j}L_{j,Z}+L_{d,Z}$

where $S(2)_{m}=\left\\{2\leq j\leq d-1|m\mid d-j\right\\}$. Let $d-1=mk$.
Then

$\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)=\left(\sum_{j\in
S(2)_{m}}(j+1)\right)+(d+1)$
$\implies\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)=\left(\sum_{j\in
S(2)_{m}}j\right)+k+d=\frac{mk(k-1)}{2}+2k+d.$

Dimension of $\textrm{C}(d)$ the moduli space of plane curves of degree $d$ is
$\frac{(d+2)(d+1)}{2}-8$. Hence we want to see when

(2.1) $\frac{mk(k-1)}{2}+2k+d>\frac{(d+2)(d+1)}{2}-10.$

The inequality (2.1) implies

(2.2) $(m-1)d^{2}+d(2m-2)-(19m-3)<0.$

Using $d\geq m+1$, we have $m^{2}+3m-20<0$ which gives $m=2$ or $m=3$.
Plugging the value of $m$ in (2.2), we have $d^{2}+2d<35$. This gives $d=5$
considering that $m\mid d-1$. Plugging in $m=3$ we have that $d=4$. Hence for
$d\geq 6$,
$\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)$
has codimension greater than or equal to $2$ in $\textrm{C}(d)$.

(2) $C\in\rho_{m,0,1}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))$ with $m\mid d$ and
any such polynomial is of the form

$Z^{d}+\sum_{j\in S(1)_{m}}Z^{d-j}L_{j,Z}+L_{d,Z}$

where $S(1)_{m}=\left\\{1\leq j\leq d-1|m\mid d-j\right\\}$. Using identical
computation we have that
$\textrm{dim}(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z})))$ has
codimension greater than or equal to $2$ in $\textrm{C}(d)$ unless
$(d,m)=(4,2)$ or $(6,2)$. Hence in this case

$\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)$

has codimension greater than or equal to $2$ if $d\geq 7$.

(3) $C\in\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))$ with $m\mid
d^{2}-3d+2$, $(a,b)\in\Gamma_{m}$ where

$\Gamma_{m}=\\{(a,b)\in\mathbb{N}^{2}\mid\textrm{gcd}(a,b)=1,1\leq a\neq b\leq
m-1\\}$

and any such polynomial is of the form

$X^{d-1}Y+Y^{d-1}Z+Z^{d-1}X+\sum_{j=2}^{d-2}\left(\sum_{j\in
S(1)^{j,X}_{m,(a,b)}}\beta_{j,i}X^{d-i}Y^{i}Z^{j-i}+\sum_{j\in
S^{j,Y}_{m,(a,b)}}\alpha_{j,i}X^{j-i}Y^{d-j}Z^{i}+\sum_{j\in
S^{j,Z}_{m,(a,b)}}\gamma_{j,i}X^{j-i}Y^{i}Z^{d-j}\right)$

where $S(1)^{j,X}_{m,(a,b)}=\left\\{1\leq i\leq j\mid ai+b(j-i)\equiv
a(\textrm{mod}(m))\right\\}$, $S^{j,Y}_{m,(a,b)}==\left\\{1\leq i\leq j\mid
bi+a(d-j)\equiv a(\textrm{mod}(m))\right\\}$ and
$S^{j,Z}_{m,(a,b)}==\left\\{1\leq i\leq j\mid ai+b(d-j)\equiv
a(\textrm{mod}(m))\right\\}$ Note that we have, $a$, $b$ and $a-b$ are coprime
to $m$ since all of these are strictly less than $m$. Hence

$\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)\leq\sum\limits_{j=2}^{d-2}\frac{3}{m}(j+1).$

So we want to see when

(2.3)
$\frac{3}{m}\left[\frac{(d-1)(d-2)}{2}+(d-4)\right]>\frac{(d+2)(d+1)}{2}-10.$

Since $m\geq 3$ (otherwise $\Gamma_{m}$ is empty), (2.3) implies that

(2.4) $\frac{(d-2)(d-1)}{2}+(d-4)>\frac{(d+2)(d+1)}{2}-10.$

This implies $d<3$.

(4) In all of these cases, noting that $a-b$ and $m$ are coprime, and that
$m\geq 3$, one can see that

$\textrm{dim}\left(\rho_{m,a,b}(M_{g}^{Pl}(\mathbb{Z}/m\mathbb{Z}))\right)\leq\frac{1}{3}\left(\left(\sum_{j=2}^{d-2}(j+1)\right)+3d-8\right).$

Hence we want to see when

$2\left(\sum_{j=2}^{d-2}(j+1)+3d-8\right)>3d^{2}+9d-54.$

But that implies $2d^{2}+6d-40<0$ which means $d\leq 3$.

(5), (6) We need $d\leq 3$ and this follows from identical computation as in
$(4)$. $\square$

######  Corollary 2.29.

There does not exist a global section $M_{2,r,d}^{0}\to\mathscr{H}_{2,r,d}$
for $rd\geq 7$.

Proof. If $3\mid rd$ but $3\nmid rd$ there cannot exist a section from any
open set. Otherwise one checks that
$\frac{r\cdot\textrm{gcd}(d,3)}{\textrm{gcd}(rd,3)}>1$. The claim now follows
from Corollary 2.25 and Theorem 2.28. $\square$

####  2.4.3. Picard group of coarse moduli of cyclic covers over the
projective space

Now we prove the final result of this section where we compute the picard
group of the coarse moduli.

######  Theorem 2.30.

Pic$(M_{n,r,d})=0$ if $rd\geq 4$.

Proof. Let $g=\textrm{gcd}(rd,n+1)$. Note that by Lemma 2.24,
$\textrm{Pic}(\mathscr{D}_{n,r,d})$ is generated by $s=\textrm{det}^{rd/g}$
satisfying the relation $s^{(rd-1)^{n}g}=1$. Now by [KKV89], Proposition
$4.2$, $\textrm{Pic}(\mathscr{M}_{n,r,d})$ injects inside
$\textrm{Pic}(\mathscr{D}_{n,r,d})$ and is contained inside the subgroup
generated by elements which restricted any non-trivial stabilizer group is the
image of the identity character. Since this is also a finite cyclic group we
have that it is generated by an element $s_{\alpha}=s^{\alpha}$. Now since
$rd\geq 4$ by [Zhe20] Lemma $2.3$ we have that there exist a smooth curve with
an element of order $(rd-1)^{n}$ in its automorphism group with presentation
$\rho=\textrm{diag}\langle\zeta,\zeta^{(1-rd)},\zeta^{(1-rd)^{2}},...,\zeta^{(1-rd)^{n}}\rangle$
where $\zeta$ is $(rd-1)^{n}$-th root of unity. Note that
$m=1+\sum_{i=1}^{n}(1-rd)^{i}$ is coprime to $(rd-1)^{n}$. Now we have that
$s_{\alpha}(\rho)=1$ and hence $\zeta^{mrd\alpha/g}=1\implies(rd-1)^{n}\mid
mrd\alpha/g$. Since $mrd$ is coprime to $(rd-1)^{n}$, we have that
$(rd-1)^{n}\mid\alpha$.

We now show that $g$ divides $\alpha$. Note that the polynomial
$\sum_{i=0}^{n}x_{0}^{d}$ is a smooth polynomial with an element order $rd$ in
its automorphism group with representation diag $\langle
1,...,\zeta_{rd}\rangle$ where $\zeta_{rd}$ is a $rd$\- th root of unity. Then
$\zeta_{rd}^{rd\alpha/g}=1\implies rd\mid rd\alpha/g\implies g\mid\alpha$.

Since $g\mid rd$ we have that $\textrm{gcd}(g,(rd-1)^{n})=1$. Hence
$(rd-1)^{n}g\mid\alpha.$ Hence $\textrm{Pic}(M_{n,r,d})=0$. $\square$

##  3\. Stacks of cyclic triple covers over $\mathbb{P}^{1}$

Throughout this section, we work with the stack of cyclic triple covers of
$\mathbb{P}^{1}$ with the objective of proving Theorem 1.2.

