# Tautological classes on Low-degree Hurwitz Spaces

Samir Canning, Hannah Larson<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus
$g$ covers of $\mathbb{P}^{1}$. We define the tautological ring of
$\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those
supported on the locus of “factoring covers,” are tautological up to
codimension roughly $g/k$ when $k\leq 5$. The set-up developed here is also
used in our subsequent work [3], wherein we prove new results about the
structure of the Chow ring for $k\leq 5$.

###### 2010 Mathematics Subject Classification:

14C15, 14C17

During the preparation of this article, S.C. was partially supported by NSF
RTG grant DMS-1502651. H.L. was supported by the Hertz Foundation and NSF GRFP
under grant DGE-1656518. This work will be part of S.C.’s and H.L.’s Ph.D.
theses.

##  1\. Introduction

When studying intersection theory of moduli spaces, one often introduces
certain natural or “tautological” classes coming from the universal family. In
the words of Mumford [22]:

> Whenever a variety or topological space is defined by some universal
> property, one expects that by virtue of its defining property, it possesses
> certain cohomology classes called tautological classes.

For example, in the case of the moduli space of curves $\mathcal{M}_{g}$, the
tautological classes Mumford proposes to study are the _kappa classes_ ,
defined as follows. Let $f:\mathcal{C}\to\mathcal{M}_{g}$ be the universal
curve; then $\kappa_{i}:=f_{*}(c_{1}(\omega_{f})^{i+1})\in
A^{i}(\mathcal{M}_{g})$, the Chow ring of $\mathcal{M}_{g}$. The _tautological
ring_ , denoted $R^{*}(\mathcal{M}_{g})\subseteq A^{*}(\mathcal{M}_{g})$, is
the subring of the rational Chow ring generated by the kappa classes.

In this paper, we study the intersection theory of the Hurwitz space
$\mathcal{H}_{k,g}$, the moduli space of degree $k$, genus $g$ covers of
$\mathbb{P}^{1}$, up to automorphisms of the target. Following Mumford’s
philosophy, let us begin by introducing a notion of tautological classes. Let
$\mathcal{C}$ be the universal curve and $\mathcal{P}$ the universal
$\mathbb{P}^{1}$-fibration over the Hurwitz space $\mathcal{H}_{k,g}$:

${\mathcal{C}}$${\mathcal{P}}$${\mathcal{H}_{k,g}.}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\scriptstyle{\pi}$

We define the tautological subring of the Hurwitz space
$R^{*}(\mathcal{H}_{k,g})\subseteq A^{*}(\mathcal{H}_{k,g})$ to be the subring
generated by classes of the form
$f_{*}(c_{1}(\omega_{f})^{i}\cdot\alpha^{*}c_{1}(\omega_{\pi})^{j})=\pi_{*}(\alpha_{*}(c_{1}(\omega_{f})^{i})\cdot
c_{1}(\omega_{\pi})^{j})$.

In general, determining the full Chow ring of a moduli space — such as
$\mathcal{M}_{g}$ or $\mathcal{H}_{k,g}$ — may be quite difficult. Having
established a notion of tautological classes, however, it makes sense to split
the study of the intersection theory of a moduli space into two parts:

  * •

Question 1: To what extent are classes tautological? If they exist, what can
we say about the support of non-tautological classes?

  * •

Question 2: What is the structure of the tautological ring? Although the full
Chow ring may be complicated, one hopes that the tautological ring has a more
easily described structure.

In this paper, we provide an answer to Question 1 for $\mathcal{H}_{k,g}$ with
$k\leq 5$. The ground work we develop here will also be important for
addressing Question 2, which we undertake in subsequent work [3].

Before stating our results, we highlight some known results about the Chow
ring of $\mathcal{M}_{g}$ related to Question 1 for context.

  1. (1a)

(codimension 1) Codimension 1 classes are tautological:
$A^{1}(\mathcal{M}_{g})=R^{1}(\mathcal{M}_{g})$ [15].

  2. (1b)

(low genus) For $g\leq 6$, all classes are tautological:
$A^{*}(\mathcal{M}_{g})=R^{*}(\mathcal{M}_{g})$ [22, 13, 14, 16, 24].

  3. (1c)

(bielliptics) In genus $12$, the fundamental class of the bielliptic locus
$B_{12}$ is _not_ tautological: $[B_{12}]\notin R^{*}(\mathcal{M}_{12})$ [26].

######  Remark 1.1.

Building upon the results for Hurwitz spaces in this paper and its sequel [3],
we extend (1b) to prove $A^{*}(\mathcal{M}_{g})=R^{*}(\mathcal{M}_{g})$ for
all $g\leq 9$ in [4].

Meanwhile, for the Hurwitz space $\mathcal{H}_{k,g}$, the previously known
results regarding tautological classes are as follows:

  1. (1a)

(codimension 1) Codimension 1 classes are tautological
$A^{1}(\mathcal{H}_{k,g})=R^{1}(\mathcal{H}_{k,g})$ for $k\leq 5$ [10] and
$k>g-1$ [21]. The general case remains an open conjecture known as the _Picard
rank conjecture_.

  2. (1b)

(low degree) For $k\leq 3$, all classes are tautological:
$A^{*}(\mathcal{H}_{k,g})=R^{*}(\mathcal{H}_{k,g})$. In the case $k=2$, it is
well-known that $A^{*}(\mathcal{H}_{2,g})=\mathbb{Q}$; the case $k=3$ is due
to Patel–Vakil [23].

Our main theorems make significant progress towards answering Question 1 for
the next open cases: the Hurwitz spaces $\mathcal{H}_{4,g}$ and
$\mathcal{H}_{5,g}$. A degree $4$ cover $C\to\mathbb{P}^{1}$ can factor as two
degree two covers $C\to C^{\prime}\to\mathbb{P}^{1}$. Let
$\mathcal{H}_{4,g}^{\mathrm{nf}}\subset\mathcal{H}_{4,g}$ denote the open
locus of _non-factoring covers_ , or equivalently covers whose monodromy group
is not contained in the dihedral group $D_{4}$. By
$R^{*}(\mathcal{H}_{4,g}^{\mathrm{nf}})$ we mean the image of
$R^{*}(\mathcal{H}_{4,g})$ under the restriction map
$A^{*}(\mathcal{H}_{4,g})\to A^{*}(\mathcal{H}_{4,g}^{\mathrm{nf}})$.

######  Theorem 1.2.

If they exist, any non-tautological classes on $\mathcal{H}_{4,g}$ are
supported on the locus of factoring covers or have codimension at least
$(g+3)/4-4$. In other words,

$A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})=R^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})\quad\text{for
all }i<(g+3)/4-4.$

######  Remark 1.3.

The fact that there may be non-tautological classes on the locus of factoring
covers should be compared with (1c). In fact, using van Zelm’s result that
$[B_{12}]$ is not tautological, we establish in [3, Remark 1.10] that
$\mathcal{H}_{4,12}$ indeed possesses non-tautological classes supported on
the factoring locus.

In degree $5$, covers cannot factor, and we obtain the following result.

######  Theorem 1.4.

If they exist, any non-tautological classes on $\mathcal{H}_{5,g}$ have
codimension at least $(g+4)/5-16$. In other words,

$A^{i}(\mathcal{H}_{5,g})=R^{i}(\mathcal{H}_{5,g})\quad\text{for all
}i<(g+4)/5-16.$

Theorems 1.2 and 1.4 are reminiscent of the Madsen–Weiss theorem [19], which
proves Mumford’s conjecture that the stable cohomology of $\mathcal{M}_{g}$ is
a polynomial ring in the kappa classes. Edidin [11, Question 3.34] asked if
the analogue of the Madsen–Weiss theorem holds in the Chow ring
$A^{*}(\mathcal{M}_{g})$, but very little is known about this question. We
view Theorems 1.2 and 1.4 as providing some evidence toward a positive answer
to Edidin’s question. Unlike the case of the stable cohomology of
$\mathcal{M}_{g}$, we will show in [3] that there are many interesting
relations among the tautological classes on $\mathcal{H}_{k,g}$ when $3\leq
k\leq 5$.

### Sketch of the proof

There are three key ingredients to proving Theorems 1.2 and 1.4. The set up we
develop will also be essential for later results determining structure of the
tautological ring in [3]. We shall therefore carry them out in the case $k=3$
as well, which fits into the same framework.

(1) Useful generators: We first explain how structure theorems of
Casnati–Ekedahl for finite covers give rise to a collection of classes on
$\mathcal{H}_{k,g}$, which we term Casnati–Ekedahl (CE) classes. In Theorem
3.10, we show that all CE classes are tautological and that they generate the
tautological ring. An interesting consequence of this is that, for fixed
$k,i$, $\dim R^{i}(\mathcal{H}_{k,g})$ is bounded above, independent of $g$.
(This part works for any $k$; see Remark 3.11).

(2) The good open: For $k=3,4,5$, we define a “good open”
$\mathcal{H}_{k,g}^{\prime}\subseteq\mathcal{H}_{k,g}$. Using our
interpretation of the Casnati–Ekedahl structure theorems, we show that this
“good open” possesses an open embedding inside a vector bundle
$\mathcal{X}_{k,g}^{\prime}$ over a moduli space $\mathcal{B}_{k,g}^{\prime}$
of pairs of vector bundles on $\mathbb{P}^{1}$. The pullbacks of classes along
$A^{*}(\mathcal{B}_{k,g}^{\prime})=A^{*}(\mathcal{X}_{k,g}^{\prime})\to
A^{*}(\mathcal{H}_{k,g}^{\prime})$ are CE classes (essentially by the
definition of CE classes). It follows that $A^{*}(\mathcal{H}_{k,g}^{\prime})$
is generated by tautological classes.

