# Étale cohomological stability of the
moduli space of stable elliptic surfaces

Oishee Banerjee, Jun–Yong Park and Johannes Schmitt

###### Abstract

We compute the (stable) étale cohomology of
$\mathrm{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$, the moduli stack of degree
$n$ morphisms from a smooth projective curve $C$ to the weighted projective
stack $\mathcal{P}(\vec{\lambda})$, the latter being a stacky quotient defined
by
$\mathcal{P}(\vec{\lambda}):=\left[\mathbb{A}^{N}-\\{0\\}/\mathbb{G}_{m}\right]$,
where $\mathbb{G}_{m}$ acts by weights
$\vec{\lambda}=(\lambda_{0},\cdots,\lambda_{N})\in\mathbb{Z}^{N}_{+}$. Our key
ingredient is formulating and proving the étale cohomological descent over the
category $\Delta S$, the symmetric (semi)simplicial category. An immediate
arithmetic consequence is the resolution of the geometric Batyrev–Manin type
conjecture for weighted projective stacks over global function fields. Along
the way, we also analyze the intersection theory on weighted projectivizations
of vector bundles on smooth Deligne–Mumford stacks.

##  1 Introduction

Fix a base field $K$ and let $C/K$ be a smooth, projective and geometrically
connected curve of genus $g$. The moduli space of morphisms of degree $n$ from
$C$ to the projective space $\mathbb{P}^{N}$, sometimes dubbed as a Hom-space
and denoted by $\operatorname{Hom}_{n}(C,\mathbb{P}^{N})$, has been studied
extensively for decades, using techniques ranging from scanning maps in the
case of $K=\mathbb{C}$ with the Euclidean topology (see e.g. [Segal, CCMM,
KS]), to more algebraic approaches in the setting of more general base fields
(see e.g. [FW16, Banerjee] and the references therein).

In this paper, we consider a generalization: what if we replace the target
space by a weighted projective stack?

To elaborate, given a vector $\vec{\lambda}=(\lambda_{0},\dotsc,\lambda_{N})$
of positive weights $\lambda_{i}\in\mathbb{Z}_{+}$, we define the
$N$-dimensional weighted projective stack

$\mathcal{P}(\vec{\lambda})\coloneqq[(\mathbb{A}_{x_{0},\dotsc,x_{N}}^{N+1}\setminus
0)/\mathbb{G}_{m}]$

where $\zeta\in\mathbb{G}_{m}$ acts by
$\zeta\cdot(x_{0},\dotsc,x_{N})=(\zeta^{\lambda_{0}}x_{0},\dotsc,\zeta^{\lambda_{N}}x_{N})$.
Now consider the Hom-stack of degree $n\in\mathbb{Z}_{+}$ morphisms from $C$
to $\mathcal{P}({\vec{\lambda}})$, which is defined as:

$\displaystyle\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))\coloneqq\left\\{f\colon
C\to\mathcal{P}(\vec{\lambda}):f^{*}\mathcal{O}_{\mathcal{P}(\vec{\lambda})}(1)\in\mathrm{Pic}^{n}C\right\\}$
$\displaystyle=\Big{\\{}\big{(}L,[s_{0}:\ldots:s_{N}]\big{)}:L\in\mathrm{Pic}^{n}C,\,\,\,\,s_{i}\in
H^{0}(C,L^{\otimes\lambda_{i}}),$ $\displaystyle s_{0},\ldots,s_{N}\text{ have
no common zeroes}\Big{\\}}/\mathbb{G}_{m}$

where $\mathbb{G}_{m}$ acts on the $i^{th}$ component
$H^{0}(C,L^{\otimes\lambda_{i}})$ by weight $\lambda_{i}$. The Hom-stack
$\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$ is a Deligne–Mumford
stack by [Olsson, Theorem 1.1].

The case of $\mathcal{P}(4,6)$ is of special interest111 See Example 2.3 and
Section 5 for other instances of weighted projective stacks of interests.:
noting that the stack $\overline{\mathcal{M}}_{1,1}$ of stable genus $1$
curves is isomorphic to $\mathcal{P}(4,6)$, one has that
$\operatorname{Hom}_{n}(C,\mathcal{P}(4,6))$ is a moduli stack of _stable
elliptic fibrations_ over $C$ (see Corollary 1.5).

Our goal in this paper is to study the $\ell$-adic cohomology of the space
$\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$. To avoid problems with
non-tame stabilizers, we henceforth assume $\mathrm{char}\,\,\,K$ is coprime
to $\mathit{l.c.m}({\vec{\lambda}})$ and that $\ell$ is a fixed prime that
does not equal $\mathit{char}\,\,\,K$.

#### Notations:

A bit of notation before we state our theorems. For a Deligne-Mumford stack
$\mathcal{X}$ over $K$, we denote the $\ell$-adic cohomology group with
rational coefficients by $H^{i}(\mathcal{X};\mathbb{Q}_{\ell})$ (warning: this
is not the étale cohomology with $\mathbb{Q}_{\ell}$ coefficients, see e.g.
[GL, Warning 3.2.1.9], or any text on étale cohomology of schemes e.g.
[Milne]); by the same token a sheaf of $\mathbb{Q}_{\ell}$ vector spaces is a
$\mathbb{Z}_{\ell}$-sheaf $\mathcal{F}=(\mathcal{F}_{n})$ and

$H^{i}(\mathcal{X};\mathcal{F}):=\varprojlim
H^{i}(\mathcal{X};\mathcal{F}_{n})\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}$

(similar to the notations set up in [Milne, Section 19]). Finally, let us
denote a vector space spanned by $\\{a_{1},\ldots,a_{k}\\}$ over
$\mathbb{Q}_{\ell}$ by $\mathbb{Q}_{\ell}\\{a_{1},\ldots,a_{k}\\}$ and write
$\operatorname{Gal}(\overline{K},K)$ for the absolute Galois group of $K$.

######  Main Theorem 1.1 (Cohomological stability).

Let $\lambda_{0},\cdots,\lambda_{N}\in\mathbb{Z}_{+}$, and let
$\mathcal{P}({\vec{\lambda}})$ be a weighted projective stack with weights
${\vec{\lambda}}=(\lambda_{0},\cdots,\lambda_{N})$. Let $C$ be a smooth
projective curve of genus $g$. Let $N$ and $n$ be fixed positive integers such
that $n\geq 2g$. Set $n_{0}:=n-2g$. Then there exists a second quadrant
spectral sequence, which converges to
$H^{*}(\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}));\mathbb{Q}_{\ell})$ as
an algebra, which has the following description. The $E_{2}$ term is a
bigraded algebra that collapses on $E_{2}^{-p,q}\Big{|}_{p\leq n_{0}}$.
Furthermore, $E_{2}^{-p,q}\Big{|}_{p\leq n_{0}}$ is a quotient of the graded
commutative $\mathbb{Q}_{\ell}$-algebra

$H^{*}(J(C);\mathbb{Q}_{\ell})[h]/h^{N}\otimes\wedge\mathbb{Q}_{\ell}\\{t\\}\otimes\operatorname{Sym}\mathbb{Q}_{\ell}\\{\alpha_{1},\ldots,\alpha_{2g}\\},$

where $H^{i}(J(C);\mathbb{Q}_{\ell})$ has degree $(0,i)$, $h$ has degree
$(0,2)$, $t$ has degree $(-1,2N+2)$ and $\alpha_{i}$ has degree $(-1,2N+1)$
for all $i$, modulo elements of degree $(-i,j)$ with $i>n_{0}$. Furthermore,
the eigenvalues of the action of $\operatorname{Gal}(\overline{K},K)$ on
$\mathbb{Q}_{\ell}\\{\alpha_{1},\ldots,\alpha_{2g}\\}$ are pure of weight
$2N+1$; $\mathbb{Q}_{\ell}\\{t\\}\cong\mathbb{Q}_{\ell}(-(N+1))$ and $h$ is a
generator of $\mathbb{Q}_{\ell}(-1)$.

The special case of rational curve $C=\mathbb{P}^{1}$ deserves a mention in
its own right:

######  Theorem 1.2.

Let $n$ be a positive integer. Then

$H^{*}(\mathrm{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}));\mathbb{Q}_{\ell})\cong\frac{\mathbb{Q}_{\ell}[h]}{h^{N}}\otimes\wedge\mathbb{Q}_{\ell}\\{t\\}$

where $h$ has cohomological degree $2$ and is a generator of
$\mathbb{Q}_{\ell}(-1)$, and $t$ has cohomological degree $2N+1$ and is a
generator $\mathbb{Q}_{\ell}(-(N+1))$. In particular, we have an isomorphism
of $Gal(\overline{K}/K)$-representations:

$\displaystyle
H^{i}(\mathrm{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}));\mathbb{Q}_{\ell})=\begin{cases}\mathbb{Q}_{\ell}(-j)&i=2j,0\leq
j\leq N-1\\\ \mathbb{Q}_{\ell}(-(j+1))&i=2j+1,N\leq j\leq 2N-1\\\ 0&\text{
otherwise.}\end{cases}$

The $\ell$-adic étale cohomology with Frobenius weights of the Hom-stack
naturally gives the following weighted point count of
$\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$ over a finite field
$\mathbb{F}_{q}$ via the Grothendieck-Lefschetz trace formula for Artin stacks
(c.f. Theorem 2.5).

######  Theorem 1.3.

Let $\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$ be the Hom stack of
degree $n\geq 2g$ morphisms from a smooth projective genus $g$ curve $C$ to
the $N$-dimensional weighted projective stack
$\mathcal{P}(\vec{\lambda})=\mathcal{P}(\lambda_{0},\dotsc,\lambda_{N})$ with
$|\vec{\lambda}|:=\sum\limits_{i=0}^{N}\lambda_{i}$. Then the weighted point
count of $\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$ over
$\mathbb{F}_{q}$ with
$\mathrm{char}(\mathbb{F}_{q})\nmid\lambda_{i}\in\mathbb{Z}_{+}$ for every $i$
is a finite sum given by

$\displaystyle\\#_{q}\left(\operatorname{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))\right)$
$\displaystyle=q^{|\vec{\lambda}|n+N-2g}+a_{\frac{1}{2}}\cdot
q^{|\vec{\lambda}|n+N-2g-\frac{1}{2}}+a_{1}\cdot q^{|\vec{\lambda}|n+N-2g-1}+$
$\displaystyle\ldots+a_{i}\cdot q^{|\vec{\lambda}|n+N-2g-i}+\ldots$

where $i\in\mathbb{Z}_{+}{[\frac{1}{2}]}$, the coefficients $a_{i}$ for
$i<n-2g$ are independent of $n$, and for $i\geq n-2g$ we have $a_{i}\cdot
q^{|\vec{\lambda}|n+N-2g-i}\ll q^{|\vec{\lambda}|n+N-2g}$ .

In the case of rational curve $C=\mathbb{P}^{1}$, the above exact étale
cohomology gives us the exact weighted point count of
$\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$ over a
finite field $\mathbb{F}_{q}$.

######  Theorem 1.4.

Let $\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$ be the
Hom stack of degree $n\geq 1$ morphisms from a smooth projective line
$\mathbb{P}^{1}$ to the $N$-dimensional weighted projective stack
$\mathcal{P}(\vec{\lambda})=\mathcal{P}(\lambda_{0},\dotsc,\lambda_{N})$ with
$|\vec{\lambda}|:=\sum\limits_{i=0}^{N}\lambda_{i}$. Then the weighted point
count of $\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$
over $\mathbb{F}_{q}$ with
$\mathrm{char}(\mathbb{F}_{q})\nmid\lambda_{i}\in\mathbb{Z}_{+}$ for every $i$
is equal to

$\begin{array}[]{ll}\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))\right)&=\left(\sum\limits_{i=0}^{N}q^{i}\right)\cdot\left(q^{|\vec{\lambda}|n}-q^{|\vec{\lambda}|n-N}\right)\\\
&\\\
&=q^{|\vec{\lambda}|n-N}\cdot\left(q^{2N}+\dotsb+q^{N+1}-q^{N-1}-\dotsb-1\right)\end{array}$

This result, in particular, effectively answers the geometric Batyrev–Manin
type conjecture (the enumeration of rational points of bounded height on
varieties over global fields, see [BM]) on weighted projective stacks over
global function fields.

As mentioned earlier, $\overline{\mathcal{M}}_{1,1}$ is of special interest;
the Deligne–Mumford stack $\overline{\mathcal{M}}_{1,1}$ of stable elliptic
curves is isomorphic to $\mathcal{P}(4,6)$ over
$\mathrm{Spec}(\mathbb{Z}[1/6])$ (c.f. Example 2.3). Consequently, the Hom
stack $\operatorname{Hom}_{n}(C,\mathcal{P}(4,6))$ over $\text{char}(K)\neq
2,3$ is isomorphic to the moduli stack $\mathcal{L}_{12n,g}$ of stable
elliptic fibrations with a section and $12n$ nodal singular fibers over the
parameterized smooth projective base curve $C_{K}$ of genus $g$ (for further
details on the formulation of the moduli stack as Hom-stack we refer to [HP,
§3] and [Park–Schmitt, §1]).

######  Corollary 1.5.

Let $\mathcal{L}_{12n,g}$ be the moduli stack of stable elliptic fibrations
with a section and discriminant degree $12n$ over the parameterized smooth
projective basecurve $C_{\mathbb{F}_{q}}$ of genus $g$. If
$\mathrm{char}(\mathbb{F}_{q})\neq 2,3$ and $n\geq 2g$, then the weighted
point count of $\mathcal{L}_{12n,g}$ over $\mathbb{F}_{q}$ is the finite sum

$\begin{array}[]{ll}\\#_{q}\left(\mathcal{L}_{12n,g}\right)&=q^{10n+1-2g}+a_{\frac{1}{2}}\cdot
q^{10n+1-2g-\frac{1}{2}}+a_{1}\cdot q^{10n+1-2g-1}+\ldots+a_{i}\cdot
q^{10n+1-2g-i}+\cdots\end{array}$

where $i\in\mathbb{Z}_{+}{[\frac{1}{2}]}$, the coefficients $a_{i}$ for
$i<n-2g$ are independent of $n$, and for $i\geq n-2g$ we have $a_{i}\cdot
q^{10n+1-2g-i}\ll q^{10n+1-2g}$ .

######  Remark 1.6.

Recall that the weighted point count of $\mathcal{L}_{12n,g}$ gives the same
number as that of the moduli of semistable elliptic surfaces (c.f. [HP,
Proposition 11]). Consequently, we have an estimate of the counting function

$\mathcal{N}(\mathbb{F}_{q}(C),0<q^{12n}\leq B),$

which counts the number of semistable (i.e., strictly multiplicative
reductions) elliptic curves over the parameterized smooth projective genus $g$
basecurve $C_{\mathbb{F}_{q}}$ ordered by height of discriminant
$0<ht(\Delta)=q^{12n}\leq B$ for $n\geq 2g$ as :

$\mathcal{N}(\mathbb{F}_{q}(C),0<q^{12n}\leq
B)=\sum\limits_{n=1}^{\left\lfloor\frac{log_{q}B}{12}\right\rfloor}2\cdot\\#_{q}\left(\mathcal{L}_{12n,g}\right)=2\cdot\frac{(q^{11-2g}-q^{9-2g})}{(q^{10}-1)}\cdot
B^{\frac{5}{6}}+o(B^{\frac{5}{6}})$

where the factor of 2 comes from the hyperelliptic involution.

This result, in particular, effectively answers the geometric Shafarevich’s
conjecture (the enumeration of families of algebraic curves (or abelian
varieties) with bounded bad reductions over global fields, see [Shafarevich])
on counting semistable elliptic curves over global function fields
$\mathbb{F}_{q}(C)$ with $\mathrm{char}(\mathbb{F}_{q})\neq 2,3$ ordered by
bounded height of discriminant.

### Methods

The central technique in our cohomology computation follows the method behind
the main theorem of [Banerjee]. We construct a suitable _$\Delta S$ object_ in
the category of Deligne-Mumford stacks, which is a simplicial object enjoying
additional properties, whose homotopy colimit is the discriminant locus

$\displaystyle\mathcal{Z}\coloneqq\left\\{f\colon
C\to\mathcal{P}(\vec{\lambda}):f^{*}\mathcal{O}_{\mathcal{P}(\vec{\lambda})}(1)\in\mathrm{Pic}^{n}C\right\\}$
$\displaystyle=\Big{\\{}\big{(}L,[s_{0}:\ldots:s_{N}]\big{)}:L\in\mathrm{Pic}^{n}C,\,\,\,\,s_{i}\in
H^{0}(C,L^{\otimes\lambda_{i}}),$ $\displaystyle s_{0},\ldots,s_{N}\text{ have
at least one common zero}\Big{\\}}/\mathbb{G}_{m}$

This category $\Delta S$, called the _symmetric simplicial category_ , is a
small category defined by Fiederowicz and Loday in [FL]. It contains $\Delta$
as a subcategory, its objects are that of $\Delta$ and it enjoys much of the
key properties of $\Delta$ (being equipped with a natural concept of face and
degeneracy maps, all compatible with those in $\Delta$, and sometimes a better
substitute for $\Delta$ for simplicial techniques in topology - evidence at
hand is Theorem 1.1). Among other things, category $\Delta S$ naturally
diminishes all the combinatorial complexities that come with $\Delta$ owing to
extra automorphisms of its objects; it gives a natural Koszul resolution in
the category of constructible $\ell$-adic sheaves on $\mathcal{Z}$ and that in
turn computes the desired cohomology.