###  3.1. Rigidification of stacks of cyclic triple covers over
$\mathbb{P}^{1}$

We formally introduce the main object of our study, namely, the fibred
category $\mathscr{H}_{1,3,d_{1},d_{2}}$. We need to fix some notations first
that we will follow throughout this section. We start with positive integers
$d_{1},d_{2}$ such that $0\leq d_{1}\leq 2d_{2},0\leq d_{2}\leq 2d_{1}$. Set

$l_{1}=2d_{1}-d_{2},\quad l_{2}=2d_{2}-d_{1}.$

If $l_{i}=0$ for any $i$ then the stack is isomorphic to the stack of simple
cyclic covers that has been treated in the previous section. We assume
throughout the rest of this section that $l_{i}\geq 4$ for $i=1,2$ and assume
that $\textrm{char}(\mathbb{k})$ does not divide $2l_{1}l_{2}$. We define the
group $\Gamma(d_{1},d_{2})$ as follows

$\Gamma(d_{1},d_{2}):=G_{m}\times
GL_{2}/(\mu_{d_{1}}\times\mu_{d_{2}})\cong\textbf{\text@underline{Aut}}(\mathbb{P}_{\mathbb{Z}}^{n},\mathscr{O}(-d_{1}))\times_{PGL_{2}}\textbf{\text@underline{Aut}}(\mathbb{P}_{\mathbb{Z}}^{n},\mathscr{O}(-d_{2})).$

Here the we denote the quotient with respect to the embedding,

$\mu_{d_{1}}\times\mu_{d_{2}}\xhookrightarrow{i}G_{m}\times GL_{2},$
$(x_{1},x_{2})\rightarrow(x_{2}/x_{1},\ x_{1}I_{2\times 2}),$

where $I_{2\times 2}$ denotes the identity in $GL_{2}$ and $\mu_{d_{i}}$
denote the $d_{i}$-th root of unity in $G_{m}.$

We now introduce the main object of our study, namely the stack
$\mathscr{H}_{1,3,d_{1},d_{2}}$.

######  Definition 3.1.

Let $\mathscr{H}_{1,3,d_{1},d_{2}}$ denote the fibred category whose objects
over any $\mathbb{k}$-scheme $S$ consists of a Brauer–Severi scheme
$P\rightarrow S$ of relative dimension 1, line bundles
$\mathscr{L}_{1},\mathscr{L}_{2}$ on $P$ which restrict to degree
$-d_{1},-d_{2}$ on a fibre of every geometric point $s$ of $S$ respectively
and injective homorphisms $i_{1}:\mathscr{L}_{1}^{\otimes
2}\rightarrow\mathscr{L}_{2}$ and $i_{2}:\mathscr{L}_{2}^{\otimes
2}\rightarrow\mathscr{L}_{1}$ that remains injective with smooth disjoint
associated divisors when restricted to any geometric fibre and morphisms
consist of Cartesian diagrams between Brauer–Severi schemes preserving the
line bundles and the injective homomorphisms.

It has been shown in [AV04], Section 6 that this stack is isomorphic to the
stack of smooth cyclic triple covers over $\mathbb{P}^{1}$. Consider the
following open subscheme

$\mathbb{U}:=\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})-Z$

where $Z$ is the closed subscheme consisting of pairs of forms with common
root. Equivalently, $\mathbb{U}$ is the open subscheme consisting of pairs of
smooth forms with no common root.

######  Theorem 3.2.

([AV04], Theorem 6.5)
$\mathscr{H}_{1,3,d_{1},d_{2}}\cong[\mathbb{U}/\Gamma(d_{1},d_{2})]$ under the
action $\Gamma(d_{1},d_{2})\times\mathbb{U}\rightarrow\mathbb{U}$ given by

$[(\alpha,A)]\cdot(f_{1}(x),f_{2}(x))=(\alpha)^{d_{2}}\cdot(f_{1}(A^{-1}x),(\alpha)^{-2d_{2}}\cdot
f_{2}(A^{-1}x))$

where $(\alpha,A)\in G_{m}\times GL_{2}\And[(\alpha,A)]\in\Gamma(d_{1},d_{2})$
denotes the corresponding class under the quotient by $i$.

Next we introduce the stack $\mathscr{D}_{1,3,d_{1},d_{2}}$ whose objects and
morphisms are introduced below.

######  Definition 3.3.

Let $\mathscr{D}_{1,3,d_{1},d_{2}}$ denote the fibred category whose objects
over any $\mathbb{k}$-scheme $S$ consists of a Brauer–Severi scheme
$P\rightarrow S$ of relative dimension 1, line bundles
$\mathscr{L}_{1},\mathscr{L}_{2}$ on $P$ which restrict to degree
$-l_{1},-l_{2}$ on a fibre of every geometric point $s$ of $S$ respectively
and injective homorphisms $i_{1}:\mathscr{L}_{1}\rightarrow\mathscr{O}_{P}$
and $i_{2}:\mathscr{L}_{2}\rightarrow\mathscr{O}_{P}$ that remains injective
with smooth disjoint associated divisors when restricted to any geometric
fibre and morphisms consist of Cartesian diagrams between Brauer–Severi
schemes preserving the line bundles and the injective homomorphisms.

We give a quotient stack structure to the stack
$\mathscr{D}_{1,3,d_{1},d_{2}}$ following the treatment of [AV04].

######  Theorem 3.4.

$\mathscr{D}_{1,3,d_{1},d_{2}}\cong[\mathbb{U}/\Gamma(l_{1},l_{2})]$ under the
action $\Gamma(l_{1},l_{2})\times\mathbb{U}\rightarrow\mathbb{U}$ given by

$[(\alpha,A)]\cdot(f_{1}(x),f_{2}(x))\rightarrow(f_{1}(A^{-1}x),(\alpha)^{-l_{2}}\cdot
f_{2}(A^{-1}x)),$

where $(\alpha,A)\in G_{m}\times GL_{2}\And[(\alpha,A)]\in\Gamma(l_{1},l_{2})$
denotes the corresponding class under the quotient by $i$.

Proof. Consider the auxiliary fibred category
$\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}$ whose objects over a
$\mathbb{k}$-scheme $S$ consist of an object $(P\to
S,\mathscr{L}_{1},\mathscr{L}_{2},i_{1}:\mathscr{L}_{1}\to\mathscr{O}_{P},i_{2}:\mathscr{L}_{2}\to\mathscr{O}_{P})$
in $\mathscr{D}_{1,3,d_{1},d_{2}}$ along with an isomorphism
$\phi_{1}:(P,\mathscr{L}_{1})\cong(\mathbb{P}_{S}^{1},\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{1}))$
and
$\phi_{2}:(P,\mathscr{L}_{2})\cong(\mathbb{P}_{S}^{1},\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{2}))$
such that $\phi_{1}\circ\phi_{2}^{-1}$ induces identity from
$\mathbb{P}^{1}_{S}\to\mathbb{P}^{1}_{S}$. Note that there is forgetful map
from
$\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}\to\mathscr{D}_{1,3,d_{1},d_{2}}$.
We define a base preserving functor from
$\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}\to\mathbb{U}$. For any object of
$\mathscr{\widetilde{D}}(1,3,d_{1},d_{2})(S)$ consider the composite
homormorphism $\phi_{j}\circ
i_{j}\circ\phi^{-1}_{j}:\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{1})\to\mathscr{O}_{\mathbb{P}_{S}^{1}},j=1,2$
which gives an element of $\mathbb{U}(S)$. Conversely given an element of
$\mathbb{U}(S)$, it can be viewed as an injective hommorphism
$j_{1}:\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{1})\to\mathscr{O}_{\mathbb{P}_{S}^{1}},j_{2}:\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{2})\to\mathscr{O}_{\mathbb{P}_{S}^{1}}$
which remains injective with smooth associated divisor when restricted to any
geometric fibre. Then consider the element $(\mathbb{P}_{S}^{1}\to
S,\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{1}),j_{1}:\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{1})\to\mathscr{O}_{\mathbb{P}_{S}^{1}},\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{2}),j_{2}:\mathscr{O}_{\mathbb{P}_{S}^{1}}(-l_{2})\to\mathscr{O}_{\mathbb{P}_{S}^{1}}).$
The above two functors are pseudo-inverses and hence
$\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}$ is isomorphic to the scheme
$\mathbb{U}$. Note that group sheaf
$\textbf{\text@underline{Aut}}(\mathbb{P}_{\mathbb{Z}}^{n},\mathscr{O}(-l_{1}))\times_{PGL_{2}}\textbf{\text@underline{Aut}}(\mathbb{P}_{\mathbb{Z}}^{n},\mathscr{O}(-l_{2}))$
which assigns to every scheme $S$ the group of isomorphisms