(3) Codimension bounds: By excision, there is a surjection
$A^{*}(\mathcal{H}_{k,g})\to A^{*}(\mathcal{H}_{k,g}^{\prime})$ whose kernel
is generated by classes supported on the complement of
$\mathcal{H}_{k,g}^{\prime}$. Thus, the final step is to bound the codimension
of the complement of $\mathcal{H}_{k,g}^{\prime}$. When $k=3$, it turns out
$\mathcal{H}_{k,g}^{\prime}=\mathcal{H}_{k,g}$, so we recover the result of
Patel–Vakil [23] that all classes are tautological. When $k=4$, the complement
of $\mathcal{H}_{k,g}^{\prime}$ contains the locus of covers that factor
through a double cover of a low-genus curve. Thus, the complement has
codimension 2. However, it turns out that the _non-factoring_ covers in the
complement of $\mathcal{H}_{k,g}^{\prime}$ have codimension at least
$(g+3)/4-4$. This leads to the proof of Theorem 1.2 at the end of Section 5.2.
Finally, for $k=5$, there are no factoring covers, and we show that the
complement of $\mathcal{H}_{k,g}^{\prime}$ has codimension at least
$(g+4)/5-16$. With this, we conclude the proof of Theorem 1.4 at the end of
Section 5.3.

### Acknowledgments

We are grateful to our advisors, Elham Izadi and Ravi Vakil, respectively, for
the many helpful conversations. We are grateful to Aaron Landesman for his
comments and insights.

##  2\. Notation and conventions

We will work over an algebraically closed field of characteristic $0$ or
characteristic $p>5$. All schemes in this paper will be taken over this fixed
field.

###  2.1. Projective bundles

We follow the subspace convention for projective bundles: given a scheme (or
stack) $X$ and a vector bundle $E$ of rank $r$ on $X$, we set

$\mathbb{P}E:=\operatorname{Proj}(\operatorname{Sym}^{\bullet}E^{\vee}),$

so we have the tautological inclusion

$\mathcal{O}_{\mathbb{P}E}(-1)\hookrightarrow\gamma^{*}E,$

where $\gamma:\mathbb{P}E\rightarrow X$ is the structure map. Set
$\zeta:=c_{1}(\mathcal{O}_{\mathbb{P}E}(1))$. With this convention, the Chow
ring of $\mathbb{P}E$ is given by

(2.1)
$A^{*}(\mathbb{P}E)=A^{*}(X)[\zeta]/\langle\zeta^{r}+\zeta^{r-1}c_{1}(E)+\ldots+c_{r}(E)\rangle.$

We call this the _projective bundle theorem_. Note that
$1,\zeta,\zeta^{2},\ldots,\zeta^{r-1}$ form a basis for $A^{*}(\mathbb{P}E)$
as an $A^{*}(X)$-module. Since

$\gamma_{*}\zeta^{i}=\begin{cases}0&\text{if $i\leq r-2$}\\\ 1&\text{if
$i=r-1$,}\end{cases}$

this determines the $\gamma_{*}$ of all classes from $\mathbb{P}E$.

###  2.2. (Equivariant) Intersection Theory

Let $X$ be a scheme and suppose $Z\subseteq X$ is a closed subscheme of
codimension $c$ and $U$ is its open complement. We denote the Chow ring of $X$
with rational coefficients by $A^{*}(X)$. The _excision property_ of Chow is
the right exact sequence

$A^{*-c}(Z)\rightarrow A^{*}(X)\rightarrow A^{*}(U)\rightarrow 0.$

If one knows $A^{*}(X)$, then to find the Chow ring of an open $U\subset X$,
one must describe the image of $A^{*-c}(Z)\rightarrow A^{*}(X)$. If
$\widetilde{Z}\to Z$ is proper and surjective, then pushforward
$A_{*}(\widetilde{Z})\to A_{*}(Z)$ is surjective, see [28, Lemma 1.2]. Given a
graded ring $R=\bigoplus R^{i}$, let

$\operatorname{Trun}^{d}R:=R/\oplus_{i\geq d}R^{d}$

denote the degree $d$ trunction. With this notation, if the complement of
$U\subseteq X$ has codimension $c$, then the excision property implies

(2.2)
$\operatorname{Trun}^{c}A^{*}(X)\xrightarrow{\sim}\operatorname{Trun}^{c}A^{*}(U).$

Chow rings also satisfy the _homotopy property_ : if $V\to X$ is a vector
bundle, then the pullback map $A^{*}(X)\to A^{*}(V)$ is an isomorphism. This
property motivates the definition of equivariant Chow groups as developed by
Edidin-Graham in [12]. Again, we will be using rational coefficients for our
equivariant Chow rings. Let $V$ be a representation of $G$ and suppose $G$
acts freely on $U\subset V$ and the codimension of $V\smallsetminus U$ is
greater than $c$. If $X$ is a smooth scheme and $G$ is a linear algebraic
group acting on $X$, Edidin and Graham defined

$A^{c}_{G}(X):=A^{c}((X\times U)/G),$

and showed that the graded ring $A^{*}_{G}(X)$ possesses an intersection
product. For quotient stacks, one has $A^{*}([X/G])\cong A^{*}_{G}(X)$ by [12,
Proposition 19], which may suffice as the definition of the Chow rings of all
stacks appearing in this paper.

By Edidin-Graham [12, Proposition 5], there is also an excision sequence for
equivariant Chow groups. Let $Z\subseteq X$ be a $G$-invariant closed
subscheme of codimension $c$ and $U$ its complement. Then there is an exact
sequence

$A^{*-c}_{G}(Z)\rightarrow A^{*}_{G}(X)\rightarrow A^{*}_{G}(U)\rightarrow 0.$

The following lemma is a useful consequence of the excision sequence. See also
[27, Theorem 2] for a much more general statement.

######  Lemma 2.1.

Suppose $P\rightarrow X$ is a principal $\mathbb{G}_{m}$-bundle. Then
$A^{*}(P)=A^{*}(X)/\langle c_{1}(L)\rangle$, where $L$ is the corresponding
line bundle.

###### Proof.

By the correspondence between principal $\mathbb{G}_{m}$-bundles and line
bundles over $X$, $P$ is the complement of the zero section of the line bundle
$L\rightarrow X$. The excision sequence gives

$A^{*-1}(X)\rightarrow A^{*}(L)\rightarrow A^{*}(P)\rightarrow 0.$

Under the identification of $A^{*}(L)$ with $A^{*}(X)$, the first map in the
above exact sequence is multiplication by $c_{1}(L)$, from which the result
follows. ∎

###  2.3. The Hurwitz space

Given a scheme $S$, an $S$ point of the _parametrized Hurwitz scheme_
$\mathcal{H}_{k,g}^{\dagger}$ is the data of a finite, flat map
$C\to\mathbb{P}^{1}\times S$, of constant degree $k$ so that the composition
$C\to\mathbb{P}^{1}\times S\to S$ is smooth with geometrically connected
fibers. (We do not impose the condition that a cover $C\to\mathbb{P}^{1}$ be
simply branched.)

The _unparametrized Hurwitz stack_ is the $\operatorname{PGL}_{2}$ quotient of
the parametrized Hurwitz scheme. There is also a natural action of
$\operatorname{SL}_{2}$ on $\mathcal{H}_{k,g}^{\dagger}$ (via
$\operatorname{SL}_{2}\subset\operatorname{GL}_{2}\to\operatorname{PGL}_{2}$).
The natural map
$[\mathcal{H}_{k,g}^{\dagger}/\operatorname{SL}_{2}]\to[\mathcal{H}_{k,g}^{\dagger}/\operatorname{PGL}_{2}]$
is a $\mu_{2}$ banded gerbe. It is a general fact that _with rational
coefficients_ , the pullback map along a gerbe banded by a finite group is an
isomorphism [24, Section 2.3]. In particular, since we work with rational
coefficients throughout,
$A^{*}([\mathcal{H}_{k,g}^{\dagger}/\operatorname{PGL}_{2}])\cong
A^{*}([\mathcal{H}^{\dagger}_{k,g}/\operatorname{SL}_{2}])$. It thus suffices
to prove all statements for the $\operatorname{SL}_{2}$ quotient
$[\mathcal{H}^{\dagger}_{k,g}/\operatorname{SL}_{2}]$, which we shall denote
by $\mathcal{H}_{k,g}$ from now on.

Explicitly, the $\operatorname{SL}_{2}$ quotient $\mathcal{H}_{k,g}$ is the
stack whose objects over a scheme $S$ are families $(C\rightarrow P\rightarrow
S)$ where $P=\mathbb{P}V\rightarrow S$ is the projectivization of a rank $2$
vector bundle $V$ with trivial determinant, $C\rightarrow P$ is a finite,
flat, finitely presented morphism of constant degree $k$, and the composition
$C\rightarrow S$ has smooth fibers of genus $g$. The benefit of working with
$\mathcal{H}_{k,g}$ is that the $\operatorname{SL}_{2}$ quotient is equipped
with a universal $\mathbb{P}^{1}$-bundle $\mathcal{P}\to\mathcal{H}_{k,g}$
that has a relative degree one line bundle $\mathcal{O}_{\mathcal{P}}(1)$ (a
$\mathbb{P}^{1}$-fibration does not). Working with this
$\mathbb{P}^{1}$-bundle simplifies our intersection theory calculations.

##  3\. The Casnati–Ekedahl structure theorem

The main objective of this section is to give a description of stacks of low-
degree covers using structure theorems of Casnati–Ekedahl. The descriptions in
Sections 3.1–3.3 are likely well-known but have not previously been spelled
out in the language of stacks except in the degree 3 case [2], as we shall
need them. On a first pass, the reader may wish to skip forward to Section
3.4, where we introduce natural classes coming from these structure theorems
and prove that they generate the tautological ring.

Generalizing earlier results of Schreyer [25] and Miranda [20],
Casnati–Ekedahl [6] proved a general structure theorem for degree $k$,
Gorenstein covers of integral schemes. Given a degree $k$ cover
$\alpha:X\rightarrow Y$ where $Y$ is integral, one obtains an exact sequence

(3.1)
$0\rightarrow\mathcal{O}_{Y}\rightarrow\alpha_{*}\mathcal{O}_{X}\rightarrow
E_{\alpha}^{\vee}\rightarrow 0,$

where $E_{\alpha}$ is a vector bundle of rank $k-1$ on $Y$. When $\alpha$ is
Gorenstein, $\alpha_{*}\mathcal{O}_{X}\cong(\alpha_{*}\omega_{\alpha})^{\vee}$
by Serre duality. Pulling back and using adjunction, we therefore obtain a map

(3.2)
$\omega_{\alpha}^{\vee}\to(\alpha^{*}\alpha_{*}\omega_{\alpha})^{\vee}\rightarrow\alpha^{*}E_{\alpha}^{\vee},$

which induces a map $X\rightarrow\mathbb{P}E^{\vee}$ that factors
$\alpha:X\rightarrow Y$.