It should be highlighted here that whereas cohomological descent for schemes
has been well studied in the pioneering work of Deligne ([Deligne]), the
translation of it to the world of algebraic stacks is technically much more
involved (see [LO] and the references therein); in fact, it is only within the
$\infty$-categorical framework that a satisfactory notion of cohomological
descent was formulated and proved in the illuminating work of Liu and Zheng
([LZ]). For our purposes, we do not need the full power of [LZ]; we can
sidestep the stacky difficulties by proving a statement that exploits
representability in algebraic spaces and the concept of proper descent on them
(see Lemma 4.2 for a precise statement and its proof).

#### Context and connection to other works.

  * •

Arithmetic statistics: In the grand scheme of things, this paper is along an
interesting connection between the Batyrev–Manin and the Shafarevich
conjectures in some very special cases. That is, understanding the arithmetic
of rational points on moduli stacks of curves (or abelian varieties) over
global fields has direct implications to the enumerations of fibrations of
curves (or abelian varieties) over global fields. In a similar regard, the
inspiring recent work of [ESZB] initiated the program of understanding the
connection between the Batyrev–Manin’s conjecture to the [Malle]’s conjecture
(the enumeration of number fields of bounded discriminant). Over global
function fields, counting semistable elliptic surfaces over
$\mathbb{P}^{1}_{\mathbb{F}_{q}}$ has been first addressed in [de Jong]; he
worked directly with the generalized Weierstrass equations, which works even
in characteristic $2$ and $3$ (unlike our method). Works like [HP] and
[Park–Spink] have similar enumerations over $\text{char}(K)\neq 2,3$ via using
motives in $K_{0}(\mathrm{Stck}_{K})$, the _Grothendieck ring of $K$-stacks_
introduced by [Ekedahl] in 2009.

The strength in our paper lies in methods that entirely deviating from the
prior literature. The attractive feature of our enumeration result is its
distinctive homotopy theoretic flavour (that techniques of homological
stability can be used to resolve geometric Manin’s conjecture has been
proposed by Ellenberg-Venkatesh, see [EV]). More precisely, taking colimits
over the _symmetric simplicial category_ $\Delta S$ (which contains $\Delta$
as a subcategory, see [Banerjee]) one gets considerable control over technical
difficulties like the class groups being nontrivial in the case of global
function fields of higher genera. Furthermore, the central theorem of our
paper - _étale cohomological stability_ , is intrinsically stronger than point
count asymptotics.

On a related note, the number of discriminants $\leq B$ of an elliptic curve
over $\mathbb{Z}$ with smooth generic fiber is estimated to be asymptotic to
$B^{\frac{5}{6}}$ by [BMc]. The lower order term of order
$B^{(7-\frac{5}{27}+\epsilon)/12}$ for counting the stable elliptic curves
over $\mathbb{Q}$ by the bounded height of squarefree discriminants was
suggested by the work of [Baier], improving upon their previous error term in
[BB].

  * •

Étale cohomological stability: Starting with Quillen’s seminal work (see
[Quillen]), homological stability has been central to the study of topology of
families of spaces/ groups, with vast applications. However, the exploration
of étale cohomological stability is relatively new and sparse in the
literature. Some works that address this are, for example, Farb-Wolfson’s work
on étale hommological stability of configuration spaces of smooth varieties
(see [FW18]), Ellenberg-Venkatesh-Westerland’s influential work on homological
stability of Hurwitz schemes having a fixed Galois group that satisfy certain
conditions (see [EVW]), Banerjee’s work on spaces admitting symmetric
semisimplicial filtration (see [Banerjee]) etc. As the branched covers of the
$\mathbb{P}^{1}$ are the fibrations with $0$-dimensional fibers, the moduli of
fibrations $f:X\to C$ on fibered surfaces $X$ over $C=\mathbb{P}^{1}$ (or even
for $C$ of higher genus) with $1$-dimensional fibers is the next natural case
to work on. Additionally, our method fits many cases: on one hand, for
example, our techniques can just as well be applied to prove étale
cohomological stability of the Hom-stack from higher dimensional smooth
projective varieties to $\mathcal{P}({\vec{\lambda}})$. And in the case of the
domain being of dimension $1$, having computed the stable étale cohomology of
$\operatorname{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ for any weight vector
${\vec{\lambda}}$ (see Example 2.3), we can, for example, make estimates on
the number of $\mathbb{F}_{q}$-points on the moduli stack of generalized
elliptic fibrations with prescribed level structures (analogous to the work of
[HS] via global fields analogy) or multiple markings as in Section 5.

### Outline of the paper

  * •

In Section 2 we review the definition, properties and examples of weighted
projective stacks. We also recall basics on arithmetic of algebraic stacks
over finite fields.

  * •

We present an in depth analysis of the intersection theory in weighted
projective bundles in Section 3. We prove that the weighted projective bundles
formula holds for the Chow ring in Theorem 1.8 as well as for the étale
cohomology in Corollary 3.6.

  * •

We prove Theorem 1.1 in Section 4 using techniques from [Banerjee, Theorem 2].

  * •

Finally, in Section 5 we show how our methods can be applied similarly to the
moduli stacks of generalized elliptic fibrations with prescribed level
structures or multiple marked points thereby extending the work of [HP] to the
étale cohomological framework of this paper with numerous applications.

### Further results

Integral Picard group. For $N=1$ case with $\mathcal{P}(a,b)$, the $\ell$-adic
rational cohomology type of the moduli stack
$\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(a,b))$ is a 3-sphere
$\mathbb{S}^{3}$ with an odd-dimensional class at $i=3$ which is independent
of both $n$ and $(a,b)$. To say something even more precise about the homotopy
type of these moduli stacks, we now compute the integral Picard group of the
moduli as it gives the torsion part of the cohomology. It turns out that it is
always cyclic, and generated by the line bundle
$\mathcal{O}_{\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))}(1)$.

######  Theorem 1.7.

If $\mathrm{char}(K)$ does not divide $\lambda_{i}\in\mathbb{Z}_{\geq 1}$ for
every $i$, then the Picard group of the Hom stack
$\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$ is a
cyclic group generated by the line bundle
$\mathcal{O}_{\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))}(1)$
and is isomorphic to

$\mathrm{Pic}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda})))\cong\begin{cases}\mathbb{Z}/(n(\lambda_{0}+\lambda_{1}))\mathbb{Z}&\text{
for }N=1,\\\ \mathbb{Z}&\text{ for }N>1.\end{cases}$

###### Proof.

The proof is at the end of §3. ∎

The torsion Picard group
$\mathrm{Pic}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(a,b)))\cong\mathbb{Z}/((a+b)n)\mathbb{Z}$
does depend on the degree $n$ and the weights $(a,b)$ (contrary to the fixed
rational cohomology type) which leads one to conjecture that its fundamental
group (for stacks as in [Noohi2]) is
$\pi_{1}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(a,b)))\cong\mathbb{Z}/((a+b)n)\mathbb{Z}$
as confirmed by the work of Épshtein in [EP] where he showed
$\pi_{1}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(1,1)\cong\mathbb{P}^{1}))\cong\mathbb{Z}/2n\mathbb{Z}$
. From the perspective of homotopy theory, it is natural to recognize that the
homotopy type of the moduli stack is a Lens space. Lens spaces are important
as they are the only 3-manifolds with non-trivial finite cyclic fundamental
group.

Rational Chow rings of weighted projectivizations of vector bundles over tame
Deligne-Mumford stacks. For this, assume we have a vector bundle
$\mathcal{E}/S$ and a splitting

$\mathcal{E}=\mathcal{E}_{0}\oplus\ldots\oplus\mathcal{E}_{N}\,,$

for vector bundles $\mathcal{E}_{i}$. Given a vector ${\vec{\lambda}}$ of
positive integers, we can again form the weighted projective bundle

$\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})=[(\mathrm{Tot}(\mathcal{E})\setminus
0)/\mathbb{G}_{m}]\to S\,,$ (1)

where the torus $\mathbb{G}_{m}$ acts on $\mathcal{E}_{i}$ with weight
$\lambda_{i}$.

To state the result for the Chow group of
$\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})$, we introduce a notion of
_twisted Chern classes_. For this recall that the (standard) Chern polynomial
is given by

$c_{t}(\mathcal{E})=1+t\cdot
c_{1}(\mathcal{E})+t^{2}c_{2}(\mathcal{E})+\ldots$

For a vector bundle $\mathcal{E}$ splitting into $n+1$ summands as above, the
Whitney sum formula implies

$c_{t}(\mathcal{E})=\prod_{i=0}^{N}c_{t}(\mathcal{E}_{i})\,.$

Given a vector $\vec{\eta}=(\eta_{0},\ldots,\eta_{N})\in\mathbb{Z}_{>0}^{N+1}$
of positive integers, we define the $\vec{\eta}$-twisted Chern polynomial of
$\mathcal{E}$ as

$c^{\vec{\eta}}_{t}(\mathcal{E})=\prod_{i=0}^{N}c_{\eta_{i}t}(\mathcal{E}_{i})\,.$

Similarly, an individual Chern class $c^{\vec{\eta}}_{j}(\mathcal{E})$ is
defined as the coefficient of $t^{j}$ in this twisted Chern polynomial.
Thinking in terms of the Chern roots of the bundle $\mathcal{E}$, the twisted
Chern classes correspond to multiplying the Chern roots of the summand
$\mathcal{E}_{i}$ by $\eta_{i}$.

######  Theorem 1.8.

Let $S$ be a smooth Deligne-Mumford stack with a vector bundle
$\mathcal{E}=\bigoplus_{i=0}^{N}\mathcal{E}_{i}$ and let
${\vec{\lambda}}\in\mathbb{Z}_{\geq 1}^{N+1}$ be a vector of positive
integers. Let $L=\mathrm{lcm}({\vec{\lambda}})$ and consider the vector
$\vec{\eta}=(L/\lambda_{0},\ldots,L/\lambda_{N})$. Then we have

$A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q}_{\ell})=A^{*}(S,\mathbb{Q}_{\ell})[\zeta]/(\zeta^{N+1}+c^{\vec{\eta}}_{1}(\mathcal{E})\zeta^{N}+\ldots+c^{\vec{\eta}}_{N+1}(\mathcal{E}))\,,$
(2)

where $\zeta=L\cdot
c_{1}(\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(1))$.

##  2 Preliminaries

We first recall the definition of a weighted projective stack
$\mathcal{P}(\vec{\lambda})$.

######  Definition 2.1.

Let ${\vec{\lambda}}=(\lambda_{0},\ldots,\lambda_{N})\in\mathbb{Z}_{\geq
1}^{N+1}$ be a vector of $N+1$ positive integers. Consider the affine space
$U_{\vec{\lambda}}=\mathbb{A}_{x_{0},\dotsc,x_{N}}^{N+1}$ endowed with the
action of $\mathbb{G}_{m}$ with weights ${\vec{\lambda}}$, i.e. an element
$\zeta\in\mathbb{G}_{m}$ acts by

$\zeta\cdot(x_{0},\ldots,x_{N})=(\zeta^{\lambda_{0}}x_{0},\ldots,\zeta^{\lambda_{N}}x_{N})\,.$
(3)

The $N$-dimensional weighted projective stack $\mathcal{P}(\vec{\lambda})$ is
then defined as the quotient stack

$\mathcal{P}(\vec{\lambda})=\left[(U_{{\vec{\lambda}}}\setminus\\{0\\})/\mathbb{G}_{m}\right]\,.$

######  Remark 2.2.

When we wish to emphasize the field $K$ of definition of
$\mathcal{P}(\vec{\lambda})$, we use the notation
$\mathcal{P}_{K}(\vec{\lambda})$. All weighted projective stacks are smooth.
The stack $\mathcal{P}(\vec{\lambda})$ is Deligne–Mumford if and only if all
weights $\lambda_{i}$ are prime to the characteristic; in this case,
$\mathcal{P}(\vec{\lambda})$ is in fact tame Deligne–Mumford as in [AOV].
Notice that $\mathcal{P}(1,p)$ is not Deligne–Mumford in characteristic $p$
since it has a point with automorphism group $\mu_{p}$ which is not formally
unramified. When $\mathcal{P}(\vec{\lambda})$ is Deligne–Mumford, it is an
orbifold if and only if $\gcd(\lambda_{0},\dotsc,\lambda_{N})=1$; this is
because $\mathcal{P}(\vec{\lambda})$ has generic stabilizer
$\mu_{\gcd(\lambda_{0},\dotsc,\lambda_{N})}$.

The natural morphism $U_{\vec{\lambda}}\to\mathcal{P}(\vec{\lambda})$ is the
total space of the _tautological line bundle_
$\mathcal{O}_{\mathcal{P}(\vec{\lambda})}(-1)$ on
$\mathcal{P}(\vec{\lambda})$. As in the classical case, we denote by
$\mathcal{O}_{\mathcal{P}(\vec{\lambda})}(1)$ the dual of this line bundle.

The fine modular curves are quintessential moduli stacks. Under mild condition
on the characteristic of the base field $K$ (i.e., a field $K$ with
$\text{char}(K)$ does not divide $\lambda_{i}\in\mathbb{N}$ for every $i$),
the genus 0 modular curves are isomorphic to the weighted projective stacks
$\mathcal{P}(a,b)$.

######  Example 2.3.

An example that will play an important role throughout this paper is the
moduli stack of stable elliptic curves. When $\text{char}(K)\neq 2,3$, we have
an explicit isomorphism

$(\overline{\mathcal{M}}_{1,1})_{K}\cong[(\mathrm{Spec}~{}K[a_{4},a_{6}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(4,6)$

given by the short Weierstrass equation $y^{2}=x^{3}+a_{4}x+a_{6}$, where
$\zeta\cdot a_{i}=\zeta^{i}a_{i}$ for $\zeta\in\mathbb{G}_{m}$ and $i=4,6$.
See, e.g., [Hassett, Proposition 3.6].

Similarly, one could consider the stack $\overline{\mathcal{M}}_{1,1}[\Gamma]$
of generalized elliptic curves with $[\Gamma]$-level structure, introduced in
the work of Deligne and Rapoport [DR] (summarized in [Conrad2, §2] and also in
[Niles, §2]) (see Proposition 5.1).

Also, one could consider the stack $\overline{\mathcal{M}}_{1,m}(m-1)$ of
$m$-marked $(m-1)$-stable curves of arithmetic genus one formulated originally
by the works of [Smyth, Smyth2] (see Proposition 5.8).

For higher genus curves, we recall the notion of quasi–admissible covers
whereby the general member of $C$ is not an admissible cover of
$\mathbb{P}^{1}$ and have been studied in depth by [Stankova, §2.4.] as the
closest covers to the original families of stable curves. In this regard,
[Fedorchuk] introduced the proper Deligne–Mumford stack
$\mathcal{H}_{2g}[2g-1]$ of quasi-admissible genus $g\geq 2$ curves. For the
case of monic odd–degree hyperelliptic curves with a generalized Weierstrass
equation
$y^{2}=x^{2g+1}+a_{4}x^{2g-1}+a_{6}x^{2g-2}+a_{8}x^{2g-3}+\cdots+a_{4g+2}$ we
have

$\mathcal{H}_{2g}[2g-1]\cong\mathcal{P}(4,6,8,\dotsc,4g+2)$

by [Fedorchuk, Proposition 4.2(1)] over $\mathrm{char}(K)=0$ and by [HP2,
Proposition 5.9] over $\mathrm{char}(K)>2g+1$. The $g=2$ case by
$y^{2}=x^{5}+a_{4}x^{3}+a_{6}x^{2}+a_{8}x+a_{10}$ with
$\mathcal{H}_{4}[3]\cong\mathcal{P}(4,6,8,10)$ is of special interest as all
genus 2 curves are hyperelliptic.

We recall the definition of weighted $\mathbb{F}_{q}$–point count of an
algebraic stack $\mathcal{X}$.

######  Definition 2.4.

The weighted point count of $\mathcal{X}$ over $\mathbb{F}_{q}$ is defined as
a sum:

$\\#_{q}(\mathcal{X})\coloneqq\sum_{x\in\mathcal{X}(\mathbb{F}_{q})/\sim}\frac{1}{|\mathrm{Aut}(x)|},$

where $\mathcal{X}(\mathbb{F}_{q})/\sim$ is the set of
$\mathbb{F}_{q}$–isomorphism classes of $\mathbb{F}_{q}$–points of
$\mathcal{X}$ (i.e. the set of non–weighted points of $\mathcal{X}$ over
$\mathbb{F}_{q}$), and we take $\frac{1}{|\mathrm{Aut}(x)|}=0$ when
$|\mathrm{Aut}(x)|=\infty$.