$\phi_{1}:(\mathbb{P}_{S}^{1},\mathscr{O}(-l_{1})))\to(\mathbb{P}_{S}^{1},\mathscr{O}(-l_{1}))),\phi_{2}:(\mathbb{P}_{S}^{1},\mathscr{O}(-l_{2})))\to(\mathbb{P}_{S}^{1},\mathscr{O}(-l_{2})))$

is isomorphic to
$GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}\cong\Gamma(l_{1},l_{2})$
(see [AV04] Theorem $6.15$). Moreover since every pair $(P\to
S,\mathscr{L}_{i},i=1,2)$ where $P\to S$ is Brauer–Severi and
$\mathscr{L}_{i},i=1,2$ is a line bundle which restricts to degree
$-l_{i},i=1,2$ in every geometric fibre is étale locally isomorphic to
$(\mathbb{P}_{S}^{1},\mathscr{O}(-l_{i}))),i=1,2$, we have that
$\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}$ is is a principal bundle over
${\mathscr{D}}(1,3,d_{1},d_{2})$. Hence
${\widetilde{\mathscr{D}}_{1,3,d_{1},d_{2}}}\cong\mathbb{U}$. Clearly the
action is given by

$A\cdot f_{1}(x)=f_{1}(A^{-1}\cdot x),\ (\alpha A)\cdot f_{2}(x)=f_{2}((\alpha
A)^{-1}(x))=(\alpha)^{-l_{2}}\cdot f_{2}(A^{-1}(x)),$

where we identify $\Gamma(l_{1},l_{2})\cong
GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}$ by sending
$[(\alpha,A)]\mapsto([A],[\alpha A])$. $\square$

From the above quotient structure we obtain the following

######  Theorem 3.5.

$\mathscr{D}_{1,3,d_{1},d_{2}}$ has trivial generic stabilizer and hence is an
orbifold.

Proof. In order to prove the result we show that the stabilizer of a general
geometric point in $\mathbb{U}$ is contained in $\mu_{l_{1}}\times\mu_{l_{2}}$
where we have the canonical embedding given by
$i:\mu_{l_{1}}\times\mu_{l_{2}}\xhookrightarrow{}G_{m}\times GL_{2}.$

Let $(f_{1},f_{2})\in\mathbb{U}$ be a general pair of smooth forms with degree
$l_{i}\geq 4,i=1,2$, and let $(\alpha,A)\in G_{m}\times GL_{2}$ be an element
of the stabilizer i.e. $(\alpha,A)\cdot(f_{1},f_{2})=(f_{1},f_{2})$ which is
equivalent to

$f_{1}(A^{-1}x)=f_{1}(x)\quad\textrm{ and }\quad
f_{2}((\alpha.A)^{-1}x)=f_{2}(x)\quad\forall x\in\mathbb{P}^{1}.$

We note that a general smooth polynomial of degree $d$ has stabilizer group
$\mu_{d}$ and consequently $A\in\mu_{l_{1}}$ and $\alpha A\in\mu_{l_{2}}$.
Thus, we obtain

$A=\zeta_{l_{1}}^{k_{1}}\quad\textrm{ and
}\quad\alpha=\zeta_{l_{2}}^{k_{2}}/\zeta_{l_{1}}^{k_{1}}$

where $\zeta_{l_{i}}$ is a primitive $l_{i}$-th root of unity for $i=1,2$.
This immediate implies $[(\alpha,A)]=[1]\in\Gamma(l_{1},l_{2})$ and hence the
action of $\Gamma(l_{1},l_{2})$ is free on some open subset of $\mathbb{U}.$
It follows that the stack $\mathscr{D}_{1,3,d_{1},d_{2}}$ is an orbifold.
$\square$

###  3.2. Gerbe Structure of $\mathscr{H}_{1,3,d_{1},d_{2}}$ over
$\mathscr{D}_{1,3,d_{1},d_{2}}$

Consider a natural morphism of stacks
$F:\mathscr{H}_{1,3,d_{1},d_{2}}\to\mathscr{D}_{1,3,d_{1},d_{2}}$ which is
defined as follows.

  * (1)

For an object $\zeta=(P\to S,\mathscr{L}_{1},\mathscr{L}_{2},i_{1},i_{2})$ in
$\mathscr{H}_{1,3,d_{1},d_{2}}(S)$,

$F(\zeta)=(P\to S,\mathscr{L}_{1}^{\otimes
2}\otimes\mathscr{L}_{2}^{\otimes{-1}},\mathscr{L}_{2}^{\otimes
2}\otimes\mathscr{L}_{1}^{\otimes{-1}},i_{1}\otimes\textrm{id}_{\mathscr{L}_{2}^{\otimes{-1}}},i_{2}\otimes\textrm{id}_{\mathscr{L}_{1}^{\otimes{-1}}}).$

  * (2)

For a morphism $\Phi=(\lambda_{1}:P\to
P^{\prime},\lambda_{2}:\mathscr{L}_{1}\to\lambda_{1}^{*}\mathscr{L}_{1}^{\prime},\lambda_{3}:\mathscr{L}_{2}\to\lambda_{1}^{*}\mathscr{L}_{2}^{\prime})$,

$F(\Phi)=(\lambda_{1}:P\to P^{\prime},\lambda_{2}^{\otimes
2}\otimes(\lambda_{3}^{*})^{\otimes{-1}},\lambda_{3}^{\otimes
2}\otimes(\lambda_{2}^{*})^{\otimes{-1}})$

where $(\lambda_{i})^{*}$ denotes the dual of $\lambda_{i}$ for $i=2,3$.

######  Theorem 3.6.

The morphism $F$ realizes $\mathscr{H}_{1,3,d_{1},d_{2}}$ as a gerbe over
$\mathscr{D}_{1,3,d_{1},d_{2}}$ with relative automorphism group $\mu_{3}$. In
particular this implies that $\mathscr{D}_{1,3,d_{1},d_{2}}$ is a
$\mu_{3}$-rigidification of $\mathscr{H}_{1,3,d_{1},d_{2}}.$

Proof. The proof that $\mathscr{H}_{1,3,d_{1},d_{2}}$ is a gerbe over
$\mathscr{D}_{1,3,d_{1},d_{2}}$ is to check the conditions $2(a)$ and $2(b)$
in Lemma 2.7 as in the proof of Theorem 2.10. We show that the relative
automorphism group is $\mu_{3}$. If $F(\Phi)=\textrm{id}$, then
$\lambda_{1}=\textrm{id}$, $\lambda_{2}^{\otimes 2}=\lambda_{3}^{*}$ and
$\lambda_{3}^{\otimes 2}=\lambda_{2}^{*}$. Then, since
$\lambda_{i}^{**}=\lambda_{i}$ for $i=2,3$, we have that

$\textrm{Aut}(\zeta/F(\zeta))=\left\\{(\textrm{id},\lambda_{2},(\lambda_{2}^{\otimes
2})^{*})|\lambda_{2}^{\otimes 3}=\textrm{id}\right\\}=\mu_{3}.$

Denote the coarse moduli scheme by $M_{1,3,d_{1},d_{2}}$. Being an orbifold
there exists an open dense schematic substack
$\mathscr{D}_{1,3,d_{1},d_{2}}^{0}$ of $\mathscr{D}_{1,3,d_{1},d_{2}}$
isomorphic to an open subscheme
$M^{0}_{1,3,d_{1},d_{2}}\xrightarrow{i}M_{1,3,d_{1},d_{2}}.$ This implies that
$\mathscr{D}_{1,3,d_{1},d_{2}}$ admits a section from
$M^{0}_{1,3,d_{1},d_{2}}$ which we denote by
$p:M^{0}\rightarrow\mathscr{D}_{1,3,d_{1},d_{2}}$ given by,

$(s:P\rightarrow M^{0},D_{1},D_{2})$

where, $\textrm{deg}(D_{1})=l_{1},\textrm{deg}(D_{2})=l_{2}$ on each geometric
fibre of the universal Brauer–Severi scheme, $D_{i}$ for $i=1,2$ being the
universal divisors on $(P\rightarrow M^{0})$. $\square$

###  3.3. Degree of line bundles on the universal Brauer–Severi $(P\rightarrow
M^{0})$

In order to understand the existence of tautological families on open sets of
$M_{1,3,d_{1},d_{2}}$, we need to understand the conditions in which there
exist line bundles $\mathscr{L}_{i},i=1,2$ of degree $-d_{1},-d_{2}$ on the
universal Brauer–Severi given by $(P\rightarrow M^{0}).$ We observe that there
exist $s_{1}:\Gamma(l_{1},l_{2})\rightarrow GL_{2}/\mu_{l_{1}}$ and
$s_{2}:\Gamma(l_{1},l_{2})\rightarrow GL_{2}/\mu_{l_{2}}$ induced by the
canonical maps $[(\alpha,A)]\rightarrow[A]$ where, $(\alpha,A)\in G_{m}\times
GL_{2}$ and maps induced by fibre products following [AV04].

