# Hurwitz spaces, Nichols algebras, and Igusa zeta functions

Kevin Chang

###### Abstract.

By constructing new quasimap compactifications of Hurwitz spaces of degrees 4
and 5, we establish a new connection between arithmetic statistics, quantum
algebra, and geometry and answer a question of Ellenberg-Tran-Westerland and
Kapranov-Schechtman. It follows from the geometry of our compactifications and
a comparison theorem of Kapranov-Schechtman that we can precisely relate the
following 3 quantities: (1) counts of $\mathbb{F}_{q}[t]$-algebras of degrees
3, 4, and 5 (2) the “invariant” part of the cohomology of certain special
Nichols algebras (3) Igusa local zeta functions for certain prehomogeneous
vector spaces.

Using Igusa’s computation of the zeta function for the space of pairs of
ternary quadratic forms, we compute the number of quartic
$\mathbb{F}_{q}[t]$-algebras with cubic resolvent of discriminant $q^{b}$ and
the part of the cohomology of a 576-dimensional Nichols algebra
$\mathfrak{B}_{4}$ invariant under a natural $\mathbf{S}_{4}$-action. From the
comparison for degree 3, we also obtain two answers to Venkatesh’s question
about the topological origin of the secondary term in the count of cubic
fields.

##  1\. Introduction

An important problem in number theory is to estimate the number of extensions
of a global field with fixed degree and bounded discriminant. So far, the main
term of this estimate has been found by Bhargava-Shankar-Wang [BSW15] in
degrees up to 5 for global fields of characteristic not 2. In the work of
Bhargava-Shankar-Wang and their predecessors [Bha05, Bha10, DW88, DH71], the
strategy to count extensions of a global field $K$ has two steps: (1) count
$\mathcal{O}_{K}$-algebras (2) sieve out the ones that do not correspond to
rings of integers of extensions of $K$.

The reason this strategy works in degrees up to 5 is the following: for $3\leq
d\leq 5$, degree $d$ $\mathcal{O}_{K}$-algebras (with extra resolvent
structure if $d\in\\{4,5\\}$) are parametrized by orbits in _prehomogeneous
vector spaces_ $(G_{d},V_{d})$. Each $G_{d}$ is a reductive group acting
linearly on a vector space $V_{d}$ with a Zariski dense open orbit
$V_{d}^{ss}$ such that the generic stabilizer is the symmetric group
$\mathbf{S}_{d}$. Each $V_{d}$ is equipped with a function $\Delta_{d}$ whose
value at a point is the discriminant of the corresponding degree $d$ algebra.
Thus, the problem of counting degree $d$ $\mathcal{O}_{K}$-algebras with
bounded discriminant is reduced to counting lattice points
$V_{d}(\mathcal{O}_{K})$ with bounded $\Delta_{d}$. Because of the
classification of prehomogeneous vector spaces, there are no parametrizations
of higher degree algebras by prehomogeneous vector spaces.

From the geometric point of view, the reason why the degrees $d\leq 5$ are
special is that for these $d$, $B\mathbf{S}_{d}$ can be presented as a GIT
quotient of a prehomogeneous vector space. For $3\leq d\leq 5$, there is a
unique (up to scalar) _discriminant_ function $\Delta_{d}$ on $V_{d}$
generating the ring of relative $G_{d}$-invariants. The dense open
$G_{d}$-orbit in $V_{d}$ is precisely the subset $V_{d}^{ss}\subset V_{d}$
where $\Delta_{d}$ does not vanish, and the quotient $[V_{d}^{ss}/G_{d}]$ is
isomorphic to $B\mathbf{S}_{d}$. By results of Wood [Woo11] and Landesman-
Vakil-Wood [LVW22], the quotients $[V_{d}/G_{d}]$ are genuinely moduli stacks
of covers. Thus, if we have a curve $C$, a map $C\to[V_{d}/G_{d}]$ that
generically lands in $B\mathbf{S}_{d}\subset[V_{d}/G_{d}]$ corresponds to a
degree $d$ cover that is generically étale. Such maps are _quasimaps_ to
$B\mathbf{S}_{d}$ in the sense of Ciocan-Fontanine-Kim-Maulik [CFKM14] and
Cheong-Ciocan-Fontanine-Kim [CCFK15].

One goal of this paper is to study the geometry of spaces of quasimaps to
$B\mathbf{S}_{d}$. To do this, it will be useful to study the local problem of
counting $G_{d}(\mathbb{F}_{q}[[t]])$-orbits in $V_{d}(\mathbb{F}_{q}[[t]])$,
which can also be thought of as counting (possibly singular) degree $d$ covers
of the formal disc over $\mathbb{F}_{q}$. This local counting problem is
related to the computation of certain Igusa local zeta functions. In many
works [Igu75, Igu84, Igu86, Igu88, Igu92, Igu94], Igusa studied $p$-adic
integrals of the form $\int_{V}|\Delta(v)|^{s}dv$, where $\Delta$ is a
compactly supported function on the $K$-vector space $V$. He computed these
_Igusa zeta functions_ for the discriminants of many prehomogeneous vector
spaces, including $(G_{3},V_{3})$ and $(G_{4},V_{4})$. It turns out that if
$d\in\\{3,4\\}$ and the size of the residue field $q$ is coprime to 6, the
Igusa zeta function $\int_{V_{d}}|\Delta_{d}(v)|^{s}dv$ is a power series in
$q$ and $t=q^{-s}$ (in fact, a rational function). A similar fact is
conjectured for all Igusa zeta functions and, in particular, for the zeta
function for $(G_{5},V_{5})$.

As the other goal of this paper, we’ll relate these zeta functions to
generating functions for counts of $\mathbb{F}_{q}[t]$-algebras and cohomology
of certain special braided Hopf algebras called _Nichols algebras_. We hope
this new connection will be interesting to both the arithmetic statistics
community and the Nichols algebra community because each side seems to have
some interesting implications for the other (see Section 1.2).

###  1.1. Main results

In this paper, we construct smooth compactifications of low degree Hurwitz
stacks as stacks of quasimaps from orbinodal curves to $B\mathbf{S}_{d}$. As
stated above, Igusa zeta functions govern a good amount of the geometry of
these compactifications. providing bounds on fiber dimensions of _branch
morphisms_ from these compactifications to symmetric powers of the base curve.
Moreover, when our base curve is $\mathbb{A}^{1}$, we can use
$\mathbb{G}_{m}$-localization to compute the cohomology of our
compactifications and obtain counts of low degree $\mathbb{F}_{q}[t]$-algebras
by reducing to the local counting problem of computing Igusa zeta functions.
The cohomology of our compactifications for base curve $\mathbb{A}^{1}$ is
also equal to the cohomology of certain Nichols algebras, so we are able to
obtain new computations of Nichols algebra cohomology and reprove some earlier
computations.

To state our results in a nice way, we define the generating functions

$I_{d}(q,t)\coloneqq C_{d}\int_{V_{d}}|\Delta_{d}(v)|^{s-1}dv$

for $d\in\\{3,4\\}$, where the $C_{d}$ are constants such that the $I_{d}$
have constant term 1. Throughout the paper, we will call these “Igusa zeta
functions,” even though Igusa zeta functions do not traditionally have the
shift $s\mapsto s-1$. Their precise values were obtained by Igusa in [Igu94]:

$\displaystyle I_{3}(q,t)$
$\displaystyle=\frac{1+t+t^{2}+t^{3}+t^{4}}{(1-t^{2})(1-qt^{6})}$
$\displaystyle I_{4}(q,t)$
$\displaystyle=\frac{f(q,t)}{(1-t)(1-t^{2})(1-qt^{6})(1-q^{2}t^{8})(1-q^{3}t^{12})},$

where
$f(q,t)=1+t^{2}+t^{3}+t^{4}-2t^{5}+2qt^{6}+(q-1)t^{7}+qt^{8}-qt^{9}+(q-1)qt^{10}-2qt^{11}+2q^{2}t^{12}-q^{2}t^{13}-q^{2}t^{14}-q^{2}t^{15}-q^{2}t^{17}$.

####  1.1.1. Algebraic geometry

The main geometric construction on which all our results are based is the _big
stack of degree $d$ covers with resolvents_ $\mathcal{R}^{d}$, where
$d\in\\{3,4,5\\}$. In this section, we give the key properties and ideas of
our construction and an overview of how properties of our stacks lead to the
other results of this paper. For this section, fix $d$, and let
$\mathbb{S}_{d}=\operatorname{Spec}\mathbb{Z}\left[\frac{1}{d!}\right]$.

Roughly, for a scheme $S$ over $\mathbb{S}_{d}$, an $S$-point of
$\mathcal{R}^{d}$ parametrizes an orbinodal curve $\mathcal{C}\to S$ along
with a representable morphism
$\chi:\mathcal{C}\to\mathcal{X}_{d}\coloneqq[V_{d}/G_{d}]$ sending all
generic, marked, and nodal points to the _étale locus_
$\mathcal{E}_{d}\coloneqq[V_{d}^{ss}/G_{d}]\cong B\mathbf{S}_{d}$. A morphism
$\mathcal{C}\to\mathcal{X}_{d}$ with this condition parametrizes a degree $d$
cover $\mathcal{D}\to\mathcal{C}$ étale above $\chi^{-1}(\mathcal{E}_{d})$
along with extra resolvent data (if $d\in\\{4,5\\}$) above the points mapping
outside $\mathcal{E}_{d}$, a.k.a. the branch points. Note that there are no
conditions on whether $\mathcal{D}$ is connected.

To translate between morphisms $\mathcal{C}\to\mathcal{X}_{d}$ and covers of
$\mathcal{C}$, we rely on results of Wood [Woo11] and Landesman-Vakil-Wood
[LVW22, §3-4] that parametrize covers of arbitrary base schemes. These results
are crucial because we need to deal with orbinodal curves over an arbitrary
base $\mathcal{C}\to S$ to construct our stacks. A discussion of Wood’s and
Landesman-Vakil-Wood’s parametrizations can be found in Section 5.

The stack $\mathcal{R}^{d}$ comes equipped with a _branch morphism_
$\operatorname{br}:\mathcal{R}^{d}\to\mathcal{M}$, where $\mathcal{M}$ is the
stack of pointed nodal curves equipped with a marked divisor disjoint from the
nodes and marked points. Let $\mathcal{M}^{\operatorname{sm}}$ denote the open
substack of smooth curves, and let $\mathcal{R}^{d,\operatorname{sm}}$ denote
its preimage under $\operatorname{br}$. Let $\mathcal{M}^{b,n}$ denote the
open and closed substack of $\mathcal{M}$ where the marked divisor has degree
$b$ and there are $n$ marked points, and let $\mathcal{R}^{d;b,n}$ denote its
preimage. We have a morphism $\mathcal{M}\to\mathcal{M}^{0,*}$ forgetting the
marked divisor.

Our main result about the stack $\mathcal{R}^{d}$ is the following:

######  Theorem 1.1 (Theorem 8.4, Corollary 8.5).

  1. (a)

$\mathcal{R}^{d}$ is a smooth algebraic stack over $\mathbb{S}_{d}$.

  2. (b)

The morphism $\operatorname{br}:\mathcal{R}^{d}\to\mathcal{M}$ is proper and
representable by Deligne-Mumford stacks.

  3. (c)

The composite
$\mathcal{R}^{d,\operatorname{sm}}\xrightarrow[]{\operatorname{br}}\mathcal{M}^{\operatorname{sm}}\to\mathcal{M}^{\operatorname{sm};0,*}$
is smooth. The restriction
$\mathcal{R}^{d,\operatorname{sm};b,*}\to\mathcal{M}^{\operatorname{sm};0,*}$
has relative dimension $b$.

Let $\mathcal{M}^{\circ}\subset\mathcal{M}$ denote the open substack where the
marked divisor is reduced. We can then define the _big Hurwitz stack of simply
branched degree $d$ covers_
$\mathcal{H}^{d}=\operatorname{br}^{-1}(\mathcal{M}^{\circ})\subset\mathcal{R}^{d}$.
It turns out that morphisms $\mathcal{C}\to\mathcal{X}_{d}$ corresponding to
simply branched covers $\mathcal{D}\to\mathcal{C}$ have unique resolvent data
above the branch points up to unique isomorphism. Thus, $\mathcal{H}^{d}$
genuinely parametrizes simply branched degree $d$ covers, as its name would
suggest.

Given a smooth $n$-pointed curve $(C;\sigma)$ over a field $k$ (over
$\mathbb{S}_{d}$), we get a map

$\operatorname{br}:\mathcal{R}_{(C;\sigma),b}^{d}\to\operatorname{Sym}^{b}(C^{\operatorname{gen}})$

by taking the base change of
$\mathcal{R}^{d;b,n}\xrightarrow[]{\operatorname{br}}\mathcal{M}^{b,n}\to\mathcal{M}^{0,n}$
along the map $\operatorname{Spec}k\to\mathcal{M}^{0,*}$ classifying
$(C;\sigma)$. Here, $C^{\operatorname{gen}}$ is the complement of the marked
points $\sigma$ in $C$. By Theorem 1.1, $\mathcal{R}_{(C;\sigma),b}^{d}$ is a
smooth Deligne-Mumford stack, and $\operatorname{br}$ is proper. The open
substack
$\mathcal{H}_{(C;\sigma),b}^{d}=\operatorname{br}^{-1}(\operatorname{Conf}^{b}(C^{\operatorname{gen}}))$
is now the familiar Hurwitz stack of (possibly disconnected) degree $d$ covers
of $C$ simply branched above $C^{\operatorname{gen}}$. From the deformation
theory of simply branched covers, we know that
$\pi:\mathcal{H}_{(C;\sigma),b}^{d}\to\operatorname{Conf}^{b}(C^{\operatorname{gen}})$
is étale.

We now explain the main applications of our construction:

  * •

Arithmetic: To obtain our count of quartic $\mathbb{F}_{q}[t]$-algebras with
cubic resolvent (Theorem 1.3), we consider the case
$(C;\sigma)=(\mathbb{P}^{1};\infty)$. In this case, there are
$\mathbb{G}_{m}$-actions on all the stacks such that the map

$\operatorname{br}:\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}\to\operatorname{Sym}^{b}(\mathbb{A}^{1})$

is $\mathbb{G}_{m}$-equivariant. Note that
$\operatorname{Sym}^{b}(\mathbb{A}^{1})$ is affine such that the
$\mathbb{G}_{m}$-action contracts everything to 0. By a
$\mathbb{G}_{m}$-localization lemma (Lemma 11.2), the
$\mathbb{Q}_{\ell}$-cohomology of
$\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}$ is isomorphic to the cohomology
of its central fiber $\mathcal{F}_{b}^{d}\coloneqq\operatorname{br}^{-1}(0)$.
Because $\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}$ is smooth and
$\operatorname{br}$ is proper, the cohomology of both of them is pure in the
sense of Deligne and can be computed by counting $\mathbb{F}_{q}$-points of
$\mathcal{F}_{b}^{d}$. This point count can be computed using the Igusa zeta
function; in fact, its generating function is exactly $I_{d}$. This tells us
the cohomology of $\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}$, and by
smoothness, we can apply Poincaré duality to get the compactly supported
cohomology and point count of $\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}$.
This gives us Theorem 1.3. From the method outlined above, it is clear that
the local-global duality of the counts comes from the $\mathbb{G}_{m}$-action
and Poincaré duality.

We take a moment to explain why we allow for stackiness above $\infty$ instead
of just considering non-stacky $\mathbb{P}^{1}$. A cover $D\to\mathbb{A}^{1}$
can be uniquely extended to a cover $\overline{D}\to\mathbb{P}^{1}$ with
$\overline{D}$ smooth, which can be made into a cover of orbicurves
$\overline{\mathcal{D}}\to\mathbb{P}_{r}^{1}$ étale over $\infty$ in a unique
way such that the classifying map
$\mathbb{P}_{r}^{1}\to\operatorname{Covers}_{d}$ is representable. Because
$\overline{\mathcal{D}}$ is smooth above $\infty$, there is no extra resolvent
data above $\infty$. Thus, a point of
$\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{d}$ corresponds to a degree $d$
cover of $\mathbb{A}^{1}$ with resolvent. If we consider
$\operatorname{br}:\mathcal{R}_{\mathbb{P}^{1},b}^{d}\to\operatorname{Sym}^{b}(\mathbb{P}^{1})$
instead, then the points of
$\operatorname{br}^{-1}(\operatorname{Sym}^{b}(\mathbb{A}^{1}))$ parametrize
degree $d$ covers of $\mathbb{A}^{1}$ with resolvent that are unramified above
$\infty$. These could be counted if we knew what part of the Igusa zeta
function counted covers with resolvents unramified above $\infty$, but for
now, we only have counts without any local conditions above $\infty$.

  * •

Intersection cohomology and Nichols algebra cohomology: We can also use the
Igusa zeta functions $I_{d}$ to prove dimension bounds on the fibers of
$\operatorname{br}$. Using these bounds, we obtain the following result:

######  Theorem 1.2 (Theorem 10.7, Theorem 10.13).

For $d\in\\{3,4\\}$,

    1. (a)

$\mathcal{H}_{(C;\sigma),b}^{d}$ is dense in $\mathcal{R}_{(C;\sigma),b}^{d}$.

    2. (b)

$\operatorname{br}$ is small.

The same results hold for $d=5$ assuming a precise conjecture about the
relevant Igusa zeta function (10.12).

Density and smallness imply that if we let
$j:\operatorname{Conf}^{b}(C^{\operatorname{gen}})\xhookrightarrow{}\operatorname{Sym}^{b}(C^{\operatorname{gen}})$
denote the open immersion, then

$j_{!*}\pi_{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{H}_{(C;\sigma),b}^{d}}\simeq
R\operatorname{br}_{*}\underline{\mathbb{Q}_{\ell}}_{\mathcal{R}_{(C;\sigma),b}^{d}}.$

If $k=\mathbb{F}_{q}$, then by proper base change, the trace function of the
right-hand side at a point
$p\in\operatorname{Sym}^{b}(C^{\operatorname{gen}})(\mathbb{F}_{q})$ counts
degree $d$ covers $D\to C^{\operatorname{gen}}$ (with a cubic resolvent if
$d=4$) with branch divisor $p$. This provides a counting interpretation to the
trace function of the intersection complex on the left-hand side, answering a
question of Ellenberg-Tran-Westerland and Kapranov-Schechtman in this case
(see Section 1.2). Moreover, the comparison theorem of Kapranov-Schechtman
[KS20, Corollary 3.3.5] lets us use the above quasi-isomorphism of complexes
to compute the $\mathbf{S}_{4}$-invariant part of the Nichols algebra
$\mathfrak{B}_{4}$ (Theorem 1.8) and provide an arithmetic interpretation of
the cohomology of the Nichols algebras $\mathfrak{B}_{d}$.

####  1.1.2. Arithmetic statistics

Our main arithmetic result is the following sharp count of quartic algebras
with cubic resolvent over $\mathbb{F}_{q}[t]$. The definition of a quartic
algebra with cubic resolvent is due to Wood [Woo11] and can be found in
Section 5.1. For this section, let $q$ be a prime power coprime to 6.

######  Theorem 1.3 (Theorem 11.9).

Let $N_{4,q,b}$ be the number of isomorphism classes of quartic
$\mathbb{F}_{q}[t]$-algebras with cubic resolvent of discriminant $q^{b}$,
inversely weighted by automorphisms over $\mathbb{F}_{q}[t]$. Then

$\sum_{b=0}^{\infty}N_{4,q,b}t^{b}=I_{4}(q^{-1},qt)=\frac{f(q^{-1},qt)}{(1-qt)(1-q^{2}t^{2})(1-q^{5}t^{6})(1-q^{6}t^{8})(1-q^{9}t^{12})},$

where
$f(q^{-1},qt)=1+q^{2}t^{2}+q^{3}t^{3}+q^{4}t^{4}-2q^{5}t^{5}+2q^{5}t^{6}-(q-1)q^{6}t^{7}+q^{7}t^{8}-q^{8}t^{9}-(q-1)q^{8}t^{10}-2q^{10}t^{11}+2q^{10}t^{12}-q^{11}t^{13}-q^{12}t^{14}-q^{13}t^{15}-q^{15}t^{17}$.