######  Example 3.1 (Covers of $\mathbb{P}^{1}$).

If $\alpha:C\to\mathbb{P}^{1}$ is a degree $k$, genus $g$ cover, then we have

$\deg(E_{\alpha}^{\vee})=\deg(\alpha_{*}\mathcal{O}_{C})=\chi(\alpha_{*}\mathcal{O}_{C})-k=\chi(\mathcal{O}_{C})-k=1-g-k,$

so $\deg(E_{\alpha})=g+k-1$. The map $C\to\mathbb{P}E_{\alpha}^{\vee}$ factors
the canonical embedding $C\hookrightarrow\mathbb{P}^{g-1}$, where the map
$\mathbb{P}E^{\vee}_{\alpha}\to\mathbb{P}^{g-1}$ is given by the line bundle
$\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}(1)\otimes\omega_{\mathbb{P}^{1}}$.
Each linear space in the image of
$\mathbb{P}E_{\alpha}^{\vee}\to\mathbb{P}^{g-1}$ is the span of the image of
the corresponding fiber of $C\to\mathbb{P}^{1}$.

The Casnati–Ekedahl structure theorem below gives a resolution of the ideal
sheaf of $X$ inside of $\mathbb{P}E^{\vee}_{\alpha}$ [6]; see also [9].

######  Theorem 3.2 (Casnati–Ekedahl, Theorem 2.1 of [6]).

Let $X$ and $Y$ be schemes, $Y$ integral and let $\alpha:X\rightarrow Y$ be a
Gorenstein cover of degree $k\geq 3$. There exists a unique
$\mathbb{P}^{k-2}$-bundle $\gamma:\mathbb{P}\rightarrow Y$ and an embedding
$i:X\hookrightarrow\mathbb{P}$ such that $\alpha=\gamma\circ i$ and
$X_{y}:=\alpha^{-1}(y)\subset\gamma^{-1}(y)\cong\mathbb{P}^{k-2}$ is a
nondegenerate arithmetically Gorenstein subscheme for each $y\in Y$. Moreover,
the following properties hold.

  1. (1)

$\mathbb{P}\cong\mathbb{P}E^{\vee}_{\alpha}$ where
$E^{\vee}_{\alpha}:=\operatorname{coker}(\mathcal{O}_{Y}\rightarrow\alpha_{*}\mathcal{O}_{X})$.

  2. (2)

The composition
$\alpha^{*}E_{\alpha}\to\alpha^{*}\alpha_{*}\omega_{\alpha}\to\omega_{\alpha}$
is surjective (dually, (3.2) does not drop rank) and the ramification divisor
$R$ satisfies
$\mathcal{O}_{X}(R)\cong\omega_{\alpha}\cong\mathcal{O}_{X}(1):=i^{*}\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}(1)$.

  3. (3)

There exists an exact sequence of locally free $\mathcal{O}_{\mathbb{P}}$
sheaves

(3.3)
$0\rightarrow\gamma^{*}F_{k-2}(-k)\rightarrow\gamma^{*}F_{k-3}(-k+2)\rightarrow\cdots\rightarrow\gamma^{*}F_{1}(-2)\rightarrow\mathcal{O}_{\mathbb{P}}\rightarrow\mathcal{O}_{X}\rightarrow
0.$

where $F_{i}$ is locally free on $Y$. The restriction of the exact sequence
above to a fiber gives a minimal free resolution of $X_{y}:=\alpha^{-1}(y)$.
This sequence is unique up to unique isomorphism. Moreover the resolution is
self-dual, meaning there is a canonical isomorphism
$\mathcal{H}om_{\mathcal{O}_{\mathbb{P}}}(F_{i},F_{k-2})\cong F_{k-2-i}$. The
ranks of the $F_{i}$ are

$\operatorname{rank}F_{i}=\frac{i(k-2-i)}{k-1}{{k}\choose{i+1}}.$

  4. (4)

If $\mathbb{P}\cong\mathbb{P}E^{\prime\vee}$, then $E^{\prime}\cong E$ if and
only if $F_{k-2}\cong\det E^{\prime}$ in the resolution (3.3) computed with
respect to the polarization $\mathcal{O}_{\mathbb{P}E^{\prime\vee}}(1)$.

######  Remark 3.3.

There is a canonical isomorphism $F_{k-2}\cong\det E_{\alpha}$, which we
describe here. Following [6, p. 446], let $A_{1}$ be the image of
$\gamma^{*}F_{1}(-2)\to\mathcal{O}_{\mathbb{P}}$, and for $2\leq i\leq k-3$,
let $A_{i}$ denote the image of
$\gamma^{*}F_{i}(-i-1)\to\gamma^{*}F_{i-1}(-i)$. We set $A_{k-2}$ to be
$\gamma^{*}F_{k-2}(-k)$. We have exact sequences

(3.4) $0\rightarrow
A_{1}\rightarrow\mathcal{O}_{\mathbb{P}}\rightarrow\mathcal{O}_{X}\rightarrow
0$

and

(3.5) $0\rightarrow A_{i+1}\rightarrow\gamma^{*}F_{i}(-i-1)\rightarrow
A_{i}\rightarrow 0.$

First, we claim that

$R^{j}\gamma_{*}\gamma^{*}F_{i}(-i-1)\cong\begin{cases}F_{k-2}\otimes\det
E^{\vee}&\text{if $i=j=k-2$}\\\ 0&\text{otherwise.}\end{cases}$

This is very similar to the calculations of [6, p. 446], but twisted up by
one. To prove the first case above, we note that the dualizing sheaf of
$\gamma$ is $\omega_{\gamma}=(\gamma^{*}\det E)(-k+1)$, and apply Serre
duality for $\gamma$, which is of relative dimension $k-2$. The other cases
follow from the theorem on cohomology and base change and the well-known
cohomology of line bundles on projective space. Tensoring the exact sequences
of (3.5) by $\mathcal{O}_{\mathbb{P}}(1)$ and pushing forward by $\gamma$, the
boundary maps provide us with isomorphisms

$\gamma_{*}A_{1}(1)\cong R^{1}\gamma_{*}A_{2}(1)\cong
R^{2}\gamma_{*}A_{3}(1)\cong\cdots\cong
R^{k-1}\gamma_{*}(\gamma^{*}F_{k-2}(-k+1))=0.$

Similarly, we have

$R^{1}\gamma_{*}A_{1}(1)\cong R^{2}\gamma_{*}A_{2}(1)\cong\cdots\cong
R^{k-2}\gamma_{*}(\gamma^{*}F_{k-2}(-k+1))\cong F_{k-2}\otimes\det E^{\vee}.$

On the other hand, tensoring (3.4) with $\mathcal{O}_{\mathbb{P}}(1)$ and
pushing forward by $\gamma$ we obtain

$0\rightarrow E\rightarrow\alpha_{*}\mathcal{O}_{X}(1)\rightarrow
R^{1}\gamma_{*}A_{1}(1)\rightarrow 0.$

Recall that $\mathcal{O}_{X}(1)\cong\omega_{\alpha}$, so dualizing (3.1) we
see that the cokernel of the left map is $\mathcal{O}_{Y}$. By the universal
property of cokernel, we obtain an isomorphism

$\mathcal{O}_{Y}\to R^{1}\gamma_{*}A_{1}(1)\cong F_{k-2}\otimes\det E^{\vee},$

or equivalently, an isomorphism $F_{k-2}\cong\det E$.

In the cases $k=3,4,5$, using self-duality, only pullbacks of the bundles
$E_{\alpha}$ and $F_{1}$ and determinants and tensor products thereof appear
in the resolution (3.3). We set $F_{\alpha}:=F_{1}$. Twisting up (3.3) by
$\mathcal{O}_{\mathbb{P}}(2)$ and pushing forward by $\gamma$, we see that

$F_{\alpha}=\ker(\operatorname{Sym}^{2}E_{\alpha}\twoheadrightarrow\alpha_{*}\omega_{\alpha}^{\otimes
2}).$

In these low degrees $k=3,4,5$, there is a special map $\delta_{\alpha}$ in
the resolution (3.3) from which one can reconstruct the cover. Furthermore, as
we shall explain, it is an _open condition_ on a space of global sections of
all such maps $\delta$ to define a finite cover. This is what distinguishes
$k=3,4,5$ and lies at the core of why our methods work in these low degrees.
Below we present an equivalence of categories between the category of degree
$k$, Gorenstein covers of a scheme $S$ and a category of certain linear
algebraic data on $S$. The main content of this step is to point out the
“essential data” of a cover, which we may remember instead of the entire
resolution. For the case of triple covers, this was done by Bolognesi–Vistoli
[2]. We give a slightly different perspective below.

###  3.1. The category of triple covers

Let $\mathrm{Trip}(S)$ denote the category of Gorenstein triple covers of a
scheme $S$: the objects are Gorenstein triple covers $\alpha:X\to S$ and the
arrows are isomorphisms over $S$. Specializing (3.3) to the case $k=3$,
associated to a cover $\alpha:X\to S$, one obtains a rank $2$ vector bundle
$E_{\alpha}$ and an exact sequence

$0\rightarrow\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}(-3)\otimes\gamma^{*}\det
E_{\alpha}\xrightarrow{\delta_{\alpha}}\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}\rightarrow\mathcal{O}_{X}\rightarrow
0.$

Conversely, from the above sequence, we can recover the cover
$\alpha:X\rightarrow S$. Indeed, the map $\delta_{\alpha}$ is a global section
in
$H^{0}(\mathbb{P}E_{\alpha}^{\vee},\mathcal{O}_{\mathbb{P}E^{\vee}_{\alpha}}(3)\otimes\gamma^{*}\det
E_{\alpha}^{\vee})$, whose zero locus inside of $\mathbb{P}E_{\alpha}^{\vee}$
is $X$. Meanwhile, given any rank $2$ vector bundle $E$ on $S$, it is an open
condition on the space of sections
$H^{0}(\mathbb{P}E^{\vee},\mathcal{O}_{\mathbb{P}E^{\vee}}(3)\otimes\gamma^{*}\det
E^{\vee})$ for the vanishing of a section $\delta$ to define a finite triple
cover: $\delta$ must not be the zero polynomial on any fiber of
$\mathbb{P}E\to S$. Equivalently, if

(3.6) $\Phi:H^{0}(S,\operatorname{Sym}^{3}E\otimes\det
E^{\vee})\xrightarrow{\sim}H^{0}(\mathbb{P}E^{\vee},\mathcal{O}_{\mathbb{P}E^{\vee}}(3)\otimes\gamma^{*}\det
E^{\vee})$

denotes the natural isomorphism, then $V(\delta)\subset\mathbb{P}E^{\vee}$ is
a Gorenstein triple cover so long as $\Phi^{-1}(\delta)$ is non-vanishing.