######  Theorem 3.7.

There exist group homomorphisms
$\phi:\Gamma(l_{1},l_{2})\rightarrow\Gamma(d_{1},d_{2})$ over $PGL_{2}.$

Proof. We can write down $\phi=(\phi_{1},\phi_{2})$ where,

$\phi_{i}:\Gamma(l_{1},l_{2})\rightarrow GL_{2}/\mu_{d_{i}},\ i=1,2$

preserving the base $PGL_{2}$. Let us consider
$\Psi:\Gamma(l_{1},l_{2})\rightarrow GL_{2}/\mu_{r}$ preserving the base
$PGL_{2}$ for some $r$. One obtains,

$\Psi([(\alpha,A)])=[f_{(\alpha,A)}A]$

for some function $f:G_{m}\times GL_{2}\rightarrow G_{m}$. We need to check
that $\Psi$ is a group homomorphism and is well-defined.

Group homomorphism property of $\Psi$: We choose
$[(\alpha,A)],[(\beta,B)]\in\Gamma(l_{1},l_{2})$ to obtain,

$\Psi([(\alpha,A)]\cdot[(\beta,B)])=\Psi([(\alpha,A)])\cdot\Psi_{1}([(\beta,B)])\implies\Psi([(\alpha\beta,AB)])=\Psi([(\alpha,A)]).\Psi([(\beta,B)])$
$\implies[f_{(\alpha\beta,AB)}\cdot AB]=[f_{(\alpha,A)}.f_{(\beta,B)}\cdot
AB]\implies f_{(\alpha\beta,AB)}=f_{(\alpha,A)}\cdot
f_{(\beta,B)}\cdot\zeta_{r}^{k},$

for some $k\in\mathbb{Z},\zeta_{r}$ denoting the primitive generator of $r$-th
root of unity.

Thus, $\Psi$ induces a group homomorphism $f:G_{m}\times GL_{2}\rightarrow
G_{m}/\mu_{r}$, consequently $(f)^{r}\in\widehat{G_{m}\times GL_{2}}$. Thus,
we obtain

$f^{r}=(\textrm{det}^{k_{1}},\textrm{det}^{k_{2}}).$

Recall that the embedding of
$i^{\prime}:\mu_{l_{1}}\times\mu_{l_{2}}\xhookrightarrow{}G_{m}\times GL_{2}$
is given by $(x_{1},x_{2})\mapsto(x_{2}/x_{1},x_{1}I_{2\times 2})$. The
generators of $\mu_{l_{1}}\times\mu_{l_{2}}$ are given by

$\left\\{\left(\dfrac{1}{\zeta_{l_{1}}},\zeta_{l_{1}}\cdot I_{2\times
2}\right),(\zeta_{l_{2}},I_{2\times 2})\right\\}$

where $\zeta_{l_{i}},i=1,2$ denote the primitive generator of $\mu_{l_{i}}$-th
root of unity.

Next we find the condition that ensures $\Psi$ is a well defined group
homomorphism.

$\Psi$ is well defined: In order for $\Psi$ to be well-defined we must have,

$\Psi\left(\left[\left(\dfrac{1}{\zeta_{l_{1}}},\zeta_{l_{1}}\cdot I_{2\times
2}\right)\right]\right)=[1]=\Psi\left(\left[\left(\zeta_{l_{2}},I_{2\times
2}\right)\right]\right)\implies\left[f_{\left(\frac{1}{\zeta_{l_{1}}},\zeta_{l_{1}}\right)}\cdot\zeta_{l_{1}}\right]=[1]=\left[f_{\left(\zeta_{l_{2}},1\right)}\right]$
$\implies\left(f_{\left(\frac{1}{\zeta_{l_{1}}},\zeta_{l_{1}}\right)}\cdot\zeta_{l_{1}}\right)^{r}=1=\left(f_{\left(\zeta_{l_{2}},1\right)}\right)^{r}\implies\left(\dfrac{1}{\zeta_{l_{1}}}\right)^{k_{1}}\cdot(\zeta_{l_{1}})^{2k_{2}}\cdot(\zeta_{l_{1}})^{r}=1=(\zeta_{l_{2}})^{k_{1}}$
$\implies(\zeta_{l_{1}})^{2k_{2}-k_{1}+r}=1=(\zeta_{l_{2}})^{k_{1}}.$

Thus one obtains the following

$l_{1}\mid 2k_{2}-k_{1}+r\quad\textrm{ and }\quad l_{2}\mid k_{1}.$

Thus in order to have $\phi:\Gamma(l_{1},l_{2})\rightarrow\Gamma(d_{1},d_{2})$
over $PGL_{2}$, we need to find
$k_{1},k_{2},k^{\prime}_{1},k^{\prime}_{2}\in\mathbb{Z}$ such that,

$l_{1}\mid(2k_{2}-k_{1}+d_{1}),\quad
l_{1}\mid(2k^{\prime}_{2}-k^{\prime}_{1}+d_{2}),\quad l_{2}\mid k_{1},\quad
l_{2}\mid k^{\prime}_{1}.$

Now we use the fact that

$l_{1}=2d_{1}-d_{2},\quad l_{2}=2d_{2}-d_{1}.$

Simplifying further we obtain that solution to the above system is equivalent
to correct choices of $n,m$ satisfying the following,

$n(2d_{1}-d_{2})=(2k_{2}-k_{1}+d_{1}),\quad(2d_{2}-d_{1})\mid k_{1},\quad
m(2d_{1}-d_{2})=(2k^{\prime}_{2}-k^{\prime}_{1}+d_{2}),\quad(2d_{1}-d_{2})\mid
k^{\prime}_{1}.$

On further notice one obtains,

$(2n-1)d_{1}-nd_{2}+k_{1}=2k_{2},\quad
2md_{1}-(m+1)d_{2}+k^{\prime}_{1}=2k^{\prime}_{2},\quad(2d_{2}-d_{1})\mid
k_{1},\quad(2d_{2}-d_{1})\mid k^{\prime}_{1}.$

To find the correct $m,n$ we deal this case by case as follows.

Case (1) $d_{1},d_{2}$ both even: choose $k_{1}=k^{\prime}_{1}=0,$ then for
choice of any $n,m$ we find $k_{2},k^{\prime}_{2}\in\mathbb{Z}$ satisfying the
above conditions.

Case (2) $d_{1},d_{2}$ both odd: choose $k_{1}=k^{\prime}_{1}=0,$ then choice
of $n,m$ both odd gives us $k_{2},k^{\prime}_{2}\in\mathbb{Z},$ solving the
above conditions.

Case (3) $d_{1}$ even, $d_{2}$ odd: choose $k_{1}=k^{\prime}_{1}=0$, then any
even $n$ and odd $m$ gives us $k_{2},k^{\prime}_{2}\in\mathbb{Z}$ solving the
above conditions.