A priori, the weighted point count can be $\infty$, but when $\mathcal{X}$ is
of finite type, then the stratification of $\mathcal{X}$ by schemes as in
[Behrend, Proof of Lemma 3.2.2] implies that
$\mathcal{X}(\mathbb{F}_{q})/\sim$ is a finite set, so that
$\\#_{q}(\mathcal{X})<\infty$.

The weighted point count $\\#_{q}(\mathcal{X})$ of an algebraic stack
$\mathcal{X}$ over $\mathbb{F}_{q}$ is algebro-topological under the framework
of the Weil conjectures as it is equal to the alternating sum of trace of
geometric Frobenius. In this regard, we recall the Grothendieck-Lefschetz
trace formula for Artin stacks by [Behrend, Sun].

######  Theorem 2.5 (Theorem 1.1. of [Sun]).

Let $\mathcal{X}$ be an Artin stack of finite type over $\mathbb{F}_{q}$. Let
$\operatorname{Frob}_{q}$ be the geometric Frobenius on $\mathcal{X}$. Let
$\ell$ be a prime number different from the characteristic of
$\mathbb{F}_{q}$, and let
$\iota:\overline{\mathbb{Q}}_{\ell}\overset{\sim}{\to}\mathbb{C}$ be an
isomorphism of fields. For an integer $i$, let
$H^{i}_{{\acute{et}},c}(\mathcal{X}_{/\overline{\mathbb{F}}_{q}};\overline{\mathbb{Q}}_{\ell})$
be the cohomology with compact support of the constant sheaf
$\overline{\mathbb{Q}}_{\ell}$ on $\mathcal{X}$. Then the infinite sum
regarded as a complex series via $\iota$

$\sum_{i\in\mathbb{Z}}~{}(-1)^{i}\cdot
tr\big{(}\operatorname{Frob}^{*}_{q}:H^{i}_{c}(\mathcal{X}_{/\overline{\mathbb{F}}_{q}};\overline{\mathbb{Q}}_{\ell})\to
H^{i}_{c}(\mathcal{X}_{/\overline{\mathbb{F}}_{q}};\overline{\mathbb{Q}}_{\ell})\big{)}$
(4)

is absolutely convergent to the weighted point count $\\#_{q}(\mathcal{X})$ of
$\mathcal{X}$ over $\mathbb{F}_{q}$.

Lastly, we recall that the cohomology in torsion-free coefficient coincide for
the fine moduli stack and its coarse moduli space. We show this for the étale
cohomology over base field $\overline{\mathbb{F}}_{q}$ in
$\mathbb{Q}_{\ell}$-coefficient by following the proof of the [Sun,
Proposition 7.3.2].

######  Lemma 2.6.

Let $\mathfrak{X}$ be a smooth separated tame Deligne–Mumford stack of finite
type over $\overline{\mathbb{F}}_{q}$ and the coarse moduli map
$c:\mathfrak{X}\to X$ giving the coarse moduli space $X$. Then for all $i$,
the pullback map

$c^{*}:H^{i}_{{\acute{et}}}({X}_{/\overline{\mathbb{F}}_{q}};\mathbb{Q}_{\ell})\cong
H^{i}_{{\acute{et}}}({\mathfrak{X}}_{/\overline{\mathbb{F}}_{q}};\mathbb{Q}_{\ell})$

is an isomorphism.

###### Proof.

As $\mathfrak{X}$ is a smooth separated tame Deligne–Mumford stack of finite
type over $\overline{\mathbb{F}}_{q}$, we can cover $X$ by étale charts $U$
such that pull-back of $U$ in $\mathfrak{X}$ is the quotient stack of an
algebraic space by a finite group [AOV, Theorem 3.2.]. The lemma follows from
the $\ell$-adic Leray spectral sequence as in [Behrend, Theorem 1.2.5] once we
have shown that the canonical map
$\mathbb{Q}_{\ell}\to\mathrm{R}c_{*}\mathbb{Q}_{\ell}$ is an isomorphism. It
suffices to show the isomorphism étale locally on $X$ and hence we assume
$\mathfrak{X}=[V/G]$ for some algebraic space $V$ under the action of finite
group $G$ where $\mathrm{char}(\overline{\mathbb{F}}_{q})$ does not divide
$|G|$. Let $q:V\to\mathfrak{X}$ be the canonical morphism. Observe that we
have $\mathbb{Q}_{\ell}\simeq(q_{*}\mathbb{Q}_{\ell})^{G}$. As both $q$ and
$c\circ q$ are finite maps and $\mathbb{Q}[G]$ for a finite group $G$ is a
semisimple $\mathbb{Q}$-algebra by the Maschke’s Theorem, we acquire

$\mathrm{R}c_{*}\mathbb{Q}_{\ell}\simeq\mathrm{R}c_{*}(q_{*}\mathbb{Q}_{\ell})^{G}\simeq((c\circ
q)_{*}\mathbb{Q}_{\ell})^{G}\simeq\mathbb{Q}_{\ell}$

∎

##  3 Intersection theory of weighted projective bundles

Many people have studied the cohomology of weighted projective spaces and
bundles, starting with [Kawasaki], where the base $S$ is a point. In [Al
Amrani] the author computes the integral cohomology of the coarse moduli space
of $\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})$ as a $\mathbb{Q}$-vector
space, and its multiplicative structure in case that $\mathcal{E}$ splits as a
sum of line bundles and ${\vec{\lambda}}$ satisfies a certain divisibility
condition (with similar results in étale cohomology). The divisibility
condition was later removed in [BFR].

Our proof of the theorem above is independent of the previous work, and
proceeds in two steps: first, in the case where $\mathcal{E}$ splits as a sum
of line bundles, the weighted projective bundle
$\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})$ admits a natural finite cover
to a _non-weighted_ projective bundle. Via the usual projective bundle
formula, this suffices to find its Chow groups. Here we explicitly use that we
work with $\mathbb{Q}$-coefficients. In the second step, we use a splitting
theorem to reduce to the first case.

Let $S$ be a smooth Deligne-Mumford stack and let

$\mathcal{E}=\mathcal{L}_{0}\oplus\cdots\oplus\mathcal{L}_{N}/S$ (5)

be a vector bundle on $S$ decomposing into a direct sum of $N+1$ line bundles
$\mathcal{L}_{i}$. We denote by $\mathrm{Tot}(\mathcal{E})\to S$ the total
space of this vector bundle, with zero section
$0\subseteq\mathrm{Tot}(\mathcal{E})$. Given a vector
${\vec{\lambda}}=(\lambda_{0},\ldots,\lambda_{N})$ of positive integers, there
exists an action of $\mathbb{G}_{m}$ on $\mathrm{Tot}(\mathcal{E})$ locally
given by
$\zeta\cdot(s_{0},\ldots,s_{N})=(\zeta^{\lambda_{0}}s_{0},\ldots,\zeta^{\lambda_{N}}s_{N})$.
In the following, we want to study the intersection theory of the _weighted
projective bundle_

$\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})=[(\mathrm{Tot}(\mathcal{E})\setminus
0)/\mathbb{G}_{m}]\to S.$ (6)

This is a (Zariski) locally trivial bundle with fibre
$\mathcal{P}({\vec{\lambda}})$ over $S$.

To present the answer, let $\ell=\mathrm{lcm}({\vec{\lambda}})$ be the least
common multiple of the $\lambda_{i}$, consider the vector
$\vec{\eta}=(\ell/\lambda_{0},\ldots,\ell/\lambda_{N})$ and the modified
bundle

$\mathcal{E}_{\eta}=\mathcal{L}_{0}^{\otimes\eta_{0}}\oplus\cdots\oplus\mathcal{L}_{N}^{\otimes\eta_{N}}/S\,.$
(7)

Denote by $\mathbb{P}(\mathcal{E}_{\eta})\to S$ the (unweighted) projective
bundle associated to this vector bundle. Then there exists a natural map

$\Phi:\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})\to\mathbb{P}(\mathcal{E}_{\eta}),[s_{0}:\ldots:s_{N}]\mapsto[s_{0}^{\otimes\eta_{0}}:\ldots:s_{N}^{\otimes\eta_{N}}]$
(8)

over $S$.

######  Theorem 3.1.

The map $\Phi$ is proper, flat and quasi-finite of degree
$d=\ell^{N}/\prod_{i=0}^{N}\lambda_{i}$ and the pullback

$\Phi^{*}:A^{*}(\mathbb{P}(\mathcal{E}_{\eta}),\mathbb{Q})\to
A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})$

induces an isomorphism on the Chow groups with $\mathbb{Q}$-coefficients,
whose inverse is given by $1/d\cdot\Phi_{*}$. In particular, we have

$A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})=A^{*}(S,\mathbb{Q})[\zeta]/(\zeta^{N+1}+c_{1}(\mathcal{E}_{\eta})\zeta^{N}+\ldots+c_{N+1}(\mathcal{E}_{\eta}))\,,$
(9)

where $\zeta=\ell\cdot
c_{1}(\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(1))$ and
where the Chern classes of $\mathcal{E}_{\eta}$ can be computed as

$c_{i}(\mathcal{E}_{\eta})=e_{i}(\eta_{0}c_{1}(\mathcal{L}_{0}),\ldots,\eta_{N}c_{1}(\mathcal{L}_{N}))\,,$

with $e_{i}$ the $i$-th elementary symmetric polynomial.

For the proof, we need some more preparatory results.

######  Lemma 3.2.

Let $G$ be an algebraic group acting on varieties $X,Y$ over $K$ such that the
action on $X$ is transitive and let $x_{0}\in X$ be a $K$-point with
stabilizer $G_{x_{0}}$. Then there exists a natural isomorphism

$f:[X\times Y/G]\xrightarrow{\sim}[Y/G_{x_{0}}]$ (10)

whose inverse is induced by the $G_{x_{0}}$-equivariant map

$Y\to\\{x_{0}\\}\times Y\subseteq X\times Y\,.$

###### Proof.

To construct $f$ consider the incidence variety

$I=\\{(g,x,y):gx=x_{0}\\}\subseteq G\times X\times Y$ (11)

Let $p:I\to X\times Y$ be the projection on the second and third factor. By
the assumption on transitivity, the map $p$ is surjective. Moreover, for the
action of the group $G_{x_{0}}$ on $I$ given by left-translation on the factor
$G$ (and the _trivial_ action on $X,Y$), we claim that $p$ is a principal
$G_{x_{0}}$-bundle. An fppf cover of $X$ trivializing this bundle is given by
the projection $p:I\to X\times Y$ itself.

Indeed, the fibre product $F=I\times_{X\times Y}I$ parameterizes tuples
$(g,g^{\prime},x,y)\in G\times G\times X\times Y$ such that $gx=x_{0}$,
$g^{\prime}x=x_{0}$. Setting $\widetilde{g}=g^{\prime}\circ g^{-1}\in
G_{x_{0}}$ this is equivalent to the data of $(g,\widetilde{g},x,y)$ such that
$gx=x_{0}$ and such that $\widetilde{g}\in G_{x_{0}}$, which defines a trivial
$G_{x_{0}}$-bundle over $I$ as desired.

Now we note that the map $\widetilde{f}:I\to Y,(g,x,y)\mapsto gy$ is
$G_{x_{0}}$-equivariant (with respect to $G_{0}$-action on the target induced
by its given $G$-action). Using that $p:I\to X\times Y$ is a principal
$G_{x_{0}}$-bundle together with the definition of the stack $[Y/G_{x_{0}}]$
it gives rise to a map $\overline{f}:X\times Y\to[Y/G_{x_{0}}]$. On the other
hand, it is not hard to see that this map is invariant under the $G$-action on
$X\times Y$ and thus factors through the quotient stack $[X\times Y/G]$ via
the desired map $f:[X\times Y/G]\to[Y/G_{x_{0}}]$. It is then straightforward
to check that the map (11) induces the inverse map of $f$. ∎

######  Proposition 3.3.

Given a vector ${\vec{\lambda}}\in\mathbb{Z}_{>0}^{N+1}$ of positive integers,
the weighted projective space $\mathcal{P}({\vec{\lambda}})$ has Chow group

$A^{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})=\mathbb{Q}[\zeta]/(\zeta^{N+1})\,,$

where $\zeta=c_{1}(\mathcal{O}_{\mathcal{P}({\vec{\lambda}})}(1))$ is the
first Chern class of the bundle
$\mathcal{O}_{\mathcal{P}({\vec{\lambda}})}(1)$. Moreover, the space
$\mathcal{P}({\vec{\lambda}})$ satisfies the _Chow-Künneth property_ , i.e.
for any smooth Deligne-Mumford stack $X$ the natural map

$\underbrace{A^{*}(X,\mathbb{Q})\otimes_{\mathbb{Q}}A^{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})}_{=A^{*}(X,\mathbb{Q})[\zeta]/(\zeta^{N+1})}\to
A^{*}(X\times\mathcal{P}({\vec{\lambda}}),\mathbb{Q})$

is an isomorphism.

Note that the intersection theory of $\mathcal{P}({\vec{\lambda}})$ was
studied before in various settings (see [Kawasaki, Al Amrani, BFR]). We give a
self-contained proof in the language of Chow groups with
$\mathbb{Q}$-coefficients, which gives us the additional Chow-Künneth property
that we need later.

###### Proof of Proposition 3.3.

For our proof we want to use the fact that $\mathcal{P}({\vec{\lambda}})$ has
a cellular decomposition (similar to the concept in [Fulton]), which in our
context means a locally closed stratification in stacks which are isomorphic
to finite quotients of affine spaces $\mathbb{A}^{n}$. The closures of these
strata then form a basis of $A^{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})$.

To make this more precise, we first note that for dimension reasons we indeed
have $\zeta^{N+1}=0$, so that there is a well-defined map from
$\mathbb{Q}[\zeta]/(\zeta^{N+1})$ to
$A^{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})$. Our goal is to show that
this map is surjective and injective.

For the surjectivity, consider the open substack

$\mathcal{U}_{0}=\\{[x_{0}:\ldots:x_{N}]:x_{0}\neq
0\\}\subseteq\mathcal{P}({\vec{\lambda}})$

with complement
$\mathcal{Z}_{0}\cong\mathcal{P}(\lambda_{1},\ldots,\lambda_{N})$. Then we
have the excision sequence

$A_{*}(\mathcal{Z}_{0},\mathbb{Q})\xrightarrow{i_{*}}A_{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})\to
A_{*}(\mathcal{U}_{0},\mathbb{Q})\to 0\,.$ (12)

By induction (starting with the trivial case $N=0$) we have
$A_{*}(\mathcal{Z}_{0},\mathbb{Q})\cong\mathbb{Q}[\zeta]/(\zeta^{N+1})$.
Moreover, since $\mathcal{Z}_{0}=\\{x_{0}=0\\}$ is the zero set of the section
$x_{0}$ of $\mathcal{O}_{\mathcal{P}({\vec{\lambda}})}(\lambda_{0})$, we have
that the image of the map $i_{*}$ equals the image of
$(\lambda_{0}\zeta)\cdot\mathbb{Q}[\zeta]/(\zeta^{N})\subseteq\mathbb{Q}[\zeta]/(\zeta^{N+1})$.
On the other hand, the open substack $\mathcal{U}_{0}$ is isomorphic to
$[\mathbb{G}_{m}\times\mathbb{A}^{N}/\mathbb{G}_{m}]$. Then observe that the
action of $\mathbb{G}_{m}$ on itself of weight $\lambda_{0}$ is transitive and
the stabilizer of $1$ equals $\mu_{\lambda_{0}}$. Thus using Lemma 3.2 we have

$\mathcal{U}_{0}\cong[\mathbb{A}^{N}/\mu_{\lambda_{0}}].$

This is a vector bundle over $B\mu_{\lambda_{0}}$ of rank $N$ and thus its
Chow group is isomorphic to
$A^{*}(B\mu_{\lambda_{0}},\mathbb{Q})=\mathbb{Q}\cdot[B\mu_{\lambda_{0}}]$ and
hence trivial. Thus the excision sequence (12) has the form

$\mathbb{Q}[\zeta]/(\zeta^{N})\xrightarrow{\cdot\lambda_{0}\zeta}A_{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})\to\mathbb{Q}\to
0\,,$

which implies that $\mathbb{Q}[\zeta]/(\zeta^{N+1})$ surjects onto the Chow
group of $\mathcal{P}({\vec{\lambda}})$. For injectivity, note that the kernel
of the natural map $\mathbb{Q}[\zeta]\to
A_{*}(\mathcal{P}({\vec{\lambda}}),\mathbb{Q})$ must contain the ideal
$(\zeta^{N+1})$ and thus be of the form $(\zeta^{m})$ for some $m\leq N+1$.
However, since $\zeta^{N}\neq 0$ as it is a zero-cycle of positive degree, the
kernel is indeed equal to $(\zeta^{N+1})$.