This “essential data” is captured by a category $\mathrm{Trip}^{\prime}(S)$ we
now define. The objects of $\mathrm{Trip}^{\prime}(S)$ are pairs $(E,\eta)$
where $E$ is a rank $2$ vector bundle and $\eta\in
H^{0}(S,\operatorname{Sym}^{3}E\otimes\det E^{\vee})$ is non-vanshing on $S$.
An arrow $(E_{1},\eta_{1})\to(E_{2},\eta_{2})$ in $\mathrm{Trip}^{\prime}(S)$
is an isomorphism $E_{1}\to E_{2}$ that sends $\eta_{1}$ into $\eta_{2}$.
There is a functor $\mathrm{Trip}(S)\to\mathrm{Trip}^{\prime}(S)$ that sends
$\alpha:X\to S$ to the pair $(E_{\alpha},\Phi^{-1}(\delta_{\alpha}))$. There
is also a functor $\mathrm{Trip}^{\prime}(S)\to\mathrm{Trip}(S)$ that sends a
pair $(E,\eta)$ to the triple cover $V(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to
S$. The following is essentially a restatement of [6, Theorem 3.4], which was
proved earlier by Miranda [20].

######  Theorem 3.4 (Miranda, Casnati–Ekedahl).

The functors above define an equivalence of categories
$\mathrm{Trip}(S)\cong\mathrm{Trip}^{\prime}(S)$.

###  3.2. The category of quadruple covers

Let $\mathrm{Quad}(S)$ denote the category whose objects are Gorenstein
quadruple covers $\alpha:X\to S$ and whose arrows are isomorphisms over $S$.
Associated to a degree $4$ cover $\alpha:X\to S$, there is a rank $3$ vector
bundle $E_{\alpha}$ and a rank $2$ vector bundle $F_{\alpha}$ and a resolution

(3.7) $0\rightarrow\gamma^{*}\det
E_{\alpha}(-4)\rightarrow\gamma^{*}F_{\alpha}(-2)\xrightarrow{\delta_{\alpha}}\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}\rightarrow\mathcal{O}_{X}\rightarrow
0.$

The section $\delta_{\alpha}\in
H^{0}(\mathbb{P}E_{\alpha}^{\vee},\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}(2)\otimes\gamma^{*}F^{\vee})$
corresponds to a relative pencil of quadrics. The cover $X$ can be recovered
as the vanishing locus of $\delta_{\alpha}$. By comparing (3.7) with the
Koszul resolution of $\delta_{\alpha}$,

(3.8) $0\rightarrow\gamma^{*}\det
F_{\alpha}(-4)\rightarrow\gamma^{*}F_{\alpha}(-2)\xrightarrow{\delta_{\alpha}}\mathcal{O}_{\mathbb{P}E^{\vee}_{\alpha}}\rightarrow\mathcal{O}_{X}\rightarrow
0,$

the uniqueness of Theorem 3.2 (3) induces a distinguished isomorphism
$\phi_{\alpha}:\det F_{\alpha}\cong\det E_{\alpha}$ (see [6, p. 450]).

We now define a category $\mathrm{Quad}^{\prime}(S)$ of the corresponding
linear algebraic data of a quadruple cover. Given vector bundles $E,F$ on $S$,
there is a natural isomorphism

(3.9)
$\Phi:H^{0}(S,F^{\vee}\otimes\operatorname{Sym}^{2}E)\xrightarrow{\sim}H^{0}(\mathbb{P}E^{\vee},\gamma^{*}F^{\vee}\otimes\mathcal{O}_{\mathbb{P}E^{\vee}}(2)).$

######  Definition 3.5.

Let $E$ and $F$ be vector bundles of ranks $3$ and $2$ respectively on $S$. We
say that a section $\eta\in H^{0}(S,F^{\vee}\otimes\operatorname{Sym}^{2}E)$
_has the right codimension_ at $s\in S$ if the vanishing locus of $\Phi(\eta)$
restricted to the fiber over $s\in S$ is zero dimensional.

The objects of $\mathrm{Quad}^{\prime}(S)$ are tuples $(E,F,\phi,\eta)$ where
$E$ and $F$ are vector bundles of ranks $3$ and $2$ respectively, $\phi:\det
F\cong\det E$ is an isomorphism and $\eta\in
H^{0}(S,F^{\vee}\otimes\operatorname{Sym}^{2}E)$ has the right codimension at
all $s\in S$. An arrow in $\mathrm{Quad}^{\prime}(S)$ is a pair of
isomorphisms $\xi:E_{1}\to E_{2}$, and $\psi:F_{1}\to F_{2}$, such that the
following diagrams commute

${F_{1}}$${\operatorname{Sym}^{2}E_{1}}$${F_{2}}$${\operatorname{Sym}^{2}E_{2}}$$\scriptstyle{\psi}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\operatorname{Sym}^{2}\xi}$$\scriptstyle{\eta_{2}}$${\det
F_{1}}$${\det E_{1}}$${\det F_{2}}$${\det
E_{2}.}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\det\psi}$$\scriptstyle{\det\xi}$$\scriptstyle{\phi_{2}}$

There is a functor $\mathrm{Quad}(S)\to\mathrm{Quad}^{\prime}(S)$ that sends
$\alpha:X\to S$ to $(E_{\alpha},F_{\alpha},\phi_{\alpha},\eta_{\alpha})$ where
$\eta_{\alpha}:=\Phi^{-1}(\delta_{\alpha})$. There is also a functor
$\mathrm{Quad}^{\prime}(S)\to\mathrm{Quad}(S)$ that sends a tuple
$(E,F,\phi,\eta)$ to the quadruple cover
$V(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$. The following is essentially a
restatement of [6, Theorem 4.4].

######  Theorem 3.6 (Casnati–Ekedahl).

The functors above define an equivalence of categories
$\mathrm{Quad}(S)\cong\mathrm{Quad}^{\prime}(S)$.

###### Proof.

Work of Casnati–Ekedahl established that the composition
$\mathrm{Quad}(S)\to\mathrm{Quad}^{\prime}(S)\to\mathrm{Quad}(S)$ is
equivalent to the identity, as $V(\delta_{\alpha})\to S$ is naturally
identified with the cover $\alpha:X\to S$.

We must provide a natural isomorphism of
$\mathrm{Quad}^{\prime}(S)\to\mathrm{Quad}(S)\to\mathrm{Quad}^{\prime}(S)$
with the identity on $\mathrm{Quad}^{\prime}(S)$. Suppose we are given
$(E,F,\phi,\eta)\in\mathrm{Quad}^{\prime}(S)$. We want to define an arrow
$(E,F,\phi,\eta)\to(E_{\alpha},F_{\alpha},\phi_{\alpha},\eta_{\alpha})$. Let
$X=V(\Phi(\eta))\subset\mathbb{P}E^{\vee}$, and $\alpha:X\to S$. The Kosul
resolution of $\Phi(\eta)$ is

$0\rightarrow(\gamma^{*}\det
F)(-4)\rightarrow\gamma^{*}F(-2)\xrightarrow{\Phi(\eta)}\mathcal{O}_{\mathbb{P}E^{\vee}}\rightarrow\mathcal{O}_{X}\rightarrow
0$

and is exact since $\eta$ has the right codimension at all $s\in S$. We break
this into two sequences

(3.10) $0\rightarrow(\gamma^{*}\det
F)(-4)\rightarrow\gamma^{*}F(-2)\rightarrow A\rightarrow 0$

and

(3.11) $0\rightarrow
A\rightarrow\mathcal{O}_{\mathbb{P}E^{\vee}}\rightarrow\mathcal{O}_{X}\rightarrow
0.$

Pushing forward (3.11) we get a short exact sequence on $S$:

$0\rightarrow\mathcal{O}_{S}\rightarrow\alpha_{*}\mathcal{O}_{X}\rightarrow
R^{1}\gamma_{*}A\rightarrow 0.$

Using (3.10), we obtain isomorphisms $R^{1}\gamma_{*}A\cong
R^{2}\gamma_{*}(\gamma^{*}\det F)(-4)\cong\det F\otimes
R^{2}\gamma_{*}\mathcal{O}_{\mathbb{P}E^{\vee}}(-4)$. Because the dualizing
sheaf of $\gamma$ is
$\omega_{\gamma}=\mathcal{O}_{\mathbb{P}E^{\vee}}(-3)\otimes\gamma^{*}\det E$,
using Serre duality, we obtain an isomorphism
$R^{2}\gamma_{*}\mathcal{O}_{\mathbb{P}E^{\vee}}(-4)\cong\det E^{\vee}\otimes
E^{\vee}$. Now the universal property of cokernel produces an isomorphism

$E_{\alpha}^{\vee}=\operatorname{coker}(\mathcal{O}_{S}\to\alpha_{*}\mathcal{O}_{X})\xrightarrow{\sim}R^{1}\gamma_{*}A\cong\det
F\otimes\det E^{\vee}\otimes E^{\vee}.$

Meanwhile $\phi$ determines an isomorphism $\det F\otimes\det
E^{\vee}\cong\mathcal{O}_{S}$. Composing with this, and dualizing, we obtain
an isomorphism $\xi:E\to E_{\alpha}$. Next, we have a commuting diagram

${0}$${F}$${\operatorname{Sym}^{2}E}$${\alpha_{*}\mathcal{O}_{X}(2)}$${0}$${0}$${F_{\alpha}}$${\operatorname{Sym}^{2}E_{\alpha}}$${\alpha_{*}\omega_{\alpha}^{\otimes
2}}$${0}$$\scriptstyle{\psi}$$\scriptstyle{\eta}$$\scriptstyle{\operatorname{Sym}^{2}\xi}$$\scriptstyle{\eta_{\alpha}}$

where the left vertical map is induced by the universal property of kernel.
Note that for any $t\in\mathcal{O}_{S}^{\times}(S)$, the diagram