Case (4) $d_{1}$ odd, $d_{2}$ even: choose $k_{1}=l_{2},k^{\prime}_{1}=0$ then
choice of any $n,m$ gives us $k_{2},k^{\prime}_{2}\in\mathbb{Z}$ solving the
above conditions. $\square$

######  Remark 3.8.

We observe that the universal Brauer–Severi $(P\rightarrow M^{0})$ is Zariski
locally trivial if and only if
$\mathscr{O}(1)\in\textrm{Pic}^{\Gamma(l_{1},l_{2})}(\mathbb{P}^{1})$, i.e.,
if and only if there exists a group homomorphism
$\Gamma(l_{1},l_{2})\rightarrow GL_{2}.$
We observe from the proof of Theorem 3.7 that for the homomorphism to exist we
can take $r=1.$ Thus the universal Brauer–Severi scheme trivializes if and
only if there exist two integers $k_{1},k_{2}\in\mathbb{Z}$ such that
$l_{1}\mid(2k_{2}-k_{1}+1)$ and $l_{2}\mid k_{1}$, or equivalently there exist
$m,n\in\mathbb{Z}$ satisfying

(3.1) $k_{1}=ml_{2},\quad\textrm{ and }\quad nl_{1}=2k_{2}-ml_{2}+1.$

Simplifying, one obtains that the above system admits a solution if and only
if

$nd_{2}+md_{1}+1$

is even. Thus (3.1) has no solution if and only if $d_{1},d_{2}$ both are
even. In other cases we find that the solution exists.

To summarize, we have non-trivial Brauer–Severi scheme when, $d_{1},d_{2}$
both are even. The Brauer–Severi scheme trivializes in all other cases of
$d_{1},d_{2}.$ $\blacksquare$

In order to compute the relative Picard group
$\textrm{Pic}(P|M^{0}):=\textrm{Pic}(P)/s^{*}\textrm{Pic}(M^{0}),$ we
introduce similar constructions as in Lemma 2.14 as follows.

######  Lemma 3.9.

Suppose that the unique section $p$ to $M^{0}\to\mathscr{D}_{1,3,d_{1},d_{2}}$
be denoted by the object $(P\to M^{0},D_{1},D_{2})$ where $P\to S$ is the
universal Brauer–Severi and $D_{i}$ is the universal divisor of degree
$l_{i},i=1,2$ on every geometric fibre. Let $U\subseteq M^{0}$ Then
$\textrm{Pic}(P|_{U}/U)\cong\textrm{Pic}^{\Gamma(l_{1},l_{2})}(\mathbb{P}^{1})$.

Proof. Let $\mathbb{D}_{i},i=1,2$ be the incidence divisors inside
$\mathbb{U}\times\mathbb{P}^{1}$ defined by
$\mathbb{D}_{1}:=\left\\{(x_{1},x_{2},y)\in\mathbb{U}\times\mathbb{P}^{1}|y\in
x_{1}\right\\}$ and similarly one defines
$\mathbb{D}_{2}:=\left\\{(x_{1},x_{2},y)\in\mathbb{U}\times\mathbb{P}^{1}|y\in
x_{2}\right\\}$. There is a natural free action of $\Gamma(l_{1},l_{2})$ on
$\mathbb{U}$ such that $M^{0}$ is its geometric quotient. Let
$\Gamma(l_{1},l_{2})$ act diagonally on $\mathbb{U}\times\mathbb{P}^{1}$ so
that the action is free and hence admits a geometric quotient $P^{\prime}$.
Now clearly $\mathbb{D}_{i},i=1,2$ are invariant under the action of
$\Gamma(l_{1},l_{2})$ and the induced action of $\Gamma(l_{1},l_{2})$ is free.
Hence there exist a geometric quotient $D^{\prime}_{i},i=1,2$. This gives us a
section $q:M^{0}\rightarrow\mathscr{D}_{1,3,d_{1},d_{2}}$ given by the object
$(P^{\prime}\rightarrow M^{0},D_{1},D_{2}).$ Following the argument of Lemma
2.14 the rest of the proof follows similarly. $\square$

###  3.4. Tautological families on Zariski open sets of the coarse moduli of
$\mathscr{H}_{1,3,d_{1},d_{2}}$

In order to lift a section of the universal Brauer–Severi $(P\rightarrow
M^{0},D_{1},D_{2})$ to $\mathscr{H}_{1,3,d_{1},d_{2}}$ over some open
subscheme of $M^{0}$ we need to need to find line bundles
$\mathscr{L}_{1},\mathscr{L}_{2}$ of degree $-d_{1},d_{2}$ on the universal
Brauer–Severi. Note that in case a section exists with a lifting then there
exists $\mathscr{L}_{i},i=1,2$ with degrees $-d_{i},i=1,2$ such that

$D_{1}\in|\mathscr{L}_{1}^{\otimes 2}\otimes\mathscr{L}_{2}^{\otimes-1}|,\quad
D_{2}\in|\mathscr{L}_{2}^{\otimes 2}\otimes\mathscr{L}_{1}^{\otimes-1}|$

with degrees $l_{i},i=1,2.$ We denote by $D^{0}$ the open schematic substack
in $\mathscr{D}_{1,3,d_{1},d_{2}}$ such that the natural course moduli map is
an isomorphism to $M^{0}.$

######  Theorem 3.10.

There always exists a tautological family of cyclic triple covers over some
open subscheme of the coarse moduli $M^{0}$ to
$\mathscr{H}_{1,3,d_{1},d_{2}}.$

Proof. Let the unique section to the natural map $p:D^{0}\to M^{0}$ be denoted
by $(\pi:P\to M_{1,3,d_{1},d_{2}}^{0},D_{i},i=1,2)$ where $(P\to
M_{1,3,d_{1},d_{2}}^{0})$ is a Brauer–Severi scheme of relative dimension $1$
and $D_{i},i=1,2$ are divisors smooth over $M_{1,3,d_{1},d_{2}}^{0}$ with
relative degree $l_{i},i=1,2$ respectively.

For $i=1,2$, we need to find line bundles $\mathscr{L}_{i}$ on
$\pi|_{U}:P|_{U}\to U$ of vertical degree $d_{i}$ on some open set $U\subseteq
M_{1,3,d_{1},d_{2}}^{0}$, such that $D_{i}|_{U}$ is in the linear system of
$|\mathscr{L}_{i}^{\otimes 2}\otimes\mathscr{L}_{j}^{\otimes-1}|$. By Theorem
3.7, we have a group homomorphism
$\phi:\Gamma(l_{1},l_{2})\to\Gamma(d_{1},d_{2})$ preserving $PGL_{2}$ for any
$d_{1},d_{2}$ which implies that
$\mathscr{O}_{\mathbb{P}^{1}}(d_{i})\in\textrm{Pic}^{\Gamma(l_{1},l_{2})}(\mathbb{P}^{1})$
for $i=1,2$. By Lemma 3.9 this is equivalent to the existence of line bundles
$\mathscr{L}_{i},i=1,2$ with degrees $d_{i}$ on $(P\to
M_{1,3,d_{1},d_{2}}^{0})$. Notice that

$\mathscr{L}_{1}^{\otimes
2}\otimes\mathscr{L}_{2}^{\otimes-1}\otimes\mathscr{O}(-D_{1})$

being a degree 0 line bundle on each geometric fibre on $(P\to
M_{1,3,d_{1},d_{2}}^{0})$, using Lemma 3.9, we have

$\mathscr{L}_{1}^{\otimes
2}\otimes\mathscr{L}_{2}^{\otimes-1}\otimes\mathscr{O}(-D_{1})\cong\pi^{*}(\mathscr{M}),$

where $\mathscr{M}\in\textrm{Pic}(M^{0}).$ On shrinking $M^{0}$ further one
can assume $D_{1}$ is in the linear system of $|\mathscr{L}_{1}^{\otimes
2}\otimes\mathscr{L}_{2}^{\otimes-1}|$. Arguing similarly one can assume
$D_{2}$ is in the linear system of $|\mathscr{L}_{2}^{\otimes
2}\otimes\mathscr{L}_{1}^{\otimes-1}|$. $\square$

####  3.4.1. Non-existence of a tautological family on
$M_{1,3,d_{1},d_{2}}^{0}$

We show that there do not exist a tautological family on
$M_{1,3,d_{1},d_{2}}^{0}$.

######  Remark 3.11.

The morphism $F$ in Theorem 3.6 is induced by the map

$\textrm{id}:\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})\to\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})$

which is equivariant under the group homomorphism
$\Phi:\Gamma(d_{1},d_{2})\to\Gamma(l_{1},l_{2})$ sending

$([A],[\alpha
A])\to\left(\left[\alpha^{-\frac{d_{2}}{l_{1}}}A\right],\left[\alpha^{\frac{2d_{2}}{l_{2}}}A\right]\right).$

Treating $\Gamma(d_{1},d_{2})\cong\mathbb{G}_{m}\times
GL_{2}/(\mu_{d_{1}}\times\mu_{d_{2}})$, we obtain

$\Phi([(\alpha,A)])=\left[\left(\alpha^{\frac{d_{2}}{l_{1}}+\frac{2d_{2}}{l_{2}}},\alpha^{\frac{-d_{2}}{l_{1}}}A\right)\right].$

Note that for any choice of $\alpha^{\frac{d_{2}}{l_{1}}}$ and
$\alpha^{\frac{2d_{2}}{l_{2}}}$,
$\left[\left(\alpha^{\frac{d_{2}}{l_{1}}+\frac{2d_{2}}{l_{2}}},\alpha^{\frac{-d_{2}}{l_{1}}}A\right)\right]$
is well defined in $\Gamma(l_{1},l_{2})$.