Given the stratification into $\mathcal{U}_{0},\mathcal{Z}_{0}$ as above, by
[BaeSchmitt, Proposition 2.10] and an inductive argument, the Chow-Künneth
property of $\mathcal{P}({\vec{\lambda}})$ follows if we can show the
analogous property for
$\mathcal{U}_{0}\cong[\mathbb{A}^{N}/\mu_{\lambda_{0}}]$. This amounts to
showing that

$A^{*}(X,\mathbb{Q})\cong
A^{*}(X\times[\mathbb{A}^{N}/\mu_{\lambda_{0}}],\mathbb{Q}).$

But again $X\times[\mathbb{A}^{N}/\mu_{\lambda_{0}}]$ is a vector bundle over
$X\times B\mu_{\lambda_{0}}$ and thus the Chow groups of the two spaces agree.
On the other hand, the coarse moduli space of $X\times B\mu_{\lambda_{0}}$
agrees with the coarse moduli space $|X|$ of $X$ and since the Chow group with
$\mathbb{Q}$-coefficients can be computed on such a coarse moduli space, we
get the desired chain of isomorphisms

$A^{*}(X\times[\mathbb{A}^{N}/\mu_{\lambda_{0}}],\mathbb{Q})\cong
A^{*}(X\times B\mu_{\lambda_{0}},\mathbb{Q})\cong A^{*}(|X|,\mathbb{Q})\cong
A^{*}(X,\mathbb{Q})\,.$

∎

###### Proof of Theorem 3.1.

Since both $\mathcal{P}(\mathcal{E},{\vec{\lambda}})$ and
$\mathbb{P}(\mathcal{E}_{\eta})$ are Zariski-locally trivial bundles over $S$,
the properties of the map $\Phi$ which are local on the target (proper, flat,
quasi-finite and the degree) can be verified for the trivial base
$S=\mathrm{Spec}(K)$. Thus consider the map
$\Phi:\mathcal{P}({\vec{\lambda}})\to\mathbb{P}^{N}$. The open subsets
$U_{i}=\\{x_{i}\neq 0\\}\subseteq\mathbb{P}^{N}$ form a Zariski cover and the
preimage of $U_{i}$ under $\Phi$ is precisely $\mathcal{U}_{i}=\\{s_{i}\neq
0\\}\subseteq\mathcal{P}({\vec{\lambda}})$, which is isomorphic to
$[\mathbb{A}^{N}/\mu_{\lambda_{i}}]$. Then we have a diagram

${\mathbb{A}^{N}}$${\mathcal{U}_{i}}$${U_{i}\cong\mathbb{A}^{N}}$

and the diagonal map $\mathbb{A}^{N}\to\mathbb{A}^{N}$ is the finite, flat map
of degree $\ell^{N}/\prod_{j\neq i}\lambda_{j}$ given by

$(s_{0},\ldots,\widehat{s_{i}},\ldots,s_{N})\mapsto(s_{0}^{\ell/\lambda_{0}},\ldots,\widehat{s_{i}^{\ell/\lambda_{i}}},\ldots,s_{N}^{\ell/\lambda_{N}})\,,$

where the hat indicates that we omit the $i$-th entry of the vector. Since the
map $\mathbb{A}^{N}\to\mathcal{U}_{i}$ is of degree $\lambda_{i}$, this shows
all local properties of the map $\Phi$.

Returning to the case of a general base $S$, it remains to show that
$\Phi^{*}$ is an isomorphism with inverse $1/d\cdot\Phi_{*}$. Indeed, once
this is established, the presentation (9) is simply the standard formula for
the Chow group of the projective bundle $\mathbb{P}(\mathcal{E}_{\eta})$,
where in addition we use that the natural line bundle
$\mathcal{O}_{\mathbb{P}(\mathcal{E}_{\eta})}(1)$ pulls back to
$\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(\ell)$ under
$\Phi$.

To complete the proof, we note that the composition
$(1/d\cdot\Phi_{*})\circ\Phi^{*}$ is clearly equal to the identity by the
projection formula (see [Fulton, Example 1.7.4] for the argument in the case
of schemes). This shows that $\Phi^{*}$ is injective, so we conclude by
proving its surjectivity.

We begin by noting, that since we have a diagram

${\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}$${\mathbb{P}(\mathcal{E}_{\eta})}$${S}$$\scriptstyle{\Phi}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$

it follows that the image of $\Phi^{*}$ contains all classes pulled back from
$S$ itself. On the other hand, the natural line bundle
$\mathcal{O}_{\mathbb{P}(\mathcal{E}_{\eta})}(1)$ pulls back to
$\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(\ell)$, so that
the class

$\widetilde{\xi}=c_{1}(\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(1))\in
A^{1}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})$

is likewise contained in the image of $\Phi^{*}$. We want to show that these
two types of classes generate
$A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})$ as a
$\mathbb{Q}$-algebra, i.e. that the natural map

$\Psi_{S}:A^{*}(S,\mathbb{Q})[\widetilde{\zeta}]\to
A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q}),\quad\alpha\cdot\widetilde{\zeta}^{m}\mapsto\pi^{*}(\alpha)\cdot\widetilde{\zeta}^{m}$

is surjective. For this, let $V\subseteq S$ be an open substack on which all
summands $\mathcal{L}_{i}$ of $\mathcal{E}$ are trivial. Then we have
$\pi^{-1}(V)\cong V\times\mathcal{P}({\vec{\lambda}})$ is a trivial product
over $V$. Denoting $K=S\setminus V$ the complement of $V$, we have two
excision sequences

$\leavevmode\hbox to298.34pt{\vbox
to52.74pt{\pgfpicture\makeatletter\hbox{\hskip
149.16808pt\lower-26.41928pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-149.16808pt}{-26.31944pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
32.29877pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.99323pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(\pi^{-1}(K),\mathbb{Q})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
32.29877pt\hfil&\hfil\hskip 59.853pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.54749pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
35.85303pt\hfil&\hfil\hskip 62.21075pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-33.90524pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(V\times\mathcal{P}({\vec{\lambda}}),\mathbb{Q})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
38.21078pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
28.5042pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.19865pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(K,\mathbb{Q})[\widetilde{\zeta}]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
28.5042pt\hfil&\hfil\hskip 51.25417pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.94865pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(S,\mathbb{Q})[\widetilde{\zeta}]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
27.2542pt\hfil&\hfil\hskip 51.37222pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.06671pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${A_{*}(V,\mathbb{Q})[\widetilde{\zeta}]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
27.37225pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-84.37054pt}{11.50003pt}\pgfsys@lineto{-61.17053pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-60.97055pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{11.33548pt}{11.50003pt}\pgfsys@lineto{34.53549pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{34.73547pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{111.75702pt}{11.50003pt}\pgfsys@lineto{134.95703pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{135.15701pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-116.86931pt}{-14.95973pt}\pgfsys@lineto{-116.86931pt}{2.24036pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-116.86931pt}{2.44034pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-128.34985pt}{-7.86804pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\Psi_{K}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-88.16512pt}{-23.81944pt}\pgfsys@lineto{-52.5717pt}{-23.81944pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-52.37172pt}{-23.81944pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{2.73665pt}{-23.81944pt}\pgfsys@lineto{45.37402pt}{-23.81944pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.574pt}{-23.81944pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-24.71754pt}{-14.95973pt}\pgfsys@lineto{-24.71754pt}{2.24036pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-24.71754pt}{2.44034pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.19809pt}{-7.86804pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\Psi_{S}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{73.34624pt}{-14.95973pt}\pgfsys@lineto{73.34624pt}{2.24036pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{73.34624pt}{2.44034pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{62.32681pt}{-7.86804pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\Psi_{V}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{100.91849pt}{-23.81944pt}\pgfsys@lineto{134.95703pt}{-23.81944pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{135.15701pt}{-23.81944pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (13)

Here the bottom row of the diagram is obtained from the excision sequence of
$V\subseteq S$ by tensoring with $\mathbb{Q}[\widetilde{\zeta}]$, which is a
right-exact operation. Now we see that the map $\Psi_{V}$ is surjective by
Proposition 3.3, whereas the surjectivity of $\Psi_{K}$ follows by Noetherian
induction. By the four lemma, it follows that $\Psi_{S}$ is surjective, which
finishes the proof. ∎

######  Claim 3.4 (Stacky Leray-Hirsch).

Let $S$ be a smooth Deligne-Mumford stack and let
$\mathcal{P}_{S}(\mathcal{E})$ be the weighted projective bundle on $S$
defined as before. Then
$H^{*}(\mathcal{P}_{S}(\mathcal{E});\mathbb{Q}_{\ell})\cong
H^{*}(S;\mathbb{Q}_{\ell})\otimes
H^{*}(\mathcal{P}({\vec{\lambda}});\mathbb{Q}_{\ell})$ as
$H^{*}(S;\mathbb{Q}_{\ell})$-modules.

###### Proof.

We first observe that the Leray-Hirsch condition is obviously satisfied; i.e.
the elements $1,\zeta,\cdots,\zeta^{N}$ of
$H^{*}(\mathcal{P}_{S}(\mathcal{E});\mathbb{Q}_{\ell})$ in cohomological
degrees $0,2,4,\cdots,2N$ restrict to form a basis of the cohomology of the
fibre $H^{*}(\mathcal{P}({\vec{\lambda}});\mathbb{Q}_{\ell})$. Therefore in
the derived category of $\ell$-adic sheaves over $S$ we have a map induced by
the cohomology classes (by the decomposition theorem):

$\oplus_{0\leq i\leq N}\underline{\mathbb{Q}_{\ell}}_{S}[-2i]\to
R\pi_{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{P}_{S}(\mathcal{E})}$

which is in fact an isomorphism thanks to the Leray-Hirsch condition. Now the
claim immediately follows by applying the global section functor to this
complex and taking cohomology. ∎

###  3.1 Proof of Theorem 1.8

Based on the theorem above, we can now also compute the Chow group of a
weighted projective bundle, where the bundle does not split into line bundles,
but rather into sub-vector bundles of possibly higher ranks.

For the proof, we need a variant of the _splitting principle_ inspired by an
answer of Angelo Vistoli on mathoverflow.

######  Lemma 3.5.

Given a Deligne-Mumford stack $S$ and a vector bundle $\mathcal{E}$ on $S$,
there exists a flat morphism $f:\widehat{S}\to S$ such that for any morphism
$W\to S$ the pullback

$f^{*}:A^{*}(W,\mathbb{Q})\to A^{*}(\widehat{S}\times_{S}W,\mathbb{Q})$

is injective and such that $f^{*}\mathcal{E}$ splits as a direct sum of line
bundles.

###### Proof.

Let $P\to S$ be the frame bundle of $\mathcal{E}$, whose fibre over $s\in S$
is the set of all bases of the vector space $\mathcal{E}_{s}$. It carries a
natural action of $\mathrm{GL}_{r}$. Let $T\subseteq
B\subseteq\mathrm{GL}_{r}$ be the maximal torus and Borel subgroup of
$\mathrm{GL}_{r}$, respectively. Then we claim that the map
$f:\widehat{S}=[P/T]\to S$ satisfies the properties of the lemma. Indeed,
since $\widehat{S}$ over any point $s\in S$ parameterizes a basis of
$\mathcal{E}_{s}$ up to scaling, the pullback $\mathcal{E}|_{\widehat{S}}$
naturally splits into line bundles. On the other hand, the map $f$ is the
composition $[P/T]\to[P/B]\to S=[P/\mathrm{GL}_{r}]$ of an affine bundle and a
surjective projective morphism, both of which have injective pullbacks in Chow
(which remains true after base change with $W\to S$). ∎

###### Proof of Theorem 1.8.

When the rank of all bundles $\mathcal{E}_{i}$ is $1$, the result is precisely
Theorem 3.1. In the more general setting, we can use the fact that the bundles
$\mathcal{E}_{i}$ split into line bundles Zariski locally and repeat the
excision sequence arguments from the proof of Theorem 3.1 to conclude that the
Chow group of $\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})$ is generated as a
$\mathbb{Q}$-algebra by classes pulled back from $S$ and by the class $\zeta$.
In other words, we have a natural surjection

$\Psi:A^{*}(S,\mathbb{Q})[\zeta]\to
A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})\,.$ (14)

Given $f:\widehat{S}\to S$ as in Lemma 3.5, we can ensure that all
$f^{*}\mathcal{E}_{i}$ split into line bundles (applying the lemma $N+1$ times
if necessary). Then we can compute the Chow group of
$\mathcal{P}_{\widehat{S}}(f^{*}\mathcal{E},{\vec{\lambda}})=\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})\times_{S}\widehat{S}$
and have a sequence of maps

$A^{*}(S,\mathbb{Q})[\zeta]\xrightarrow{\Psi}A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})\xrightarrow{f^{*}}A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})=A^{*}(\widehat{S},\mathbb{Q})[\zeta]/(\zeta^{N+1}+\ldots+f^{*}c_{N+1}^{\vec{\eta}}(\mathcal{E}))$

with $f^{*}$ injective. It follows that the element

$Q=\zeta^{N+1}+\zeta^{N}c_{1}^{\vec{\eta}}(\mathcal{E})\ldots+c_{N+1}^{\vec{\eta}}(\mathcal{E})$

must be in the kernel of $\Psi$. To show that it generates the kernel (as an
ideal), note that the quotient $A^{*}(S,\mathbb{Q})[\zeta]/(Q)$ has a direct
sum decomposition

$A^{*}(S,\mathbb{Q})[\zeta]/(Q)=\bigoplus_{i=0}^{N}A^{*}(S,\mathbb{Q})\cdot\zeta^{i}$
(15)

into $N+1$ copies of $A^{*}(S,\mathbb{Q})$. Let
$\pi:\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})\to S$ be the projection to
the base, then we have a linear map

$G:A^{*}(S,\mathbb{Q})[\zeta]/(Q)\to\bigoplus_{i=0}^{N}A^{*}(S,\mathbb{Q}),\alpha\mapsto\left(\pi_{*}(\Psi(\alpha)\cdot\zeta^{i})\right)_{i=0,\ldots,N}\,.$

We claim that with respect to the direct sum decomposition (15), the map $G$
is given by multiplication with a matrix of elements in $A^{*}(S,\mathbb{Q})$
with entries $1$ on the anti-diagonal, and vanishing entries above this anti-
diagonal. Assuming the claim, the map $G$ is injective (up to reordering it is
a triangular base change) and since it factors via
$\Psi:A^{*}(S,\mathbb{Q})[\zeta]/(Q)\to
A^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q})$, the map
$\Psi$ must also be injective. We have already seen it to be surjective, so it
is an isomorphism and thus the theorem is proven.

To show the claim, note that $\pi_{*}\zeta^{N}=[S]$, whereas
$\pi_{*}\zeta^{i}=0$ for $i<N$. It follows that for an element
$\alpha=\sum_{j=0}^{N}\alpha_{j}\zeta^{j}$ in the domain of $G$ (with
$\alpha_{j}\in A^{*}(S,\mathbb{Q})$), we have that the $i$th component of
$G(\alpha)$ is given by

$G(\alpha)_{i}=\sum_{j=0}^{N}\pi_{*}(\alpha_{j}\cdot\zeta^{j+i})=\sum_{j=0}^{N}\alpha_{j}\cdot\underbrace{\pi_{*}(\zeta^{j+i})}_{=0\text{
for }j<N-i}\,.$

But as stated before, we have $\pi_{*}(\zeta^{j+i})=[S]$ for $j=N-i$ and
$\pi_{*}(\zeta^{j+i})=0$ for $j<N-i$, which is precisely the shape of the
matrix describing $G$ that was claimed. ∎

The weighted projective bundle formula in the rational Chow ring holds in the
$\ell$-adic rational étale cohomology ring as well.

######  Corollary 3.6.

Let $S$ be a smooth Deligne-Mumford stack with a vector bundle
$\mathcal{E}=\bigoplus_{i=0}^{N}\mathcal{E}_{i}$ and let
${\vec{\lambda}}\in\mathbb{Z}_{\geq 1}^{N+1}$ be a vector of positive
integers. Let $L=\mathrm{lcm}({\vec{\lambda}})$ and consider the vector
$\vec{\eta}=(L/\lambda_{0},\ldots,L/\lambda_{N})$. Then we have

$H^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q}_{\ell})=H^{*}(S,\mathbb{Q}_{\ell})[\zeta]/(\zeta^{N+1}+c^{\vec{\eta}}_{1}(\mathcal{E})\zeta^{N}+\ldots+c^{\vec{\eta}}_{N+1}(\mathcal{E}))\,,$
(16)

where $\zeta=L\cdot
c_{1}(\mathcal{O}_{\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}})}(1))$.

###### Proof.

The rational cohomology of the smooth separated tame Deligne–Mumford stack is
the same as its coarse space by Lemma 2.6. This allows us to use the
description of the cohomology ring of a weighted projective bundle as an
algebra over the cohomology ring of the base as in [BFR, Theorem 6.2] (see
also [Al Amrani, §III]) which are proved via the application of Leray-Hirsch
Theorem.