(3.12)
${F}$${\operatorname{Sym}^{2}E}$${F_{\alpha}}$${\operatorname{Sym}^{2}E_{\alpha}}$$\scriptstyle{t^{2}\cdot\psi}$$\scriptstyle{\eta}$$\scriptstyle{\operatorname{Sym}^{2}(t\cdot\xi)}$$\scriptstyle{\eta_{\alpha}}$

also commutes. Finally, the cover $\alpha$ determines an isomorphism
$\phi_{\alpha}:\det F_{\alpha}\cong\det E_{\alpha}$. It may not agree with
$\phi$, but since the maps below involve isomorphisms of line bundles, there
exists some $t\in\mathcal{O}_{S}^{\times}(S)$ such that the following diagram
commutes

${\det F}$${\det E}$${\det F_{\alpha}}$${\det
E_{\alpha}.}$$\scriptstyle{\phi}$$\scriptstyle{t\cdot\det\psi}$$\scriptstyle{\det\xi}$$\scriptstyle{\phi_{\alpha}}$

Since $E$ is rank $3$ and $F$ is rank $2$, this implies the diagram

(3.13) ${\det F}$${\det E}$${\det F_{\alpha}}$${\det
E_{\alpha}}$$\scriptstyle{\phi}$$\scriptstyle{\det(t^{2}\cdot\psi)}$$\scriptstyle{\det(t\cdot\xi)}$$\scriptstyle{\phi_{\alpha}}$

also commutes. Thus, the pair of isomorphisms $t\cdot\xi:E\to E_{\alpha}$ and
$t^{2}\cdot\psi:F\to F_{\alpha}$ determine an arrow
$(E,F,\phi,\eta)\to(E_{\alpha},F_{\alpha},\phi_{\alpha},\eta_{\alpha})$. ∎

###  3.3. The category of regular pentagonal covers

By the Casnati–Ekedahl theorem, each degree $5$ Gorenstein cover $\alpha:X\to
S$ determines a resolution

(3.14) $0\rightarrow\gamma^{*}\det
E_{\alpha}(-5)\rightarrow\gamma^{*}(F_{\alpha}^{\vee}\otimes\det
E_{\alpha})(-3)\xrightarrow{\delta_{\alpha}}\gamma^{*}F_{\alpha}(-2)\rightarrow\mathcal{O}_{\mathbb{P}}\rightarrow\mathcal{O}_{X}\rightarrow
0,$

where $E_{\alpha}$ has rank $4$ and $F_{\alpha}$ has rank $5$. Casnati showed
that the map $\delta_{\alpha}$ is alternating in the sense that it can be
identified with a section of $\wedge^{2}\pi^{*}F_{\alpha}\otimes\gamma^{*}\det
E_{\alpha}^{\vee}(1)$. For any pair of vector bundles $E$ and $F$, via push-
pull, we have an identification

(3.15) $\Phi:H^{0}(S,\mathcal{H}om(E^{\vee}\otimes\det
E,\wedge^{2}F))\xrightarrow{\sim}H^{0}(\mathbb{P}E^{\vee},\gamma^{*}(\wedge^{2}F\otimes\det
E^{\vee})(1)).$

Hence, $\delta_{\alpha}$ corresponds to a map
$\eta_{\alpha}:=\Phi^{-1}(\delta_{\alpha}):E_{\alpha}^{\vee}\otimes\det
E_{\alpha}\to\wedge^{2}F_{\alpha}$. Throughout this section we shall write
$E^{\prime}:=E^{\vee}\otimes\det E$. A degree $5$ cover $\alpha:X\to S$ is
called _regular_ if $\eta_{\alpha}$ is injective as a map of vector bundles
(i.e. the cokernel of $\eta_{\alpha}$ is locally free). Casnati notes that if
$\alpha^{-1}(s)$ is a local complete intersection scheme for all $s\in S$,
then $\alpha$ is regular, so all covers we need will be regular. We let
$\mathrm{Pent}(S)$ denote the category whose objects are regular, degree $5$
Gorenstein covers $\alpha:X\to S$ and arrows are isomorphisms over $S$.

Regular degree $5$ covers have a nice geometric description. Indeed, if the
cover is regular, then $\eta_{\alpha}$ corresponds to an injective map
$E^{\prime}_{\alpha}\rightarrow\wedge^{2}F_{\alpha}$, which induces an
embedding

(3.16)
$\mathbb{P}E_{\alpha}^{\prime}\hookrightarrow\mathbb{P}(\wedge^{2}F_{\alpha}).$

Given a section $\delta\in
H^{0}(\mathbb{P}E^{\vee},\gamma^{*}(\wedge^{2}F\otimes\det E^{\vee})(1))$, we
let $D(\delta)\subset\mathbb{P}E^{\vee}$ be the subscheme defined by the
vanishing of $4\times 4$ Pfaffians of $\delta$. When $\alpha$ is regular, we
can recover $X=D(\delta_{\alpha})$, which is also the same as the scheme
defined by the $3\times 3$ minors of $\delta_{\alpha}$ (Proposition 3.5 of
[7]). These $3\times 3$ minors are pullbacks to
$\mathbb{P}E_{\alpha}^{\prime}$ along (3.16) of the equations that define the
Grassmannian bundle $G(2,F_{\alpha})\subset\mathbb{P}(\wedge^{2}F_{\alpha})$
under its relative Plücker embedding. Using a resolution of the relative
Grassmannian, Casnati obtains another resolution of $\mathcal{O}_{X}$ in
equation (3.5.2) of [7]. Comparing this resolution with (3.14), the uniqueness
of Theorem 3.2 (2) induces a distinguished isomorphism
$\epsilon:F_{\alpha}\otimes\det F_{\alpha}^{\vee}\otimes(\det
E_{\alpha})^{\otimes 2}\to F_{\alpha}$ (see p. 467 of [7]). Moreover, both of
these vector bundles arise as subbundles of $\operatorname{Sym}^{2}E_{\alpha}$
and the projectivization of $\epsilon$ induces the identity on points (as it
must be the restriction of the identity on
$\mathbb{P}(\operatorname{Sym}^{2}E_{\alpha})$). Hence, we obtain an
isomorphism of line bundles

$\mathcal{O}_{\mathbb{P}F_{\alpha}}(1)\otimes\det
F_{\alpha}^{\vee}\otimes(\det E_{\alpha})^{\otimes
2}\cong\mathcal{O}_{\mathbb{P}(F_{\alpha}\otimes\det
F_{\alpha}^{\vee}\otimes\det
E_{\alpha}^{2})}(1)\cong\epsilon^{*}\mathcal{O}_{\mathbb{P}F_{\alpha}}(1)=\mathcal{O}_{\mathbb{P}F_{\alpha}}(1).$

which induces a distinguished isomorphism $\phi_{\alpha}:(\det
E_{\alpha})^{\otimes 2}\cong\det F_{\alpha}$.

Now we define a category $\mathrm{Pent}^{\prime}(S)$ that keeps track of the
associated linear algebraic data of regular degree $5$ covers.

######  Definition 3.7.

Suppose we are given vector bundles $E$ and $F$ on $S$ of ranks $4$ and $5$.
Let $\eta\in H^{0}(S,\mathcal{H}om(E^{\prime},\wedge^{2}F))$ be a global
section. We say $\eta$ has the _right codimension_ if every fiber of
$D(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$ is $0$-dimensional and
$\eta:E^{\prime}\to\wedge^{2}F$ is injective with locally free cokernel.

We define $\mathrm{Pent}^{\prime}(S)$ to be the category whose objects are
tuples $(E,F,\phi,\eta)$ where $E$ and $F$ are vector bundles on $S$ of ranks
$4$ and $5$ respectively, $\phi$ is an isomorphism $(\det E)^{\otimes
2}\cong\det F$ and $\eta\in H^{0}(S,\mathcal{H}om(E^{\vee}\otimes\det
E,\wedge^{2}F))$ has the right codimension. An arrow
$(E_{1},F_{1},\phi_{1},\eta_{1})\to(E_{2},F_{2},\phi_{2},\eta_{2})$ in
$\mathrm{Pent}^{\prime}(S)$ is pair of isomorphisms $\xi:E_{1}\to E_{2}$ and
$\psi:F_{1}\to F_{2}$ such that the following two diagrams commute

${E_{1}^{\prime}}$${\wedge^{2}F_{1}}$${E_{2}^{\prime}}$${\wedge^{2}F_{2}}$$\scriptstyle{\det\xi\otimes(\xi^{-1})^{\vee}}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\wedge^{2}\psi}$$\scriptstyle{\eta_{2}}$${\det
E_{1}^{\otimes 2}}$${\det F_{1}}$${\det E_{2}^{\otimes 2}}$${\det
F_{2}.}$$\scriptstyle{(\det\xi)^{\otimes
2}}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\det\psi}$$\scriptstyle{\phi_{2}}$

There is a functor $\mathrm{Pent}(S)\to\mathrm{Pent}^{\prime}(S)$ that sends
$\alpha:X\to S$ to the tuple
$(E_{\alpha},F_{\alpha},\phi_{\alpha},\eta_{\alpha})$. There is also a functor
$\mathrm{Pent}^{\prime}(S)\to\mathrm{Pent}(S)$ that sends a tuple
$(E,F,\phi,\eta)$ to the degree $5$ cover
$D(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$. The following is essentially a
restatement of [7, Theorem 3.8].

######  Theorem 3.8 (Casnati).

The above functors define an equivalence of categories
$\mathrm{Pent}(S)\cong\mathrm{Pent}^{\prime}(S)$.