Proof. It is clear that $F$ is induced by the identity morphism under some
$\Phi:\Gamma(d_{1},d_{2})\to\Gamma(l_{1},l_{2})$. We prove the description of
$\Phi$. Suppose $\Phi([A],[\alpha A])=([\lambda_{1}A],[\lambda_{2}A])$. The
condition of equivariance of the identity map gives

$((\alpha)^{d_{2}}\cdot f_{1}(A^{-1}x),(\alpha)^{-2d_{2}}\cdot
f_{2}(A^{-1}x))=(\lambda_{1}^{-l_{1}}\cdot
f_{1}(A^{-1}x),\lambda_{2}^{-l_{2}}\cdot f_{2}(A^{-1}x))$

Hence $[\lambda_{1}]=\left[\alpha^{-\frac{d_{2}}{l_{1}}}\right]$ in
$GL_{2}/\mu_{l_{1}}$ and
$[\lambda_{2}]=\left[\alpha^{\frac{2d_{2}}{l_{2}}}\right]$ in
$GL_{2}/\mu_{l_{2}}$. Hence our assertion follows. $\blacksquare$

######  Lemma 3.12.

The natural map:
$\textrm{Pic}(\mathscr{D}_{1,3,l_{1},l_{2}})\to\textrm{Pic}(\mathscr{H}_{1,3,d_{1},d_{2}})$
is an injective homomorphism which is not an isomorphism.

Proof. We have a commutative diagram of exact sequences,

${0}$${\langle\Delta_{1},\Delta_{2},Z\rangle}$${\widehat{\Gamma(d_{1},d_{2})}}$${\textrm{Pic}(\mathscr{H}_{1,3,d_{1},d_{2}})}$${0}$${0}$${\langle\Delta_{1},\Delta_{2},Z\rangle}$${\widehat{\Gamma(l_{1},l_{2})}}$${\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}})}$${0}$$\scriptstyle{\widehat{\Phi}}$

where the morphism $\widehat{\Phi}$ is induced by the morphism $\Phi$ as in
Remark 3.11, $\Delta_{i}$ is the inverse image of the discriminant divisor in
$\mathbb{A}_{sm}(1,l_{i})$ inside
$\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})$ and $Z$ is the
divisor in $\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})$ consisting
of pairs of forms with a common zero. This shows that the map right vertical
map is injective since $\widehat{\Phi}$ is injective (since $\Phi$ is
surjective) and its index is same as the index of $\widehat{\Phi}$.

Suppose that $l_{1}$ and $l_{2}$ are both even. Note that by [AV04], Section
6, $\widehat{\Gamma(l_{1},l_{2})}$ is generated by $\langle
v_{1},v_{2}\rangle$ such that $v_{1}=l_{2}e_{1}+\frac{l_{2}}{2}e_{2}$ and
$v_{2}=\frac{l_{1}}{2}e_{2}$ where $e_{1}([(\alpha,A)])=\alpha$ and
$e_{2}([(\alpha,A)])=\textrm{det}(A)$.

$\widehat{\Phi}(v_{1})([(\alpha,A)])=(v_{1}\circ\Phi)([(\alpha,A)])=v_{1}\left(\left[\left(\alpha^{\frac{d_{2}}{l_{1}}+\frac{2d_{2}}{l_{2}}},\alpha^{\frac{-d_{2}}{l_{1}}}A\right)\right]\right)=\alpha^{\frac{d_{2}l_{2}}{l_{1}}+2d_{2}}\alpha^{-\frac{-d_{2}l_{2}}{l_{1}}}\alpha^{-d}(\textrm{det}(A))^{\frac{l_{2}}{2}}=\alpha^{2d_{2}}(\textrm{det}(A))^{\frac{l_{2}}{2}}.$

Note that since both $l_{1}$ and $l_{2}$ are even we have that both $d_{1}$
and $d_{2}$ are even as well. Hence by [AV04], Section 6,
$\widehat{\Gamma(d_{1},d_{2})}$ is generated by $\langle u_{1},u_{2}\rangle$
such that $u_{1}=d_{2}e_{1}+\frac{d_{2}}{2}e_{2}$ and
$v_{2}=\frac{d_{1}}{2}e_{2}$ Suppose
$\widehat{\Phi}(v_{1})=a_{1}u_{1}+a_{2}u_{2}$. Then

$(a_{1}u_{1}+a_{2}u_{2})([(\alpha,A)])=\alpha^{d_{2}a_{1}}(\textrm{det}(A))^{\frac{d_{2}a_{1}}{2}}(\textrm{det}(A))^{\frac{d_{1}a_{2}}{2}}=\alpha^{d_{2}a_{1}}(\textrm{det}(A))^{\frac{d_{2}a_{1}+d_{1}a_{2}}{2}}.$

Hence we have $2d_{2}=d_{2}a_{1}\implies a_{1}=2$ and
$d_{2}a_{1}+d_{1}a_{2}=l_{2}=2d_{2}-d_{1}\implies a_{2}=-1$. Therefore
$\widehat{\Phi}(v_{1})=2u_{1}-u_{2}$. Similarly,

$\widehat{\Phi}(v_{2})([(\alpha,A)])=(v_{2}\circ\Phi)([(\alpha,A)])=v_{2}\left(\left[\left(\alpha^{\frac{d_{2}}{l_{1}}+\frac{2d_{2}}{l_{2}}},\alpha^{\frac{-d_{2}}{l_{1}}}A\right)\right]\right)=\alpha^{-d_{2}}(\textrm{det}(A))^{\frac{l_{1}}{2}}$

Suppose $\widehat{\Phi}(v_{2})=b_{1}u_{1}+b_{2}u_{2}$. Then

$(b_{1}u_{1}+b_{2}u_{2})([(\alpha,A)])=\alpha^{d_{2}b_{1}}(\textrm{det}(A))^{\frac{d_{2}b_{1}}{2}}(\textrm{det}(A))^{\frac{d_{1}b_{2}}{2}}=\alpha^{d_{2}b_{1}}(\textrm{det}(A))^{\frac{d_{2}b_{1}+d_{1}b_{2})}{2}}$

Hence we have $-d_{2}=d_{2}b_{1}\implies b_{1}=-1$ and
$d_{2}b_{1}+d_{1}b_{2}=l_{1}=2d_{1}-d_{2}\implies b_{2}=2$. Therefore
$\widehat{\Phi}(v_{2})=-u_{1}+2u_{2}$.

Now the fact that the determinant of the matrix

$\begin{vmatrix}$2$&$-1$\\\ $-1$&$2$\\\ \end{vmatrix}$

is equal to $5$ which is not a unit in $\mathbb{Z}$, we have that the map
$\widehat{\Gamma(l_{1},l_{2})}\to\widehat{\Gamma(d_{1},d_{2})}$ is strictly
injective and hence the map
$\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}})\to\textrm{Pic}(\mathscr{H}_{1,3,d,d})$
is also strictly injective.

When $l_{1}$ odd, $d_{2}$ is odd. In this case generators of
$\widehat{\Gamma(l_{1},l_{2})}$ are given by
$v_{1}=l_{2}e_{1}+\frac{l_{2}(l_{1}+1)}{2}e_{2}$ and $v_{2}=l_{1}e_{2}$ and
after applying the automorphism
$\widehat{\Gamma(d_{1},d_{2})}\to\widehat{\Gamma(d_{1},d_{2})}$ given by
$[(\alpha,A)]\mapsto[(\alpha^{-1},\alpha A)]$, the generators of
$\Gamma(d_{1},d_{2})$ are given by
$u_{1}=d_{1}e_{1}+\frac{d_{1}(d_{2}+1)}{2}e_{2}$ and $u_{2}=d_{2}e_{2}$. A
similar computation again shows that the map
$\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}})\to\textrm{Pic}(\mathscr{H}_{1,3,d_{1},d_{2}})$
is strictly injective and that completes the proof. $\square$

The following proposition its corollary show that there does not exist a
tautological family over $M_{1,3,d_{1},d_{2}}^{0}$.