These references finish the case when the bundle $\mathcal{E}$ splits as a sum
of line bundles. As before we can conclude the general case by applying the
splitting principle. For this, we repeat the arguments of the section above,
replacing Chow groups with cohomology with $\mathbb{Q}_{\ell}$-coefficients.
The global shape of the argument is the same, but we replace various technical
sub-claims as follows:

  * •

For the map $f:\widehat{S}\to S$ on which the pullback of $\mathcal{E}$
splits, we take the same construction $f:\widehat{S}=[P/T]\to S$ as in Lemma
3.5. Again, the pullback in cohomology via $f$, which is a composition of an
affine bundle and a full flag bundle, is injective (using the Leray-Hirsch
theorem).

  * •

Instead of using excision theorems, the surjectivity of the map

$\Psi:H^{*}(S,\mathbb{Q}_{\ell})[\zeta]\to
H^{*}(\mathcal{P}_{S}(\mathcal{E},{\vec{\lambda}}),\mathbb{Q}_{\ell}),$

defined analogously to (14), follows from Claim 3.4. The same claim also
proves that $\Psi$ becomes an isomorphism when dividing its domain by the
ideal generated by $Q$.

This last statement finishes the proof. ∎

We conclude this Section by proving Theorem 1.7.

###  3.2 Proof of Theorem 1.7

###### Proof.

We have that
$\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$ is
naturally an open substack of the weighted projective stack
$\mathcal{P}({\vec{\Lambda}})$ of dimension $|\vec{\lambda}|n+N$, whose
complement $Z\subseteq\mathcal{P}({\vec{\Lambda}})$ is the locus of points
$[s_{0}:\ldots s_{N}]$ where all $s_{i}$ _do_ vanish simultaneously at some
$q\in\mathbb{P}^{1}$. We have an isomorphism
$\mathrm{Pic}(\mathcal{P}({\vec{\Lambda}}))=\mathbb{Z}$ by [Noohi, Proposition
6.4.] and for the Picard group of
$\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda}))$, we claim
that the codimension of $Z$ in $\mathcal{P}({\vec{\Lambda}})$ is precisely
$N$, implying that for $N\geq 2$ we have

$\mathrm{Pic}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\vec{\lambda})))=\mathrm{Pic}(\mathcal{P}({\vec{\Lambda}})\setminus
Z)=\mathrm{Pic}(\mathcal{P}({\vec{\Lambda}}))=\mathbb{Z}$

by [Fringuelli, (ii) Theorem 2.1.4.]. To bound the codimension of $Z$,
consider the incidence variety

$\widehat{Z}=\\{(q,[s_{0},\ldots,s_{N}])\in\mathbb{P}^{1}\times\mathcal{P}({\vec{\Lambda}}):s_{i}(q)=0\text{
for
}i=0,\ldots,N\\}\subseteq\mathbb{P}^{1}\times\mathcal{P}({\vec{\Lambda}}).$

For a fixed $q\in\mathbb{P}^{1}$, the set of points
$(s_{0},\ldots,s_{N})\in{\vec{\Lambda}}$ such that all $s_{i}$ vanish at $q$
is a linear subspace of codimension $N+1$ in ${\vec{\Lambda}}$, and in fact,
the projection $\widehat{Z}\to\mathbb{P}^{1}$ is a projective bundle of the
corresponding rank. In particular, we have that
$\widehat{Z}\subset\mathbb{P}^{1}\times\mathcal{P}({\vec{\Lambda}})$ is
irreducible of codimension $N+1$. On the other hand, we have that
$Z\subset\mathcal{P}({\vec{\Lambda}})$ is the image of $\widehat{Z}$ under the
projection
$\mathbb{P}^{1}\times\mathcal{P}({\vec{\Lambda}})\to\mathcal{P}({\vec{\Lambda}})$.
Since the fibres of $\widehat{Z}\to Z$ are finite, it follows that the
codimension of $Z$ is indeed $N$.

On the one hand, for $N=1$ with $\mathcal{P}(\lambda_{0},\lambda_{1})$ we have
that the subvariety $Z$ of $\mathcal{P}({\vec{\Lambda}})$ is cut out by the
resultant $\mathrm{Res}(s_{0},s_{1})$. The resultant is an irreducible
polynomial in the coefficients of $s_{0},s_{1}$ ([GKZ, Chapter 8, Proposition-
Definition 1.1]), which is homogeneous of degree $n(\lambda_{0}+\lambda_{1})$.

Then, by [Fringuelli, (iv) Theorem 2.1.4.] we have an exact sequence

$\mathbb{Z}\to\mathrm{Pic}(\mathcal{P}({\vec{\Lambda}}))\to\mathrm{Pic}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\lambda_{0},\lambda_{1})))\to
0$

where the first map is defined by $1\mapsto 1\cdot V(\mathrm{Res})$. As
$\mathrm{Pic}(\mathcal{P}({\vec{\Lambda}}))\cong\mathbb{Z}$, we have that

$\mathrm{Pic}(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(\lambda_{0},\lambda_{1})))\cong\mathbb{Z}/\mathrm{Deg(Res)}\mathbb{Z}=\mathbb{Z}/((\lambda_{0}+\lambda_{1})n)\mathbb{Z}.$

∎

##  4 Stable cohomology of the Hom-stacks with weights

Recall that for a Deligne-Mumford stack $\mathcal{X}$ over $K$, we denote the
$\ell$-adic cohomology group with rational coefficients by
$H^{i}(\mathcal{X};\mathbb{Q}_{\ell})$ (warning: this is not the étale
cohomology with $\mathbb{Q}_{\ell}$ coefficients, see e.g. [GL, Warning
3.2.1.9], or any text on étale cohomology of schemes e.g. [Milne]); by the
same token a sheaf of $\mathbb{Q}_{\ell}$ vector spaces is a
$\mathbb{Z}_{\ell}$-sheaf $\mathcal{F}=(\mathcal{F}_{n})$ and

$H^{i}(\mathcal{X};\mathcal{F}):=\varprojlim
H^{i}(\mathcal{X};\mathcal{F}_{n})\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}$

(similar to the notations set up in [Milne, Section 19].) For brevity and
convenience, and noting there is no scope of confusion since we are always
working over rational coefficients, we will write $H^{i}(\mathcal{X})$ to
stand for $H^{i}(\mathcal{X};\mathbb{Q}_{\ell})$.

For ${\vec{\lambda}}=(\lambda_{0},\cdots,\lambda_{N})\in\mathbb{Z}^{N+1}_{>0}$
and positive integers $a,b$, we denote by $a{\vec{\lambda}}+b$ the vector
$(a\lambda_{0}+b,a\lambda_{1}+b,\cdots,a\lambda_{N}+b)$. Furthermore, given
${\vec{\lambda}}=(\lambda_{0},\cdots,\lambda_{N})$ with
$|\vec{\lambda}|\coloneqq\sum\limits_{i=0}^{N}\lambda_{i}$ and a vector space
$V\coloneqq V_{0}\oplus\cdots\oplus V_{N}$ of dimension
$|\vec{\lambda}|n+(N+1)-(N+1)g$, we denote by $\mathcal{P}(\oplus
V_{i},{\vec{\lambda}})$ the weighted projective stack where $\mathbb{G}_{m}$
acts on the direct summand $V_{i}$ of $V$ by weight $\lambda_{i}$. If
$\mathcal{E}$ is a vector bundle on a space $\mathcal{X}$ such that
$\mathcal{E}=\mathcal{E}_{0}\oplus\cdots\oplus\mathcal{E}_{N}$ then by
$\mathcal{P}_{\mathcal{X}}(\oplus_{i}\mathcal{E}_{i},{\vec{\lambda}})$ we
denote the weighted projectivization of $\mathcal{E}$ where $\mathbb{G}_{m}$
acts on the sub-bundle $\mathcal{E}_{i}$ by weight $\lambda_{i}$. Note that
when we write
$\mathcal{P}_{\mathcal{X}}(\oplus_{i}\mathcal{E}_{i},{\vec{\lambda}})$ we
implicitly assume that $i$ runs from 0 to $N$ and that
${\vec{\lambda}}\in\mathbb{Z}^{N+1}_{>0}$.

###  4.1 Proof of Main Theorem 1.1

Main Theorem 1.1 (and in turn Theorem 1.2) is a direct consequence of
[Banerjee, Theorem 2]. In particular, we construct an object over $\Delta S$
(called the _symmetric simplicial category_ , see [Banerjee, Definition 2.10])
in the category of (smooth proper) Deligne-Mumford stacks.

Now a degree $n$ morphism $C\to\mathcal{P}({\vec{\lambda}})$ is equivalent to
the following data:

  * •

a line bundle $L$ of degree $n$ on $C$,

  * •

an $(N+1)$-tuple $(s_{0},\ldots,s_{N})$ where $s_{i}\in
H^{0}(C,L^{\otimes\lambda_{i}})$

  * •

the sections $s_{0},\ldots,s_{N}$ satisfy the condition that they have no
common zeroes (also known as $\\{s_{0},\ldots,s_{N}\\}$ is _basepoint free_).

Then, $\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ is a Zariski open
dense subset of $\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$
defined by

$\displaystyle\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))\coloneqq\\{L,~{}[s_{0}:\ldots:s_{N}]:L\in\mathrm{Pic}^{n}(C),~{}s_{i}\in
H^{0}(C,L^{\otimes\lambda_{i}})\text{ for all }i\\}.$

Note that $\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ is
isomorphic to the weighted projectivization of a vector bundle $\mathcal{E}$
on $\mathrm{Pic}^{n}(C)$ (assuming $n\geq 2g$) whose fibre over $L$ is given
by $V_{L}\coloneqq\oplus_{i}H^{0}(C,L^{\otimes\lambda_{i}}),$ where via the
Riemann-Roch theorem we have
$H^{0}(C,L^{\otimes\lambda_{i}})\cong\mathbb{A}^{\lambda_{i}n-g+1}$ and
$\mathbb{G}_{m}$ acts by weight $\lambda_{i}$ on
$H^{0}(C,L^{\otimes\lambda_{i}})$; fibre over $L\in\mathrm{Pic}^{n}C$
isomorphic to the weighted projective stack
$\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n)^{\otimes\lambda_{i}}),{\vec{\lambda}})$.

More precisely, if

$\nu:C\times\mathrm{Pic}^{n}C\to\mathrm{Pic}^{n}C$

is the projection to the second factor and $P(n)$ denotes a Poincare bundle of
degree $n$ on $C\times\mathrm{Pic}^{n}C$, then we let
$\Gamma_{\lambda_{i}}(n)\coloneqq\nu_{*}P(n)^{\otimes\lambda_{i}}$ (and when
$\lambda_{i}=1$ we write $\Gamma(n)$). Then
$\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ is the weighted
projectivization of $\oplus_{i}\Gamma_{\lambda_{i}}(n)$ where $\mathbb{G}_{m}$
acts on $\Gamma_{\lambda_{i}}(n)$ by weight $\lambda_{i}$. By Theorem 1.8 we
know the cohomology of
$\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$\- a key player
in the proof of Theorem 1.1 as we shall soon see.

Even though the exact cohomology of Theorem 1.2 follows from stable cohomology
of Theorem 1.1 when we take the curve $C$ to be of genus $0$, literature shows
that not only is the genus 0 case worth proving in it’s own right, in this
manuscript it also portrays how the basic strategy of proving the higher genus
case is essentially the same as that of the genus $0$ case; in other words if
the reader understands the proof of Theorem 1.2, he knows the proof of Theorem
1.1 as well. Keeping this in mind, we first prove Theorem 1.2.

###### Proof of Theorem 1.2.

First observe that
$\mathrm{\widetilde{Hom}}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}))$ is
a weighted projective stack isomorphic to
$\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n)^{\lambda_{i}},{\vec{\lambda}})$.
Define a symmetric semisimplicial space (see [Banerjee, Defintion 2.10]) as
follows. Let

$\displaystyle\mathcal{T}_{0}\coloneqq\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1)),{\vec{\lambda}})$
(17)

and let

$\displaystyle\mathcal{T}_{p}\coloneqq(\mathbb{P}^{1})^{p+1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-p)),{\vec{\lambda}}).$
(18)

Letting $\mathcal{T}_{-1}$ denote
$\mathrm{\widetilde{Hom}}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}))$,
we observe that there are natural _face maps_ (which are finite morphisms of
smooth proper Deligne-Mumford stacks)
$f_{i}:\mathcal{T}_{p}\to\mathcal{T}_{p-1}$ for all $p\geq 0$ given by _adding
a basepoint_ from the $i^{th}$ factor i.e.

$\displaystyle
f_{i}:(\mathbb{P}^{1})^{p+1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-(p+1)),{\vec{\lambda}})\to(\mathbb{P}^{1})^{p}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-p),{\vec{\lambda}})$
$\displaystyle\big{(}[a_{0}:b_{0}],\ldots,[a_{p}:b_{p}]\big{)},\,\,[s_{0}:\ldots:s_{N}]\mapsto$
$\displaystyle\big{(}[a_{0},b_{0}],\ldots,\widehat{[a_{i}:b_{i}]},\ldots[a_{p}:b_{p}]\big{)},\,\,[(b_{i}x-a_{i}y)s_{0}:\ldots:(b_{i}x-a_{i}y)s_{N}]$
(19)

where $\widehat{[a_{i}:b_{i}]}$ denotes removing the $i^{th}$ entry
$[a_{i}:b_{i}]$. In other words, the hypercover under consideration is the
following:

$\displaystyle\cdots(\mathbb{P}^{1})^{3}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-3)),{\vec{\lambda}})\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-2.15277pt]
\textstyle\rightarrow\\\\[-2.15277pt]
\textstyle\rightarrow\end{subarray}}(\mathbb{P}^{1})^{2}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-2)),{\vec{\lambda}})$
$\displaystyle\rightrightarrows\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1)),{\vec{\lambda}})\to\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i})),{\vec{\lambda}})$

with the unlabelled arrows denoting the face maps $f_{i}$.

Let
$\mathcal{Z}\subset\mathrm{\widetilde{Hom}}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}))$
be the “discriminant locus" given by

$\displaystyle\mathcal{Z}=\\{[s_{0}:\cdots:s_{N}]:s_{i}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}(n)^{\otimes\lambda_{i}})\text{ for all
}i=0,\ldots,N,$ $\displaystyle s_{0},\ldots,s_{N}\text{ have at least one
common root}\\}.$

Now we make the following observation (an almost immediate consequence of
[Conrad, Theorem 7.1] or [Deligne])):

######  Claim 4.1.

The hypercovering $\mathcal{T}_{\bullet}\to\mathcal{Z}$ is universally of
cohomological descent.

To prove it we show a more general lemma:

######  Lemma 4.2.

Let $\pi_{\bullet}:\mathcal{T}_{\bullet}\to\mathcal{Z}$ be a proper
hypercovering in the category of Deligne-Mumford stacks, with the étale
topology, such that for all $p$, and a morphism

$x:\mathrm{Spec}\,\,K\to\mathcal{T}_{p}$

the kernel
$\mathrm{Isom}_{\mathcal{T}_{p}}(x,x)\to\mathrm{Isom}_{\mathcal{Z}}(x,x)$ is
trivial, then it is universally of cohomological descent.

###### Proof of Lemma 4.2.

The proof of this lemma follows immediately by noting that the claim is about
étale sheaves on the base Deligne-Mumford stack $\mathcal{Z}$, so it suffices
to check the claim étale locally. Now thanks to [Stacks, Lemma 100.6.2] the
face maps are representable by algebraic spaces, so étale locally we can
simply apply the result for schemes, which is precisely [Conrad, Theorem 7.1]:
that a proper hypercovering in the category of schemes is universally of
cohomological descent. ∎

###### Proof of 4.1.

In the special case when $\mathcal{T}_{\bullet}$ is the specific simplicial
space defined above (4.1) and $\mathcal{Z}$ is the discriminant locus, the
condition in the claim of the kernel being trivial is clearly satisfied, and
thus the observation follows. ∎

Having settled that $T_{\bullet}\to\mathcal{Z}$ as a semisimplicial space is
indeed of cohomological descent, by virtue of having the additional structure
of a $\Delta_{inj}S$ object, it further satisfies the conditions of [Banerjee,
Theorem 2] 222In fact one can make a more general statement at no extra cost,
simply by translating cohomological descent in the $\infty$-category of
$\ell$-adic sheaves $\mathrm{Sh}_{\ell}(\mathcal{Z})$ (as defined, for
example, by Gaitsgory-Lurie in [GL]) over to the indexing category $\Delta S$,
essentially giving an $\infty$-categorical version of [Banerjee, Lemma 2.11],
as follows. Given a $S_{\bullet}$ hypercovering
$\pi_{\bullet}:\mathcal{T}_{\bullet}\to\mathcal{Z}$ satisfying the conditions
of Lemma 4.2, there is an equivalence of endofunctors
$\mathrm{id}\to\big{(}\mathrm{R}{\pi_{\bullet}}_{*}\pi^{*}_{\bullet}\otimes\mathrm{sgn}_{S_{\bullet}}\big{)}^{S_{\bullet}}$
in the $\infty$-category
$\mathrm{Fun}(\mathrm{Sh}_{\ell}(\mathcal{Z}),\mathrm{Sh}_{\ell}(\mathcal{Z}))$,
where $S_{\bullet}$ denotes the symmetric simplicial group given by $S_{n}$
denoting the symmetric group on $(n+1)$ elements. However, we do not need the
full power of this statement in for our immediate computation. The interested
reader can refer to [Banerjee2]. and results in a second quadrant spectral
sequence whose $E_{1}$ page reads as

$\displaystyle
E_{1}^{-p,q}=\begin{cases}H^{q}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i})),{\vec{\lambda}}))(0)&p=0,\\\
H^{q-2N}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1))(-1)&p=1,\\\
H^{0}(\mathbb{P}^{1})\otimes H^{2}(\mathbb{P}^{1})\otimes
H^{q-4N-2}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-2))(-2)&p=2,\\\
0&\text{ otherwise},\end{cases}$

with the differentials given by the alternating sum of the Gysin pushforwards
induced by the face maps, which is what we shall compute now.