######  Corollary 1.4 (Corollary 11.10).

There are rational functions
$A_{b},B_{b},C_{b},D_{b},E_{b}\in\mathbb{Q}(q^{1/12})$ depending only on
$b\pmod{24}$ such that

$N_{4,q,b}=(A_{b}b+B_{b})q^{b}+C_{b}q^{5b/6}+(D_{b}b+E_{b})q^{3b/4}$

for all $q,b$.

If we let $N_{3,q,b}$ be the number of isomorphism classes of cubic
$\mathbb{F}_{q}[t]$-algebras of discriminant $q^{b}$, also inversely weighted
by automorphisms over $\mathbb{F}_{q}[t]$, then

$\sum_{b=0}^{\infty}N_{3,q,b}t^{b}=I_{3}(q^{-1},qt)=\frac{1+qt+q^{2}t^{2}+q^{3}t^{3}+q^{4}t^{4}}{(1-q^{2}t^{2})(1-q^{5}t^{6})}.$

Though this can be computed using our method in the same way as $N_{4,q,b}$,
there is a much easier and more direct method to compute $N_{3,q,b}$, which is
described in Remark 11.13.

Using $\mathbb{G}_{m}$-localization and Poincaré duality, we show that the
above formulas for $N_{3,q,b}$ and $N_{4,q,b}$ come from the following local
counts, which fall directly out of Igusa’s computations of $I_{d}$.

######  Proposition 1.5 (Special case of Corollary 6.4).

  1. (a)

Let $N_{3,q,b}^{\operatorname{loc}}$ be the number of isomorphism classes of
cubic $\mathbb{F}_{q}[[t]]$-algebras of discriminant $q^{b}$, inversely
weighted by automorphisms over $\mathbb{F}_{q}[[t]]$. Then

$\sum_{b=0}^{\infty}N_{3,q,b}^{\operatorname{loc}}t^{b}=I_{3}(q,t)=\frac{1+t+t^{2}+t^{3}+t^{4}}{(1-t^{2})(1-qt^{6})}.$

  2. (b)

Let $N_{4,q,b}^{\operatorname{loc}}$ be the number of isomorphism classes of
quartic $\mathbb{F}_{q}[[t]]$-algebras with cubic resolvent of discriminant
$q^{b}$, inversely weighted by automorphisms over $\mathbb{F}_{q}[[t]]$. Then

$\sum_{b=0}^{\infty}N_{4,q,b}^{\operatorname{loc}}t^{b}=I_{4}(q,t)=\frac{f(q,t)}{(1-t)(1-t^{2})(1-qt^{6})(1-q^{2}t^{8})(1-q^{3}t^{12})},$

where
$f(q,t)=1+t^{2}+t^{3}+t^{4}-2t^{5}+2qt^{6}+(q-1)t^{7}+qt^{8}-qt^{9}+(q-1)qt^{10}-2qt^{11}+2q^{2}t^{12}-q^{2}t^{13}-q^{2}t^{14}-q^{2}t^{15}-q^{2}t^{17}$.

In [GGW21, Problem 5], Venkatesh asks the following question: “What is the
topological meaning of secondary terms appearing in asymptotic counts in
number theory?” The example he considers is the counting problem for cubic
extensions of $\mathbb{Q}$, where the number of cubic fields of absolute
discriminant less than $X$ is currently known to be
$CX+KX^{5/6}+O(X^{2/3+\epsilon})$ for constants $C,K$ by Bhargava-Taniguchi-
Thorne [BTT21]. The terms $X$ and $X^{5/6}$ can be seen in the less refined
count of cubic algebras of bounded discriminant, which was done much earlier
by Shintani [Shi72].

From the “local-global duality” above, we can provide one answer to
Venkatesh’s question: the lower order terms in the global counts correspond to
the higher order terms in the local counts. To put this answer in more
geometric terms, we will show (Proposition 11.6) that the Igusa zeta function
$I_{3}$ is roughly the generating function for the point counts of the central
fibers $\mathcal{F}_{b}^{3}$ of the branch maps
$\mathcal{R}_{(\mathbb{P}^{1};\infty),b}^{3}\to\operatorname{Sym}^{b}(\mathbb{A}^{1})$.
The $1-qt^{6}$ in the denominator of $I_{3}(q,t)$ tells us that the dimension
of $\mathcal{F}_{b}^{3}$ is roughly $\frac{b}{6}$. This means that the leading
term $q^{b/6}$ in the point counts of the central fibers corresponds under
Poincaré duality to the $q^{5b/6}$ term in the count of cubic
$\mathbb{F}_{q}[t]$-algebras of discriminant $q^{b}$ [Igu88].

######  Remark 1.6.

The fact that the central fibers $\mathcal{F}_{b}^{3}$ have dimension roughly
$\frac{b}{6}$ was actually observed by Deopurkar in [Deo13, §4], where he
considers the stratification of $\mathcal{F}_{b}^{3}$ by Maroni invariant and
shows that the strata are quotients of affine spaces by finite groups by
describing all the possible generically étale triple covers of the formal
disc. This approach has the advantage of being much more explicit than our
computation of the dimension, which just uses the point count given by the
Igusa zeta function. However, it is much more difficult to generalize this
explicit approach to describe all the points of $\mathcal{F}_{b}^{4}$, which
correspond to generically étale quartic covers of the formal disc with cubic
resolvent. Our approach of expressing the generating function of the point
count as an Igusa zeta function lets us compute not only the dimension of
$\mathcal{F}_{b}^{4}$ (roughly $\frac{b}{4}$) but also the cohomology and the
exact point count.

####  1.1.3. Quantum algebra

In contrast to the strategy of Ellenberg-Venkatesh-Westerland [EVW16] and
Ellenberg-Tran-Westerland [ETW17], we use number theory and algebraic geometry
to deduce something on the topological/quantum algebraic side. The theorem
that allows us to make this translation is due to Kapranov-Schechtman:

######  Theorem 1.7 ([KS20, Corollary 3.3.5]).

There is an isomorphism

$\operatorname{Ext}_{T_{!*}(V(c,-1))}^{n-j,n}(k,k)\cong
H^{j}(\operatorname{Sym}^{n}(\mathbb{A}^{1}),j_{!*}\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}})^{\vee}.$

Here, $G$ is an arbitrary finite group, $T_{!*}(V(c,-1))$ is a braided graded
Hopf algebra associated to a union of conjugacy classes $c\subset G$ known as
the _Nichols algebra_ , and
$\pi:\operatorname{Hur}_{G,n}^{c}\to\operatorname{Conf}^{n}(\mathbb{A}^{1})$
is a covering space corresponding to a natural action of the $n$th braid group
$\mathbf{B}_{n}$ on $c^{\times n}$. The Hurwitz space
$\operatorname{Hur}_{G,n}^{c}$ parametrizes marked $G$-covers of the (closed)
unit disc with $n$ branch points whose local monodromy is in $c$, where a
marking is a $G$-equivariant isomorphism between the fiber above 1 and $G$.
Our Hurwitz stacks $\mathcal{H}_{(C;\sigma),b}^{d}$, which correspond to the
case $G=\mathbf{S}_{d}$ and $c=\tau_{d}$, the class of transpositions, are
unmarked. To make better use of the theorem of Kapranov-Schechtman, we
construct marked Hurwitz spaces as certain $\mathbf{S}_{d}$-torsors
$\mathcal{H}_{M,(C;\sigma),b}^{d}\to\mathcal{H}_{(C;\sigma),b}^{d}$ along with
marked compactifications
$\mathcal{R}_{M,(C;\sigma),b}^{d}\to\mathcal{R}_{(C;\sigma),b}^{d}$, and we
show that the space
$\mathcal{H}_{M,(\mathbb{P}^{1};\infty),b}^{d}(\mathbb{C})$ is homeomorphic to
$\operatorname{Hur}_{\mathbf{S}_{d},b}^{\tau_{d}}$.

Our result concerns the cohomology of the Nichols algebra $\mathfrak{B}_{4}$
associated to the Yetter-Drinfeld module
$k\tau_{4}\otimes\operatorname{sgn}\in\mathcal{YD}_{\mathbf{S}_{4}}^{\mathbf{S}_{4}}$,
where $k$ is a field of characteristic 0 and $\operatorname{sgn}$ is the sign
representation of $\mathbf{S}_{4}$. This 576-dimensional Nichols algebra is
significant because it is one of very few known examples of finite-dimensional
elementary Nichols algebras. The cohomology of the previous Nichols algebra on
this list, the 12-dimensional $\mathfrak{B}_{3}$, was computed algebraically
by Ştefan-Vay in [ŞV16]. We prove the following theorem using the Kapranov-
Schechtman theorem and the density/smallness from Theorem 1.2. We can also
recover the $\mathbf{S}_{3}$-invariant part of the Ştefan-Vay computation by
the same method and recover the entire computation by applying
$\mathbb{G}_{m}$-localization to the marked compactification (Remark 11.16).

######  Theorem 1.8 (Theorem 11.14 for $d=4$).

  1. (a)

There are $\mathbf{S}_{4}$-equivariant isomorphisms

$\operatorname{Ext}_{\mathfrak{B}_{4}}^{a,b}(k,k)\cong
H^{b-a}(\mathcal{R}_{M,(\mathbb{P}^{1};\infty),b}^{4}(\mathbb{C}),k)^{\vee}\cong
H^{b-a}(\mathcal{F}_{M,b}^{4}(\mathbb{C}),k)^{\vee},$

where the $\mathbf{S}_{4}$-action on the $\operatorname{Ext}$ groups is the
geometric action.

  2. (b)

Let
$E^{a,b}=\dim\operatorname{Ext}_{\mathfrak{B}_{4}}^{a,b}(k,k)^{\mathbf{S}_{4}}$.
Then

$\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}E^{a,b}q^{a}t^{b}=I_{4}(q^{-2},qt)=\frac{f(q^{-2},qt)}{(1-qt)(1-q^{2}t^{2})(1-q^{4}t^{6})(1-q^{4}t^{8})(1-q^{6}t^{12})},$

where
$f(q^{-2},qt)=1+q^{2}t^{2}+q^{3}t^{3}+q^{4}t^{4}-2q^{5}t^{5}+2q^{4}t^{6}-(q^{2}-1)q^{5}t^{7}+q^{6}t^{8}-q^{7}t^{9}-(q^{2}-1)q^{6}t^{10}-2q^{9}t^{11}+2q^{8}t^{12}-q^{9}t^{13}-q^{10}t^{14}-q^{11}t^{15}-q^{13}t^{17}$.

We clarify what we mean by “geometric action” in part (a). There are two
$\mathbf{S}_{4}$-actions on $k\tau_{4}\otimes\operatorname{sgn}$. The first is
the _standard action_ , which is the action on
$k\tau_{4}\otimes\operatorname{sgn}$ as a Yetter-Drinfeld module: for
$t\in\tau_{4}$ and $g\in\mathbf{S}_{4}$, we have
$(g,t)\mapsto\operatorname{sgn}(g)gtg^{-1}$. The second is the _geometric
action_ : for $t\in\tau_{4}$ and $g\in\mathbf{S}_{4}$, we have $(g,t)\mapsto
gtg^{-1}$. Each of these actions induces an $\mathbf{S}_{4}$-action on
$\mathfrak{B}_{4}$. Although the standard action may seem more natural from an
algebraic point of view, the geometric action is the one that appears in our
work since it corresponds to the $\mathbf{S}_{4}$-action on
$\mathcal{H}_{M,(C;\sigma),b}^{d}$.

Unfortunately, we cannot say anything about the Yoneda ring structure of
$\operatorname{Ext}_{\mathfrak{B}_{4}}^{*,*}(k,k)^{\mathbf{S}_{4}}$ with our
methods, as the multiplication on the Hurwitz space side is topological rather
than algebro-geometric. However, we do note that the terms in the denominator
suggest the presence of polynomial generators in bidegrees $(1,1)$, $(2,2)$,
$(4,6)$, $(4,8)$, and $(6,12)$. In the $d=3$ case, the terms in the
denominator $(1-q^{2}t^{2})(1-q^{4}t^{6})$ indeed appear as polynomial
generators in $\operatorname{Ext}_{\mathfrak{B}_{3}}^{2,2}$ and
$\operatorname{Ext}_{\mathfrak{B}_{3}}^{4,6}$ by the computation of the
cohomology ring by Ştefan-Vay [ŞV16], so it is likely that the
$\mathfrak{B}_{4}$ picture looks like this too.

We get another answer to Venkatesh’s question about the topological origin of
secondary terms in arithmetic counts: the secondary terms correspond to
polynomial generators of the $\mathbf{S}_{d}$-invariant cohomology of
$\mathfrak{B}_{d}$. In particular, the secondary term for the count of cubic
fields/algebras comes from the generator in
$\operatorname{Ext}_{\mathfrak{B}_{3}}^{4,6}$. Since the denominator terms in
the Nichols algebra cohomology really come from the denominator of the Igusa
zeta function, this answer is secretly the same as the one about local counts.

###  1.2. Motivation: Arithmetic statistics and Nichols algebras

This paper originated from a question asked by Ellenberg-Tran-Westerland (ETW)
in their paper proving the upper bound in the Malle conjecture over
$\mathbb{F}_{q}(t)$ [ETW17]. In that paper, the authors study the cohomology
of the quantum shuffle algebra $T_{*}(V(c,-1))$, which contains the Nichols
algebra $T_{!*}(V(c,-1))$ as a natural subalgebra. Whereas the Nichols algebra
cohomology is dual to the cohomology of the intersection complex
$j_{!*}\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}}$ on
$\operatorname{Sym}^{n}(\mathbb{A}^{1})$, the quantum shuffle algebra
cohomology is dual to the cohomology of the Hurwitz space itself, so its
associated trace function counts extensions of $\mathbb{F}_{q}(t)$. In
contrast to the simplicity of its topological and arithmetic interpretation,
the quantum shuffle algebra is generally much more complicated than the
Nichols algebra and much harder to work with.

In [ETW17], the authors ask whether the trace of Frobenius on the Nichols
algebra/intersection cohomology counts anything, particularly in the case
$(G,c)=(\mathbf{S}_{3},\tau_{3})$, where the cohomology has been computed by
Ştefan-Vay [ŞV16]. In a similar vein, in [KS20], Kapranov-Schechtman remark
that it would be interesting to describe the intersection cohomology as the
cohomology of some natural (partial) compactification of
$\operatorname{Hur}_{G,n}^{c}$. By general facts about perverse sheaves, the
cohomology of such a compactification is the same as the intersection
cohomology as long as it is smooth and admits a small extension of the branch
map to $\operatorname{Sym}^{n}(\mathbb{A}^{1})$. There are many previously
studied Hurwitz space compactifications such as twisted admissible covers
compactifications [ACV03] and Deopurkar’s quasimap compactifications [Deo14],
but we have to construct new compactifications in this paper because these
prior compactifications either fail to be small (twisted admissible covers) or
smooth (Deopurkar’s for $d\geq 4$).

We note that the Nichols algebra $T_{!*}(V(c,-1))$ and corresponding
intersection complexes
$j_{!*}\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}}$ on
$\operatorname{Sym}^{n}(\mathbb{A}^{1})$ only depend on the pair $(G,c)$. This
means that the cohomology and $\mathbb{F}_{q}$-point counts of a smooth small
compactification are intrinsic to the pair $(G,c)$ as well. In other words,
whatever counting problem is governed by a smooth small compactification of
$\operatorname{Hur}_{G,n}^{c}$ is a canonical approximation to the problem of
counting $G$-extensions of $\mathbb{F}_{q}(t)$. By constructing the smooth
small compactifications in this paper, we show that the approximations for the
cases $(G,c)=(\mathbf{S}_{d},\tau_{d})$ with $3\leq d\leq 5$ are rank $d$
algebras with resolvent. In other words, the parametrizations of low degree
algebras used in arithmetic statistics, which seem to fall out of the
classification of prehomogeneous vector spaces, are actually intrinsic to the
groups $\mathbf{S}_{3},\mathbf{S}_{4},\mathbf{S}_{5}$. This suggests some
heuristics relating arithmetic statistics, Nichols algebras, and geometry:

  * •

Heuristic 1: Constructing smooth small compactifications of
$\operatorname{Hur}_{G,n}^{c}$ may tell us how to parametrize $G$-extensions.
It would be very interesting to try to find smooth small compactifications for
other $(G,c)$ or to prove that such compactifications cannot exist. We remark
that the approach taken in this paper of compactifying Hurwitz spaces by
spaces of quasimaps to $BG$ cannot lead to anything beyond what we do in this
paper with prehomogeneous vector spaces (see Remark 8.29 for details), but it
may still be possible to find smooth small compactifications by other means.

  * •

Heuristic 2: The complexity of the count of $G$-extensions should match the
complexity of the relevant Nichols algebras over $G$. In particular, when the
relevant Nichols algebra is finite-dimensional, the count should be simpler in
some way. For instance, for $G$ abelian, the Nichols algebras are exterior
algebras, which are finite-dimensional and have extremely simple cohomology.

In this paper, we are concerned with Nichols algebras over symmetric groups
$\mathbf{S}_{d}$. One observation that we have no explanation for is the
similarity between the following two facts:

    1. (1)

$\mathbf{S}_{d}$ can be realized as the generic stabilizer of a prehomogeneous
vector space precisely when $d\leq 5$.

    2. (2)

The Nichols algebras $\mathfrak{B}_{d}$ for $(G,c)=(\mathbf{S}_{d},\tau_{d})$
are finite-dimensional when $d\leq 5$ and conjectured to be infinite-
dimensional for $d\geq 6$.

This paper matches prehomogeneous vector spaces and Nichols algebras by
equating the coefficients of the Igusa zeta function for $(G_{d},V_{d})$ with
the $\mathbf{S}_{d}$-invariant cohomology of $\mathfrak{B}_{d}$. Given this
connection, the non-existence of prehomogeneous vector spaces with generic
stabilizer $\mathbf{S}_{d}$ for $d\geq 6$ can be seen as evidence for the
conjectured infinite-dimensionality of $\mathfrak{B}_{d}$ for $d\geq 6$.
However, our connection is rather indirect, so many aspects remain mysterious,
e.g. what it means geometrically if the Nichols algebra is finite-dimensional.
For more context about the algebras $\mathfrak{B}_{d}$, see Section 3.4.

###  1.3. Relation to previous work

We owe a large debt to Anand Deopurkar and his thesis work. Much of this paper
came from trying to interpret the geometry of his stacks of triple covers
arithmetically and find higher degree compactifications with similar
properties. Many details of our construction of stacks of degree $d$ covers
with resolvents carry over identically from his construction of stacks of
degree $d$ covers [Deo14]. Both of us construct stacks of quasimaps to
$B\mathbf{S}_{d}$. The difference is that we consider $B\mathbf{S}_{d}$ as an
open substack of $\mathcal{X}_{d}$ for $d\in\\{3,4,5\\}$, whereas Deopurkar
uses the stacks $\operatorname{Covers}_{d}$, which exist for all $d$ and fail
to be smooth for $d\geq 4$. There are maps
$\mathcal{X}_{d}\to\operatorname{Covers}_{d}$ inducing maps from our stacks to
Deopurkar’s stacks; this map is an isomorphism for $d=3$, so we end up
constructing the same triple cover compactification in this case. When $d\geq
4$, Deopurkar’s stacks are singular, and for our applications, we need our
compactifications to be smooth. The contributions of this paper are to use
resolvent data to construct resolutions of Deopurkar’s stacks for
$d\in\\{4,5\\}$ and to connect the geometry of these low degree Hurwitz spaces
to the broader picture of prehomogeneous vector spaces and Nichols algebras.

Because we use the presentation of $B\mathbf{S}_{d}$ as the GIT quotient stack
$[V_{d}^{ss}/G_{d}]$, our construction over $\mathbb{C}$ is also a special
case of the stacks of orbifold quasimaps constructed by Cheong-Ciocan-
Fontanine-Kim [CCFK15]. However, we do everything in mixed characteristic, so
we rewrite everything.