###### Proof.

The fact that
$\mathrm{Pent}(S)\to\mathrm{Pent}^{\prime}(S)\to\mathrm{Pent}(S)$ is
equivalent to the identity was established by Casnati. We provide further
details here that
$\mathrm{Pent}^{\prime}(S)\to\mathrm{Pent}(S)\to\mathrm{Pent}^{\prime}(S)$ is
naturally isomorphic to the identity on $\mathrm{Pent}^{\prime}(S)$. Let
$(E,F,\phi,\eta)\in\mathrm{Pent}(S)$ be given and let $X=D(\Phi(\eta))$ and
$\alpha:X\to S$. By (3.5.2) of [7], $\mathcal{O}_{X}$ admits a resolution

$\displaystyle 0\rightarrow\gamma^{*}(\det F^{-2}\otimes\det
E^{5}(-5)\rightarrow\gamma^{*}(F^{\vee}\otimes\det F^{-1}\otimes\det
E^{3})(-3)$ $\displaystyle\qquad\qquad\qquad\rightarrow\gamma^{*}(F\otimes\det
F^{-1}\otimes\det
E^{2})(-2)\rightarrow\mathcal{O}_{\mathbb{P}E^{\vee}}\rightarrow\mathcal{O}_{X}\rightarrow
0.$

Let $A_{1}$ be the image of $\gamma^{*}(F\otimes\det F^{-1}\otimes\det
E^{2})(-2)\to\mathcal{O}_{\mathbb{P}E^{\vee}}$. When we push forward the above
equation by $\gamma$, we obtain

$0\rightarrow\mathcal{O}_{S}\rightarrow\alpha_{*}\mathcal{O}_{X}\rightarrow
R^{1}\gamma_{*}A_{1}\rightarrow 0.$

We use a similar method as in Remark 3.3 to produce isomorphisms

$E_{\alpha}\cong R^{1}\gamma_{*}A_{1}\cong R^{2}\gamma_{*}A_{2}\cong
R^{3}\gamma_{*}(\gamma^{*}(\det F^{-2}\otimes\det E^{5}))(-5)\cong\det
F^{-2}\det E^{4}\otimes E^{\vee}.$

Using $\phi$, we turn this into an isomorphism $E_{\alpha}^{\vee}\cong
E^{\vee}$, which we dualize to define $\xi:E\cong E_{\alpha}$. Using the
uniqueness of the CE resolution, we also get an isomorphism $F\otimes\det
F^{-1}\otimes\det E^{2}\to F_{\alpha}$. Making use of $\phi$ again, we obtain
an isomorphism $\psi:F\cong F_{\alpha}$. This in turn induces a map $G(2,F)\to
G(2,F_{\alpha})$ which sends $X$ into $X$. Since the points of $X$ span each
fiber of $\mathbb{P}E^{\prime}\cong\mathbb{P}E_{\alpha}^{\prime}$, the
following diagram of linear maps of spaces commutes

${\mathbb{P}E^{\prime}}$${\mathbb{P}(\wedge^{2}F)}$${\mathbb{P}E_{\alpha}^{\prime}}$${\mathbb{P}(\wedge^{2}F_{\alpha}).}$

In other words, there exists $t\in\mathcal{O}_{S}^{\times}(S)$ such that the
first diagram below commutes, and, since $E$ has rank $4$, so does the second:

(3.17)
${E^{\prime}}$${\wedge^{2}F}$${E_{\alpha}^{\prime}}$${\wedge^{2}F_{\alpha}}$$\scriptstyle{t\cdot\det\xi\otimes(\xi^{-1})^{\vee}}$$\scriptstyle{\eta}$$\scriptstyle{\wedge^{2}\psi}$$\scriptstyle{\eta_{\alpha}}$
${E^{\prime}}$${\wedge^{2}F}$${E_{\alpha}^{\prime}}$${\wedge^{2}F_{\alpha}.}$$\scriptstyle{\det(t\cdot\xi)\otimes((t\cdot\xi)^{-1})^{\vee}}$$\scriptstyle{\eta}$$\scriptstyle{\wedge^{2}(t\cdot\psi)}$$\scriptstyle{\eta_{\alpha}}$

Finally, we must compare $\phi$ and $\phi_{\alpha}$. Since all the maps
involved are isomorphisms of line bundles, there exists some
$x\in\mathcal{O}_{S}^{\times}(S)$ such that the first diagram below commutes;
recalling that $E$ is rank $4$ and $F$ is rank $5$, hence so does the second:

${\det E^{\otimes 2}}$${\det F}$${\det E_{\alpha}^{\otimes 2}}$${\det
F_{\alpha}}$$\scriptstyle{x\cdot\det(t\cdot\xi)^{2}}$$\scriptstyle{\phi}$$\scriptstyle{\det(t\cdot\psi)}$$\scriptstyle{\phi_{\alpha}}$${\det
E^{\otimes 2}}$${\det F}$${\det E_{\alpha}^{\otimes 2}}$${\det
F_{\alpha}.}$$\scriptstyle{\det(x^{2}t\cdot\xi)^{2}}$$\scriptstyle{\phi}$$\scriptstyle{\det(x^{3}t\cdot\psi)}$$\scriptstyle{\phi_{\alpha}}$

Finally, note that

${E^{\prime}}$${\wedge^{2}F}$${E_{\alpha}^{\prime}}$${\wedge^{2}F_{\alpha}.}$$\scriptstyle{\det(x^{2}t\cdot\xi)\otimes((x^{2}t\cdot\xi)^{-1})^{\vee}}$$\scriptstyle{\eta}$$\scriptstyle{\wedge^{2}(x^{3}t\cdot\psi)}$$\scriptstyle{\eta_{\alpha}}$

also commutes, as it just rescales both vertical maps of the second diagram in
(3.17) by $x^{6}$. Hence, pair of isomorphisms $x^{2}t\cdot\xi:E\to
E_{\alpha}$ and $x^{3}t\cdot\psi:F\to F_{\alpha}$ define an arrow
$(E,F,\phi,\eta)\to(E_{\alpha},F_{\alpha},\phi_{\alpha},\eta_{\alpha})$ in
$\mathrm{Pent}^{\prime}(S)$. ∎

###  3.4. Casnati–Ekedahl classes

We now define some preferred generators for $R^{*}(\mathcal{H}_{k,g})$ using
the Chern classes of vector bundles appearing in the Casnati–Ekedahl
resolution. Let $\pi:\mathcal{P}\to\mathcal{H}_{k,g}$ denote the universal
$\mathbb{P}^{1}$-bundle and $\alpha:\mathcal{C}\to\mathcal{P}$ the universal
degree $k$ cover. We define
$z:=-\frac{1}{2}c_{1}(\omega_{\pi})=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$ and

(3.18)
$c_{2}:=c_{2}(\pi_{*}\mathcal{O}_{\mathcal{P}}(1))\qquad\Rightarrow\qquad
z^{2}+\pi^{*}c_{2}=0,$

where the equality on the right follows from (2.1). Define
$\mathcal{E}^{\vee}:=E_{\alpha}^{\vee}$ to be the cokernel of
$\mathcal{O}_{\mathcal{P}}\to\alpha_{*}\mathcal{O}_{\mathcal{C}}$, which is a
rank $k-1$ vector bundle on $\mathcal{P}$. For $i=1,\ldots,k-1$, we define
classes $a_{i}\in A^{i}(\mathcal{H}_{k,g})$ and $a_{i}^{\prime}\in
A^{i-1}(\mathcal{H}_{k,g})$ by the formula

(3.19) $a_{i}:=\pi_{*}(z\cdot c_{i}(\mathcal{E})),\quad
a_{i}^{\prime}:=\pi_{*}(c_{i}(\mathcal{E}))\qquad\Rightarrow\qquad
c_{i}(\mathcal{E})=\pi^{*}a_{i}+\pi^{*}a_{i}^{\prime}z.$

By Example 3.1, $\mathcal{E}$ has relative degree $g+k-1$ on the fibers of
$\mathcal{P}\to\mathcal{H}_{k,g}$, so $a_{1}^{\prime}=g+k-1$. By the
Casnati–Ekedahl structure theorem, the universal curve $\mathcal{C}$ embeds in
$\mathbb{P}\mathcal{E}^{\vee}$. We have the associated Casnati–Ekedahl
resolution

$0\rightarrow\gamma^{*}\mathcal{F}_{k-2}(-k)\rightarrow\gamma^{*}\mathcal{F}_{k-3}(-k+2)\rightarrow\cdots\rightarrow\gamma^{*}\mathcal{F}_{1}(-2)\rightarrow\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}\rightarrow\mathcal{O}_{\mathcal{C}}\rightarrow
0.$

For each bundle $\mathcal{F}_{j}$, we define

$f_{j,i}:=\pi_{*}(z\cdot c_{i}(\mathcal{F}_{j})),\quad
f_{j,i}^{\prime}:=\pi_{*}(c_{i}(\mathcal{F}_{j}))\qquad\Rightarrow\qquad
c_{i}(\mathcal{F}_{j})=\pi^{*}f_{j,i}+\pi^{*}f_{j,i}^{\prime}z.$

######  Definition 3.9.

We define $c_{2},a_{i},a_{i}^{\prime},f_{j,i},f_{j,i}^{\prime}$ to be the
_Casnati–Ekedahl classes_ , abbreviated CE classes.

######  Theorem 3.10.

The CE classes are tautological and they generate the tautological ring
$R^{*}(\mathcal{H}_{k,g})$.

######  Remark 3.11.

The ranks of the $\mathcal{F}_{i}$ depend only on $i$ and $k$, so this bounds
the number of generators of $R^{*}(\mathcal{H}_{k,g})$ and their degrees in
terms of $k$ (independent of $g$).

###### Proof.

First, we show that the Casnati–Ekedahl classes are tautological. Let us call
a class on $\mathcal{P}$ _pre-tautological_ if it is a polynomial in $z$ and
classes of the form $\alpha_{*}(c_{1}(\omega_{f})^{j})$. By the push-pull
formula, the $\pi$ pushforward of a pre-tautological class is tautological.
Therefore, our goal is to show that the Chern classes of $\mathcal{E}$ and
$\mathcal{F}_{i}$ are pre-tautological.

By Grothendieck–Riemann–Roch and the splitting principle, we have that the
Chern classes of $\alpha_{*}(\omega_{\alpha}^{\otimes
i})=\alpha_{*}(\omega_{f}^{\otimes i})\otimes(\omega_{\pi}^{\vee})^{\otimes
i}$ are pre-tautological. In particular, the Chern classes of $\mathcal{E}$
are pre-tautological by its defining exact sequence. By the construction of
the Casnati–Ekedahl sequence, $\mathcal{F}_{1}$ is the kernel of a surjective
map
$\operatorname{Sym}^{2}\mathcal{E}\twoheadrightarrow\alpha_{*}(\omega_{\alpha}^{\otimes
2})$, so the Chern classes of $\mathcal{F}_{1}$ are pre-tautological.
Similarly, following the construction of $\mathcal{F}_{i}$ on [6, p. 445-446]
and using the splitting principle, we inductively see that the Chern classes
of all $\mathcal{F}_{i}$ are pre-tautological.