  * •

Computing $d_{1}^{1,q}:E_{1}^{-1,q}\to E_{1}^{0,q}$.

For simplicity we denote the differential by $d_{1}^{1}$. Let

$\iota:\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1)),{\vec{\lambda}})\hookrightarrow\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i})),{\vec{\lambda}})$

denote the inclusion given by adding a basepoint.

Choose generators $\mathbf{1}\in H^{0}(\mathbb{P}^{1})$ and $e\in
H^{2}(\mathbb{P}^{1})$, and let $h$ denote the hyperplane class in
$\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}})$.
Then we claim that:

$\displaystyle
d_{1}^{1}={f_{0}}_{*}:H^{*-2N}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1)),{\vec{\lambda}}))\to$
$\displaystyle
H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}))$
$\displaystyle\mathbf{1}\otimes\iota^{*}\alpha+e\otimes\iota^{*}\beta\mapsto\alpha
h^{N}+\beta h^{N+1}$

is a map of
$H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}))$-modules,
where

$\alpha,\beta\in
H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}).$

To see this, first note that

$\iota^{*}:H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}}))\to
H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}))$

is a surjection; next, the image of the fundamental class

$[\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}})]\in
H^{0}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}}))$

is the locus of elements in
$\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}})$
that has a basepoint i.e. $\mathcal{Z}$, which is rationally equivalent, and
thus cohomologous, to (a multiple of) $h^{N}$; and finally, for a fixed point
$[a:b]\in\mathbb{P}^{1}$, the locus given by

$\\{[s_{0}:\ldots:s_{N}]\in\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}):s_{i}([a:b])=0\\}$

is rationally equivalent, and in turn cohomologous, to (a multiple of)
$h^{N+1}$. For the sake of simplicity we won’t bother ourselves with the
scalar multiples, which is fine because we’re working over $\mathbb{Q}$. The
Gysin pushforward $d_{1}^{1}={f_{0}}_{*}$ surjects onto the ideal generated by
$h^{N}$ in
$H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i},{\vec{\lambda}}))$.
Indeed, the preimage of $h^{N+i}$ is given by

$\displaystyle
d_{1}^{1}(\mathbf{1}\otimes\iota^{*}h^{i})=h^{N+i}=d_{1}^{1}(e\otimes\iota^{*}h^{i-1})\text{
for }i\geq 1$ $\displaystyle
d_{1}^{1}([\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}}))=h^{N},$

which shows that the image of $d_{1}^{1}$ is the ideal generated by $h^{N}$ in
$H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}))$.
The kernel of $d_{1}^{1}$ is given by elements of the form
$\iota^{*}(\alpha)(h-e)$ for all $\alpha\in
H^{*}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}}))$.

The upshot is that on the $E_{2}$ page, for $p=0$ we have:

$\displaystyle E_{2}^{0,q}=\begin{cases}\mathbb{Q}(0)&q=2j,\,\,0\leq j\leq
2(N-1)\\\ 0&\text{ otherwise.}\end{cases}$ (20)

  * •

Computing $d_{1}^{2,q}:E_{1}^{-2,q}\to E_{1}^{-1,q}$. For simplicity, we
denote the differential by $d_{1}^{2}$. Like before, let
$\iota:\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}})\hookrightarrow\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}),{\vec{\lambda}})$
denote the inclusion given by adding a basepoint, and let $h$ denote the
hyperplane class in
$\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}})$.
Then the way we computed ${f_{0}}_{*}$ above works verbatim, and we have

$\displaystyle{f_{0}}_{*}:H^{0}(\mathbb{P}^{1})\otimes
H^{2}(\mathbb{P}^{1})\otimes
H^{*-2N-2}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-2)),{\vec{\lambda}}))$
$\displaystyle\to
H^{*}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}})$
$\displaystyle\mathbf{1}\otimes e\otimes\alpha\mapsto e\otimes\alpha h^{N}$

and

$\displaystyle{f_{1}}_{*}:H^{0}(\mathbb{P}^{1})\otimes
H^{2}(\mathbb{P}^{1})\otimes
H^{*-2N-2}(\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-2),{\vec{\lambda}}))$
$\displaystyle\to
H^{*}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}}))$
$\displaystyle\mathbf{1}\otimes e\otimes\alpha\mapsto\mathbf{1}\otimes\alpha
h^{N+1},$

and therefore

$d_{1}^{2}(\mathbf{1}\otimes e\otimes\alpha)=\mathbf{1}\otimes\alpha
h^{N+1}-e\otimes\alpha h^{N}.$

Note that $d_{1}^{2}$ is injective, and the image is generated by $h^{N}$ in

$H^{*}(\mathbb{P}^{1}\times\mathcal{P}(\oplus_{i}H^{0}(\mathbb{P}^{1},\mathcal{O}(n\lambda_{i}-1),{\vec{\lambda}})).$

Consequently, on the $E_{2}$ page we have:

$\displaystyle
E_{2}^{-1,q}=\begin{cases}\mathbb{Q}_{\ell}(-N)&p=1,q=2j+2N+2,\,\,0\leq j\leq
2(N-1)\\\ 0&\text{ otherwise}\end{cases},$ $\displaystyle
E_{2}^{-2,q}=0,\text{ for all }q.$

In effect on the $E_{2}$ page all differentials vanish; the spectral sequence
degenerates and we obtain

$H^{*}(\mathrm{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}({\vec{\lambda}}));\mathbb{Q})\cong\frac{\mathbb{Q}[h]}{h^{N}}\otimes\wedge\mathbb{Q}\\{t\\}$

where $h$ has cohomological degree $2$, and $t$ (which corresponds to $e-h\in
Ker(d_{1}^{1})$) has cohomological degree $2N+1$. Furthermore, over a field
$\kappa$, with algebraic closure $\overline{\kappa}$, we have an isomorphism
of $Gal(\overline{\kappa}/\kappa)$-representations:

$\displaystyle
H^{i}_{{\acute{et}}}(\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}));\mathbb{Q}_{\ell})=\begin{cases}{\mathbb{Q}_{\ell}}(-j)&i=2j,0\leq
j\leq N-1\\\ \mathbb{Q}_{\ell}(-(j+1))&i=2j+1,N\leq j\leq 2N-1\\\ 0&\text{
otherwise.}\end{cases}$

This completes the proof of Theorem 1.2.

∎

Following the same plan, we now proof Main Theorem 1.1.

###### Proof of Theorem 1.1..

As already observed above, the space
$\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ is open and dense in
$\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$, and we have

$\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))\cong\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}).$

Let $\mathcal{Z}$ denote the discriminant locus i.e. the complement of
$\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))$ in
$\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))$.

Now we construct a hypercover that is equipped with the additional structure
of a $\Delta S$ object in the category of Deligne-Mumford stacks, and which
admits universal cohomological descent.

To this end, we define spaces $\mathcal{T}_{0}$ and $\mathcal{I}_{0}$ as
certain fibre products. First, consider the following commutative diagram:

${\mathcal{T}_{0}}$${C\times\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-1}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n-1),{\vec{\lambda}})}$${\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})}$${\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n),{\vec{\lambda}})}$${\mathcal{I}_{0}}$${C\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-1}C}$${\mathrm{Pic}^{n}C}$${\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}$$\scriptstyle{\pi_{0}}$$\scriptstyle{A}$$\scriptstyle{A}$$\scriptstyle{A}$$\scriptstyle{\otimes^{{\vec{\lambda}}}}$

where the maps above are defined by:

$\displaystyle\otimes^{{\vec{\lambda}}}:\mathrm{Pic}^{n}\to\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C$
$\displaystyle L\mapsto(L^{\otimes\lambda_{0}},\cdots,L^{\otimes\lambda_{N}})$
(21)

henceforth often denoting
$(L^{\otimes\lambda_{0}},\cdots,L^{\otimes\lambda_{N}})$ by
$L^{\otimes{\vec{\lambda}}}$;

$\displaystyle
A:C\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-1}C\to\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C$
$\displaystyle x,(L_{0},\ldots,L_{N})\mapsto
L_{0}\otimes\mathcal{O}_{C}(x),\cdots,L_{N}\otimes\mathcal{O}_{C}(x)$ (22)

which is the map of ‘adding a point’:

$\mathcal{I}_{0}\coloneqq\mathrm{Pic}^{n}\times_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}(C\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-1}C)$

completes the commutative square at the ‘lower face’ of the cube and we abuse
notation and still denote the resulting ‘adding a point’ map by
$A:\mathcal{I}_{0}\to\mathrm{Pic}^{n}C$; the ‘upper face’ of the cube consists
of spaces which are essentially the space of global sections of suitable
Poincare bundles, i.e. the vertical arrows all correspond to taking fibrewise
global sections over the moduli of line bundles; and $\mathcal{T}_{0}$, quite
like $\mathcal{I}_{0}$, is defined to complete the square on the upper face of
the cube, and admits natural map to $\mathcal{I}_{0}$ so that each of the side
faces are also naturally commutative squares. Whereas the ‘adding a point’ map
on the lower face of the cube adds points to line bundles, on the upper level
they effectively add ‘basepoints’ to global sections; in other words

$\pi_{0}:\mathcal{T}_{0}\to\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$

simply captures the notion of adding a basepoint.

We define spaces $\mathcal{T}_{p}$ for all $p\geq 0$ likewise. Consider the
following commutative diagram:

${\mathcal{T}_{p}}$${C^{p+1}\times\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n-(p+1),{\vec{\lambda}})}$${\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})}$${\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n),{\vec{\lambda}})}$${\mathcal{I}_{p}}$${C^{p+1}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C}$${\mathrm{Pic}^{n}C}$${\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}$$\scriptstyle{\pi_{p}}$$\scriptstyle{f_{1}}$$\scriptstyle{h}$$\scriptstyle{A}$$\scriptstyle{A}$$\scriptstyle{\otimes^{{\vec{\lambda}}}}$
where $\otimes^{{\vec{\lambda}}}$ is defined like before; $A$, by abusing
notation, now denotes the addition of $(p+1)$ points to a line bundle i.e.

$\displaystyle
A:C^{p+1}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C\to\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C$
$\displaystyle(x_{0},\ldots,x_{p}),(L_{0},\ldots,L_{N})\mapsto L_{0}(\sum
x_{i}),\cdots,L_{N}(\sum x_{i});$ (23)

and moreover observe that we have commutative cubes of the following form for
all $p$:

${\mathcal{T}_{p}}$${C^{p+1}\times\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n-(p+1),{\vec{\lambda}})}$${\mathcal{T}_{p-1}}$${C^{p}\times\mathcal{P}_{\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-p}C}(\oplus_{i}\nu_{*}P(\lambda_{i}n-p,{\vec{\lambda}})}$${\mathcal{I}_{p}}$${C^{p+1}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C}$${\mathcal{I}_{p-1}}$${C^{p}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-p}C}$$\scriptstyle{f_{i}}$$\scriptstyle{A}$$\scriptstyle{A}$$\scriptstyle{\otimes^{{\vec{\lambda}}}}$
and the all the down-right arrows are maps corresponding to adding a point
from the $i^{th}$ factor of $C^{p+1}$ i.e.

$\displaystyle A:C^{p+1}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-(p+1)}C\to
C^{p}\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-p}C$
$\displaystyle(x_{0},\ldots,x_{p}),(L_{0},\ldots,L_{N})\mapsto$
$\displaystyle(x_{0},\ldots,\widehat{x_{i}},\ldots,x_{p}),(L_{0}(x_{i}),\ldots,L_{N}(x_{i}))$

where $\widehat{x_{i}}$ implies we delete the $i^{th}$ entry; most
importantly, we obtain $f_{i}:\mathcal{T}_{p}\to\mathcal{T}_{p-1}$ as the
$i^{th}$ face map of the semisimplicial space $\mathcal{T}_{\bullet}$.

It is easy to see that $\mathcal{I}_{p}\cong C^{p+1}\times\mathrm{Pic}^{n}C$
for all $p\geq 0$: for $p=0$ the map

$\displaystyle\mathrm{Pic}^{n}C{\times_{i}\Pi_{i}\mathrm{Pic}^{\lambda_{i}n}C}.C\times\Pi_{i}\mathrm{Pic}^{\lambda_{i}n-1}C\to
C\times\mathrm{Pic}^{n}C$ $\displaystyle
L,(x,(L_{0},\cdots,L_{N}))\mapsto(x,L)$

has a natural inverse thereby giving an isomorphism; for higher values of $p$
the observation follows likewise. Along those lines, it is not hard to show
for $p\leq n-2g+1$ that $\mathcal{T}_{p}\cong
C^{p+1}\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\mathcal{E}_{i},\lambda_{i})$
(the isomorphism is not canonical) for some sub-vector budnles
$\mathcal{E}_{i}\subset\nu_{*}P(n)^{\otimes\lambda_{i}}$ for each $i$, of rank
$\lambda_{i}n-g+1-(p+1)$, in the following way. For any line bundle $L$ on $C$
and any effective divisor $D$ on $C$ there is a natural injection

$\displaystyle H^{0}(C,L)\to H^{0}(C,L(D))$ $\displaystyle s\mapsto s(D)$

coming from the short exact sequence of the corresponding locally free sheaves

$0\to L\to L(D)\to\mathcal{O}_{D}\to 0;$

by definition we have

$\displaystyle\mathcal{T}_{p}:=\\{[s_{0}:\ldots:s_{N}],((x_{0},\cdots,x_{p}),[\widetilde{s_{0}}:\ldots:\widetilde{s_{N}})):s_{i}\in
H^{0}(C,L^{\lambda_{i}})\text{ where }L\in\mathrm{Pic}^{n}C,$
$\displaystyle\widetilde{s_{i}}\in H^{0}(C,L_{i})\text{ where
}L_{i}\in\mathrm{Pic}^{\lambda_{i}n-(p+1)}C,$
$\displaystyle\widetilde{s_{i}}(\sum x_{j})=t^{\lambda_{i}}s_{i}\text{ for
some }t\in\mathbb{G}_{m}\text{ and for all }i\\}.$

Now fix a general unordered $(p+1)$ subset $\\{z_{0},\cdots,z_{p}\\}\in C$ and
define

$\displaystyle
M_{p}:=\\{[\widetilde{s_{0}}:\ldots:\widetilde{s_{N}}]:\widetilde{s_{i}}\in
H^{0}(C,L_{i})\text{ where }L_{i}\in\mathrm{Pic}^{\lambda_{i}n-(p+1)}C,\text{
there exists }t\in\mathbb{G}_{m}$ $\displaystyle\text{ such that for each
}i,\,\,\,\widetilde{s_{i}}(\sum_{j}c_{j})=t^{\lambda_{i}}s_{i},$
$\displaystyle\text{ where }s_{i}\in
H^{0}(C,L^{\lambda_{i}}),\,\,\,\,L\in\mathrm{Pic}^{n}C\\}$

Then one can define a bijective (easy to check) morphism that only depends on
the choice of the points $z_{0},\ldots,z_{p}\in C$:

$\displaystyle C^{p+1}\times M_{p}\to\mathcal{T}_{p}$
$\displaystyle(x_{0},\ldots,x_{p}),[\widetilde{s_{0}}:\ldots:\widetilde{s_{N}}]\mapsto$
$\displaystyle[\widetilde{s_{0}}(\sum_{j}x_{j}):\ldots:\widetilde{s_{N}}(\sum_{j}x_{j})],((x_{0},\cdots,x_{p}),[\widetilde{s_{0}}:\ldots:\widetilde{s_{N}}])$
(24)

where the only non-trivial part is to check that the map is well-defined and
that fact follows from the observation that degree $0$ line bundles always
have $\lambda^{th}$ roots for all positive integers $\lambda$. Now $M_{p}$ is
naturally a fibre bundle over $\mathrm{Pic}^{n}C$, in fact it is a weighted
projectivization of the sub-vector bundle of
$\oplus\nu_{*}P(n)^{\otimes\lambda_{i}}$ whose fibres are given by
$(N+1)$-tuples of sections in $\oplus\nu_{*}P(n)^{\otimes\lambda_{i}}$ having
common zeroes at $z_{0},\ldots,z_{p}$; we denote that sub-vector bundle by
$\oplus\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p}}$. In other words

$M_{p}\cong\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p}},{\vec{\lambda}}),$

and for all $0\leq p\leq n-2g+1$ we abuse notation and denote the fibre bundle
by $\rho$ i.e.