We point out the recent work of Han-Park [HP23] and Bejleri-Park-Satriano
[BPS22], where the authors provide alternative answers to Venkatesh’s question
about secondary terms in separate contexts. In particular, Han-Park treat the
$d=2$ of the problem we study in this paper by enumerating quasi-admissible
odd-degree hyperelliptic curves over $\mathbb{P}^{1}$. It would be interesting
to see if their explanation for their lower order terms has the same
underlying reason as our lower order terms. However, the techniques in both of
these papers are totally different from ours. Their counting problems are
equivalent to counting points of a mapping stack from $\mathbb{P}^{1}$ or a
stacky $\mathbb{P}^{1}$ to a weighted projective space, so they are able to
compute the motives of these stacks and extract the point counts. On the other
hand, we count maps to $\mathcal{X}_{d}=[V_{d}/G_{d}]$, and our point counts
are based on the point counts of the central fiber $\mathcal{F}_{b}^{d}$,
which encode local data (e.g. the Igusa zeta function).

###  1.4. Further questions

We hope that the present work will serve as motivation for interested readers
to compute the Igusa zeta function for $(G_{5},V_{5})$, as we will get many
results for $d=5$ if $I_{5}$ is computed (see Section 10.3.2 for specific
conjectures). Igusa was unable to do so, but maybe the advances in both
mathematics and technology will make this computation feasible nowadays. We
point to the recent work of Tajima-Yukie [TY20] on the GIT stratification of
$V_{5}$.

Another result that would be useful to study the case $d=5$ would be a
parametrization of quintic algebras with sextic resolvents over a general base
analogous to Wood’s theorem [Woo11] parametrizing quartic algebras with cubic
resolvents over a general base. Because we have parametrizations of cubics and
quartics over a general base, $\mathcal{R}^{3}$ and $\mathcal{R}^{4}$
genuinely are moduli stacks of cubic and quartic covers with resolvents. Right
now, we do not know that $\mathcal{R}^{5}$ is actually a moduli stack of
quintic covers with sextic resolvents.

###  1.5. Outline of the paper

Readers who are interested only in the arithmetic or algebro-geometric aspects
of this paper should start at Section 5.

Section 3 is a brief introduction to Nichols algebras, focusing on the
examples $\mathfrak{B}_{d}$. Section 4 is a topological introduction to
Hurwitz spaces, recapping the material of [EVW16, §2] and stating the theorem
of Kapranov-Schechtman. Section 5 discusses the prehomogeneous vector spaces
$(G_{d},V_{d})$ and the relationship between
$\mathcal{X}_{d}\coloneqq[V_{d}/G_{d}]$ and the stack of degree $d$ covers
$\operatorname{Covers}_{d}$. Section 6 translates the values of Igusa zeta
functions to counts of $G_{d}(\mathcal{O}_{K})$-orbits in
$V_{d}(\mathcal{O}_{K})$ for a local field $K$. Section 7 is an introduction
to orbinodal curves. Many important stacks and their properties can be found
here. Section 8 contains the main geometric content of this paper. In it, we
construct our stacks $\mathcal{R}^{d}$ and prove their important properties
(properness of the branch morphism, smoothness, relative dimension over
$\mathcal{M}$, etc.). Section 9 uses Igusa zeta functions to bound the
dimensions of the fibers of the branch morphism $\operatorname{br}$. Section
10 constructs and proves properties of several important geometric objects
derived from $\mathcal{R}^{d}$. These include Hurwitz stacks
$\mathcal{H}^{d}$, marked variants of $\mathcal{R}^{d}$ and $\mathcal{H}^{d}$,
and stacks $\mathcal{R}_{(C;\sigma),b}^{d}$ for a single pointed curve
$(C;\sigma)$. We present some applications of our constructions to
smoothability of curves in Section 10.3.1. Section 11 specializes to the case
of $(C;\sigma)=(\mathbb{P}^{1};\infty)$. This is where we use
$\mathbb{G}_{m}$-localization to prove all of our main results about counts of
$\mathbb{F}_{q}[t]$-algebras and cohomology of Nichols algebras.

###  1.6. Acknowledgments

I am indebted to my advisor Will Sawin for suggesting the problem that led to
this paper and for introducing me to the beautiful theory of prehomogeneous
vector spaces and parametrizations of low degree covers. I am grateful to
Will, Aaron Landesman, Jordan Ellenberg, and Craig Westerland for providing
helpful comments on a draft of this paper. I would like to thank Melissa Liu
for introducing me to quasimaps and the utility of
$\mathbb{G}_{m}$-localization early on in graduate school. Finally, I thank
Will, Aaron, Jordan, Craig, Andres Fernandez Herrero, Johan de Jong, and Anand
Deopurkar for very helpful conversations related to the work and Alan Peng for
valuable typesetting advice.

##  2\. Notation and conventions

  1. (1)

$\mathbf{S}_{d}$ will denote the $d$th symmetric group. In the context of
$\mathbf{S}_{d}$, we will denote the conjugacy class of transpositions in
$\mathbf{S}_{d}$ by $\tau_{d}$.

  2. (2)

$\mu_{n}$ will denote the group scheme of $n$th roots of unity.

  3. (3)

Given a groupoid $X$, let $X/\sim$ denote its set of isomorphism classes. If
$X$ has finitely many isomorphism classes, $\\#X$ will denote the sum
$\sum_{x}\frac{1}{\\#\operatorname{Aut}(x)}$, where $x$ ranges over the
isomorphism classes of $X$. We will often call $\\#X$ the count _inversely
weighted by automorphisms_.

  4. (4)

Given a finite rank locally free sheaf $\mathcal{E}$ on a scheme $X$, let
$\mathbb{P}\mathcal{E}\coloneqq\operatorname{Proj}\operatorname{Sym}^{\bullet}\mathcal{E}$.

  5. (5)

We will use the term _finite cover_ to refer to a finite locally free morphism
of constant positive degree. We will use _degree $d$ cover_ to refer to a
finite cover of degree $d$. For a finite cover $f:Y\to S$, we will often write
$\mathcal{O}_{Y}$ in place of $f_{*}\mathcal{O}_{Y}$.

  6. (6)

Whenever we have a _curve over a scheme $S$_ written $C\to S$, we mean that
$C$ is an algebraic space and $C\to S$ is a flat proper morphism whose
geometric fibers are purely 1-dimensional.

  7. (7)

For a separated finite type scheme $X/S$, $\operatorname{PConf}^{n}(X)$ will
denote the ordered configuration space given by removing the image of all the
diagonal embeddings $X^{n-1}\xhookrightarrow{}X^{n}$ (a.k.a. the _big
diagonal_) from $X^{n}$,where the fiber product is over $S$). Then
$\operatorname{Conf}^{n}(X)$ will denote the unordered configuration space
$\operatorname{PConf}^{n}(X)/S_{n}$.

  8. (8)

We can work with $\ell$-adic and perverse sheaves on Deligne-Mumford stacks
the same way we would on schemes, thanks to the work of Olsson [Ols07b] and
Laszlo-Olsson [LO08a, LO08b, LO09]. We always take $\ell$ to be a prime
different from the characteristic of the base field.

  9. (9)

For an open immersion of a smooth subvariety $j:U\xhookrightarrow{}X$ and a
local system $\mathcal{L}$ on $U$, $j_{!*}\mathcal{L}$ will denote the shifted
perverse sheaf $j_{!*}(\mathcal{L}[\dim X])[-\dim X]$.

##  3\. Nichols algebras

This section is a brief introduction to Nichols algebras. For a more detailed
introduction, see [ETW17, §2] and [KS20, §2-3]. For a general survey, see
[AS02].

For this section, let $k$ be a field of characteristic 0. All tensor products
of $k$-vector spaces will be over $k$.

###  3.1. Braided vector spaces and braided monoidal categories

A _braided vector space_ is a pair $(V,c)$, where $V$ is a finite-dimensional
$k$-vector space and $c:V\otimes V\xrightarrow[]{\sim}V\otimes V$ is an
automorphism satisfying the braid equation

$(c\otimes\operatorname{id})(\operatorname{id}\otimes
c)(c\otimes\operatorname{id})=(\operatorname{id}\otimes
c)(c\otimes\operatorname{id})(\operatorname{id}\otimes c)$

as automorphisms of $V\otimes V\otimes V$.

Recall that the _$n$ th braid group_ $\mathbf{B}_{n}$ is a group with
presentation

$\displaystyle\mathbf{B}_{n}\coloneqq\langle\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}$
$\displaystyle=\sigma_{i+1}\sigma_{i}\sigma_{i+1},\forall 1\leq i\leq n-2;$
$\displaystyle\sigma_{i}\sigma_{j}$
$\displaystyle=\sigma_{j}\sigma_{i},\forall|i-j|\geq 2\rangle.$

A braiding $c$ on a vector space $V$ naturally makes $V^{\otimes n}$ a
$\mathbf{B}_{n}$-representation for each $n$: each $\sigma_{i}$ acts on
$V^{\otimes n}$ as $1\otimes\cdots\otimes 1\otimes c\otimes 1\otimes 1$ with
$c$ acting on the $i$th and $(i+1)$th copies of $V$, and the braid equation
ensures that the braid relations between the $\sigma_{i}$ are satisfied.

Let $(\mathcal{V},\otimes,1,R)$ be a braided monoidal $k$-linear abelian
category where $\otimes$ is exact with respect to both entries. Here,
$(\mathcal{V},\otimes,1)$ is the underlying monoidal category, and for any
$V_{1},V_{2}\in\mathcal{V}$, we have the $R$-matrix

$R_{V_{1},V_{2}}:V_{1}\otimes V_{2}\xrightarrow[]{\sim}V_{2}\otimes V_{1}$

satisfying the braiding axioms [HS20, Definition 3.2.1]. The braiding axioms
imply that $R_{V,V}:V\otimes V\xrightarrow[]{\sim}V\otimes V$ satisfies the
braid equation. If $\mathcal{V}$ is equipped with a faithful $k$-linear strict
monoidal functor to the category of $k$-vector spaces, then the underlying
vector space of an object $V$ is a braided vector space with braiding
$c=R_{V,V}$.

The main examples of braided monoidal categories relevant to us are categories
of Yetter-Drinfeld modules:

######  Definition 3.1.

For a group $G$, a (right, right) _Yetter-Drinfeld module_ 111Yetter-Drinfeld
modules can be defined over any Hopf algebra [HS20, §3.4], but we will only
deal with group algebras (of finite groups). $M$ is a $G$-graded right
$G$-module with the property that $M_{g}h\subset M_{h^{-1}gh}$ for all $g,h\in
G$. A map of Yetter-Drinfeld modules is a $G$-equivariant map respecting the
$G$-grading.

A _graded Yetter-Drinfeld module_ is a $\mathbb{Z}$-graded collection of
Yetter-Drinfeld modules. A map of graded Yetter-Drinfeld modules is a map of
Yetter-Drinfeld modules respecting the $\mathbb{Z}$-grading.

Note that if we have a Yetter-Drinfeld module $M$ and a character $\psi:G\to
k^{\times}$, we can define the twisted Yetter-Drinfeld module $M^{\psi}$, in
which the $G$-action is multiplied by $\psi$.

######  Example 3.2.

The most important example of a Yetter-Drinfeld module we will work with is
$M=kc$, where $c$ is a union of conjugacy classes in $G$. Here, the basis
element corresponding to an element $g\in c$ is in $M_{g}$.

######  Definition 3.3.

We define two braided monoidal structures on the category of Yetter-Drinfeld
modules:

  1. (a)

For $m\in M$ and $n\in N_{g}$, we will write $m^{n}\coloneqq mg$. Then the
$R$-matrix $R_{M,N}:M\otimes N\to N\otimes M$ is defined by $R_{M,N}(m\otimes
n)=n\otimes m^{n}$, where $m\in M$ and $n\in N_{g}$ for some $g\in G$. We will
use $\mathcal{YD}_{G}^{G}$ to denote the braided monoidal category of Yetter-
Drinfeld modules with braiding $R$.

  2. (b)

We define an alternate braided monoidal structure on $\mathcal{YD}_{G}^{G}$:
$R_{M,N}^{\epsilon}(m\otimes n)=-n\otimes m^{n}$. We will use
$\mathcal{YD}_{G,\epsilon}^{G}$ to denote the braided monoidal category
obtained from the braiding $R^{\epsilon}$.

###  3.2. Braided bialgebras

With the above examples in mind, we return to the setting of an arbitrary
braided monoidal $k$-linear abelian category $\mathcal{V}$. We will refer to
monoid objects $(A,\mu,\eta)$ as algebras and comonoid objects
$(A,\Delta,\epsilon)$ as coalgebras. Given two algebras $A,B\in\mathcal{V}$,
we define $A\underline{\otimes}B$ as the algebra having underlying object
$A\otimes B$, multiplication

$\mu_{A\underline{\otimes}B}:A\otimes B\otimes A\otimes
B\xrightarrow[]{1\otimes R_{B,A}\otimes 1}A\otimes A\otimes B\otimes
B\xrightarrow[]{\mu_{A}\otimes\mu_{B}}A\otimes B$

and unit

$\eta_{A\underline{\otimes}B}:1\xrightarrow[]{\sim}1\otimes
1\xrightarrow[]{\eta_{A}\otimes\eta_{B}}A\otimes B.$

######  Definition 3.4.

A _braided bialgebra_ in $\mathcal{V}$ is a tuple
$(A,\mu,\eta,\Delta,\epsilon)$, where

  1. (1)

$(A,\mu,\eta)$ is an algebra.

  2. (2)

$(A,\Delta,\epsilon)$ is a coalgebra.

  3. (3)

$\Delta:A\to A\underline{\otimes}A$ and $\epsilon:1\to A$ are morphisms of
algebras.

A braided bialgebra is a _braided Hopf algebra_ if there exists a morphism
$\chi:A\to A$ such that

$\eta\epsilon=\mu(1\otimes\chi)\Delta=\mu(\chi\otimes 1)\Delta.$

Such a $\chi$ is called an antipode, and it is unique if it exists.

A braided graded bialgebra is a braided bialgebra $A$ equipped with an
$\mathbb{N}$-grading $A=\bigoplus_{n=0}^{\infty}A^{n}$ that makes
$(A,\mu,\eta)$ a graded algebra and $(A,\Delta,\epsilon)$ a graded bialgebra.

Given any object $V$ of a braided monoidal category $\mathcal{V}$, there are
three braided graded Hopf algebras naturally associated to it:

  1. (1)

The _tensor algebra_ $T_{!}(V)$ has underlying object

$T_{!}(V)=\bigoplus_{n=0}^{\infty}V^{\otimes n}$

with multiplication induced by the identity map

$V^{\otimes m}\otimes V^{\otimes n}\to V^{\otimes m+n}.$

We give $T_{!}(V)$ the unique coalgebra structure that makes $T_{!}^{1}(V)=V$
_primitive_ , i.e.
$\Delta|_{T_{!}^{1}(v)}=1\otimes\operatorname{id}+\operatorname{id}\otimes
1:V\to V\otimes V$.

  2. (2)

The _cotensor algebra_ $T_{*}(V)$ has underlying object

$T_{*}(V)=\bigoplus_{n=0}^{\infty}V^{\otimes n}$

with comultiplication induced by the identity map

$V^{\otimes m+n}\to V^{\otimes m}\otimes V^{\otimes n}.$

To define the multiplication, we first recall that a _shuffle_
$w:\\{1,\ldots,m\\}\sqcup\\{1,\ldots,n\\}\to\\{1,\ldots,m+n\\}$ is a bijection
that respects the orders on $\\{1,\ldots,m\\}$ and $\\{1,\ldots,n\\}$. For
each shuffle $w$, there is a natural morphism $R_{w}:V^{\otimes m}\otimes
V^{\otimes n}\to V^{\otimes m+n}$ that uses $R_{V,V}$ to shuffle the copies of
$V$ according to $w$. The multiplication $\star$ on $T_{*}(V)$, called the
_shuffle product_ , is defined as $\star=\sum_{w}R_{w}$. In [ETW17],
$T_{*}(V)$ is called the _quantum shuffle algebra_ , and its cohomology is
bounded, related to the cohomology of Hurwitz spaces (Theorem 3.5), and used
to prove the upper bound in Malle’s conjecture over $\mathbb{F}_{q}(t)$.

  3. (3)

Because $T_{!}(V)$ is the free algebra on $V$ (or alternatively, because
$T_{*}(V)$ is the free coalgebra on $V$), there is a natural map $T_{!}(V)\to
T_{*}(V)$. This is a map of coalgebras because $T_{*}(V)^{1}=V$ is primitive
and $T_{!}(V)$ is generated by the primitives $T_{!}(V)^{1}=V$. Hence, we have
a map of braided Hopf algebras $T_{!}(V)\to T_{*}(V)$. We define the _Nichols
algebra_ $T_{!*}(V)$ as the braided Hopf algebra

$T_{!*}(V)=\operatorname{im}(T_{!}(V)\to T_{*}(V)).$

Details (e.g. proofs that these are braided Hopf algebras) can be found in
[KS20, §2.4, §3.1]. Of course, if $\mathcal{V}$ is a category of vector spaces
(e.g. $\mathcal{YD}_{G}^{G}$), then all the above braided Hopf algebras have
underlying $k$-algebras/coalgebras. It is possible to define a Nichols algebra
of any braided vector space in the same way as above, and the underlying
$k$-algebra/coalgebra of a Nichols algebra $T_{!*}(V)\in\mathcal{V}$ is
determined by the underlying braided vector space of $V$.

The notation comes from the correspondence of Kapranov-Schechtman [KS20,
Theorem 3.3.1, Theorem 3.3.3] between certain bialgebras and certain sequence
of perverse sheaves on $\operatorname{Sym}^{n}(\mathbb{A}^{1})$. According to
this correspondence, an object $V$ corresponds to a sequence of local systems
$\mathcal{L}_{n}(V[1])$ on $\operatorname{Conf}^{n}(\mathbb{A}^{1})$. If we
let
$j_{n}:\operatorname{Conf}^{n}(\mathbb{A}^{1})\xhookrightarrow{}\operatorname{Sym}^{n}(\mathbb{A}^{1})$
denote the usual open immersion, then $T_{!}(V)$ (resp. $T_{*}(V)$)
corresponds to the sequence of perverse sheaves
$j_{n!}\mathcal{L}_{n}(V[1])[n]$ (resp. $j_{n*}\mathcal{L}_{n}(V[1])[n]$), and
the Nichols algebra $T_{!*}(V)$ corresponds to the intermediate extension

$j_{!*}\mathcal{L}_{n}(V[1])[n]=\operatorname{im}(j_{n!}\mathcal{L}_{n}(V[1])[n]\to
j_{n*}\mathcal{L}_{n}(V[1])[n]).$

We will be interested in the cohomological consequences of this correspondence
(Theorem 4.6).

###  3.3. Quantum shuffle algebras and homology of braid groups

The Nichols algebras and quantum shuffle algebras that are relevant to Hurwitz
spaces and arithmetic statistics are those arising from the Yetter-Drinfeld
modules $V(c,-1)\coloneqq kc\in\mathcal{YD}_{G,\epsilon}^{G}$ (note the sign
$\epsilon$). The reason is the following key result, proven independently by
Ellenberg-Tran-Westerland [ETW17, Corollary 3.7] and Kapranov-Schechtman
[KS20, Corollary 3.3.4]:

######  Theorem 3.5 ([ETW17, KS20]).

Consider a finite-dimensional Yetter-Drinfeld module
$V_{\epsilon}\in\mathcal{YD}_{G,\epsilon}^{G}$. Then

$H_{j}(\mathbf{B}_{n},(V_{\epsilon}^{\vee})^{\otimes
n}\otimes\operatorname{sgn})\cong\operatorname{Ext}_{T_{*}(V_{\epsilon})}^{n-j,n}(k,k),$

where $\operatorname{sgn}:\mathbf{B}_{n}\to\\{\pm 1\\}$ is the sign
representation.