Next, we must show that all tautological classes are polynomials in
Casnati–Ekedahl classes. We have a diagram

(3.20)
${\mathcal{C}}$${\mathbb{P}\mathcal{E}^{\vee}}$${\mathcal{P}}$${\mathcal{H}_{k,g}}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

First, note that

$f_{*}(c_{1}(\omega_{f})^{i}\cdot\alpha^{*}(\omega_{\pi})^{j})=\pi_{*}(\alpha_{*}(c_{1}(\omega_{\alpha})+\alpha^{*}c_{1}(\omega_{\pi}))^{i}\cdot
c_{1}(\omega_{\pi})^{j}),$

so using push-pull, it will suffice to show that
$\pi_{*}(\alpha_{*}(c_{1}(\omega_{\alpha})^{i})\cdot z^{j})$ is a polynomial
in CE classes for all pairs $i,j$. Now, let
$\zeta:=c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1))$ and note that
$\iota^{*}\zeta=c_{1}(\omega_{\alpha})$. We have

$\displaystyle\alpha_{*}(c_{1}(\omega_{\alpha})^{i})=\gamma_{*}\iota_{*}(\iota^{*}\zeta^{i})=\gamma_{*}([\mathcal{C}]\cdot\zeta^{i}).$

Grothendieck–Riemann–Roch for
$\iota:\mathcal{C}\hookrightarrow\mathbb{P}\mathcal{E}^{\vee}$ tells us that
$[\mathcal{C}]=\operatorname{ch}_{k-2}(\iota_{*}\mathcal{O}_{\mathcal{C}})$.
By additivity of Chern characters in exact sequences, the later is a
polynomial in $\zeta$ and the Chern classes of $\mathcal{F}_{i}$. Using the
projective bundle theorem (2.1), $\gamma_{*}([\mathcal{C}]\cdot\zeta^{i})$ is
therefore a polynomial in the Chern classes of $\mathcal{E}$ and the
$\mathcal{F}_{i}$. The $\pi$ push forward of such a polynomial times any power
of $z$ is a polynomial in the CE classes (essentially from the definition of
the CE classes). ∎

Using the idea in the proof above, we explain how to rewrite the
$\kappa$-classes in terms of CE classes.

######  Example 3.12 ($\kappa$-classes).

Let us retain notation as in (3.20). Writing $\zeta$ for the hyperplane class
of $\mathbb{P}\mathcal{E}^{\vee}$ and $z$ for the hyperplane class on
$\mathcal{P}$, we have

$c_{1}(\omega_{f})=c_{1}(\omega_{\alpha})+c_{1}(\omega_{\pi})=\iota^{*}(\zeta-2z).$

By the push-pull formula, we have

(3.21)
$\kappa_{i}=f_{*}(c_{1}(\omega_{f})^{i+1})=\pi_{*}\gamma_{*}\iota_{*}(\iota^{*}(\zeta-2z)^{i+1})=\pi_{*}\gamma_{*}([\mathcal{C}]\cdot(\zeta-2z)^{i+1}).$

Meanwhile, the fundamental class of
$\mathcal{C}\subset\mathbb{P}\mathcal{E}^{\vee}$ is

(3.22) $\displaystyle[\mathcal{C}]$
$\displaystyle=\sum_{i=1}^{k-3}(-1)^{i-1}\operatorname{ch}_{k-2}(\mathcal{F}_{i}(-i-1))+(-1)^{k-2}\operatorname{ch}_{k-2}(\mathcal{F}_{k-2}(-k))$
$\displaystyle=\begin{cases}-\operatorname{ch}_{1}(\det\mathcal{E}(-3))=c_{1}(\det\mathcal{E}^{\vee}(3))&\text{if
$k=3$}\\\
-\operatorname{ch}_{2}(\mathcal{F}(-2))+\operatorname{ch}_{2}(\det\mathcal{E}(-4))=c_{2}(\mathcal{F}^{\vee}(2))&\text{if
$k=4$}\\\
-\operatorname{ch}_{3}(\mathcal{F}(-2))+\operatorname{ch}_{3}((\mathcal{F}^{\vee}\otimes\det\mathcal{E})(-3))-\operatorname{ch}_{3}(\det\mathcal{E}(-5))&\text{if
$k=5$.}\end{cases}$

Using (3.21) and (3.22), it is straightforward to compute $\kappa_{i}$ in
terms of the CE classes using a computer.

In degree $3$, the CE classes are $c_{2},a_{1},a_{2},a_{2}^{\prime}$. In
degrees $k=4,5$, self-duality of the Casnati–Ekedahl resolution implies that
all CE classes are expressible in terms of $c_{2}$ the $a_{i},a_{i}^{\prime}$
and the $b_{i}:=f_{1,i}$ and $b_{i}^{\prime}:=f_{1,i}^{\prime}$. These classes
are all pulled back from a moduli space $\mathcal{B}_{k,g}$ of (pairs of)
vector bundles on $\mathbb{P}^{1}$, which we shall construct in the next
section.

##  4\. Pairs of vector bundles on $\mathbb{P}^{1}$-bundles

By the results of Casnati–Ekedahl and Casnati in the previous section, there
is a correspondence between covers of $\mathbb{P}^{1}$ and certain linear
algebraic data. In this section, following ideas of Bolognesi–Vistoli [2], we
construct moduli stacks parametrizing the associated linear algebraic data and
describe the Chow rings of these stacks. In [2], Bolognesi–Vistoli gave a
quotient stack presentation for the moduli stack parametrizing globally
generated vector bundles on $\mathbb{P}^{1}$-fibrations. As explained in
Section 2.3, we will instead make use of $\operatorname{SL}_{2}$ quotients,
since they have the same rational Chow ring as the $\operatorname{PGL}_{2}$
quotient.

######  Definition 4.1.

Let $r,d$ be nonnegative integers.

  1. (1)

The objects of $\mathcal{V}_{r,d}^{\dagger}$ are pairs $(S,E)$ where $E$ is a
locally free sheaf of rank $r$ on $\mathbb{P}^{1}\times S$ whose restriction
to each of the fibers of $\mathbb{P}^{1}\times S\rightarrow S$ is globally
generated of degree $d$. A morphism between objects $(S,E)$ and
$(S^{\prime},E^{\prime})$ is a Cartesian diagram

${\mathbb{P}^{1}\times S^{\prime}}$${\mathbb{P}^{1}\times
S}$${S^{\prime}}$${S}$$\scriptstyle{F}$

together with an isomorphism $\phi:F^{*}E\rightarrow E^{\prime}$.

  2. (2)

We define $\mathcal{V}_{r,d}$ to be the $\operatorname{SL}_{2}$ quotient of
$\mathcal{V}_{r,d}^{\dagger}$. Explicitly, the objects of $\mathcal{V}_{r,d}$
are triples $(S,V,E)$ where $S$ is a $k$-scheme, $V$ is a rank $2$ vector
bundle on $S$ with trivial determinant, and $E$ is a rank $r$ vector bundle on
$\mathbb{P}V$ whose restrictions to the fibers of $\mathbb{P}V\rightarrow S$
are globally generated of degree $d$. A morphism between objects $(S,V,E)$ and
$(S^{\prime},V^{\prime},E^{\prime})$ is a Cartesian diagram

${\mathbb{P}V^{\prime}}$${\mathbb{P}V}$${S^{\prime}}$${S}$$\scriptstyle{F}$

together with an isomorphism $\phi:F^{*}E\rightarrow E^{\prime}$.

Bolognesi–Vistoli gave a presentation for $\mathcal{V}_{r,d}^{\dagger}$ as a
quotient stack, which we briefly summarize here. Let $M_{r,d}$ be the affine
space that represents the functor which sends a scheme $S$ to the set of
matrices of size $(r+d)\times d$ with entries in
$H^{0}(\mathbb{P}^{1}_{S},\mathcal{O}_{\mathbb{P}^{1}_{S}}(1))$. We can
identify such a matrix with the associated map

$\mathcal{O}_{\mathbb{P}^{1}_{S}}(-1)^{d}\rightarrow\mathcal{O}_{\mathbb{P}^{1}_{S}}^{r+d}.$

Let $\Omega_{r,d}\subset M_{r,d}$ denote the open subscheme parametrizing
injective maps with locally free cokernel. The group $\operatorname{GL}_{d}$
acts $M_{r,d}$ by multiplication on the left, $\operatorname{GL}_{r+d}$ by
multiplication on the right. Bolognesi–Vistoli establish [2, Theorem 4.4] that

$\mathcal{V}_{r,d}^{\dagger}\cong[\Omega_{r,d}/\operatorname{GL}_{d}\times\operatorname{GL}_{d+r}].$

The group $\operatorname{SL}_{2}$ acts by change of coordinates on
$H^{0}(\mathbb{P}^{1}_{S},\mathcal{O}_{\mathbb{P}^{1}_{S}}(1))$. This commutes
with the $\operatorname{GL}_{d}\times\operatorname{GL}_{d+r}$ action. Thus we
obtain the following.

######  Proposition 4.2.

There is an isomorphism of fibered categories

$\mathcal{V}_{r,d}\cong[\mathcal{V}_{r,d}^{\dagger}/\operatorname{SL}_{2}]\cong[\Omega_{r,d}/\operatorname{GL}_{d}\times\operatorname{GL}_{r+d}\times\operatorname{SL}_{2}].$

######  Remark 4.3.

Bolognesi–Vistoli also describe the $\operatorname{PGL}_{2}$ quotient of
$\mathcal{V}_{r,d}^{\dagger}$, which is slightly more subtle. This distinction
is important in their work which concerns _integral_ coefficients.