$\rho:M_{p}\to\mathrm{Pic}^{n}C.$

Deviating from the norm, we set
$\mathcal{T}_{-1}\coloneqq\mathrm{\widetilde{Hom}}_{n}(C,\mathcal{P}({\vec{\lambda}}))\cong\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$)
(whereas the norm would be to set $\mathcal{T}_{-1}$ as $\mathcal{Z}$\- this a
minor deviation just to simplify the notations involved in the subsequent
computation). From [Banerjee, Definition 2.1], $\mathcal{T}_{\bullet}$
satisfies the conditions of being a $\Delta S$ object. We therefore use
[Banerjee, Theorem 1.2] to get a second quadrant spectral sequence that reads
as:

$\displaystyle
E_{1}^{-p,q}=H^{q-2pN}\big{(}\mathcal{T}_{p}\otimes\mathit{sgn}_{p+1}\big{)}^{{S}_{(p+1)}}(-(p+1)N)\implies
H^{q+p}(\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}})).$

where $\mathit{sgn}_{p+1}$ denote the sign action of the symmetric group
$S_{p+1}$ on the $(p+1)$ factors of $T_{p}$ by permutation; and the
differentials of this spectral sequence are given by the alternating sum of
the Gysin pushforwards induced by the face maps. So we split the rest of the
proof into the following parts:

  1. 1.

Computing the $E_{1}$ terms. To this end note that a complete understanding of

$H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p}},{\vec{\lambda}}))$

(at least in a stable range) gives us full knowledge of the $E_{1}$ terms (in
that range). The Chern classes of $\nu_{*}P(n)^{\otimes\lambda_{i}}$ can be
computed for example, directly using Grothendieck-Riemann-Roch, or via ad-hoc
methods:

$c_{i}(\mathcal{E}_{\lambda_{i}})=(-1)^{i}{\theta^{i}\over
i!}\,\,\,i=0,\ldots,g$

where $\theta$ is the fundamental class of the theta divisor (several proofs
are available in [ACGH, Sections 4, 5, Chapter VII and Section 1, Chapter
VIII]). Using the Whitney sum formula we can express the twisted Chern classes
in terms of $\theta$.

In turn, let $N_{0}\coloneqq(n-g+1)(N+1)$, which is the dimension of the
fibres of $\mathcal{E}\to\mathrm{Pic}^{n}(C)$, and let $h$ denote the relative
hyperplane class i.e. $h=c_{1}(\mathcal{O}_{\rho}(1))\in
H^{2}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$,
then
$H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$,
which is an algebra on $H^{*}(\mathrm{Pic}^{n}(C))\cong\wedge(H^{1}(C))$, is
given by (using (16)):

$\displaystyle
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$
$\displaystyle\cong$
$\displaystyle{H^{*}(\mathrm{Pic}^{n}(C))[h]\over{h^{N_{0}}+\rho^{*}c^{\vec{\eta}}_{1}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})h^{N_{0}-1}+\ldots+\rho^{*}c^{\vec{\eta}}_{g}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})h^{N_{0}-g}}}.$
(25)

Let $p$ be such that $n-p\geq 2g$ and let
$N_{p}\coloneqq(n-p-g+1)(N+1)=N_{0}-p(N+1)$, the dimension of the fibres of
$\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p}},{\vec{\lambda}})\to\mathrm{Pic}^{n}(C)$,
then combining (16) and (1) we have a complete description of the $E_{1}$
terms of the spectral sequence above. We remark here that since $n-p\geq 2g$,
we have that $N_{p}-g=(n-p-g+1)(N+1)-g\geq N$. This remark will be useful
later.

  2. 2.

Computing the differentials $d_{1}^{p}:E_{1}^{-p,*}\to E_{1}^{-(p-1),*+2N}$.

Following previously introduced notations, let
$h=c_{1}(\mathcal{O}_{\rho_{n}}(1))$, and for all $p$ satisfying $n-p\geq 2g$,
let

$\iota:\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p-1}},{\vec{\lambda}})\to\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+\cdots+z_{p}},{\vec{\lambda}})$

denote the closed embedding induced by adding a basepoint $x$ (an abuse of
notation that won’t cause any confusion down the way). Note that $\iota$ is
fibrewise linear embedding, up to translation of $\mathrm{Pic}^{n}(C)$ by $x$.
Finally, let $e\in H^{2}(C)$ be the class of a point, $\mathbb{1}$ the
fundamental class of $C$, and let $\gamma_{1},\ldots,\gamma_{2g}$ be the
standard basis of $H^{1}(C)$ and because
$H^{*}(\mathrm{Pic}^{n}(C))\cong\wedge H^{1}(C)$, let
$\overline{\gamma_{1}},\ldots,\overline{\gamma_{2g}}$ be the image of
$\gamma_{1},\ldots,\gamma_{2g}$ under the aforementioned isomorphism.

First, we observe that

$\displaystyle
d_{1}^{1}:H^{*}(C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))\to
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))$
$\displaystyle[C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}})]\mapsto
h^{N}$ $\displaystyle e\mapsto h^{N+1}$
$\displaystyle\gamma_{i}\mapsto\overline{\gamma_{i}}h^{N},$
$\displaystyle\text{ for all }i.$

is a map of
$H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))$-modules,
and in turn

$\iota^{*}\alpha+e\iota^{*}\beta+\sum_{i=1}^{2g}\gamma_{i}\iota^{*}\gamma_{i}\xmapsto{d_{1}^{1}}\alpha
h^{N}+\beta h^{N+1}+\sum_{i=1}^{2g}\overline{\gamma_{i}}\gamma_{i}h^{N},$

where $\alpha,\beta,\gamma_{1},\ldots,\gamma_{2g}\in
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))$.
Indeed, the justification for the formula for $d_{1}^{1}$ in the previous case
of $C=\mathbb{P}^{1}$ holds almost verbatim here. We know

$\iota^{*}:H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))\to
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))$

is a surjection; next, for a fixed point $x\in C$, the image
$t_{x}^{r}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}})$)
is rationally equivalent, and in turn cohomologous, to (a multiple of)
$h^{N+1}$, and finally, that the image of the fundamental class
$[C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}})]\in
H^{0}(C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))$
is rationally equivalent, and thus cohomologous, to (a multiple of) $h^{N}$,
can be seen as in the following way. Recall that a Poincaré bundle $P(n)$ is
$\nu$-relatively very ample for all $n\geq 2g-1$, which in turn induces a
relative embedding of
$C\times\mathrm{Pic}^{n}(C)\xrightarrow{i_{n}}\mathbb{P}(\nu_{*}P(n))$ over
$\mathrm{Pic}^{n}(C)$ and we have a natural sequence of maps over
$\mathrm{Pic}^{n}C$

${C\times\mathrm{Pic}^{n}C}$${\mathbb{P}(\nu_{*}P(n))}$${\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\lambda_{i}},{\vec{\lambda}})}$${\mathrm{Pic}^{n}C}$

and we continue to denote the composition mapping $C\times\mathrm{Pic}^{n}C$
to
$\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\lambda_{i}},{\vec{\lambda}})$
over $\mathrm{Pic}^{n}C$ by $i_{n}$. This makes
$i_{n}(C\times\mathrm{Pic}^{n}(C))$ in
$\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$
homologous to (a scalar multiple of) the (relative, over the base
$\mathrm{Pic}^{n}C$) Poincaré dual of $h\in
H^{2}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))$.
In turn, the image of the
$[C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}})]$
under the Gysin map ${f_{0}}_{*}$ is given by

$\displaystyle{f_{0}}_{*}\big{(}[C\times\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}})]\big{)}=h^{N+1}\frown
i_{n}(C\times\mathrm{Pic}^{n}(C))$ $\displaystyle=h^{N}.$

Yet again, for the sake of simplicity we won’t bother ourselves with the
scalar multiples, which is fine because we’re working over $\mathbb{Q}$.
Noting that

$\overline{\gamma_{i}}(e-h)+h(\gamma_{i}-\overline{\gamma_{i}})=\gamma_{i}h-e\overline{\gamma_{i}},$

it is now easy to check that the kernel of $d_{1}^{1}$ is given by:

$\displaystyle
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))(e-h)[2N]$
$\displaystyle\bigoplus
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))(\gamma_{i}-\overline{\gamma_{i}})[2N],$
$\displaystyle(i=1,\ldots,2g)$

where $[2N]$ denotes a shift in the cohomological degree by $2N$, and which is
viewed as a
$\iota^{*}H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))\cong
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}},{\vec{\lambda}}))$-module.
The cokernel of $d^{1}_{1}$, which forms $E_{2}^{0,*}$ is given by

$H^{*}(\mathrm{Pic}^{n}(C))[h]\over h^{N}$

(where note that as remarked before $r<N_{0}-g$, see (1)).

Now we work out the differential for $p=2$ by computing the Gysin pushfowards
by each of the face maps:

$\displaystyle{f_{0}}_{*}(\mathbb{1}\otimes
e)=eh^{N},\,\,\,\,{f_{1}}_{*}(\mathbb{1}\otimes e)=h^{N+1}\implies
d_{1}^{2}(\mathbb{1}\otimes e)=(e-h)h^{N},$
$\displaystyle{f_{0}}_{*}(e\otimes\gamma_{i})=\gamma_{i}h^{N+1},\,\,\,\,{f_{1}}_{*}(e\otimes\gamma_{i})=e\overline{\gamma_{i}}h^{N}\implies
d^{2}_{1}(e\otimes\gamma_{i})=(\gamma_{i}h-e\overline{\gamma_{i}})h^{N},$
$\displaystyle{f_{0}}_{*}(\mathbb{1}\otimes\gamma_{i})=\gamma_{i}h^{N},\,\,\,\,{f_{1}}_{*}(\mathbb{1}\otimes\gamma_{i})=\overline{\gamma_{i}}h^{N}\implies
d^{2}_{1}(\mathbb{1}\otimes\gamma_{i})=(\gamma_{i}-\overline{\gamma_{i}})h^{N},$
$\displaystyle d_{1}^{2}(\gamma_{i}\gamma_{j})=0,$

where the last equality follows form the fact that on
$\operatorname{Sym}^{p}H^{1}(C)$ for $p\geq 2$, the alternating sum of face
maps is, by definition, $0$. Recalling our earlier remark that $N<N_{1}-g$, we
see that the $E_{2}^{-1,*}$ terms, as an
$H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+z_{1}},{\vec{\lambda}}))$-module,
are given by:

$\displaystyle{H^{*}(\mathrm{Pic}^{n}(C);\mathbb{Q}(-N))[h]\over
h^{N}}(e-h)[2N]$ $\displaystyle\bigoplus_{1\leq i\leq
2g}{H^{*}(\mathrm{Pic}^{n}(C);\mathbb{Q}(-r))[h]\over
h^{N}}(\gamma_{i}-\overline{\gamma_{i}})[2N].$

Whereas the kernel of $d_{1}^{2}$ is generated by exactly what one expects: as
a
$H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+z_{1}},{\vec{\lambda}}))$-module,
we have

$\displaystyle\mathrm{Ker}(d_{1}^{2})=\bigoplus_{1\leq i\leq
2g}H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+z_{1}},{\vec{\lambda}}))\big{(}e\otimes\gamma_{i}-1\otimes\gamma_{i}h+\mathbb{1}\otimes
e\overline{\gamma_{i}}\big{)}[4N]$ $\displaystyle\bigoplus_{1\leq i,j\leq
2g}H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}}_{z_{0}+z_{1}},{\vec{\lambda}}))(\gamma_{i}\gamma_{j})[4N].$

For $p=3$ we have $d_{1}^{3}:E_{1}^{-3,*}\to E_{1}^{-2,*}$ given by:

$\displaystyle d_{1}^{3}(\mathbb{1}\otimes
e\otimes\gamma_{i})=e\otimes\gamma_{i}h^{N}-\mathbb{1}\otimes\gamma_{i}h^{N+1}+\mathbb{1}\otimes
e\overline{\gamma_{i}}h^{N}\impliedby\begin{cases}{f_{0}}_{*}(\mathbb{1}\otimes
e\otimes\gamma_{i})=e\otimes\gamma_{i}h^{N},\\\ {f_{1}}_{*}(\mathbb{1}\otimes
e\otimes\gamma_{i})=\mathbb{1}\otimes\gamma_{i}h^{N+1}\\\
{f_{2}}_{*}(\mathbb{1}\otimes e\otimes\gamma_{i})=\mathbb{1}\otimes
e\overline{\gamma_{i}}h^{N}\end{cases}$ $\displaystyle
d_{1}^{3}(e\otimes\gamma_{i}\gamma_{j})=\gamma_{i}\gamma_{j}h^{N+1},$
$\displaystyle
d_{1}^{3}(\mathbb{1}\otimes\gamma_{i}\gamma_{j})=\gamma_{i}\gamma_{j}h^{N},$
$\displaystyle d_{1}^{3}(\gamma_{i}\gamma_{j}\gamma_{k})=0,$

where, for the last three equalities, recall again that on
$\operatorname{Sym}^{p}H^{1}(C)$ for $p\geq 2$, the alternating sum of face
maps is, by definition, $0$. Therefore the $E_{1}^{-2,*}$ terms defined by
$\mathrm{Ker}(d^{2}_{1})/\mathrm{Coker}(d^{3}_{1})$ is given by:

$\displaystyle\bigoplus_{1\leq i\leq
2g}{H^{*}(\mathrm{Pic}^{n}(C);\mathbb{Q}(-2N))[h]\over
h^{N}}\big{(}e\otimes\gamma_{i}-1\otimes\gamma_{i}h+\mathbb{1}\otimes
e\overline{\gamma_{i}}\big{)}[4N]$ $\displaystyle\bigoplus_{1\leq i,j\leq
2g}{H^{*}(\mathrm{Pic}^{n}(C);\mathbb{Q}(-2N))[h]\over
h^{N}}(\gamma_{i}\gamma_{j})[4N].$

The formula for the differentials in the case of $p\geq 3$ mimics that of
$p=3$, and we have:

$\displaystyle\mathbb{1}\otimes e\otimes c^{\vec{\eta}}_{1}\ldots
c^{\vec{\eta}}_{p-2}\mapsto\Big{(}(e\otimes c^{\vec{\eta}}_{1}\ldots
c^{\vec{\eta}}_{p-2})-(\mathbb{1}\otimes c_{1}\ldots c_{p-2})h\Big{)}h^{N},$
$\displaystyle e\otimes c^{\vec{\eta}}_{1}\ldots c^{\vec{\eta}}_{p-1}\mapsto
c^{\vec{\eta}}_{1}\ldots c^{\vec{\eta}}_{p-1}h^{N+1},$
$\displaystyle\mathbb{1}\otimes c^{\vec{\eta}}_{1}\ldots
c^{\vec{\eta}}_{p-1}\mapsto c^{\vec{\eta}}_{1}\ldots
c^{\vec{\eta}}_{p-1}h^{r}$ $\displaystyle c^{\vec{\eta}}_{1}\ldots
c^{\vec{\eta}}_{p}\mapsto 0$

It is now easy to check that

$\displaystyle\mathrm{Ker}(d_{1}^{p})/\mathrm{Coker}(d_{1}^{p+1})=$
$\displaystyle\bigoplus_{1\leq i\leq
2g}{H^{*}(\mathrm{Pic}^{n}(C);\mathbb{Q}(-pN))[h]\over h^{N}}\big{(}e\otimes
c^{\vec{\eta}}_{1}\ldots c^{\vec{\eta}}_{p-1}-\mathbb{1}\otimes
c^{\vec{\eta}}_{1}\ldots c^{\vec{\eta}}_{p-1}\big{)}[2pN]$
$\displaystyle\bigoplus_{1\leq i,j\leq
2g}{H^{*}(\mathrm{Pic}^{n}(C);\overline{\mathbb{Q}}_{\ell}(-pN))[h]\over
h^{N}}(c^{\vec{\eta}}_{1}\ldots c^{\vec{\eta}}_{p})[2pN].$

Now we are left with analysing the resulting $E_{2}$ page. That the
differentials on the $E_{2}$ page vanish for $p\leq n-2g$ follow simply from
weight considerations- the space $\mathcal{T}_{\bullet}$ consists of smooth
projective varieties and thus their $n^{th}$ cohomology is pure of weight $n$.
Now observe the following: we have an equality

$\displaystyle\mathrm{R}\Gamma_{c}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}),C^{\bullet}(\underline{\mathbb{Q}_{\ell}}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})}))=$
$\displaystyle\mathrm{R}\Gamma_{c}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}),j_{!}\underline{\mathbb{Q}_{\ell}}_{\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))})$

in the derived category of constructible sheaves over
$\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$
where
$C^{\bullet}(\mathbb{Q}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})})$
denotes the complex

$\displaystyle 0\to
j_{!}j^{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{T}_{-1}}\to\underline{\mathbb{Q}_{\ell}}_{\mathcal{T}_{-1}}\to\pi_{0_{*}}\pi_{0}^{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{T}_{-1}}\to(\pi_{1_{*}}\pi_{1}^{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{T}_{-1}}\otimes\mathit{sgn}_{2})^{{S}_{2}}\cdots\to$
$\displaystyle\cdots\to(\pi_{p_{*}}\pi_{p}^{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{T}_{-1}}\otimes\mathit{sgn}_{p+1})^{{S}_{p+1}}\to\cdots$
(26)

on the other hand, for any $m\in\mathbb{N}$ we have

$\displaystyle\mathrm{R}^{i}\Gamma_{c}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}),C^{\bullet}(\underline{\mathbb{Q}_{\ell}}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})}))\cong$
$\displaystyle\mathrm{R}^{i}\Gamma_{c}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}),C^{\bullet}(\underline{\mathbb{Q}_{\ell}}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})})/\tau_{\geq
m}C^{\bullet}(\underline{\mathbb{Q}_{\ell}}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})})$

for all $i\geq 2(m+1)-2N$, where $\tau_{\geq
m}C^{\bullet}(\mathbb{Q}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})})$
denotes the truncated complex up to the $(N-1)$ term and this is because
$\tau_{\geq
m}C^{\bullet}(\mathbb{Q}_{\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})})$
is supported on complex codimension $m$ in
$\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}})$.
Therefore the cohomology of $\mathrm{Hom}_{n}(C,\mathcal{P}({\vec{\lambda}}))$
up to degree $n-2g$ is solely dictated by the $E_{2}$ page.