When $V_{\epsilon}=V(c,-1)$ for some $c\subset G$, then the left-hand side
$H_{j}(\mathbf{B}_{n},(V_{\epsilon}^{\vee})^{\otimes
n}\otimes\operatorname{sgn})\cong H_{j}(\mathbf{B}_{n},kc^{\times n})$ is the
homology of the Hurwitz space $\operatorname{Hur}_{G,n}^{c}$, which we will
discuss in Section 4. The quantum shuffle algebras $T_{*}(V(c,-1))$ are very
complicated in general but their cohomology groups have a straightforward
arithmetic interpretation in terms of point counts of Hurwitz spaces. On the
other hand, the Nichols algebras $T_{!*}(V(c,-1))$ are simpler, but we do not
yet know when their cohomology groups admit point counting interpretations.

###  3.4. The Nichols algebras $\mathfrak{B}_{d}$

The main objects from quantum algebra featured in this paper are the Nichols
algebras $\mathfrak{B}_{d}\coloneqq T_{!*}(V(\tau_{d},-1))$ for $3\leq d\leq
5$, which are rather special.

First off, for arbitrary $d\geq 3$, we can consider $\mathfrak{B}_{d}$ as the
underlying $k$-algebra of the Nichols algebra
$T_{!*}(k\tau_{d}\otimes\operatorname{sgn})\in\mathcal{YD}_{\mathbf{S}_{d}}^{\mathbf{S}_{d}}$,
where $\operatorname{sgn}:\mathbf{S}_{d}\to\\{\pm 1\\}$ is the sign
representation, since
$V(\tau_{d},-1)=k\tau_{d}\in\mathcal{YD}_{\mathbf{S}_{d},\epsilon}^{\mathbf{S}_{d}}$
and
$k\tau_{d}\otimes\operatorname{sgn}\in\mathcal{YD}_{\mathbf{S}_{d}}^{\mathbf{S}_{d}}$
have the same underlying braided vector space. In particular,
$\mathfrak{B}_{d}$ is an _elementary_ Nichols algebra: the Yetter-Drinfeld
module $k\tau_{d}\otimes\operatorname{sgn}$ is finite-dimensional and
absolutely irreducible, and its support (the subset of $\mathbf{S}_{d}$ with
nonzero graded pieces) generates $\mathbf{S}_{d}$. For $3\leq d\leq 5$,
$\mathfrak{B}_{d}$ is finite-dimensional, and for $d\geq 6$, it is unknown but
expected that $\mathfrak{B}_{d}$ is infinite-dimensional.

The fact that $\mathfrak{B}_{d}$ is finite-dimensional for $d\in\\{3,4,5\\}$
is significant, as the list of known finite-dimensional elementary Nichols
algebras [HLV15, Table 9.1] is extremely short. Letting
$(a)_{t}=1+t+\cdots+t^{a-1}$, we present some of the numerical data associated
to $\mathfrak{B}_{d}$ for $3\leq d\leq 5$:

  * •

$\mathfrak{B}_{3}$ has dimension 12, Hilbert series $(2)_{t}^{2}(3)_{t}$, and
top degree 4.

  * •

$\mathfrak{B}_{4}$ has dimension 576, Hilbert series
$(2)_{t}^{2}(3)_{t}^{2}(4)_{t}^{2}$, and top degree 12.

  * •

$\mathfrak{B}_{5}$ has dimension 8294400, Hilbert series
$(4)_{t}^{4}(5)_{t}^{2}(6)_{t}^{4}$, and top degree 40.

We will see in Section 5 that the numbers 4, 12, and 40 appear as the
dimensions of certain prehomogeneous vector spaces. We do not know why these
numbers match or what the other numerical data of $\mathfrak{B}_{d}$
correspond to on the prehomogeneous side. These numbers appear as the number
of indecomposable representations of the preprojective algebra of type
$A_{d-1}$, as noted by Majid [Maj05, §3], and equivalently as the number of
cluster variables in the coordinate ring of the group of upper unitriangular
matrices $N_{d}\subset\operatorname{SL}_{d}$ by work of Geiß-Leclerc-Schröer
[GLS06].

Another distinguishing feature of $\mathfrak{B}_{d}$ for $d\in\\{3,4,5\\}$ is
that the algebras are known to be defined by quadratic relations in this range
and not known to be quadratic outside it, though they are expected to be. For
concreteness, we give the defining relations in the generators
$(ij)\in\mathfrak{B}_{d}^{1}=k\tau_{d}$:

$\displaystyle(ij)^{2}$ $\displaystyle=0,\quad\forall i,j$
$\displaystyle(ij)(kl)+(kl)(ij)$
$\displaystyle=0,\quad\forall\\{i,j\\}\cap\\{k,l\\}=\emptyset$
$\displaystyle(ij)(jk)+(jk)(ik)+(ik)(ij)$ $\displaystyle=0,\quad\forall i<j<k$
$\displaystyle(jk)(ij)+(ik)(jk)+(ij)(ik)$ $\displaystyle=0,\quad\forall
i<j<k.$

We end by noting that there are two different $\mathbf{S}_{d}$-actions on
$\mathfrak{B}_{d}$, one coming from
$k\tau_{d}\otimes\operatorname{sgn}\in\mathcal{YD}_{\mathbf{S}_{d}}^{\mathbf{S}_{d}}$
and one coming from
$k\tau_{d}\in\mathcal{YD}_{\mathbf{S}_{d},\epsilon}^{\mathbf{S}_{d}}$. We call
the former the _standard $\mathbf{S}_{d}$-action_ and the latter the
_geometric $\mathbf{S}_{d}$-action_.

##  4\. Topology of Hurwitz spaces

In this section, we will review the topological models for Hurwitz spaces
studied in [ETW17, EVW16] and state the Kapranov-Schechtman theorem comparing
the cohomology of certain Nichols algebra and the cohomology of certain
intersection cohomology complexes on $\operatorname{Sym}^{n}(\mathbb{A}^{1})$
(Theorem 4.6). For this section, $G$ will denote a finite group, and $c$ will
denote a union of conjugacy classes of $G$. Again, $k$ will be a field of
characteristic 0 for this section. Just for this section, all schemes will be
over $\mathbb{C}$ and will be identified with their spaces of
$\mathbb{C}$-points.

###  4.1. Hurwitz spaces of the disc

Following [EVW16, §2], we will define Hurwitz spaces
$\operatorname{Hur}_{G,n}^{c}$ parametrizing marked $G$-covers of the disc
with monodromy in $c$ around $n$ branch points.

Let $\overline{D}\subset\mathbb{C}$ be the closed unit disc, and let
$D\subset\overline{D}$ be the open unit disc. The configuration space
$\operatorname{Conf}^{n}(D)$ is the space of $n$ unordered points in $D$. Let
$c_{n}$ be the set of $n$ points $\\{p_{1},\ldots,p_{n}\\}$ where
$p_{j}=\frac{j-1}{n}i\in D$. There is a well-known description of the braid
group $\mathbf{B}_{n}=\pi_{1}(\operatorname{Conf}^{n}(D),c_{n})$, where
$\sigma_{j}$ corresponds to the loop in $\operatorname{Conf}^{n}(D)$ where all
the points remain the same except $p_{j}$ and $p_{j+1}$, which move according
to

$\displaystyle\widetilde{p}_{j}(t)$
$\displaystyle=\frac{2j-1}{n}i-\frac{1}{2n}ie^{\pi it}$
$\displaystyle\widetilde{p}_{j+1}(t)$
$\displaystyle=\frac{2j-1}{n}i+\frac{1}{2n}ie^{\pi it}$

for $0\leq t\leq 1$. Note that

$\displaystyle\widetilde{p}_{j}(0)$ $\displaystyle=p_{j}$
$\displaystyle\widetilde{p}_{j}(1)$ $\displaystyle=p_{j+1}$
$\displaystyle\widetilde{p}_{j+1}(0)$ $\displaystyle=p_{j+1}$
$\displaystyle\widetilde{p}_{j+1}(1)$ $\displaystyle=p_{j},$

so $\sigma_{j}$ corresponds to the loop moving $p_{j}$ and $p_{j+1}$
counterclockwise around each other until they swap positions.

Because $\pi_{1}(\operatorname{Conf}^{n}(D),c_{n})=\mathbf{B}_{n}$, any
$\mathbf{B}_{n}$-set $T$ gives rise to a covering space of
$\operatorname{Conf}^{n}(D)$ by taking
$\widetilde{\operatorname{Conf}}^{n}(D)\times_{\mathbf{B}_{n}}T\to\operatorname{Conf}^{n}(D)$,
where $\widetilde{\operatorname{Conf}}^{n}(D)\to\operatorname{Conf}^{n}(D)$ is
the universal cover. In particular, we will consider $S=c^{\times n}$, where
$\mathbf{B}_{n}$ acts on $\underline{g}=(g_{1},\ldots,g_{n})\in c^{\times n}$
by

$\sigma_{i}(\underline{g})_{j}=\begin{cases}g_{j+1}&\text{if }j=i\\\
g_{j+1}^{-1}g_{j}g_{j+1}&\text{if }j=i+1\\\
g_{j}&\text{otherwise}.\end{cases}$

######  Definition 4.1.

The _$n$ th Hurwitz space of the disc associated to $(G,c)$_ is

$\operatorname{Hur}_{G,n}^{c}\coloneqq\widetilde{\operatorname{Conf}}^{n}(D)\times_{\mathbf{B}_{n}}c^{\times
n}.$

If $c=G$, we will simply write $\operatorname{Hur}_{G,n}$.

The Hurwitz space $\operatorname{Hur}_{G,n}^{c}$ has a natural moduli
description in terms of marked covers of the disc. Note that in the following
definition, we do not restrict to connected covers.

######  Definition 4.2.

A _marked $n$-branched $G$-cover of the disc_ is a quintuple
$(Y,p,\bullet,S,\alpha)$, where

  1. (1)

$S\subset D$ is a set of $n$ distinct points.

  2. (2)

$p:Y\to\overline{D}-S$ is a covering map.

  3. (3)

$\alpha:G\to\operatorname{Aut}(p)$ is a homomorphism inducing a simply
transitive action of $G$ on each fiber of $p$.

  4. (4)

$\bullet$ is a point in the fiber $p^{-1}(1)$.

######  Proposition 4.3 ([EVW16, §2.3]).

There are bijections between the following sets:

  1. (a)

Points of $\operatorname{Hur}_{G,n}$.

  2. (b)

Pairs $(S,f)$, where $S\in\operatorname{Conf}^{n}(D)$ and
$f:\pi_{1}(\overline{D}-S,1)\to G$ is a homomorphism.

  3. (c)

Isomorphism classes of marked $n$-branched $G$-covers of the disc.

Given a pair $(S,f)$ as in (b) and a point $s\in S$, we can get a well-defined
conjugacy class in $G$, which we call the _monodromy around $s$_. This is
defined by taking a small counterclockwise loop around $s$, identifying it
with a conjugacy class in $\pi_{1}(\overline{D}-S,1)$ by picking a path from 1
to a point on the loop, and taking the image of this conjugacy class in $G$
under $f$. Under the bijections above,
$\operatorname{Hur}_{G,n}^{c}\subset\operatorname{Hur}_{G,n}$ corresponds to
pairs $(S,f)$ with monodromy in $c$ around each $s\in S$.

Note that there is a $G$-action on $\operatorname{Hur}_{G,n}^{c}$ because the
$G$-action and $\mathbf{B}_{n}$-action on $c^{\times n}$ commute. This
$G$-action corresponds to the action of $G$ on the marking $\bullet\in
p^{-1}(1)$. Thus, the points of the quotient
$\operatorname{Hur}_{G,n}^{c}/G=\widetilde{\operatorname{Conf}}^{n}(D)\times_{\mathbf{B}_{n}}(c^{\times
n}/G)$ correspond to unmarked $n$-branched $G$-covers of the disc.

####  4.1.1. The case $G=\mathbf{S}_{d}$

Recall the equivalence between $\mathbf{S}_{d}$-covering spaces and degree $d$
covering spaces. Starting with an $\mathbf{S}_{d}$-covering space $Y\to S$, we
get a degree $d$ covering space $Y/\mathbf{S}_{d-1}\to S$. Starting with a
degree $d$ covering space $X\to S$, we get a $\mathbf{S}_{d}$-covering space
by taking the $d$th fiber product $X^{d}\to S$ and deleting the big diagonal.

With this equivalence in mind, we make the following definition:

######  Definition 4.4.

A _marked $n$-branched degree $d$ cover of the disc_ is a quintuple
$(X,p,\beta,S)$, where

  1. (1)

$S\subset D$ is a set of $n$ distinct points.

  2. (2)

$p:X\to\overline{D}-S$ is a degree $d$ covering map.

  3. (3)

$\beta:p^{-1}(1)\xrightarrow[]{\sim}\\{1,\ldots,d\\}$ is an isomorphism of
sets.

Note that $\beta$ plays the role of the marking $\bullet$. From the
equivalence of $\mathbf{S}_{d}$-covering spaces and degree $d$ covering
spaces, we get the following corollary of Proposition 4.3:

######  Corollary 4.5.

There is a bijection between points of $\operatorname{Hur}_{G,n}$ and
isomorphism classes of marked $n$-branched degree $d$ covers of the disc.

For a marked $n$-branched degree $d$-cover of the disc $(X,p,\beta,S)$, each
point $s\in S$ gives us a well-defined conjugacy class of $\mathbf{S}_{d}$:
take a small counterclockwise loop around $s$, and see how it permutes a
nearby fiber. Hence, the points $\operatorname{Hur}_{\mathbf{S}_{d},n}^{c}$
parametrize marked $n$-branched degree $d$ covers of the disc with monodromy
in $c$ around each point in $S$. We will be particularly interested in the
case $c=\tau_{d}$, the conjugacy class of transpositions, as
$\operatorname{Hur}_{\mathbf{S}_{d},n}^{\tau_{d}}$ parametrizes simply
branched covers.

####  4.1.2. Hurwitz spaces of the affine line

For the rest of this paper, fix a radial homeomorphism $D\cong\mathbb{A}^{1}$
induced by a homeomorphism $[0,1)\xrightarrow[]{\sim}[0,\infty)$. This induces
homeomorphisms
$\operatorname{Conf}^{n}(D)\cong\operatorname{Conf}^{n}(\mathbb{A}^{1})$, so
that $\operatorname{Hur}_{G,n}^{c}\to\operatorname{Conf}^{n}(\mathbb{A}^{1})$
and $\operatorname{Hur}_{G,n}^{c}/G\to\operatorname{Conf}^{n}(\mathbb{A}^{1})$
are covering spaces. In fact, by Riemann’s existence theorem,
$\operatorname{Hur}_{G,n}^{c}$ and $\operatorname{Hur}_{G,n}^{c}/G$ are the
spaces of $\mathbb{C}$-points of finite étale covers of
$\operatorname{Conf}^{n}(\mathbb{A}^{1})$. For the rest of Section 4, we will
not distinguish between the spaces $\operatorname{Hur}_{G,n}^{c}$ and the
corresponding varieties over $\mathbb{C}$.

###  4.2. Nichols algebra cohomology and intersection cohomology

Because

$\pi:\operatorname{Hur}_{G,n}^{c}\to\operatorname{Conf}^{n}(\mathbb{A}^{1})$

is the finite étale cover (or equivalently, finite covering space)
corresponding to the $\mathbf{B}_{n}$-set $c^{\times n}$, the pushforward
$\pi_{*}\underline{k}$ is the local system on
$\operatorname{Conf}^{n}(\mathbb{A}^{1})$ corresponding to the
$\mathbf{B}_{n}$-representation $kc^{\times n}$.

Note that there in an open immersion
$j:\operatorname{Conf}^{n}(\mathbb{A}^{1})\xhookrightarrow{}\operatorname{Sym}^{n}(\mathbb{A}^{1})$.
The following theorem of Kapranov-Schechtman is a major motivation for this
paper. We present it in its dual formulation, as the $\operatorname{Ext}$
algebra seems more natural to us than the $\operatorname{Tor}$ coalgebra
[KS20, Remark 3.3.6].

######  Theorem 4.6 ([KS20, Corollary 3.3.5]).

There is an isomorphism

$\operatorname{Ext}_{T_{!*}(V(c,-1))}^{n-j,n}(k,k)\cong
H^{j}(\operatorname{Sym}^{n}(\mathbb{A}^{1}),j_{!*}\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}})^{\vee}.$

Note that $\pi$ factors as
$\operatorname{Hur}_{G,n}^{c}\to\operatorname{Hur}_{G,n}^{c}/G\xrightarrow[]{\pi^{\prime}}\operatorname{Conf}^{n}(\mathbb{A}^{1})$.
Then

$\pi_{*}^{\prime}\underline{k}_{\operatorname{Hur}_{G,n}^{c}/G}\cong(\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}})^{G}.$

Because $k$ has characteristic 0,
$(\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}})^{G}$ is naturally a
direct summand of $\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}}$, so

$j_{!*}\pi_{*}^{\prime}\underline{k}_{\operatorname{Hur}_{G,n}^{c}/G}\cong(j_{!*}\pi_{*}\underline{k}_{\operatorname{Hur}_{G,n}^{c}})^{G}.$

From the proof of [KS20, Theorem 3.3.1], we see that this action corresponds
to the $G$-action on $T_{!*}(V(c,-1))$, so we get the following corollary.

######  Corollary 4.7.

There is an isomorphism

$\operatorname{Ext}_{T_{!*}(V(c,-1))}^{n-j,n}(k,k)^{G}\cong
H^{j}(\operatorname{Sym}^{n}(\mathbb{A}^{1}),j_{!*}\pi_{*}^{\prime}\underline{k}_{\underline{k}_{\operatorname{Hur}_{G,n}^{c}/G}})^{\vee}.$

######  Remark 4.8.

Note that when $G=\mathbf{S}_{d}$ and $c=\tau_{d}$ for $d\in\\{3,4,5\\}$, the
$\mathbf{S}_{d}$-action in Corollary 4.7 comes from the geometric
$\mathbf{S}_{d}$-action on $\mathfrak{B}_{d}$, as the name suggests. On the
other hand, by [ŞV16], the invariants of the cohomology of $\mathfrak{B}_{d}$
under the standard $\mathbf{S}_{d}$-action correspond to the cohomology of a
Hopf algebra, the bosonization $\mathfrak{B}_{d}\\#k[\mathbf{S}_{d}]$.

We find it a bit unsatisfying that these two actions for which we take
invariants are not the same, and we wonder if there is any quantum algebraic
interpretation of the geometric $\mathbf{S}_{d}$-invariants or geometric
interpretation of the standard $\mathbf{S}_{d}$-invariants. On the bright
side, the actions agree on
$\operatorname{Ext}_{\mathfrak{B}_{d}}^{n-j,n}(k,k)$ with $n$ even. In Theorem
11.14, we will compute the geometric $\mathbf{S}_{4}$-invariant cohomology of
$\mathfrak{B}_{4}$, which agrees with the cohomology of
$\mathfrak{B}_{4}\\#k[\mathbf{S}_{4}]$ in even internal degrees by Corollary
11.15.

##  5\. Prehomogeneous vector spaces and low degree covers

In this section, we will review some of the connections between prehomogeneous
vector spaces, parametrizations of low degree covers, and arithmetic
statistics that we will need to construct our stacks of covers with
resolvents.

Recall that a _prehomogeneous vector space_ (over some base field $k$) is a
triple $(G,\rho,V)$ such that $G$ is a connected linear algebraic group, $V$
is a vector space, and $\rho:G\to\operatorname{GL}(V)$ is a representation
such that there is a Zariski dense orbit $V^{ss}\subset V$.

Throughout this paper, we will only ever consider three prehomogeneous vector
spaces, which we will denote $(G_{d},V_{d})$ for $d\in\\{3,4,5\\}$. We use
$\operatorname{std}_{n}$ to denote the standard $n$-dimensional representation
of $\operatorname{GL}_{n}$:

  * •

$d=3$: $G_{3}=\operatorname{GL}_{2}$,
$V_{3}=\operatorname{Sym}^{3}(\operatorname{std}_{2})\otimes\det\operatorname{std}_{2}^{\vee}$.

  * •

$d=4$:
$G_{4}=\\{(g_{3},g_{2})\in\operatorname{GL}_{3}\times\operatorname{GL}_{2}|\det
g_{3}=\det g_{2}\\}$,
$V_{4}=\operatorname{Sym}^{2}(\operatorname{std}_{3})\otimes\operatorname{std}_{2}^{\vee}$.