######  Proposition 3.13.

The codimension of $M_{1,3,d_{1},d_{2}}-M_{1,3,d_{1},d_{2}}^{0}$ is greater
than or equal to two.

Proof. Let $\mathbb{U}^{0}\subset\mathbb{U}$ denote the open set of forms
$(f_{1},f_{2})$ with trivial stabilizer under the action of
$\Gamma(l_{1},l_{2})$. It is enough to show that the complement of
$\mathbb{U}^{0}$ has codimension greater than or equal to two in $\mathbb{U}$.
Let $U\subset\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})$ be the
open set consisting of forms $(f_{1},f_{2})$ with trivial stabilizer under the
action of $\Gamma(l_{1},l_{2})$. We have shown in the proof of Proposition
2.26 that the complement of the open set $\mathbb{A}_{sm}(1,l_{i})^{0}$
consisting of forms with trivial stabilzer inside $GL_{2}/\mu_{l_{i}}$ has
codimension greater than or equal to two inside $\mathbb{A}_{sm}(1,l_{i})$.
Consider $\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}$.
Complement of this open set is a union of two closed subloci, each of
codimension greater than or equal to two inside
$\mathbb{A}_{sm}(1,l_{1})\times\mathbb{A}_{sm}(1,l_{2})$. Now consider
$(f_{1},f_{2})\in\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}$.
Suppose $([A],[\alpha A])$ be in the stabilizer of $(f_{1},f_{2})$. Then
$(f_{1}(A^{-1}x),f_{2}((\alpha A)^{-1}x))=(f_{1}(x),f_{2}(x))$. Hence
$f_{1}(x)=f_{1}(A^{-1}x)\implies
A\in\mu_{l_{1}}\implies[A]=[\textrm{id}_{GL_{2}/\mu_{l_{1}}}]$. Also
$f_{2}(x)=f_{2}((\alpha A)^{-1}x)\implies\alpha A\in\mu_{l_{2}}\implies[\alpha
A]=[\textrm{id}_{GL_{2}/\mu_{l_{2}}}]$. Hence $(f_{1},f_{2})$ has trivial
stabilizer in $\Gamma(l_{1},l_{2})$ and
$\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}\in U$. Hence
the complement of $U$ has codimension greater than or equal to two. Since
$\mathbb{U}^{0}=U\bigcap\mathbb{U}$, its complement is of codimension greater
than or equal to two in
$\mathbb{U}=\mathbb{U}\bigcap(\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0})$.
$\square$

######  Corollary 3.14.

There does not exist a tautological section on $M_{1,3,d_{1},d_{2}}^{0}$.

Proof. Suppose there exist such a section. Then we have a map
$M_{1,3,d_{1},d_{2}}^{0}\cong\mathscr{D}_{1,3,d_{1},d_{2}}^{0}\to\mathscr{H}_{1,3,d_{1},d_{2}}$.
Then there exist a map
$\textrm{Pic}(\mathscr{H}_{1,3,d_{1},d_{2}})\to\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}}^{0})\cong\textrm{Cl}(\mathscr{D}_{1,3,d_{1},d_{2}}^{0})\cong\textrm{Cl}(\mathscr{D}_{1,3,d_{1},d_{2}})\cong\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}})$
which is a section to the natural map
$\textrm{Pic}(\mathscr{D}_{1,3,d_{1},d_{2}})\to\textrm{Pic}(\mathscr{H}_{1,3,d_{1},d_{2}})$.
The first isomorphism in the chain is due to smoothness of
$\mathscr{D}_{1,3,d_{1},d_{2}}^{0}$, the second is because the complement of
$\mathscr{D}_{1,3,d_{1},d_{2}}^{0}$ in $\mathscr{D}_{1,3,d,d}$ has codimension
greater than or equal to two by Proposition 3.13 and third is due to
smoothness of the stack $\mathscr{D}_{1,3,d_{1},d_{2}}$. However the latter
map is strictly injective by Lemma 3.12 and cannot have a section. $\square$

We turn to the question of rationality of $\mathscr{H}_{1,3,d_{1},d_{2}}$ and
describe the geometry of $M_{1,3,d_{1},d_{2}}$.

######  Proposition 3.15.

$\mathscr{H}_{1,3,d_{1},d_{2}}$ is unirational. The coarse moduli
$M_{1,3,d_{1},d_{2}}$ is unirational and is fibred over a rational base by
fibres which are birational to the homogeneous space
$\frac{GL_{2}/\mu_{l_{1}}\times
GL_{2}/\mu_{l_{2}}}{GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}}$. If
$\textrm{char}(\mathbb{k})=0$, $M_{1,3,d_{1},d_{2}}$ is rationally fibred over
a rational base.

Proof. Since,
$\mathscr{H}_{1,3,d_{1},d_{2}}\cong[\mathbb{U}/\Gamma(d_{1},d_{2})]$, where
$\mathbb{U}$ is an open subscheme of
$\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}$, we have that
$\mathscr{H}_{1,3,d_{1},d_{2}}$ admits a dominant one morphism from
$\mathbb{U}$ which is rational. Hence $\mathscr{H}_{1,3,d_{1},d_{2}}$ is
unirational.

Since we are dealing with the birational properties of $M_{1,3,d_{1},d_{2}}$,
we can replace $\mathbb{U}$ by
$\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}$.
Unirationality of $M_{1,3,d_{1},d_{2}}$ then follows since there is a dominant
rational map from
$\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0}\to
M_{1,3,d_{1},d_{2}}$. Consider the natural map

$(\mathbb{A}_{sm}(1,l_{1})^{0}\times\mathbb{A}_{sm}(1,l_{2})^{0})/\Gamma(l_{1},l_{2})\to\mathbb{A}_{sm}(1,l_{1})^{0}/(GL_{2}/\mu_{l_{1}})\times\mathbb{A}_{sm}(1,l_{2})^{0}/(GL_{2}/\mu_{l_{2}})$

sending $[(f_{1},f_{2})]\to([f_{1}],[f_{2}])$. This map is surjective and the
fibres are isomorphic to $\frac{GL_{2}/\mu_{l_{1}}\times
GL_{2}/\mu_{l_{2}}}{GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}}$.
Now the conclusion follows from part $(2)$ of the next remark. $\square$

######  Remark 3.16.

Two remarks are in order.

  * (1)

The group
$GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}\cong\frac{\mathbb{G}_{m}\times
GL_{2}}{\mu_{l_{1}}\times\mu_{l_{2}}}$ is not solvable. Note that solvability
of the denominator would guarantee the rationality of the fibres irrespective
of the characteristic of the base field (see [CZ17]).

  * (2)

The homogeneous space $\frac{GL_{2}/\mu_{l_{1}}\times
GL_{2}/\mu_{l_{2}}}{GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}}$ is
rational if $\textrm{char}(\mathbb{k})=0$.

Proof. (1) Assume, $\frac{\mathbb{G}_{m}\times
GL_{2}}{\mu_{l_{1}}\times\mu_{l_{2}}}$ is solvable which implies
$\mathbb{G}_{m}\times GL_{2}$ is solvable and implies $GL_{2}$ is solvable and
we arrive at a contradiction.

(2) We observe that the dimension of
$GL_{2}/\mu_{l_{1}}\times_{PGL_{2}}GL_{2}/\mu_{l_{2}}$ is the dimension of
$PGL_{2}$ which is 3. The lemma follows from Theorem 4.4 [CZ17].
$\blacksquare$

##  4\. Appendix

As a consequence of our study on least degree linearzation, we aim to prove a
Mestrano-Ramanan type results. We start with a definition.

######  Definition 4.1.

Let $\pi:\mathscr{F}\to U$ be a family of smooth projective varieties of
dimension $n$. A $G^{n}_{r}$ on $\mathscr{F}$ is a linear system on
$\mathscr{F}$ which restricted to $\mathscr{F}_{s}$ is a linear system
inducing a finite $r:1$ morphism
$\varphi_{s}:\mathscr{F}_{s}\to\mathbb{P}^{n}$ for each $s\in S$. We say that
such a $G^{n}_{r}$ is Galois (resp. simple cyclic) with Galois group $G$, if
each $\varphi_{s}$ is Galois (resp. simple cyclic) with Galois group $G$.