To this end, let

$t:=(e-h)$

which has degree $(-1,2N+2)$ and let

$\alpha_{i}:=\gamma_{i}-\overline{\gamma_{i}},\,\,\,\,\,i=1,\ldots,2g$

which has degree $(-1,2N+1)$. Clearly for $3\leq p\leq n-2g$, the element
$t\alpha_{i_{1}}\ldots\alpha_{i_{p}}$, which is of degree
$(-(p+1),2N+2+p(2N+1))$, when expanded, gives us

$\displaystyle t\alpha_{i_{1}}\ldots\alpha_{i_{p}}$
$\displaystyle=(e-h)(c_{i_{1}}-\overline{c_{i_{1}}})\ldots(c_{i_{p}}-\overline{c_{i_{p}}})$
$\displaystyle=(e-h)\prod_{j=1}^{p}c_{i_{j}}+\Big{\\{}\text{lower order terms
as a polynomial on }c_{i_{1}},\ldots,c_{i_{p}}\Big{\\}}$
$\displaystyle=(e-h)\prod_{j=1}^{p}c_{i_{j}}$

because the lower order terms are all $0$ in

$\big{(}H^{2}(C)\oplus
H^{0}(C)\big{)}\bigotimes\operatorname{Sym}^{p}H^{1}(C)\otimes
H^{*}(\mathrm{Pic}^{n-(p+1)}(C))[h]/h^{N},$

thanks to the alternating action of ${S}_{p+1}$. Whereas
$\alpha_{i_{1}}\ldots\alpha_{i_{p+1}}$, which is of degree
$(-(p+1),(p+1)(2N+1))$, when expanded, gives us

$\displaystyle\alpha_{i_{1}}\ldots\alpha_{i_{p+1}}$
$\displaystyle=(c_{i_{1}}-\overline{c_{i_{1}}})\ldots(c_{i_{p+1}}-\overline{c_{i_{p+1}}})$
$\displaystyle=\prod_{j=1}^{p+1}c_{i_{j}}+\Big{\\{}\text{lower order terms as
a polynomial on }c_{i_{1}},\ldots,c_{i_{p+1}}\Big{\\}}$
$\displaystyle=\prod_{j=1}^{p+1}c_{i_{j}}$

because again, the lower order terms are all $0$ for the exact same reason
cited above.

Now as for $p=2$, we have

$\displaystyle
t\alpha_{i}=(e-h)(\gamma_{i}-\overline{\gamma_{i}})=e\gamma_{i}-\gamma_{i}h+e\overline{\gamma_{i}}+h\overline{\gamma_{i}}$
$\displaystyle=e\gamma_{i}-\gamma_{i}h+e\overline{\gamma_{i}}$

because the alternating action of ${S_{2}}$ kills $H^{0}(C^{2})\otimes
H^{*}(\mathcal{P}_{\mathrm{Pic}^{n}C}(\oplus_{i}\nu_{*}P(n)^{\otimes\lambda_{i}},{\vec{\lambda}}))$,
and in turn, $h\overline{\gamma_{i}}$. This give us the algebra structure on
the $E_{2}$ page for $p\leq n-2g$ and thus completes the proof of Theorem 1.1.

∎

##  5 Arithmetic moduli of generalized elliptic surfaces with prescribed
structures

In this section, we prove new sharp enumerations on the number of generalized
elliptic fibrations over $C=\mathbb{P}^{1}_{\mathbb{F}_{q}}$ with prescribed
level structures or multiple marked points by applying the exact étale
cohomology Theorem 1.2 (followed by the exact point count Theorem 1.4) to the
relevant moduli stacks formulated as Hom-stacks similar to [HP, §3]. In all
our applications, results over higher genus $C$ where we acquire the
corresponding stable étale cohomology followed by stable point counts as in
Corollary 1.5 is straightforward.

### Arithmetic moduli of generalized elliptic fibrations over $C$ with
prescribed level structures

We enumerate the number of generalized elliptic curves over global function
fields with prescribed level structures by first extending the notion of
(nonsingular) elliptic curves that admits desired level structures. By the
work of Deligne and Rapoport [DR] (summarized in [Conrad2, §2] and also in
[Niles, §2]), we consider the generalized elliptic curves over $C_{K}$ with
$[\Gamma]$–level structures over a field $K$ (focusing on $K=\mathbb{F}_{q}$).

Recall that a level structure $[\Gamma_{1}(m)]$ on an elliptic curve $E$ is a
choice of point $P\in E$ of exact order $m$ in the smooth part of $E$ such
that over every geometric point of the base scheme every irreducible component
of $E$ contains a multiple of $P$ (see [KM, §1.4]). And a level structure
$[\Gamma(2)]$ on an elliptic curve $E$ is a choice of isomorphism
$\phi:\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\to E(2)$ where $E(2)$
is the scheme of 2-torsion Weierstrass points (i.e., kernel of the
multiplication-by-2 map $[2]:E\to E$) (see [DR, II.1.18 & IV.2.3]).

The following Proposition shows that the fine modular curves of the above
level structures are isomorphic to the weighted projective stacks
$\mathcal{P}(a,b)$ under mild condition on the characteristic of the base
field $K$.

######  Proposition 5.1.

The moduli stack $\overline{\mathcal{M}}_{1,1}[\Gamma]$ of generalized
elliptic curves with $[\Gamma]$-level structure is isomorphic to the following
over a field $K$:

  1. 1.

if $\mathrm{char}(K)\neq 2$, the tame Deligne–Mumford moduli stack of
generalized elliptic curves with $[\Gamma_{1}(2)]$-structures is isomorphic to

$(\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])_{K}\cong[(\mathrm{Spec}~{}K[a_{2},a_{4}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(2,4),$

  2. 2.

if $\mathrm{char}(K)\neq 3$, the tame Deligne–Mumford moduli stack of
generalized elliptic curves with $[\Gamma_{1}(3)]$-structures is isomorphic to

$(\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(3)])_{K}\cong[(\mathrm{Spec}~{}K[a_{1},a_{3}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(1,3),$

  3. 3.

if $\mathrm{char}(K)\neq 2$, the tame Deligne–Mumford moduli stack of
generalized elliptic curves with $[\Gamma_{1}(4)]$-structures is isomorphic to

$(\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(4)])_{K}\cong[(\mathrm{Spec}~{}K[a_{1},a_{2}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(1,2),$

  4. 4.

if $\mathrm{char}(K)\neq 2$, the tame Deligne–Mumford moduli stack of
generalized elliptic curves with $[\Gamma(2)]$-structures is isomorphic to

$(\overline{\mathcal{M}}_{1,1}[\Gamma(2)])_{K}\cong[(\mathrm{Spec}~{}K[a_{2},a_{2}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(2,2),$

  5. 5.

if $\mathrm{char}(K)\nmid m$, the fine moduli space of generalized elliptic
curves with $[\Gamma_{1}(m)]$-structures for
$m=5,6,7,8,9,10\mathrm{~{}or~{}}12$ is isomorphic to

$(\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(m)])_{K}\cong[(\mathrm{Spec}~{}K[a_{1},a_{1}]-(0,0))/\mathbb{G}_{m}]=\mathcal{P}_{K}(1,1)\cong\mathbb{P}^{1},$

where $\lambda\cdot a_{i}=\lambda^{i}a_{i}$ for $\lambda\in\mathbb{G}_{m}$ and
$i=1,2,3,4$. Thus, the $a_{i}$’s have degree $i$ respectively. Moreover, the
discriminant divisors of
$(\overline{\mathcal{M}}_{1,1}[\Gamma])_{K}\cong\mathcal{P}_{K}(a,b)$ as above
have degree 12.

###### Proof.

The moduli stack $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]$ of generalized
elliptic curves with $[\Gamma_{1}(2)]$-level structure has an isomorphism
$\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]\cong\mathcal{P}(2,4)$ over
$\mathrm{Spec}(\mathbb{Z}[1/2])$ as in [Behrens, §1.3] through the universal
equation

$Y^{2}Z=X^{3}+a_{2}X^{2}Z+a_{4}XZ^{2}\,.$

And the moduli stack $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(3)]$ of
generalized elliptic curves with $[\Gamma_{1}(3)]$-level structure has an
isomorphism $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(3)]\cong\mathcal{P}(1,3)$
over $\mathrm{Spec}(\mathbb{Z}[1/3])$ as in [HMe, Proposition 4.5] through the
universal equation

$Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}=X^{3}\,.$

And the moduli stack $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(4)]$ of
generalized elliptic curves with $[\Gamma_{1}(4)]$-level structure has an
isomorphism $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(4)]\cong\mathcal{P}(1,2)$
over $\mathrm{Spec}(\mathbb{Z}[1/2])$ as in [Meier, Examples 2.1] through the
universal equation

$Y^{2}Z+a_{1}XYZ+a_{1}a_{2}YZ^{2}=X^{3}+a_{2}X^{2}Z\,.$

And the moduli stack $\overline{\mathcal{M}}_{1,1}[\Gamma(2)]$ of generalized
elliptic curves with $[\Gamma(2)]$-level structure has an isomorphism
$\overline{\mathcal{M}}_{1,1}[\Gamma(2)]\cong\mathcal{P}(2,2)$ over
$\mathrm{Spec}(\mathbb{Z}[1/2])$ as in [Stojanoska, Proposition 7.1] through
the universal equation (where the degree of each $\lambda_{i}$ is 2)

$Y^{2}Z=X^{3}+(\lambda_{1}+\lambda_{2})X^{2}Z+\lambda_{1}\lambda_{2}XZ^{2}\,.$

Finally, the fine moduli space $\overline{\mathcal{M}}_{1,1}[\Gamma(m)]$ of
generalized elliptic curves with $[\Gamma(m)]$-level structure for
$m=5,6,7,8,9,10\mathrm{~{}or~{}}12$ has an isomorphism
$\overline{\mathcal{M}}_{1,1}[\Gamma(m)]\cong\mathbb{P}^{1}$ over
$\mathrm{Spec}(\mathbb{Z}[1/m])$ as in [Meier, Example 2.5].

By Remark 2.2, the respective $\overline{\mathcal{M}}_{1,1}[\Gamma]$ as
weighted projective stacks are tame Deligne–Mumford as well, and in fact,
smooth.

For the degree of the discriminant, it suffices to find the weight of the
$\mathbb{G}_{m}$-action. First, the four papers cited above explicitly
construct universal families of elliptic curves over the schematic covers
$(\mathrm{Spec}\;K[a_{i},a_{j}]-(0,0))\rightarrow\mathcal{P}_{K}(i,j)$ of the
corresponding moduli stacks. The explicit defining equation of the respective
universal family implies that the $\lambda\in\mathbb{G}_{m}$ also acts on the
discriminant of the universal family by multiplying $\lambda^{12}$. Therefore,
the discriminant has degree 12. ∎

A generalized elliptic curve $X$ over $C_{K}$ can be thought of as a flat
family of semistable elliptic curves admitting a group structure, such that a
finite group scheme $\mathcal{G}\rightarrow C_{K}$ (determined by $\Gamma$)
embeds into $X$ and its image meets every irreducible component of every
geometric fibers of $X$.

We now consider the moduli stack
$\mathcal{L}_{12n,g}^{[\Gamma]}:=\operatorname{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma])$
of generalized elliptic fibrations over $C$ with $[\Gamma]$-level structures
and $12n$ nodal singular fibers.

######  Proposition 5.2.

Let $n\in\mathbb{Z}_{+}$ and $K$ be a field with $\mathrm{char}(K)=0$ or
$\mathrm{char}(K)\nmid m$ where $m$ depends on $[\Gamma]$ as Proposition 5.1.
Then the moduli stack $\mathcal{L}_{12n,g}^{[\Gamma]}$ of generalized elliptic
fibrations over the parameterized smooth projective basecurve $C_{K}$ of genus
$g$ with discriminant degree $12n>0$ and $[\Gamma]$-level structures is the
tame Deligne–Mumford stack
$\operatorname{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma])$
parameterizing the $K$-morphisms
$f:C\rightarrow\overline{\mathcal{M}}_{1,1}[\Gamma]$ such that
$f^{*}\mathcal{O}_{\overline{\mathcal{M}}_{1,1}[\Gamma]}(1)\in\mathrm{Pic}^{n}C$
.

###### Proof.

Without the loss of generality, we prove the
$\operatorname{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$ case
over a field $K$ with $\mathrm{char}(K)\neq 2$. The proof for the other cases
are analogous. By the definition of the universal family $p$, any generalized
elliptic curves $\pi:Y\rightarrow C$ with $[\Gamma_{1}(2)]$-structures comes
from a morphism $f:C\rightarrow\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]$
and vice versa. As this correspondence also works in families, the moduli
stack of generalized elliptic curves over $C$ with
$[\Gamma_{1}(2)]$-structures is isomorphic to
$\mathrm{Hom}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$.

Since the discriminant degree of $f$ is $12\deg
f^{*}\mathcal{O}_{\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]}(1)$ by
Proposition 5.1, the substack
$\mathrm{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$
parametrizing such $f$’s with $\deg
f^{*}\mathcal{O}_{\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]}(1)=n$ is the
desired moduli stack. Since $\deg
f^{*}\mathcal{O}_{\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]}(1)=n$ is an
open condition,
$\mathrm{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$ is an open
substack of $\mathrm{Hom}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$,
which is tame Deligne–Mumford as $\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)]$
itself is tame Deligne–Mumford by Proposition 5.1. Thus
$\mathrm{Hom}_{n}(C,\overline{\mathcal{M}}_{1,1}[\Gamma_{1}(2)])$ satisfies
the desired properties. ∎

We now acquire the exact number
$|\mathcal{L}_{12n,0}^{[\Gamma]}(\mathbb{F}_{q})/\sim|$ of
$\mathbb{F}_{q}$–isomorphism classes of $\mathbb{F}_{q}$–points (i.e., the
non–weighted point count) of the moduli stack $\mathcal{L}_{12n,0}^{[\Gamma]}$
.

######  Theorem 5.3.

If $\mathrm{char}(\mathbb{F}_{q})\neq 2$, then

$\displaystyle|\mathcal{L}_{12n,0}^{[\Gamma_{1}(2)]}(\mathbb{F}_{q})/\sim|$
$\displaystyle=2\cdot\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(2,4))\right)=2(q^{6n+1}-q^{6n-1})$
$\displaystyle|\mathcal{L}_{12n,0}^{[\Gamma_{1}(4)]}(\mathbb{F}_{q})/\sim|$
$\displaystyle=\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(1,2))\right)=q^{3n+1}-q^{3n-1}$
$\displaystyle|\mathcal{L}_{12n,0}^{[\Gamma(2)]}(\mathbb{F}_{q})/\sim|$
$\displaystyle=2\cdot\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(2,2))\right)=2(q^{4n+1}-q^{4n-1})$

If $\mathrm{char}(\mathbb{F}_{q})\neq 3$, then

$|\mathcal{L}_{12n,0}^{[\Gamma_{1}(3)]}(\mathbb{F}_{q})/\sim|=\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(1,3))\right)=q^{4n+1}-q^{4n-1}$

If $\mathrm{char}(\mathbb{F}_{q})\nmid m$, then

$|\mathcal{L}_{12n,0}^{[\Gamma_{1}(m)]}(\mathbb{F}_{q})/\sim|=\\#_{q}\left(\operatorname{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(1,1)\cong\mathbb{P}^{1})\right)=q^{2n+1}-q^{2n-1}$

$m$ is for $m=5,6,7,8,9,10\mathrm{~{}or~{}}12$.

###### Proof.

Fix $n\in\mathbb{Z}_{\geq 1}$. Since any
$\varphi_{g}\in\mathrm{Hom}_{n}(\mathbb{P}^{1},\mathcal{P}(a,b))$ is