  * •

$d=5$:
$G_{5}=\\{(g_{4},g_{5})\in\operatorname{GL}_{4}\times\operatorname{GL}_{5}|(\det
g_{4})^{2}=\det g_{5}\\}$,
$V_{5}=\operatorname{std}_{4}\otimes\det\operatorname{std}_{4}^{\vee}\otimes\Lambda^{2}\operatorname{std}_{5}$.

If $d\in\\{4,5\\}$, we will write a $G_{d}$-bundle as a tuple
$(\mathcal{F},\mathcal{F}^{\prime},\alpha)$, where

  * •

$d=4$: $\mathcal{F}$ is a rank 3 vector bundle, $\mathcal{F}^{\prime}$ is a
rank 2 vector bundle, and
$\alpha:\det(\mathcal{F})\xrightarrow[]{\sim}\det(\mathcal{F}^{\prime})$ is an
isomorphism.

  * •

$d=5$: $\mathcal{F}$ is a rank 4 vector bundle, $\mathcal{F}^{\prime}$ is a
rank 5 vector bundle, and
$\alpha:\det(\mathcal{F})^{2}\xrightarrow[]{\sim}\det(\mathcal{F}^{\prime})$
is an isomorphism.

For the rest of this paper, $d$ will denote either 3, 4, or 5. We will use
$G_{d}$ and $V_{d}$ to denote the corresponding group schemes and affine
spaces over $\mathbb{Z}$. We will use $\mathfrak{g}_{d}$ to denote the Lie
algebra of $G_{d}$ as an affine space over $\mathbb{Z}$. Define the quotient
stack $\mathcal{X}_{d}=[V_{d}/G_{d}]$.

Each $V_{d}$ comes with a (unique up to a scalar) homogeneous polynomial
$\Delta_{d}:V_{d}\to S$ called the _discriminant_ that scales with respect to
the $G_{d}$-action according to the square of a generator
$\chi_{d}:G_{d}\to\mathbb{G}_{m}$ of the character group of $G_{d}$:

  * •

$d=3$: For a binary cubic form
$f(x,y)=ax^{3}+bx^{2}y+cxy^{2}+dy^{3}\otimes(x\wedge y)^{\vee}\in V_{3}$, we
have

$\Delta_{3}(f)=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd.$

For $g\in G_{3}$, we have $\chi_{3}(g)=\det(g)$.

  * •

$d=4$: For a pair of ternary quadratic forms $(\sum_{0\leq i\leq j\leq
2}a_{ij}x_{i}x_{j},\sum_{0\leq i\leq j\leq 2}b_{ij}x_{i}x_{j})$ represented as
a pair of $3\times 3$ symmetric matrices

$(A,B)=\left(\begin{pmatrix}a_{00}&\frac{1}{2}a_{01}&\frac{1}{2}a_{02}\\\
\frac{1}{2}a_{01}&a_{11}&\frac{1}{2}a_{12}\\\
\frac{1}{2}a_{02}&\frac{1}{2}a_{12}&a_{22}\end{pmatrix},\begin{pmatrix}b_{00}&\frac{1}{2}b_{01}&\frac{1}{2}b_{02}\\\
\frac{1}{2}b_{01}&b_{11}&\frac{1}{2}b_{12}\\\
\frac{1}{2}b_{02}&\frac{1}{2}b_{12}&b_{22}\end{pmatrix}\right),$

we have222Even though there are $\frac{1}{2}$’s in the expressions for $A,B$,
the expression $4\det(Ax-By)$ has only integer coefficients. Thus, it makes
sense even when 2 is not invertible.

$\Delta_{4}(A,B)=\Delta_{3}(4\det(Ax-By)\otimes(x\wedge y)^{\vee}).$

For $(g_{3},g_{2})\in G_{4}$, we have $\chi_{4}(g_{3},g_{2})=\det(g_{3})$.

  * •

$d=5$: The function $\Delta_{5}$ is a complicated degree 40 polynomial, so we
refer the reader to [Kim03, Example 2.11] and [Bha08, §4]. For
$(g_{4},g_{5})\in G_{5}$, we have $\chi_{5}(g_{4},g_{5})=\det(g_{4})$.

We will call the open subset of $V_{d}$ where $\Delta_{d}$ does not vanish the
_semistable locus_ $V_{d}^{ss}$. It is well-known (e.g. [Kim03, Example 2.4,
Example 2.8, Example 2.11]) that in each case,
$\Delta_{d}(gv)=\chi_{d}(g)^{2}v$. Thus, $\Delta_{d}$ descends to a section of
a line bundle $\mathcal{L}_{d}$ on $\mathcal{X}_{d}$, which we call the
_discriminant line bundle_. Imitating [Deo14, §2.1], we will call the zero
locus in $\mathcal{X}_{d}$ of $\Delta_{d}$ the _branch locus_ $\Sigma_{d}$ and
its complement the _étale locus_ $\mathcal{E}_{d}$. We will justify this
terminology later.

For a field $k$ of characteristic not dividing $d!$, the ring of relative
invariant functions on $(V_{d})_{k}$ is a polynomial ring generated by
$(\Delta_{d})_{k}$. This means that $(V_{d}^{ss})_{k}$ really is the
$\chi_{d}$-semistable locus in the sense of GIT (see [Kin94, §2] for
definitions). In fact, $(V_{d}^{ss})_{k}$ is also the $\chi_{d}$-stable locus
because $(G_{d})_{k}$ acts transitively on $(V_{d}^{ss})_{k}$ with finite
stabilizers. Thus, we will sometimes write $V_{d}^{s}$ for $V_{d}^{ss}$.

###  5.1. Covers and resolvents

Our motivation for studying prehomogeneous vector spaces comes from their
relationship with parametrizations of low degree covers and arithmetic
statistics. In the work of Davenport-Heilbronn [DH71] and Bhargava [Bha04,
Bha08], the authors count degree $d$ number fields (for $3\leq d\leq 5$) by
counting certain unrefined objects and sieving out the ones that do not
contribute to the number field count. These unrefined objects are parametrized
by orbits in prehomogeneous vector spaces:

  * •

$V_{3}(\mathbb{Z})/G_{3}(\mathbb{Z})$ parametrizes cubic
$\mathbb{Z}$-algebras.

  * •

$V_{4}(\mathbb{Z})/G_{4}(\mathbb{Z})$ parametrizes quartic
$\mathbb{Z}$-algebras with cubic resolvents.

  * •

$V_{5}(\mathbb{Z})/G_{5}(\mathbb{Z})$ parametrizes quintic
$\mathbb{Z}$-algebras with sextic resolvents.

The idea is that the quotient stacks $\mathcal{X}_{d}$ should parametrize
versions of these objects for general base. For a fixed base scheme $S$, we
make the following definitions:

  * •

A _cubic cover of $S$_ is a degree 3 finite locally free morphism $C\to S$ (to
define an _$n$ -ic cover_ or _degree $n$ cover_, replace 3 with $n$). It is
shown in [Woo11, §2] that the stack of cubic covers is isomorphic to the stack
$\mathcal{X}_{3}$.

  * •

A _quartic cover of $S$ with cubic resolvent_, as defined by Wood in [Woo11,
§3], is a tuple $(Q,C,\phi,\delta)$, where

    * –

$Q$ is a quartic cover of $S$.

    * –

$C$ is a cubic cover of $S$.

    * –

The _resolvent map_ $\phi$ is a map
$\operatorname{Sym}_{2}(\mathcal{O}_{Q}/\mathcal{O}_{S})\to\mathcal{O}_{C}/\mathcal{O}_{S}$
of vector bundles on $S$, which we can view as a quadratic map
$\mathcal{O}_{Q}/\mathcal{O}_{S}\to\mathcal{O}_{C}/\mathcal{O}_{S}$. Here, as
in [Woo11], for a locally free sheaf $\mathcal{F}$, we use
$\operatorname{Sym}_{n}\mathcal{F}$ to denote the subsheaf of symmetric
elements of $\mathcal{F}^{\otimes n}$, as opposed to the usual quotient
$\operatorname{Sym}^{n}\mathcal{F}$ of $\mathcal{F}^{\otimes n}$. By [Woo11,
Lemma A.4],
$(\operatorname{Sym}_{n}\mathcal{F})^{\vee}\cong\operatorname{Sym}^{n}\mathcal{F}^{\vee}$.
If 2 is invertible (as it will be for most of this paper), the map
$\operatorname{Sym}_{2}\mathcal{F}\to\operatorname{Sym}^{2}\mathcal{F}$ is an
isomorphism, so there is no real distinction.

    * –

The _orientation_ $\delta$ is an isomorphism
$\det(\mathcal{O}_{Q}/\mathcal{O}_{S})\xrightarrow[]{\sim}\det(\mathcal{O}_{C}/\mathcal{O}_{S})$.

such that

    1. (1)

For local sections $x,y$ of $\mathcal{O}_{Q}/\mathcal{O}_{S}$, we have333Here,
$xy$ does not make sense as a section of $\mathcal{O}_{Q}/\mathcal{O}_{S}$,
but it does make sense when wedged with $x\wedge y$. $\delta(x\wedge y\wedge
xy)=\phi(x)\wedge\phi(y)$.

    2. (2)

If we pick local frames for our vector bundles, we can write $\phi$ as a pair
of $3\times 3$ symmetric matrices $(A,B)$ and define the binary cubic form
$(x,y)\mapsto 4\det(Ax-By)\otimes(x\wedge y)^{\vee}$. These local expressions
glue to a global section $\det\phi\in
H^{0}(S,\operatorname{Sym}^{3}(\mathcal{O}_{C}/\mathcal{O}_{S})^{\vee}\otimes\det(\mathcal{O}_{C}/\mathcal{O}_{S}))$.
We require that $C$ is the cubic cover of $S$ corresponding to the binary
cubic form $\det\phi$ with underlying bundle
$(\mathcal{O}_{C}/\mathcal{O}_{S})^{\vee}$.

The main result of [Woo11] is that there is an isomorphism from the stack of
quartic covers with cubic resolvents to the stack $\mathcal{X}_{4}$ that sends
$(Q,C,\phi,\delta)$ to the section $\phi$ of the $V_{4}$-bundle associated to
the $G_{4}$-bundle
$((\mathcal{O}_{Q}/\mathcal{O}_{S})^{\vee},(\mathcal{O}_{C}/\mathcal{O}_{S})^{\vee},(\delta^{\vee})^{-1})$.

  * •

_Quintic covers with sextic resolvent_ have only been studied for Dedekind
domains. In [O’D22], it is shown that for a Dedekind domain $R$, there is a
bijection between $\mathcal{X}_{5}(R)/\sim$ and isomorphism classes of quintic
$R$-algebras with sextic resolvents, and it is also shown that each quintic
$R$-algebra admits a sextic resolvent. However, unlike the cases
$d\in\\{3,4\\}$, an isomorphism between $\mathcal{X}_{5}$ and a suitable stack
of quintic covers with sextic resolvents has not been found. The closest
result is the Casnati-Ekedahl/Landesman-Vakil-Wood parametrization of quintic
Gorenstein covers, which we will discuss in Section 5.5.

###  5.2. The stack of covers

######  Definition 5.1.

The _stack of degree $d$ covers_ $\operatorname{Covers}_{d}$ is the stack over
$\mathbb{Z}$ whose objects over $S$ are degree $d$ finite locally free maps
$C\to S$.

The following facts about $\operatorname{Covers}_{d}$ are well-known and can
be found in [Poo08]:

  * •

$\operatorname{Covers}_{d}$ has a presentation as a quotient stack
$[B_{d}/\operatorname{GL}_{d}]$, where $B_{d}$ is the finite type affine
scheme over $\mathbb{Z}$ parametrizing _based degree $d$ covers_: the points
$B_{d}(S)$ correspond to $\mathcal{O}_{S}$-algebra structures on
$\mathcal{O}_{S}^{\oplus d}$, and $\operatorname{GL}_{d}(S)$ acts on
$B_{d}(S)$ by acting on $\mathcal{O}_{S}^{\oplus d}$. The coordinates of
$B_{d}$ can be thought of as the coordinates of the identity as a linear
combination of the standard basis of $\mathcal{O}_{S}^{\oplus d}$ and the
elements in the multiplication table for the algebra structure in terms of the
standard basis. $B_{d}$ is cut out of the affine space with these coordinates
by equations corresponding to the identity, associativity, and commutativity
axioms.

  * •

There is a function $\Delta_{d}^{\operatorname{Cov}}$ on $B_{d}$ called the
_discriminant_ which computes the usual discriminant of a based degree $d$
algebra. This function has the property that for
$g\in\operatorname{GL}_{d}(S)$ and $b\in B_{d}(S)$,
$\Delta_{d}^{\operatorname{Cov}}(gb)=\det(g)^{-2}\Delta_{d}^{\operatorname{Cov}}(b)$.
Thus, $\Delta_{d}^{\operatorname{Cov}}$ descends to a section of a line bundle
$\mathcal{L}_{d}^{\operatorname{Cov}}$ on $\operatorname{Covers}_{d}$. We will
use $\Delta_{d}^{\operatorname{Cov}}$ to denote this section (in addition to
the function above). We call $\mathcal{L}_{d}^{\operatorname{Cov}}$ the
_discriminant line bundle_.

  * •

The complement of the zero locus of $\Delta_{d}^{\operatorname{Cov}}$ is
called the _étale locus_ , and it is naturally isomorphic to $B\mathbf{S}_{d}$
because it parametrizes degree $d$ finite étale covers.

Note that we have used $\Delta_{d}$ for the discriminant on both
$\mathcal{X}_{d}$ and $\operatorname{Covers}_{d}$. We will justify this by
exhibiting morphisms $\mathcal{X}_{d}\to\operatorname{Covers}_{d}$ and then
showing that the pullback of the discriminant on $\operatorname{Covers}_{d}$
agrees with the discriminant on $\mathcal{X}_{d}$.

###  5.3. The morphisms $\phi_{d}:\mathcal{X}_{d}\to\operatorname{Covers}_{d}$

We define natural morphisms
$\phi_{d}:\mathcal{X}_{d}\to\operatorname{Covers}_{d}$. Fix a base scheme $S$
throughout. Whenever we have a projective bundle $\mathbb{P}\mathcal{F}\to S$,
let $\pi$ denote the projection. The cases $d\in\\{3,4\\}$ are done in
[Woo11], so we will recall the construction in those cases without providing
too many details. This discussion is similar to [LVW22, §3], except that they
consider only the Gorenstein case:

  * •

$\phi_{3}:\mathcal{X}_{3}\xrightarrow[]{\sim}\operatorname{Covers}_{3}$: Let
$\mathcal{F}_{2}$ be a rank 2 bundle on $S$, along with a section $\eta\in
H^{0}(S,\operatorname{Sym}^{3}\mathcal{F}_{2}\otimes\det\mathcal{F}_{2}^{\vee})$.
There is a natural isomorphism
$\Phi_{3}:H^{0}(S,\operatorname{Sym}^{3}\mathcal{F}_{2}\otimes\det\mathcal{F}_{2}^{\vee})\xrightarrow[]{\sim}H^{0}(\mathbb{P}\mathcal{F}_{2},\pi^{*}\det\mathcal{F}_{2}^{\vee}(3))$.
Using $\Phi_{3}(\eta)$, we can form a cdga $K$ of bundles on
$\mathbb{P}\mathcal{F}_{2}$:

(1) $\displaystyle\pi^{*}\det\mathcal{F}_{2}(-3)$
$\displaystyle\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{2}}$

Here $\mathcal{O}_{\mathbb{P}\mathcal{F}_{2}}$ is in degree 0. Then
$\mathcal{O}_{C}\coloneqq R^{0}\pi_{*}K$ is the structure sheaf of a cubic
cover $C\to S$. This construction commutes with base change on $S$, and we can
recover $\mathcal{F}_{2}$ from $C$ as
$\mathcal{F}_{2}=(\mathcal{O}_{C}/\mathcal{O}_{S})^{\vee}$. In this case (and
only this case), $\phi_{3}$ is an isomorphism.

  * •

$\phi_{4}:\mathcal{X}_{4}\to\operatorname{Covers}_{4}$: Let $\mathcal{F}_{3}$
and $\mathcal{F}_{2}$ be bundles on $S$ of respective ranks 3 and 2, along
with an isomorphism
$\delta:\det\mathcal{F}_{3}\xrightarrow[]{\sim}\det\mathcal{F}_{2}$ and a
section $\eta\in
H^{0}(S,\operatorname{Sym}^{2}\mathcal{F}_{3}\otimes\mathcal{F}_{2}^{\vee})$.
There is a natural isomorphism
$\Phi_{4}:H^{0}(S,\operatorname{Sym}^{2}\mathcal{F}_{3}\otimes\mathcal{F}_{2}^{\vee})\xrightarrow[]{\sim}H^{0}(\mathbb{P}\mathcal{F}_{3},\pi^{*}\mathcal{F}_{2}^{\vee}(2))$.
Taking the Koszul complex of $\Phi_{4}(\eta)$, we can form a cdga $K$ of
bundles on $\mathbb{P}\mathcal{F}_{3}$:

(2) $\displaystyle\pi^{*}\det\mathcal{F}_{2}(-4)\to\pi^{*}\mathcal{F}_{2}(-2)$
$\displaystyle\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{3}}$

Here $\mathcal{O}_{\mathbb{P}\mathcal{F}_{3}}$ is in degree 0. Then
$\mathcal{O}_{Q}\coloneqq R^{0}\pi_{*}K$ is the structure sheaf of a quartic
cover $Q\to S$. This construction commutes with base change on $S$, and we can
recover $\mathcal{F}_{3}$ from $Q$ as
$\mathcal{F}_{3}=(\mathcal{O}_{Q}/\mathcal{O}_{S})^{\vee}$.

  * •

$\phi_{5}:\mathcal{X}_{5}\to\operatorname{Covers}_{5}$: Let $\mathcal{F}_{4}$
and $\mathcal{F}_{5}$ be bundles on $S$ of respective ranks 4 and 5, along
with an isomorphism
$\delta:(\det\mathcal{F}_{4})^{2}\xrightarrow[]{\sim}\det\mathcal{F}_{5}$ and
a section $\eta\in
H^{0}(S,\mathcal{F}_{4}\otimes\det\mathcal{F}_{4}^{\vee}\otimes\Lambda^{2}\mathcal{F}_{5})$.
There is a natural isomorphism
$\Phi_{5}:H^{0}(S,\mathcal{F}_{4}\otimes\det\mathcal{F}_{4}^{\vee}\otimes\Lambda^{2}\mathcal{F}_{5})\xrightarrow[]{\sim}H^{0}(\mathbb{P}\mathcal{F}_{3},\pi^{*}\det\mathcal{F}_{4}^{\vee}\otimes\Lambda^{2}\pi^{*}\mathcal{F}_{5}(1))$.
We can form a complex of bundles $K$ on $\mathbb{P}\mathcal{F}_{4}$ as in
[LVW22, Lemma 3.15]:

(3)
$\displaystyle\pi^{*}\det\mathcal{F}_{4}(-5)\xrightarrow[]{\beta_{3}}\pi^{*}\det\mathcal{F}_{4}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}(-3)\xrightarrow[]{\beta_{2}}\pi^{*}\mathcal{F}_{5}(-2)$
$\displaystyle\xrightarrow[]{\beta_{1}}\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$

Here, $\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$ is in degree 0. The map
$\beta_{2}$ comes from the section $\Phi_{5}(\eta)$. If we use $\delta$ to
write
$\pi^{*}\det\mathcal{F}_{4}(-5)\cong\pi^{*}\det\mathcal{F}_{4}^{\vee}\otimes\pi^{*}\det\mathcal{F}_{5}(-5)$,
then $\beta_{3}$ is obtained by taking the five $4\times 4$ Pfaffians of
$\beta_{2}$. We obtain $\beta_{1}$ by taking the dual of $\beta_{3}$ and
tensoring with $\pi^{*}\det\mathcal{F}_{4}(-5)$. We now describe the
$\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$-algebra structure on $K$ term by
term:

    * –

$\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}\otimes K\to K$: This is just the
$\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$-module structure on $K$.