The existence of a global $g^{1}_{2}$ on families of hyperelliptic curves was
studied by Mestrano and Ramanan in [MR85]. In [GV06], the authors studied the
existence of global $g^{1}_{3}$ for trigonal curves.

######  Theorem 4.2.

The following statements hold.

  * (1)

Suppose $\textrm{gcd}(rd,n+1)\mid d$ and $\pi:\mathscr{F}\to U$ be a
tautological family over $U\subset M_{n,r,d}^{0}$ open. Then

    * (i)

there exists a simple cyclic $G^{n}_{r}$ on $\mathscr{F}$ if and only if
$\textrm{gcd}(rd,n+1)=1$,

    * (ii)

there always exist a $\textrm{gcd}(rd,n+1)G^{n}_{r}$ on $\mathscr{F}$ which is
simple cyclic.

  * (2)

Suppose $\pi:\mathscr{F}\to U$ be a tautological family over $U\subset
M_{1,3,d_{1},d_{2}}^{0}$ open. Then

    * (i)

there exist a cyclic $G^{1}_{3}$ on $\mathscr{F}$ if and only if either
$d_{1}$ or $d_{2}$ is odd,

    * (ii)

there exists a cyclic $2G^{1}_{3}$ on $\mathscr{F}$.

Proof. (1)(i) and (2)(i). In either case, let a $G^{n}_{r}$ on $\mathscr{F}$
be given a line bundle $\mathscr{L}$ and a space of global sections
$\mathscr{V}\subseteq H^{0}(\mathscr{L})$. Note that
$H^{0}(\mathscr{V}_{s})=n+1$ for all $s$. Let
$\mathscr{M}\subseteq\mathscr{L}$ be the subsheaf generated by $\mathscr{V}$.
Then $\pi_{*}(\mathscr{M})$ is locally free of rank $n+1$ and
$\pi:\mathscr{F}\to U$ factors as
$\pi:\mathscr{F}\to\mathbb{P}(\pi_{*}(\mathscr{M}))\to U$. If the $G^{n}_{r}$
is simple cyclic in $(1)(i)$ or Galois (cyclic) in $(2)(i)$ then the
Brauer–Severi scheme associated to the tautological section is trivial and
hence the conclusion follows from Remark 2.16 and Remark 3.8.

(1)(ii) and (2)(ii). A line bundle of fibrewise degree $k$ on the
Brauer–Severi scheme $P\to U$ associated to the tautological section pulls
back to give a $kG^{n}_{r}$ which is simple cyclic in the first case and
cyclic in the second case. Now the assertion follows from the fact that the
minimum fibrewise degree of a line bundle on $P\to U$ is
$\textrm{gcd}(rd,n+1)$ in the first case (Lemma 2.15) and $2$ in the second
case (Remark 3.8). $\square$

Finally, we include the proof of Proposition 2.26 for the sake of
completeness.

Proof of Proposition 2.26. Let us for simplicity, in this proof denote by
$M_{1,r,d}$ and $M_{1,r,d}^{0}$, the closed points. Define
$M^{\textrm{aut}}_{1,r,d}=M_{1,r,d}-M_{1,r,d}^{0}$ to be isomorphism classes
of the set

$\left\\{(\mathbb{P}^{1}\to\mathbb{k},D)|\exists\sigma:\mathbb{P}^{1}\to\mathbb{P}^{1},\sigma\neq\textrm{id},\sigma(D)=D\right\\}$

where $D$ is a reduced divisor of degree $rd$. It is known that automorphisms
of $\mathbb{P}^{1}$ that fix a smooth divisor form a finite group and hence
$\sigma$ has finite order. We will try to estimate the dimension of
$M^{\textrm{aut}}_{1,r,d}$. Consider the following stratification of
$M^{\textrm{aut}}_{1,r,d}$,

$M^{\textrm{aut}}_{1,r,d}=\bigcup_{p}M^{\textrm{aut},p}_{1,r,d}$

where $M^{\textrm{aut},p}_{1,r,d}$ parametrizes isomorphism classes of

$\left\\{(\mathbb{P}^{1}\to\mathbb{k},D)|\exists\sigma:\mathbb{P}^{1}\to\mathbb{P}^{1},\sigma^{p}=\textrm{id},\sigma\neq\textrm{id},\sigma(D)=D\right\\}$

where $p$ is a prime. Consider the map
$\tau:\mathbb{P}^{1}\xrightarrow{p:1}\mathbb{P}^{1}/\sigma\cong\mathbb{P}^{1}$.
Let $\\{x_{1},\cdots,x_{n}\\}$ be the reduced points of the ramification
divisor. Hence they are fixed under the action of $\sigma$. Consider
$D-\\{x_{1},\cdots,x_{n}\\}$. Since $\sigma(D)=D$, we have
$\sigma(D-\\{x_{1},\cdots,x_{n}\\})=D-\\{x_{1},\cdots,x_{n}\\}$. Now the
action of $\sigma$ is free and transitive on
$\mathbb{P}^{1}-\\{x_{1},\cdots,x_{n}\\}$. So $D-\\{x_{1},\cdots,x_{n}\\}$ is
a disjoint union of orbits, each of cardinality $p$. Hence length of
$D-\\{x_{1},\cdots,x_{n}\\}=kp$ for some $k$ and hence $p\leq rd$. Hence the
stratification

$M^{\textrm{aut}}_{1,r,d}=\bigcup_{p\leq rd}M^{\textrm{aut},p}_{1,r,d}$

is a finite union. We need to estimate
$\textrm{dim}(M^{\textrm{aut},p}_{1,r,d})$ for each $p\leq rd$.

Now consider the moduli $M^{\textrm{aut},p-\textrm{fixed}}_{1,r,d}$
parameterizing isomorphism classes of the set

$\left\\{(\mathbb{P}^{1}\to\mathbb{k},D,\sigma:\mathbb{P}^{1}\to\mathbb{P}^{1})|\sigma^{p}=\textrm{id},\sigma\neq\textrm{id},\sigma(D)=D\right\\}$

There is a finite surjective map from
$M^{\textrm{aut},p-\textrm{fixed}}_{1,r,d}\to M^{\textrm{aut},p}_{1,r,d}$. So
we estimate $\textrm{dim}(M^{\textrm{aut},p-\textrm{fixed}}_{1,r,d})$ for each
$p\leq rd$. Now we use the well known fact that the map $\tau$ has a cyclic
ramification of order $p$ at two points $x_{1}$ and $x_{2}$ and that $\tau$ is
uniquely determined by it. Now there could be three possible cases,

  * (1)

$D$ contains none of $x_{1}$ and $x_{2}$,

  * (2)

$D$ contains one of $X_{1}$ and $X_{2}$,

  * (3)

$D$ contains both of $x_{1}$ and $x_{2}$.

Now let $M^{\textrm{aut},p-\textrm{fixed},i}_{1,r,d}$ be defined as the
isomorphism classes of

$\left\\{\mathbb{P}^{1}\to\mathbb{k},D,\sigma:\mathbb{P}^{1}\to\mathbb{P}^{1}|\sigma^{p}=\textrm{id},\sigma\neq\textrm{id},\sigma(D)=D,\\\
D\hskip 1.42271pt\textrm{contains exactly}\hskip 1.42271pti\hskip
1.42271pt\textrm{of the two ramification points of $\tau$}\right\\}.$

In other words, $M^{\textrm{aut},p-\textrm{fixed},i}_{1,r,d}$ parametrizes as
the isomorphism classes of

$\left\\{\mathbb{P}^{1}\to\mathbb{k},(R,E)\textrm{ where $R$ and $E$ are
disjoint reduced divisors of degree $2$ and $\frac{rd-i}{p}$
respectively}\right\\}.$

Hence,
$\textrm{dim}(M^{\textrm{aut},p-\textrm{fixed},i}_{1,r,d})=2+\frac{rd-i}{p}-3=\frac{rd-i}{p}-1\leq\frac{rd}{2}-1$.
Now $\textrm{dim}(M_{1,r,d})=rd-3$. Hence if $rd\geq 8$, our statement
follows. $\square$

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