    * –

$\pi^{*}\mathcal{F}_{5}(-2)\otimes\pi^{*}\mathcal{F}_{5}(-2)\to\pi^{*}\det\mathcal{F}_{4}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}(-3)$:
Because of our orientation $\delta$, there is a natural isomorphism

$\pi^{*}\det\mathcal{F}_{4}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}\cong\pi^{*}\det\mathcal{F}_{4}^{\vee}\otimes\pi^{*}\det\mathcal{F}_{5}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}$

Moving the $\pi^{*}\mathcal{F}_{5}^{\vee}$ to the other side, we see that we
want to provide a map

$\pi^{*}\mathcal{F}_{5}^{\otimes
3}(-4)\to\pi^{*}\det\mathcal{F}_{4}^{\vee}\otimes\pi^{*}\det\mathcal{F}_{5}$

This map is given by $(x,y,z)\mapsto\Phi_{5}(\eta)\wedge(x\wedge y\wedge z)$.

    * –

$\pi^{*}\mathcal{F}_{5}(-2)\otimes(\pi^{*}\det\mathcal{F}_{4}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}(-3))\to\pi^{*}\det\mathcal{F}_{4}(-5)$:
This comes from the pairing
$\pi^{*}\mathcal{F}_{5}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}(-3)\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$.

    * –

All the other pairings are 0 for degree reasons.

By construction, our multiplication on $K$ is graded commutative. The proof of
[BE77, Theorem 4.1] shows that the multiplication we have defined is
associative and compatible with the differential. Thus, we have given $K$ a
cdga structure, and its derived pushforward $\mathcal{O}_{Q}\coloneqq
R^{0}\pi_{*}K$ is an $\mathcal{O}_{S}$-algebra. By cohomology and base change,
we have natural quasi-isomorphisms

$\displaystyle R\pi_{*}\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}}$
$\displaystyle\simeq\mathcal{O}_{S}[0]$ $\displaystyle
R\pi_{*}\left(\pi^{*}\mathcal{F}_{5}(-2)\right)$ $\displaystyle\simeq 0$
$\displaystyle
R\pi_{*}\left(\pi^{*}\det\mathcal{F}_{4}\otimes\pi^{*}\mathcal{F}_{5}^{\vee}(-3)\right)$
$\displaystyle\simeq 0$ $\displaystyle
R\pi_{*}\left(\pi^{*}\det\mathcal{F}_{4}(-5)\right)$
$\displaystyle\simeq\mathcal{F}_{4}^{\vee}[-3]$

The hypercohomology spectral sequence then shows that
$R\pi_{*}K\simeq\mathcal{O}_{Q}[0]$ and that there is a natural short exact
sequence

$0\to\mathcal{O}_{S}\to\mathcal{O}_{Q}\to\mathcal{F}_{4}^{\vee}\to 0$

Thus, $\mathcal{O}_{Q}$ is the structure sheaf of a quintic cover $Q\to S$.
Because all the $R^{i}\pi_{*}K$ are flat, the cohomology and base change
theorem implies that the formation of $Q$ is compatible with base change. We
also have a natural isomorphism
$\mathcal{F}_{4}\cong(\mathcal{O}_{Q}/\mathcal{O}_{S})^{\vee}$.

We will need the following definition when we discuss Gorenstein covers in
Section 5.5.

######  Definition 5.2 ([LVW22, Definition 3.12]).

Fix a base scheme $S$ and a map $S\to\mathcal{X}_{d}$. Let $\eta$ be the
corresponding section of

  * •

$H^{0}(S,\operatorname{Sym}^{3}\mathcal{F}_{2}\otimes\det\mathcal{F}_{2}^{\vee})$
if $d=3$.

  * •

$H^{0}(S,\operatorname{Sym}^{2}\mathcal{F}_{3}\otimes\mathcal{F}_{2}^{\vee})$
if $d=4$.

  * •

$H^{0}(S,\mathcal{F}_{4}\otimes\det\mathcal{F}_{4}^{\vee}\otimes\Lambda^{2}\mathcal{F}_{5})$
if $d=5$.

Define a closed subscheme $\Psi_{d}(\eta)\subset\mathbb{P}\mathcal{F}_{d-1}$
as follows. Here, the maps to $\mathcal{O}_{\mathbb{P}\mathcal{F}_{d-1}}$ are
the ones obtained from $\eta$ in the construction of $\phi_{d}$ above:

  * •

$\mathcal{O}_{\Psi_{3}(\eta)}\coloneqq\operatorname{coker}(\pi^{*}\det\mathcal{F}_{2}(-3)\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{2}})$
if $d=3$.

  * •

$\mathcal{O}_{\Psi_{4}(\eta)}\coloneqq\operatorname{coker}(\pi^{*}\mathcal{F}_{2}(-2)\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{3}})$
if $d=4$.

  * •

$\mathcal{O}_{\Psi_{5}(\eta)}\coloneqq\operatorname{coker}(\pi^{*}\mathcal{F}_{5}(-2)\to\mathcal{O}_{\mathbb{P}\mathcal{F}_{4}})$
if $d=5$.

We say that $\eta$ has the _right codimension_ at a point $s\in S$ if the
fiber $\Psi_{d}(\eta)_{s}$ has dimension 0. If $\eta$ has the right
codimension at all points $s\in S$, we say that $\eta$ has the _right
codimension_.

###  5.4. The discriminants agree

We check that the discriminants on $\mathcal{X}_{d}$ and
$\operatorname{Covers}_{d}$ agree.

######  Proposition 5.3.

The pullback $\phi_{d}^{*}\Delta_{d}^{\operatorname{Cov}}$ agrees with
$\Delta_{d}$. More precisely, there is an isomorphism of the relevant line
bundles on $\mathcal{X}_{d}$ that takes one section to the other.

###### Proof.

Recall that in all the above morphisms
$\mathcal{X}_{d}\to\operatorname{Covers}_{d}$, we have a natural isomorphism
$\mathcal{F}_{d-1}\cong(\mathcal{O}_{Y}/\mathcal{O}_{S})^{\vee}$. Thus,
$\det\mathcal{O}_{Y}^{\operatorname{univ}}$ and
$(\det\mathcal{F}_{d-1}^{\operatorname{univ}})^{\vee}$ are isomorphic, and the
pullback
$\phi_{d}^{*}\mathcal{L}_{d}^{\operatorname{Cov}}=\phi^{*}(\det\mathcal{O}_{Y}^{\operatorname{univ}})^{-2}$
is isomorphic to the discriminant line bundle
$\mathcal{L}_{d}=(\det\mathcal{F}_{d-1}^{\operatorname{univ}})^{2}$ on
$\mathcal{X}_{d}$. Here, $\mathcal{O}_{Y}^{\operatorname{univ}}$ denotes the
structure sheaf of the universal degree $d$ cover on
$\operatorname{Covers}_{d}$, and $\mathcal{F}_{d-1}^{\operatorname{univ}}$
denotes the universal $\mathcal{F}_{d-1}$-bundle on $\mathcal{X}_{d}$.

It remains to show that the
$\phi_{d}^{*}\Delta_{d}^{\operatorname{Cov}}=\Delta_{d}$ as sections of
$\mathcal{L}_{d}$. By the definition of $\mathcal{L}_{d}$, we can identify the
global sections $H^{0}(\mathcal{X}_{d},\mathcal{L}_{d})$ with the functions
$f$ on $V_{d}$ such that for $g\in G_{d}(S)$ and $v\in V_{d}(S)$,
$f(gv)=\chi_{d}(g)^{2}f(v)$. Recall that $\Delta_{d}$ generates the
1-dimensional $\mathbb{C}$-vector space of functions with this property on
$(V_{d})_{\mathbb{C}}$. Because the coordinate ring of $V_{d}$ injects into
that of $(V_{d})_{\mathbb{C}}$, we see that
$\phi_{d}^{*}\Delta_{d}^{\operatorname{Cov}}=a\Delta_{d}$ for some
$a\in\mathbb{C}$. To show that $a=1$, we just need to check that
$\phi_{d}^{*}\Delta_{d}^{\operatorname{Cov}}$ and $\Delta_{d}$ agree and are
nonzero when evaluated at a single element $v\in V_{d}(\mathbb{Z})$. In all
the following examples, $v$ has the right codimension, so the cover of
$\operatorname{Spec}\mathbb{Z}$ corresponding to $v$ is actually just
$\Psi_{d}(v)$ by Theorem 5.5(b) below:

  * •

$d=3$: The binary cubic form $v=(x^{2}y-xy^{2})\otimes(x\wedge y)^{\vee}$ cuts
out the finite étale $\mathbb{Z}$-scheme
$\Psi_{3}(v)=\\{(1:0),(0:1),(1:1)\\}\subset\mathbb{P}_{\mathbb{Z}}^{2}$. We
check by direct computation that $\Delta_{3}(v)=1$. Because
$\Psi_{3}(v)\to\operatorname{Spec}\mathbb{Z}$ is étale, its discriminant is 1,
independent of basis444This just means that to calculate the discriminant, we
do not have to check what basis of $\mathcal{O}_{\Psi_{3}(v)}/\mathbb{Z}$ we
get from the frame $x,y$ of $\mathcal{F}_{2}$. This is because changing the
$\mathbb{Z}$-basis multiplies the discriminant by the square of a unit, which
must be 1.. Thus, $a=1$.

  * •

$d=4$: The pair of ternary quadratic forms
$v=(x_{0}x_{2}-x_{1}x_{2},x_{0}x_{1}-x_{1}x_{2})$ cuts out the finite étale
$\mathbb{Z}$-scheme
$\\{(1:0:0),(0:1:0),(0:0:1),(1:1:1)\\}\subset\mathbb{P}_{\mathbb{Z}}^{2}$. We
check by direct compuation that $\Delta_{4}(v)=1$. Because
$\Psi_{4}(v)\to\operatorname{Spec}\mathbb{Z}$ is étale, its discriminant is 1,
independent of basis. Thus, $a=1$.

  * •

$d=5$: The $5\times 5$ alternating matrix

$v=\begin{pmatrix}0&x_{0}-x_{3}&0&x_{3}&x_{2}\\\
-x_{0}+x_{3}&0&0&x_{3}&x_{3}\\\ 0&0&0&x_{1}&x_{2}\\\
-x_{3}&-x_{3}&-x_{1}&0&-x_{3}\\\
-x_{2}&-x_{3}&-x_{2}&x_{3}&0\end{pmatrix}\otimes(x_{0}\wedge x_{1}\wedge
x_{2}\wedge x_{3})^{\vee}$

has $4\times 4$ Pfaffians
$x_{1}x_{3}-x_{2}x_{3},-x_{1}x_{2}+x_{2}x_{3},-x_{0}x_{3}+x_{2}x_{3},-x_{0}x_{2}+x_{2}x_{3},x_{0}x_{1}-x_{1}x_{3}$,
which cut out the finite étale $\mathbb{Z}$-scheme
$\\{(1:0:0:0),(0:1:0:0),(0:0:1:0),(0:0:0:1),(1:1:1:1)\\}\subset\mathbb{P}_{\mathbb{Z}}^{3}$.
We check using the description of $\Delta_{5}$ in [Bha08, §4] that
$\Delta_{5}(v)=1$. Because $\Psi_{5}(v)\to\operatorname{Spec}\mathbb{Z}$ is
étale, its discriminant is 1, independent of basis. Thus, $a=1$.

We conclude that $\phi_{d}^{*}\Delta_{d}^{\operatorname{Cov}}=\Delta_{d}$. ∎

######  Remark 5.4.

One important advantage of $\mathcal{X}_{d}$ over $\operatorname{Covers}_{d}$
is that $\mathcal{X}_{d}$ is smooth. For $d\geq 4$, the stacks
$\operatorname{Covers}_{d}$ are not smooth by [Poo08, Proposition 8.4], so the
moduli stacks of maps from curves to $\operatorname{Covers}_{d}$, as studied
in [Deo14], are no longer nice (as far as we know). If we instead consider
stacks of maps from curves to $\mathcal{X}_{d}$, we get smoothness for our
stacks, among other useful properties (Theorem 8.4).

###  5.5. Parametrization of low degree Gorenstein covers

We say that a finite locally free map of schemes $Y\to S$ is _Gorenstein_ if
all its fibers are Gorenstein. In particular, if $S$ is Gorenstein, a finite
cover $Y\to S$ is Gorenstein if $Y$ is Gorenstein. When we restrict to
Gorenstein covers, we do not need to deal with resolvents.

By [LVW22, Lemma 4.4], there is an open substack
$\operatorname{Covers}_{d}^{\operatorname{Gor}}\subset\operatorname{Covers}_{d}$
parametrizing degree $d$ Gorenstein covers. By the upper semicontinuity of
fiber dimension, there is a $G_{d}$-invariant open subset $U_{d}\subset V_{d}$
parametrizing sections with the right codimension. We have the following
beautiful parametrization of low degree Gorenstein covers, first proved by
Casnati-Ekedahl over a Noetherian integral base and then extended by
Landesman-Vakil-Wood to an arbitrary base.

######  Theorem 5.5 (Casnati-Ekedahl, Landesman-Vakil-Wood).

  1. (a)

Over $\operatorname{Covers}_{d}^{\operatorname{Gor}}$, $\phi_{d}$ restricts to
an isomorphism
$[U_{d}/G_{d}]\xrightarrow[]{\sim}\operatorname{Covers}_{d}^{\operatorname{Gor}}$.

  2. (b)

Suppose we have a morphism $S\to[U_{d}/G_{d}]$ classifying a section $\eta$ of
the right codimension. Then the map from the complex (1) (if $d=3$), (2) (if
$d=4$), or (3) (if $d=5$) to its 0th homology $\mathcal{O}_{\Psi_{d}(\eta)}$
is a quasi-isomorphism, i.e. the complex is a resolution of
$\mathcal{O}_{\Psi_{d}(\eta)}$.

###### Proof.

Part (a) is [LVW22, Proposition 4.7]. Part (b) follows from the proofs of
[LVW22, Theorem 3.13], [LVW22, Theorem 3.14], and [LVW22, Theorem 3.16] for
$d=3$, $d=4$, and $d=5$, respectively. ∎

######  Corollary 5.6.

The morphism $\phi_{d}:\mathcal{X}_{d}\to\operatorname{Covers}_{d}$ restricts
to an isomorphism
$\mathcal{E}_{d}=[V_{d}^{ss}/G_{d}]\xrightarrow[]{\sim}B\mathbf{S}_{d}$. In
particular, for any scheme $S$ and any vector $v\in V_{d}^{ss}(S)$, the map
$(G_{d})_{S}\to(V_{d})_{S}$ given by $g\mapsto gv$ is étale, and the
differential $(\mathfrak{g}_{d})_{S}\to(V_{d})_{S}$ given by $x\mapsto xv$ is
an isomorphism.

###### Proof.

Because étale schemes over a field are Gorenstein, the étale locus
$B\mathbf{S}_{d}\subset\operatorname{Covers}_{d}$ is contained in
$\operatorname{Covers}_{d}^{\operatorname{Gor}}$. By Proposition 5.3, the
preimage $\phi_{d}^{-1}(B\mathbf{S}_{d})$ is the complement of the zero locus
of $\Delta_{d}$, which is $\mathcal{E}_{d}$. By Theorem 5.5(a), $\phi_{d}$
restricts to an isomorphism
$\mathcal{E}_{d}\xrightarrow[]{\sim}B\mathbf{S}_{d}$.

The fact that $(G_{d})_{S}\to(V_{d})_{S}$ is étale follows because it is the
top map in the following Cartesian square, where the morphism
$S\to(B\mathbf{S}_{d})_{S}$ is the composite
$S\xrightarrow[]{v}(V_{d}^{ss})_{S}\to[(V_{d}^{ss})_{S}/(G_{d})_{S}]\xrightarrow[]{\sim}B\mathbf{S}_{d}$:

${(G_{d})_{S}}$${(V_{d}^{ss})_{S}}$${{[(V_{d}^{ss})_{S}/(G_{d})_{S}]}}$${S}$${(B\mathbf{S}_{d})_{S}}$$\scriptstyle{\sim}$

Then the differential $(\mathfrak{g}_{d})_{S}\to(V_{d})_{S}$ is a map of
vector bundles on $S$ that is an isomorphism on each fiber by the étaleness of
$(G_{d})_{S}\to(V_{d})_{S}$, so $(\mathfrak{g}_{d})_{S}\to(V_{d})_{S}$ is an
isomorphism. ∎

######  Corollary 5.7.

If $Q\to S$ is a quartic Gorenstein cover, then there is a cubic resolvent
$(Q,C,\phi,\delta)$, unique up to unique isomorphism inducing the identity on
$Q$.

###### Proof.

By construction, the morphism
$\phi_{d}:\mathcal{X}_{4}\to\operatorname{Covers}_{4}$ factors as
$\mathcal{X}_{4}\xrightarrow[]{\sim}\operatorname{CovRes}_{4}\to\operatorname{Covers}_{4}$,
where the first morphism is Wood’s isomorphism from [Woo11] and the second
morphism forgets the cubic resolvent. By Theorem 5.5(a), the second morphism
becomes an isomorphism above $\operatorname{Covers}_{4}^{\operatorname{Gor}}$,
proving the corollary. ∎

##  6\. Prehomogeneous vector spaces over local fields

In this section, we will discuss certain integrals on prehomogeneous vector
spaces over local fields related to Igusa zeta functions. In this paper, we
will mainly work with local fields of positive characteristic $>d$, but the
results below apply equally to $p$-adic fields with $p>d$. The first part of
our discussion follows [BSW15, §7.1].

Fix a nonarchimedean local field $K$ of characteristic not dividing $d!$. Fix
the following notation:

  * •

$\mathcal{O}_{K}$ is the ring of integers of $K$.

  * •

$k$ is the residue field of $K$.

  * •

$q$ is the cardinality of $k$.

  * •

$v_{K}$ is the valuation on $K$ with image $\mathbb{Z}$.

  * •

$|\cdot|=q^{-v_{K}(\cdot)}$ is the normalized absolute value on $K$.

We will use $G_{d}$ and $V_{d}$ to denote $(G_{d})_{K}$ and $(V_{d})_{K}$,
respectively. Pick nonzero left-invariant top forms $\omega_{G_{d}}$ and
$\omega_{V_{d}}$ on $G_{d}$ and $V_{d}$ giving rise to Haar measures
$\mu_{G_{d}}$ and $\mu_{V_{d}}$, respectively. Because $G_{d}$ is reductive,
$\omega_{G_{d}}$ is in fact bi-invariant; because $V_{d}$ is commutative,
$\omega_{V_{d}}$ is bi-invariant as well. For any element $v\in
V_{d}(\mathcal{O}_{K})$ with $\Delta_{d}(v)\neq 0$, consider the map
$\pi_{v}:G_{d}\to V_{d}$ given by $g\mapsto gv$, which is étale by Corollary
5.6. The top form $g\mapsto(\pi_{v}^{*}\omega_{V})(g)/\chi_{d}(g)^{2}$ on
$G_{d}$ is a nonzero left-invariant form because $G_{d}$ acts on $\det V_{d}$
by $\chi_{d}^{2}$. Thus,
$(\pi_{v}^{*}\omega_{V_{d}})(g)=J_{d}(v)\chi_{d}(g)^{2}\omega_{G_{d}}(g)$ for
a rational function $J_{d}$ on $V_{d}$. We have
$(\pi_{hv}^{*}\omega_{V_{d}})(g)=(\pi_{v}^{*}\omega_{V_{d}})(gh)$. Meanwhile,

$\displaystyle(\pi_{hv}^{*}\omega_{V_{d}})(g)$
$\displaystyle=J_{d}(hv)\chi_{d}(g)^{2}\omega_{G_{d}}(g)$
$\displaystyle(\pi_{v}^{*}\omega_{V_{d}})(gh)$
$\displaystyle=J_{d}(v)\chi_{d}(gh)^{2}\omega_{G_{d}}(gh)$
$\displaystyle=\chi_{d}(h)^{2}J_{d}(v)\chi_{d}(g)^{2}\omega_{G_{d}}(g).$

Thus, $J_{d}(hv)=\chi_{d}(h)^{2}J_{d}(v)$, and $J_{d}$ must be equal to
$\mathcal{J}_{d}\Delta_{d}$ for a nonzero constant $\mathcal{J}_{d}\in
K^{\times}$. Thus,

$\pi_{v}^{*}\omega_{V_{d}}=\mathcal{J}_{d}\Delta_{d}(v)\chi_{d}^{2}\omega_{G_{d}}.$

We pick $\omega_{G_{d}}$ so that $\omega_{G_{d}}(1)=e_{1}\wedge\cdots\wedge
e_{r}$ for a basis $e_{1},\ldots,e_{r}$ of
$\mathfrak{g}_{d}^{*}(\mathcal{O}_{K})$ and do the same thing for
$\omega_{V_{d}}$. This normalization makes it so that
$d\mu_{V_{d}}(V_{d}(\mathcal{O}_{K}))=1$. Moreover, if we pick $v\in
V_{d}(\mathcal{O}_{K})$ such that $v_{K}(\Delta_{d}(v))=0$, then the
differential $d\pi_{v}:\mathfrak{g}_{d}(\mathcal{O}_{K})\to
V_{d}(\mathcal{O}_{K})$ is an isomorphism because it induces an isomorphism
$\mathfrak{g}_{d}(k)\to V_{d}(k)$ by Corollary 5.6. Thus, with this
normalization, $v_{K}(\mathcal{J}_{d})=0$.

For each $b\in\mathbb{N}$, let $V_{d}(\mathcal{O}_{K})_{b}$ denote the open
subset of $v\in V_{d}(\mathcal{O}_{K})$ such that $v_{K}(\Delta_{d}(v))=b$.
Note that each $V_{d}(\mathcal{O}_{K})_{b}$ is
$G_{d}(\mathcal{O}_{K})$-invariant. Moreover, each
$G_{d}(\mathcal{O}_{K})$-orbit in $V_{d}^{ss}(K)$ is open, and
$V_{d}(\mathcal{O}_{K})_{b}$ is compact, so $V_{d}(\mathcal{O}_{K})_{b}$ is a
finite union of $G_{d}(\mathcal{O}_{K})$-orbits.

######  Proposition 6.1.

For $s\in\mathbb{C}$ with $\mathrm{Re}(s)\geq 1$, we have

$\sum_{b=0}^{\infty}\\#\left(V_{d}(\mathcal{O}_{K})_{b}//G_{d}(\mathcal{O}_{K})\right)q^{-bs}=\frac{1}{\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))}\int_{V_{d}(\mathcal{O}_{K})}|\Delta_{d}(v)|^{s-1}d\mu_{V_{d}}(v).$

###### Proof.

We can write

$\int_{V_{d}(\mathcal{O}_{K})}|\Delta_{d}(v)|^{s-1}d\mu_{V_{d}}(v)=\sum_{b=0}^{\infty}\int_{V_{d}(\mathcal{O}_{K})_{b}}q^{-b(s-1)}d\mu_{V_{d}}(v).$

Each $V_{d}(\mathcal{O}_{K})_{b}$ is a finite union of
$G_{d}(\mathcal{O}_{K})$-orbits
$V_{d}(\mathcal{O}_{K})_{b}=\bigsqcup_{u}G_{d}(\mathcal{O}_{K})u$, so

$\int_{V_{d}(\mathcal{O}_{K})_{b}}q^{-b(s-1)}d\mu_{V_{d}}(v)=q^{-b(s-1)}\sum_{u}\int_{G_{d}(\mathcal{O}_{K})u}d\mu_{V_{d}}.$

Recall the formula
$\pi_{v}^{*}\omega_{V_{d}}=\mathcal{J}_{d}\Delta_{d}(v)\chi_{d}^{2}\omega_{G_{d}}$.
Since $|\mathcal{J}_{d}|=1$ and $|\chi_{d}(G_{d}(\mathcal{O}_{K}))|=1$,

$\int_{G_{d}(\mathcal{O}_{K})u}d\mu_{V_{d}}=\frac{1}{\\#\operatorname{stab}_{G_{d}(\mathcal{O}_{K})}(u)}\int_{G_{d}(\mathcal{O}_{K})}q^{-b}d\mu_{G_{d}}=\frac{q^{-b}\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))}{\\#\operatorname{stab}_{G_{d}(\mathcal{O}_{K})}(u)}.$

Putting this all together, we see that

$\displaystyle\int_{V_{d}(\mathcal{O}_{K})}|\Delta_{d}(v)|^{s-1}d\mu_{V_{d}}(v)$
$\displaystyle=\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))\sum_{b=0}^{\infty}\sum_{u}\frac{q^{-bs}}{\\#\operatorname{stab}_{G_{d}(\mathcal{O}_{K})}(u)}$
$\displaystyle=\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))\sum_{b=0}^{\infty}\\#\left(V_{d}(\mathcal{O}_{K})_{b}//G_{d}(\mathcal{O}_{K})\right)q^{-bs}.$

Note that the right-hand side converges for $\mathrm{Re}(s)\geq 1$ because
$V_{d}(\mathcal{O}_{K})$ has finite measure and $|\Delta_{d}(v)|\leq 1$ for
all $v\in V_{d}(\mathcal{O}_{K})$. Since all the terms on both sides are
nonnegative, we have established convergence for the left-hand side as well
for $\mathrm{Re}(s)\geq 1$. ∎

######  Proposition 6.2.

Let $(a)$ denote $1-q^{-a}$. With $\mu_{G_{d}}$ and $\mu_{V_{d}}$ normalized
as above,

$\displaystyle\mu_{G_{3}}(G_{3}(\mathcal{O}_{K}))$ $\displaystyle=(1)(2)$
$\displaystyle\mu_{G_{4}}(G_{4}(\mathcal{O}_{K}))$
$\displaystyle=(1)(2)^{2}(3)$
$\displaystyle\mu_{G_{5}}(G_{5}(\mathcal{O}_{K}))$
$\displaystyle=(1)(2)^{2}(3)^{2}(4)^{2}(5)$

###### Proof.

By the proof of Proposition 6.1,

$\\#(V_{d}(\mathcal{O}_{K})_{0}//G_{d}(\mathcal{O}_{K}))=\frac{1}{\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))}\int_{V_{d}(\mathcal{O}_{K})_{0}}|\Delta_{d}(v)|^{-1}d\mu_{V_{d}}(v)=\frac{\mu_{V_{d}}(V_{d}(\mathcal{O}_{K})_{0})}{\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))}.$

Because $V_{d}(\mathcal{O}_{K})_{0}=V_{d}^{ss}(\mathcal{O}_{K})$ and
$[V_{d}^{ss}/G_{d}]\cong B\mathbf{S}_{d}$ (Corollary 5.6), the left-hand side
$\\#(V_{d}(\mathcal{O}_{K})_{0}//G_{d}(\mathcal{O}_{K}))$ counts degree $d$
étale $\mathcal{O}_{K}$-algebras inversely weighted by automorphisms, which is
equal to the weighted count of degree $d$ étale $k$-algebras. For any finite
field, this is equal to 1. This can be proven either by direct counting in the
cases $d\in\\{3,4,5\\}$ or by a generating function argument for any $d$.
Hence,
$\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))=\mu_{V_{d}}(V_{d}(\mathcal{O}_{K})_{0})$.

Because of the normalization $\mu_{V_{d}}(V_{d}(\mathcal{O}_{K}))=1$ and the
condition that $v_{K}(\Delta_{d}(v))=0$ only depends on the image of $v$ in
$V_{d}(k)$, $\mu_{V_{d}}(V_{d}(\mathcal{O}_{K})_{0})$ is equal to the fraction
of points $\overline{v}\in V_{d}(k)$ for which $\Delta_{d}(\overline{v})\neq
0$. This density has been computed for degrees $d\in\\{3,4,5\\}$ in [DH71,
Lemma 1], [Bha04, Lemma 23], and [Bha08, Lemma 20], so we get the proposition.
In each paper, the authors work with $\mathbb{F}_{p}$ because they are
interested in counting extensions of $\mathbb{Q}$, but their methods extend to
$\mathbb{F}_{q}$. ∎

Alternatively, as suggested to us by Will Sawin, Proposition 6.2 can be proven
by counting the $k$-points of $G_{d}$ and dividing by $q^{\dim G_{d}}$.

###  6.1. Igusa zeta functions

For the rest of the paper, let

$I_{d}(s)=\frac{1}{\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))}\int_{V_{d}(\mathcal{O}_{K})}|\Delta_{d}(v)|^{s-1}d\mu_{V_{d}}(v)$

with the normalizations above. In particular,
$\mu_{V_{d}}(V_{d}(\mathcal{O}_{K}))=1$, and
$\mu_{G_{d}}(G_{d}(\mathcal{O}_{K}))$ is the relevant constant from
Proposition 6.2.

We recall Igusa’s computations of two local zeta functions relevant to us. The
coefficients of these Igusa zeta functions, which count algebras over local
fields, will be relevant later when we want to count algebras over global
fields. Let $t=q^{-s}$ and $(a)=1-q^{-a}$. Let
$f(q,t)=1+t^{2}+t^{3}+t^{4}-2t^{5}+2qt^{6}+(q-1)t^{7}+qt^{8}-qt^{9}+(q-1)qt^{10}-2qt^{11}+2q^{2}t^{12}-q^{2}t^{13}-q^{2}t^{14}-q^{2}t^{15}-q^{2}t^{17}$.
We note that $f(q,t)$ is irreducible.

######  Theorem 6.3 ([Igu94]).

$\displaystyle I_{3}(s)$
$\displaystyle=\frac{1+t+t^{2}+t^{3}+t^{4}}{(1-t^{2})(1-qt^{6})}$
$\displaystyle I_{4}(s)$
$\displaystyle=\frac{f(q,t)}{(1-t)(1-t^{2})(1-qt^{6})(1-q^{2}t^{8})(1-q^{3}t^{12})}$

Note that for $d\in\\{3,4\\}$, $I_{d}(s)$ is dependent only on $q$ and $t$, so
we will write $I_{d}(q,t)$ instead. By some computation, it can be shown that
all the coefficients of the power series $I_{d}(q,t)$ are nonnegative
integers.

By Proposition 6.1 and Proposition 6.2, $I_{d}(q,t)$ is equal to
$\sum_{b=0}^{\infty}\\#\left(V_{d}(\mathcal{O}_{K})_{b}//G_{d}(\mathcal{O}_{K})\right)t^{b}$,
so we know exactly what the counts
$\\#\left(V_{d}(\mathcal{O}_{K})_{b}//G_{d}(\mathcal{O}_{K})\right)$ are for
$d\in\\{3,4\\}$. From the discussion in Section 5.1, we get the following:

######  Corollary 6.4.

  1. (a)

Let $N_{3,\mathcal{O}_{K},b}^{\operatorname{loc}}$ be the number of
isomorphism classes of cubic $\mathcal{O}_{K}$-algebras of discriminant
$q^{b}$, inversely weighted by automorphisms over $\mathcal{O}_{K}$. Then

$\sum_{b=0}^{\infty}N_{3,\mathcal{O}_{K},b}^{\operatorname{loc}}t^{b}=I_{3}(q,t)=\frac{1+t+t^{2}+t^{3}+t^{4}}{(1-t^{2})(1-qt^{6})}.$

  2. (b)

Let $N_{4,\mathcal{O}_{K},b}^{\operatorname{loc}}$ be the number of
isomorphism classes of quartic $\mathcal{O}_{K}$-algebras with cubic resolvent
of discriminant $q^{b}$, inversely weighted by automorphisms over
$\mathcal{O}_{K}$. Then

$\sum_{b=0}^{\infty}N_{4,\mathcal{O}_{K},b}^{\operatorname{loc}}t^{b}=I_{4}(q,t)=\frac{f(q,t)}{(1-t)(1-t^{2})(1-qt^{6})(1-q^{2}t^{8})(1-q^{3}t^{12})},$

where
$f(q,t)=1+t^{2}+t^{3}+t^{4}-2t^{5}+2qt^{6}+(q-1)t^{7}+qt^{8}-qt^{9}+(q-1)qt^{10}-2qt^{11}+2q^{2}t^{12}-q^{2}t^{13}-q^{2}t^{14}-q^{2}t^{15}-q^{2}t^{17}$.

Unfortunately, the Igusa zeta function for $(G_{5},V_{5})$ has not been
computed. Thus, many of the results of this paper will be for $d\in\\{3,4\\}$.
However, as we will explain in Section 10.3.2, computing the Igusa zeta
function for the degree 5 prehomogeneous vector space would enable us to
extend these results to $d=5$.

######  Remark 6.5.

As we hinted towards in Section 3.4, there is a mysterious “coincidence” where
the dimension of $V_{d}$ is the top degree of $\mathfrak{B}_{d}$. We will see
in Theorem 11.14 that for $d\in\\{3,4\\}$, $I_{d}(q^{-2},qt)$ is the
generating function for the geometric $\mathbf{S}_{d}$-invariant cohomology of
$\mathfrak{B}_{d}$, providing a connection between $(G_{d},V_{d})$ and
$\mathfrak{B}_{d}$. The strange thing is that the dimension of $V_{d}$ and top
degree of $\mathfrak{B}_{d}$ are not apparently related to the Igusa zeta
function and cohomology, respectively.

##  7\. Orbinodal curves

Following [Deo14], we want our stacks of covers with resolvents
$\mathcal{R}^{d}$ to include covers of orbinodal curves. In this section, we
will recall some information about orbinodal curves and the relevant stacks.
Some references for orbinodal curves include [AV02, §4] and [Ols07a], where
they are referred to as “twisted curves.”

We first recall the definition of an orbinodal curve. Recall that a Deligne-
Mumford stack $\mathcal{Y}$ is _tame_ if the automorphism group of any
geometric point $x\in\mathcal{Y}(k)$ has order not divisible by
$\mathrm{char}(k)$. Note that we require orbinodal curves to be balanced and
do not require them to have geometrically connected fibers.

######  Definition 7.1.

Let $S$ be a scheme. An _$n$ -pointed orbinodal curve_ over $S$ consists of
the data $(\mathcal{C}\to C\to S;\sigma_{1},\ldots,\sigma_{n})$, where

  1. (1)

$C\to S$ is a nodal curve, and the $\sigma_{i}$ are pairwise disjoint sections
$S\to C$.

  2. (2)

$\mathcal{C}\to S$ is a tame Deligne-Mumford stack with coarse moduli space
$\mathcal{C}\to C$. The morphism $\mathcal{C}\to C$ is an isomorphism over the
open subset $C^{\operatorname{gen}}\subset C$ defined as the complement of the
nodes and the marked points.

  3. (3)

Let $c$ be a geometric point of $\mathcal{C}$ mapping to a node of $C\to S$,
and let $s$ be its image in $S$. Then there exists an open neighborhood
$T\subset S$ of $s$, an étale neighborhood $U\to\mathcal{C}$ of $c$ over $T$,
a function $t\in\mathcal{O}_{T}$, and an integer $n\geq 1$ for which we have
the following Cartesian diagram:

${\mathcal{C}\times_{C}U}$${U}$${{[\mathrm{Spec}\left(\frac{\mathcal{O}_{T}[u,v]}{uv-t}\right)/\mu_{n}]}}$${{\mathrm{Spec}\left(\frac{\mathcal{O}_{T}[x,y]}{xy-t^{n}}\right)}}$étét

Here, the $\mu_{n}$-action on $\mathrm{Spec}\mathcal{O}_{T}[u,v]/(uv-t)$ is
given by $(\zeta,u)\mapsto\zeta u$ and $(\zeta,v)\mapsto\zeta^{-1}v$, and the
bottom morphism is induced by the the map sending $x\mapsto u^{n}$ and
$y\mapsto v^{n}$.

  4. (4)

Let $s$ be a geometric point of $S$, and let $c=\sigma_{i}(s)$. Then there
exists an étale neighborhood $U\to C$ of $c$ and an integer $n\geq 1$ for
which we have the following Cartesian diagram:

${\mathcal{C}\times_{C}U}$${U}$${{[\mathrm{Spec}\mathcal{O}_{S}[u]/\mu_{n}]}}$${{\mathrm{Spec}\mathcal{O}_{S}[x]}}$étét

Here, the $\mu_{n}$-action on $\mathrm{Spec}\mathcal{O}_{S}[u]$ is given by
$(\zeta,u)\mapsto\zeta u$, and the bottom morphism is induced by the map
sending $x\mapsto u^{n}$.

We will abbreviate $(\mathcal{C}\to C\to S;\sigma_{1},\ldots,\sigma_{n})$ as
$(\mathcal{C}\to C\to S;\sigma)$ or $(\mathcal{C}\to C;\sigma)$. We will often
use the term “smooth orbicurve” or “orbicurve” when $\mathcal{C}\to S$ is
smooth.

A 1-morphism $(\mathcal{C}_{1}\to
C_{1};\\{\sigma_{1j}\\})\to(\mathcal{C}_{2}\to C_{2};\\{\sigma_{2j}\\})$ is a
1-morphism $\mathcal{C}_{1}\to\mathcal{C}_{2}$ over $S$ that takes
$\sigma_{1j}$ to $\sigma_{2j}$. A 2-morphism of two 1-morphisms
$(\mathcal{C}_{1}\to
C_{1};\\{\sigma_{1j}\\})\rightrightarrows(\mathcal{C}_{2}\to
C_{2};\\{\sigma_{2j}\\})$ is a 2-morphism of the underlying 1-morphisms
$\mathcal{C}_{1}\rightrightarrows\mathcal{C}_{2}$.

######  Remark 7.2.

Conveniently, [AV02, Lemma 4.2.2] states that the 2-category of orbinodal
curves is equivalent to a 1-category, i.e. all 2-automorphism groups of a
1-morphism are trivial. Thus, from now on, we will use “morphism” to refer to
an isomorphism class of 1-morphisms, and we will make no reference to
2-morphisms.

Because the morphism $\mathcal{C}\to C$ is an isomorphism over
$C^{\operatorname{gen}}$, we will often write $C^{\operatorname{gen}}$ to
denote its preimage in $\mathcal{C}$. We wil use this convention for subsets
of $C^{\operatorname{gen}}$ as well, such as the $U_{i}$ not containing any
nodes or marked points in Section 8.6.

###  7.1. Stacks of nodal and orbinodal curves

Our constructions of stacks of covers with resolvents will lie over the
following stacks of nodal and orbinodal curves.

######  Definition 7.3 ([Deo14, Definition 3.1]).

Define the stack $\mathcal{M}^{*,n}$ of divisorially marked, $n$-pointed nodal
curves as the stack over $\mathbb{Z}$ whose objects over $S$ are tuples $(C\to
S;\Sigma;\sigma_{1},\ldots,\sigma_{n})$, where

  1. (1)

$C$ is an algebraic space and $C\to S$ is a nodal curve with geometrically
connected fibers.

  2. (2)

$\Sigma\subset C$ is a relative effective Cartier divisor over $S$ lying in
the smooth locus of $C\to S$.

  3. (3)

$\sigma_{j}:S\to C$ are relatively pairwise disjoint sections lying in the
smooth locus of $C\to S$ that do not intersect $\Sigma$.

We will abbreviate $(C\to S;\Sigma;\sigma_{1},\ldots,\sigma_{n})$ as $(C\to
S;\Sigma;\sigma)$. Let $\mathcal{M}$ denote the disjoint union of all the
$\mathcal{M}^{*,n}$, and let
$\mathcal{M}^{\operatorname{sm}}\subset\mathcal{M}$ denote the open substack
parametrizing smooth curves. Let $\mathcal{M}^{b,n}$ denote the open and
closed substack of $\mathcal{M}$ where the degree of the marked divisor is $b$
and there are $n$ marked points.

Deopurkar’s proof of the following result occurs over $\mathbb{C}$, but it
works exactly the same over $\mathbb{Z}$.

######  Proposition 7.4 ([Deo14, Proposition 3.2]).

$\mathcal{M}$ is a smooth algebraic stack over $\mathbb{Z}$.

######  Definition 7.5.

Define the stack $\mathcal{M}^{\operatorname{orb};*,n}$ of divisorially
marked, $n$-pointed orbinodal curves as the stack over $\mathbb{Z}$ whose
objects over $S$ are tuples $(\mathcal{C}\to C\to
S;\Sigma;\sigma_{1},\ldots,\sigma_{n}))$, where

  1. (1)

$(\mathcal{C}\to C\to S;\sigma)$ is an $n$-pointed orbinodal curve with
geometrically connected fibers.

  2. (2)

$\Sigma\subset C$ is a relative effective Cartier divisor over $S$ lying in
the smooth locus of $C\to S$ and disjoint from the sections $\sigma$.

Let $\mathcal{M}^{\operatorname{orb}}$ denote the disjoint union of all the
$\mathcal{M}^{\operatorname{orb};*,n}$. For $N\in\mathbb{N}_{>0}$, let
$\mathcal{M}^{\operatorname{orb}\leq N}$ denote the full subcategory of
$\mathcal{M}^{\operatorname{orb}}$ where all the automorphism groups of
$\mathcal{C}$ have order bounded above by $N$. Given an $n$-tuple of positive
integers $\underline{a}=(a_{1},\ldots,a_{n})$, let
$\mathcal{M}^{\operatorname{orb};*,n}(\underline{a})$ denote the open and
closed substack of $\mathcal{M}^{\operatorname{orb};*,n}$ where the
automorphism groups of the geometric points of $\mathcal{C}$ mapping to
$\sigma_{i}$ have order $a_{i}$.

######  Remark 7.6.

For convenience, from now on, whenever we write a curve $C\to S$ or an
orbinodal curve $\mathcal{C}\to C\to S$, we will implicitly require the fibers
to be geometrically connected unless specified otherwise. This is because many
of the stacks we will construct will be over $\mathcal{M}$.

Given any stack $\mathcal{Y}$ over $\mathcal{M}$, we write $\mathcal{Y}^{b,n}$
to denote the preimage of $\mathcal{M}^{b,n}$ and
$\mathcal{Y}^{\operatorname{sm}}$ to denote the preimage of
$\mathcal{M}^{\operatorname{sm}}$. If $\mathcal{Y}$ is over
$\mathcal{M}^{\operatorname{orb}}$, we write
$\mathcal{Y}^{*,n}(\underline{a})$ to denote the preimage of
$\mathcal{M}^{\operatorname{orb};*,n}(\underline{a})$. We have a morphism
$\mathcal{M}\to\mathcal{M}^{0,*}$ forgetting the marked divisor. For the
stacks of orbinodal curves, we have a morphism
$\mathcal{M}^{\operatorname{orb}}\to\mathcal{M}^{\operatorname{orb};0,*}$
forgetting the marked divisor and a morphism
$\mathcal{M}^{\operatorname{orb}}\to\mathcal{M}$ sending $(\mathcal{C}\to C\to
S;\Sigma;\sigma)\to(C\to S;\Sigma;\sigma)$. We observe that
$\mathcal{M}^{\operatorname{orb}}$ is the open substack of the 2-fiber product
$\mathcal{M}^{\operatorname{orb};0,*}\times_{\mathcal{M}^{0,*}}\mathcal{M}$
where the marked divisor does not intersect the marked points. Similarly,
given any stack $\mathcal{Y}\to\mathcal{M}^{\operatorname{orb}}$, we write
$\mathcal{Y}^{\leq N}$ for the preimage of
$\mathcal{M}^{\operatorname{orb}\leq N}$.

The following result of Olsson will be very useful when we prove algebraicity
and boundedness for our stacks of covers with resolvents in Section 8.

######  Theorem 7.7 ([Ols07a, Theorem 1.9, Corollary 1.11]).

  1. (a)

$\mathcal{M}^{\operatorname{orb};0,*}$ is a smooth algebraic stack over
$\mathbb{Z}$. The morphism
$\mathcal{M}^{\operatorname{orb};0,*}\to\mathcal{M}^{0,*}$ is representable by
Deligne-Mumford stacks.

  2. (b)

The morphism $\mathcal{M}^{\operatorname{orb}\leq N;0,*}\to\mathcal{M}^{0,*}$
is of finite type.

######  Corollary 7.8.

  1. (a)

$\mathcal{M}^{\operatorname{orb}}$ is a smooth algebraic stack over
$\mathbb{Z}$. The morphism $\mathcal{M}^{\operatorname{orb}}\to\mathcal{M}$ is
representable by Deligne-Mumford stacks.

  2. (b)

For any $N$, the morphism $\mathcal{M}^{\operatorname{orb}\leq
N}\to\mathcal{M}$ is of finite type.

###### Proof.

The smoothness in (a) follows from the smoothness of
$\mathcal{M}^{\operatorname{orb};0,*}$ and the fact that the marked divisor
must remain in the smooth locus, so that its deformations are unobstructed.
Each remaining statement follows from the corresponding part of Theorem 7.7
and the fact that $\mathcal{M}^{\operatorname{orb}}$ is an open substack of
$\mathcal{M}^{\operatorname{orb};0,*}\times_{\mathcal{M}^{0,*}}\mathcal{M}$. ∎

###  7.2. Stacky structures on smooth curves

Smooth orbicurves admit a simple description: a smooth orbicurve is
characterized by its underlying $n$-pointed smooth curve and $n$ positive
integers corresponding to the automorphism groups above the marked points. The
construction that lets us recover the smooth orbicurve from these data is the
root stack, introduced in [Cad07]. We will only discuss root stacks in the
context of smooth orbicurves, but the construction works in much more general
contexts.

We describe the construction. Fix an $n$-tuple
$\underline{a}=(a_{1},\ldots,a_{n})$ of positive integers, and let
$N=a_{1}\cdots a_{n}$. Suppose we have a smooth $n$-pointed curve $(C\to
S;\sigma)$, where $S$ is a scheme with $N$ invertible. The $n$ relative
effective Cartier divisors $D_{i}=\sigma_{i}(S)$ induce a morphism
$C\to[\mathbb{A}^{1}/\mathbb{G}_{m}]^{n}\cong[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$,
where the $n$th morphism $C\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ corresponds to
the line bundle $\mathcal{O}_{C}(D_{i})$ with section dual to the inclusion
$\mathcal{O}_{C}(-D_{i})\to\mathcal{O}_{C}$. There is an $\underline{a}$th
power map
$\theta_{\underline{a}}:[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]\to[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$,
which is the $a_{i}$th power map
$\theta_{a_{i}}:[\mathbb{A}^{1}/\mathbb{G}_{m}]\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$
on the $i$th coordinate. The _$\underline{a}$ th root stack associated to
$(C\to S;\sigma)$_ is the 2-fiber product
$\mathcal{C}_{\underline{a}}\coloneqq
C\times_{[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}],\theta_{\underline{a}}}[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$.

######  Theorem 7.9 ([Cad07, Theorem 4.1]).

The $\underline{a}$th root stack $\mathcal{C}_{\underline{a}}$ is a smooth
orbicurve over $S$ with coarse moduli space $\mathcal{C}_{\underline{a}}\to
C$. The functor $C\mapsto\underline{C}_{\underline{a}}$ induces an isomorphism
$\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{sm};0,*}\xrightarrow[]{\sim}\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{orb},\operatorname{sm};0,*}(\underline{a})$
inverse to the map
$\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{orb},\operatorname{sm};0,*}(\underline{a})\to\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{sm};0,*}$.

###### Proof.

We note that [Cad07, Theorem 4.1] is stated only for a Noetherian base $S$.
However, we can generalize to arbitrary $S$ by using [Sta23, Tag 062Y] in
place of [Cad07, Lemma 5.1] in the proof. ∎

Because the marked divisor of an orbinodal curve is disjoint from the stacky
part, we get the following corollary.

######  Corollary 7.10.

The map
$\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{orb},\operatorname{sm}}(\underline{a})\to\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{sm}}$
is an isomorphism.

######  Remark 7.11.

Implicit in Theorem 7.9 is the fact that a smooth orbicurve $\mathcal{C}\to S$
has no nontrivial automorphisms that induce the identity on the coarse moduli
space $C\to S$. This can also be seen from [ACV03, Proposition 7.1.1] and the
fact that $\mathcal{M}^{\operatorname{orb};0,*}\to\mathcal{M}^{0,*}$ is
representable by Deligne-Mumford stacks. In later sections, we will make use
of this fact without explicit reference.

####  7.2.1. Framings for smooth orbicurves

We finish this section by introducing stacks of smooth orbicurves/curves with
framings of the normal bundles to the marked points. These stacks will be
useful when we construct stacks of marked covers with resolvents and marked
Hurwitz spaces in Section 10.

######  Definition 7.12.

Define the stack $\mathcal{M}^{\operatorname{fr};*,n}$ as the stack over
$\mathcal{M}^{*,n}$ whose objects over $S$ consist of the following data:

  1. (1)

An object $(C\to S;\Sigma;\sigma)\in\mathcal{M}^{*,n}(S)$.

  2. (2)

For each $i$, a trivialization
$\alpha_{i}:\sigma_{i}^{-1}\mathcal{O}_{C}(D_{i})\xrightarrow[]{\sim}\mathcal{O}_{S}$
on $S$, where $D_{i}=\sigma_{i}(S)$.

We will abbreviate these data as $(C\to S;\Sigma;\sigma;\alpha)$. A morphism
in $\mathcal{M}^{\operatorname{fr};*,n}$ is a morphism of objects of
$\mathcal{M}^{*,n}$ compatible with the framings $\alpha$. Let
$\mathcal{M}^{\operatorname{fr}}$ denote the disjoint union of all the
$\mathcal{M}^{\operatorname{fr};*,n}$.

Note that $\mathcal{M}^{\operatorname{fr};*,n}\to\mathcal{M}^{*,n}$ is the
principal $\mathbb{G}_{m}^{n}$-bundle associated to the direct sum of the $n$
universal line bundles $\sigma_{i}^{-1}\mathcal{O}_{C}(D_{i})$. Thus,
$\mathcal{M}^{\operatorname{fr};*,n}$ is a smooth algebraic stack over
$\mathbb{Z}$ by Proposition 7.4.

Let $\underline{a}=(a_{1},\ldots,a_{n})$, and let $S$ be a scheme on which
$N=a_{1}\cdots a_{n}$ is invertible. The reason we consider
$\mathcal{M}^{\operatorname{fr}}$ is that for an object $(C\to
S;\Sigma;\sigma;\alpha)\in\mathcal{M}^{\operatorname{fr},\operatorname{sm};*,n}(S)$,
the framing data gives us $n$ natural sections
$S\to\mathcal{C}_{\underline{a}}$, which are defined as follows. Consider the
two composites

$\displaystyle S\xrightarrow[]{\sigma_{i}}C$
$\displaystyle\to[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$ $\displaystyle
S\to[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$
$\displaystyle\xrightarrow[]{\theta_{\underline{a}}}[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$

Here, the map $C\to[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$ is the one considered
above and the map $S\to[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$ in the second
composite classifies $n$ trivial line bundle with the zero section in the
$i$th bundle and the identity sections in the other bundles. The first
composite classifies the $n$-tuple of line bundles
$(\mathcal{O}_{S},\ldots,\mathcal{O}_{S},\sigma_{i}^{-1}\mathcal{O}_{C}(D_{i}),\mathcal{O}_{S},\ldots,\mathcal{O}_{S})$
where $\sigma_{i}^{-1}\mathcal{O}_{C}(D_{i})$ is in the $i$th position, along
with the zero section in the $i$th bundle and the identity sections in the
other bundles. The second composite classifies
$(\mathcal{O}_{S},\ldots,\mathcal{O}_{S})$ with the zero section in the $i$th
bundle and the identity sections in the other bundles. Thus, $\alpha$ provides
a 2-isomorphism of the two composites and gives rise to a section
$S\to\mathcal{C}_{\underline{a}_{i}}=C\times_{[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]}[\mathbb{A}^{n}/\mathbb{G}_{m}^{n}]$.
This construction yields a canonical morphism
$\psi_{i,\underline{a}}^{\operatorname{univ}}:\mathcal{M}_{\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{fr},\operatorname{sm};*,n}\to\mathcal{C}_{\underline{a},\mathbb{Z}\left[\frac{1}{N}\right]}^{\operatorname{univ}}$,
where $\mathcal{C}_{\underline{a}}^{\operatorname{univ}}$ is the universal
orbicurve over
$\mathcal{M}^{\operatorname{orb},\operatorname{sm};*,n}(\underline{a})$.

##  8\. The stacks of covers with resolvents $\mathcal{R}^{d}$

In this section, we construct the main geometric objects of this paper
$\mathcal{R}^{d}$ and prove some important geometric properties about them.
Unless specified otherwise, for the rest of the paper, we will work over
$\mathbb{S}_{d}\coloneqq\operatorname{Spec}\mathbb{Z}\left[\frac{1}{d!}\right]$,
as the invertibility of $d!$ will ensure that we only need to deal with tame
ramification and tame stacks. The following definition is nearly identical to
[Deo14, Definition 3.3], except that we replace $\operatorname{Covers}_{d}$
with $\mathcal{X}_{d}$.

######  Definition 8.1.

Define the _big stack of degree $d$ covers with resolvents_ $\mathcal{R}^{d}$
as the disjoint union of the stacks $\mathcal{R}^{d;*,n}$ over
$\mathbb{S}_{d}$ whose objects over $S$ are

$\mathcal{R}^{d;*,n}(S)\coloneqq\\{(\mathcal{C}\to C\to
S;\sigma_{1},\ldots,\sigma_{n};\chi:\mathcal{C}\to\mathcal{X}_{d})\\},$

where

  1. (1)

$(\mathcal{C}\to C\to S;\sigma)$ is an $n$-pointed orbinodal curve.

  2. (2)

$\chi:\mathcal{C}\to\mathcal{X}_{d}$ is a representable morphism that maps the
following to $\mathcal{E}_{d}$ for each fiber $\mathcal{C}_{s}$ of
$\mathcal{C}\to S$: the generic points of the components of $\mathcal{C}_{s}$,
the nodes of $\mathcal{C}_{s}$, and the preimages of the marked points in
$\mathcal{C}_{s}$.

A morphism between $(\mathcal{C}_{1}\to P_{1}\to
S_{1};\\{\sigma_{1j}\\};\chi_{1}:\mathcal{C}_{1}\to\mathcal{X}_{d})$ and
$(\mathcal{C}_{2}\to P_{2}\to
S_{2};\\{\sigma_{2j}\\};\chi_{2}:\mathcal{C}_{2}\to\mathcal{X}_{d})$ over a
morphism $S_{1}\to S_{2}$ consists of a pair $(F,\alpha)$, where

  1. (1)

$F$ is a morphism of pointed orbinodal curves:
$F:\mathcal{C}_{1}\to\mathcal{C}_{2}$ such that we get a Cartesian diagram

${\mathcal{C}_{1}}$${\mathcal{C}_{2}}$${S_{1}}$${S_{2}}$$\scriptstyle{F}$

  2. (2)

$\alpha:\chi_{1}\to\chi_{2}\circ F$ is a 2-morphism.

We abbreviate $(\mathcal{C}\to C\to
S;\sigma_{1},\ldots,\sigma_{n};\chi:\mathcal{C}\to\mathcal{X}_{d})$ by
$(\mathcal{C}\to C;\sigma;\chi)$.

Given an object $(\mathcal{C}\to C\to S;\sigma;\chi)$ of
$\mathcal{R}^{d;*,n}(S)$, the branch locus $\chi^{-1}(\Sigma_{d})$ is disjoint
from the stacky points in $\mathcal{C}$, so we can identify it with its image
in $C$, which we will denote $\operatorname{br}\chi$. Because
$\operatorname{br}\chi$ is the zero locus of a section of a line bundle and
its intersection with each fiber $P_{s}$ is an effective Cartier divisor,
$\operatorname{br}\chi$ is a relative effective Cartier divisor in $C\to S$ by
[Sta23, Tag 062Y]. Thus, we have a _branch morphism_
$\operatorname{br}:\mathcal{R}^{d}\to\mathcal{M}$ defined by

$(\mathcal{C}\to C\to S;\sigma;\chi)\mapsto(C\to
S;\operatorname{br}\chi;\sigma).$

We will also refer to the map
$\mathcal{R}^{d}\to\mathcal{M}^{\operatorname{orb}}$ as $\operatorname{br}$ as
an abuse of notation.

######  Remark 8.2.

The stack $\mathcal{X}_{3}$ is equivalent to the stack of degree 3 covers
$\mathcal{A}_{3}$ considered in [Deo14] and [Poo08], so our construction is
nothing new when $d=3$ (other than extending Deopurkar’s construction from
$\mathbb{Q}$ to $\mathbb{Z}\left[\frac{1}{6}\right]$). However, we do get a
new proof of smoothness for Deopurkar’s degree 3 big Hurwitz stack
$\mathcal{H}^{3}$ that also works for $\mathcal{R}^{4}$ and $\mathcal{R}^{5}$.

######  Remark 8.3.

Later in this paper, we will consider points $p\in\mathcal{M}(k)$
corresponding to curves of the form $(\mathbb{A}_{k}^{1};\Sigma;\infty)$, i.e.
$\infty$ will be the unique marked point and
$\Sigma\in\operatorname{Sym}^{n}(\mathbb{A}^{1})(k)$ will be some marked
divisor. The fiber $\operatorname{br}^{-1}(p)$ will parametrize covers
$D\to\mathbb{A}^{1}$ (if $d=3$) or covers with resolvents
$(D,R)\to\mathbb{A}^{1}$ (if $d\in\\{4,5\\}$) such that
$\pi:D\to\mathbb{A}^{1}$ is étale outside $\pi^{-1}(\Sigma)$.

In the rest of Section 8, we will prove the following theorem concerning
$\mathcal{R}^{d}$ and the branch morphism
$\operatorname{br}:\mathcal{R}^{d}\to\mathcal{M}$. The proof is very similar
to that of [Deo14, Theorem 3.8]. However, Deopurkar works over a field of
characteristic 0, so we will rewrite everything over $\mathbb{S}_{d}$.

######  Theorem 8.4.

  1. (a)

$\mathcal{R}^{d}$ is a smooth algebraic stack over $\mathbb{S}_{d}$.

  2. (b)

The morphism $\operatorname{br}:\mathcal{R}^{d}\to\mathcal{M}$ is proper and
representable by Deligne-Mumford stacks.

  3. (c)

The composite
$\mathcal{R}^{d}\xrightarrow[]{\operatorname{br}}\mathcal{M}^{\operatorname{orb}}\to\mathcal{M}^{\operatorname{orb};0,*}$
is smooth. The restriction
$\mathcal{R}^{d;b,*}\to\mathcal{M}^{\operatorname{orb};0,*}$ has relative
dimension $b$.

From Theorem 7.9 and Theorem 8.4(c), we get the following.

######  Corollary 8.5.

The composite
$\mathcal{R}^{d,\operatorname{sm}}\xrightarrow[]{\operatorname{br}}\mathcal{M}^{\operatorname{sm}}\to\mathcal{M}^{\operatorname{sm};0,*}$
is smooth. The restriction
$\mathcal{R}^{d,\operatorname{sm};b,*}\to\mathcal{M}^{\operatorname{sm};0,*}$
has relative dimension $b$.

The proof of Theorem 8.4 is broken down into parts.

###  8.1. Generating sheaves for tame Deligne-Mumford stacks

A large part of our proof of Theorem 8.4(b) is based on Deopurkar’s proof of
[Deo14, Theorem 3.8] for his stacks of degree $d$ covers. Deopurkar works over
a field of characteristic 0, whereas we will work over $\mathbb{S}_{d}$. We
will show that much of his proof works the same way for our situation.

We will need some results from [Nir08] concerning moduli of sheaves on tame
Deligne-Mumford stacks. We recall the notion of a generating sheaf, as
generating sheaves are a necessary part of Deopurkar’s proof of [Deo14,
Proposition 4.12], which appears in our paper as Proposition 8.15. For more
information, see [OS03, §5] or [Kre09, §5.2].

######  Definition 8.6.