To parametrize the linear algebraic data associated to a low degree cover of
$\mathbb{P}^{1}$, we are interested in products of the form
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$, which
parametrize a pair of vector bundles on the same $\mathbb{P}^{1}$-bundle. Let
$G_{r,d,s,e}:=\operatorname{GL}_{d}\times\operatorname{GL}_{r+d}\times\operatorname{GL}_{e}\times\operatorname{GL}_{s+e}$.
The group $G_{r,d,s,e}\times\operatorname{SL}_{2}$ acts on $M_{r,d}$ via the
projection
$G_{r,d,s,e}\times\operatorname{SL}_{2}\to\operatorname{GL}_{d}\times\operatorname{GL}_{r+d}\times\operatorname{SL}_{2}$;
and similarly on $M_{s,e}$ via the projection
$G_{r,d,s,e}\times\operatorname{SL}_{2}\to\operatorname{GL}_{e}\times\operatorname{GL}_{s+e}\times\operatorname{SL}_{2}$.
By Proposition 4.2, it follows that

(4.1)
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}=[\Omega_{r,d}\times\Omega_{s,e}/G_{r,d,s,e}\times\operatorname{SL}_{2}].$

Let $T_{d}$ and $T_{r+d}$ denote the universal vector bundles on
$\operatorname{BGL}_{d}$ and $\operatorname{BGL}_{r+d}$; similarly, let
$S_{e}$ and $S_{s+e}$ be the universal vector bundles on
$\operatorname{BGL}_{e}$ and $\operatorname{BGL}_{s+e}$. The Chow ring of
$\mathrm{B}(G_{r,d,s,e}\times\operatorname{SL}_{2})$ is the free
$\mathbb{Q}$-algebra on the Chern classes of $T_{d},T_{r+d},S_{e},S_{s+e}$,
together with the universal second Chern class $c_{2}$ on
$\operatorname{BSL}_{2}$. Let us denote these classes by

$\displaystyle t_{i}$ $\displaystyle=c_{i}(T_{d})\qquad\text{and}\qquad
u_{i}=c_{i}(T_{r+d})$ $\displaystyle v_{i}$
$\displaystyle=c_{i}(S_{e})\qquad\text{and}\qquad w_{i}=c_{i}(S_{s+e}).$

Since $\Omega_{r,d}\times\Omega_{s,e}$ is open inside the affine space
$M_{r,d}\times M_{s,e}$, the excision and homotopy properties imply

(4.2)
$A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})$ is
generated by the restrictions of the $t_{i},u_{i},v_{i},w_{i}$.

We now identify the restrictions of the tautological bundles $T_{d}$ and
$T_{d+r}$ in terms of the universal rank $r$, degree $d$ vector bundle on
$\mathbb{P}^{1}$. Let $\pi:\mathcal{P}\rightarrow\mathcal{V}_{r,d}$ be the
universal $\mathbb{P}^{1}$-bundle. We write
$z:=c_{1}(\mathcal{O}_{\mathcal{P}}(1))\in A^{1}(\mathcal{P})$. We have
$c_{2}=c_{2}(\pi_{*}\mathcal{O}_{\mathcal{P}}(1))\in
A^{2}(\mathcal{V}_{r,d})$, the universal second Chern class (pulled back via
the natural map $\mathcal{V}_{r,d}\rightarrow\operatorname{BSL}_{2}$). Note
that $c_{1}(\pi_{*}\mathcal{O}_{\mathcal{P}}(1))=0$, so by Equation (2.1),

$A^{*}(\mathcal{P})=A^{*}(\mathcal{V}_{r,d})[z]/(z^{2}+\pi^{*}c_{2}).$

Let $\mathcal{E}$ be the universal rank $r$, degree $d$ vector bundle on
$\mathcal{P}$. The Chern classes of $\mathcal{E}$ may thus be written as

$c_{i}(\mathcal{E})=\pi^{*}a_{i}+(\pi^{*}a_{i}^{\prime})z\qquad\text{where}\qquad
a_{i}\in A^{i}(\mathcal{V}_{r,d}),\quad a_{i}^{\prime}\in
A^{i-1}(\mathcal{V}_{r,d}).$

Note that $a_{1}^{\prime}=d$. Let
$\gamma:\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}\to
B(G_{r,d,s,e}\times\operatorname{SL}_{2})$ be the structure map. Then by [17,
Lemma 3.2] (noting that $\det(\pi_{*}\mathcal{O}_{\mathcal{P}}(1))$ is
trivial), we have

(4.3)
$\gamma^{*}T_{d}=\pi_{*}\mathcal{E}(-1)\qquad\text{and}\qquad\gamma^{*}T_{r+d}=\pi_{*}\mathcal{E}.$

Since $R^{1}\pi_{*}\mathcal{E}(-1)$ and $R^{1}\pi_{*}\mathcal{E}$ are zero,
Grothendieck–Riemann–Roch says that the Chern characters of
$\pi_{*}\mathcal{E}(-1)$ and $\pi_{*}\mathcal{E}$ are push forwards by $\pi$
of polynomials in the $c_{i}(\mathcal{E})$ and $z$. The push forward of such a
polynomial is a polynomial in the $a_{i},a_{i}^{\prime}$ and $c_{2}$. In
particular, the restrictions of $t_{i}$ and $u_{i}$ to
$A^{*}(\mathcal{V}_{r,d})$ are polynomials in
$a_{1},\ldots,a_{r},a_{2}^{\prime},\ldots,a_{r}^{\prime}$ and $c_{2}$.

###  4.1. The rational Chow ring

Let us denote the universal rank $s$ vector bundle from the second factor of
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ by
$\mathcal{F}$ on $\mathcal{P}$ and its Chern classes by

$c_{i}(\mathcal{F})=\pi^{*}b_{i}+(\pi^{*}b_{i}^{\prime})z\qquad\text{where}\qquad
b_{i}\in A^{i}(\mathcal{V}_{s,e}),\quad b_{i}^{\prime}\in
A^{i-1}(\mathcal{V}_{s,e}).$

It follows from Equation (4.2) and the discussion following (applied to both
factors of the product) that the $a_{i},a_{i}^{\prime},b_{i},b_{i}^{\prime}$
and $c_{2}$ are generators for
$A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,d})$. We
now show that there are no relations among these generators in low degrees.
This is a generalization of [17, Theorem 4.1], which shows
$A^{*}(\mathcal{V}_{r,d})$ is generated by the $a_{i},a_{i}^{\prime},$ and
$c_{2}$ with no relations in degrees less than $d+1$.

######  Theorem 4.4.

The rational Chow ring of
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ is
generated as a $\mathbb{Q}$-algebra by

$c_{2},a_{1},\ldots,a_{r},a_{2}^{\prime},\ldots,a_{r}^{\prime},b_{1},\ldots,b_{s},b_{2}^{\prime},\ldots,b_{s}^{\prime},$

and all relations have degree at least $\min(d,e)+1$. In the notation of
Equation (2.2),

$\displaystyle\operatorname{Trun}^{\min(d,e)+1}A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})$
$\displaystyle\qquad\qquad=\operatorname{Trun}^{\min(d,e)+1}\mathbb{Q}[c_{2},a_{1},\ldots,a_{r},a_{2}^{\prime},\ldots,a_{r}^{\prime},b_{1},\ldots,b_{s},b_{2}^{\prime},\ldots,b_{s}^{\prime}].$

######  Remark 4.5.

(1) The codimension of the complement of $\Omega_{r,d}\subset M_{r,d}$ is $r$,
so the Theorem does _not_ follow immediately from dimension counting and
excision if $\min(d,e)>\min(r,s)$.

(2) If $s=0$, the fact that there are no $b_{i}$ classes follows immediately
from [17, Theorem 4.1].

###### Proof.

Let

$M:=[M_{r,d}/G_{r,d,s,e}\times\operatorname{SL}_{2}]\quad\text{and}\quad
N:=[M_{s,e}/G_{r,d,s,e}\times\operatorname{SL}_{2}].$

Equation (4.1) says that
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ is an open
inside the vector bundle $M\oplus N$ over
$B:=\mathrm{B}(G_{r,d,s,e}\times\operatorname{SL}_{2})$. The complement
consists of two components, namely

$X:=[\Omega_{r,d}^{c}\times
M_{s,e}/G_{r,d,s,e}\times\operatorname{SL}_{2}]\quad\text{and}\quad
Y:=[M_{r,d}\times\Omega_{s,e}^{c}/G_{r,d,s,e}\times\operatorname{SL}_{2}].$

One readily checks that
$X\subset\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ is
irreducible of codimension $r$ and
$Y\subset\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ is
irreducible of codimension $s$ (see [2, Remark 4.3]). Excision gives a right-
exact sequence

(4.4) $A^{*-r}(X)\oplus A^{*-s}(Y)\rightarrow A^{*}(M\oplus N)\rightarrow
A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})\rightarrow
0.$

From this it is clear that there are no relations among the restrictions to
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ of the
Chern classes of $T_{d},T_{d+r},S_{e}$ and $S_{s+e}$ in degrees less than
$\min(r,s)$. We now describe relations among the restrictions of these Chern
classes in degrees $\min(r,s)$ up to $\min(d,e)$. (If $\min(d,e)<\min(r,s)$ we
are already done.) In particular, shall conclude that

(4.5)
$\displaystyle\operatorname{Trun}^{\min(d,e)+1}A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})$
$\displaystyle\qquad=\operatorname{Trun}^{\min(d,e)+1}\mathbb{Q}[c_{2},t_{1},\ldots,t_{r-1},u_{1},\ldots,u_{r},v_{1},\ldots
v_{s-1},w_{1},\ldots,w_{s}].$

Since the classes in the statement of the theorem are generators and have the
same degrees as those above, the statement in the theorem must hold for
dimension reasons.

It suffices to understand the image of $A^{*-r}(X)\to A^{*}(M\oplus N)$, the
other factor being similar. For this we resolve $X$ (resp. $Y$) as in the
proof of [17, Theorem 4.1], taking the $N$ factor (resp. $M$ factor) “along
for the ride”.

Using the excision sequence (4.4) and arguing exactly as in [17, Theorem 4.1],
we have

(4.6)
$A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})=\frac{\mathbb{Q}[c_{2},t_{1},\ldots,t_{d},u_{1},\ldots,u_{r+d},v_{1},\ldots,v_{e},w_{1},\ldots,w_{s+e}]}{\langle
f_{i,j}:0\leq i\leq 1,0\leq j\leq d-1\rangle+\langle g_{i,j}:0\leq i\leq
1,0\leq j\leq e-1\rangle}.$

where

$\displaystyle f_{1,j-1}$ $\displaystyle=-t_{j+r}+u_{j+r}+\ldots$
$\displaystyle f_{0,j}$ $\displaystyle=-(r+d)t_{j+r}+(d-j)u_{j+r}+\ldots$
$\displaystyle g_{1,j-1}$ $\displaystyle=-v_{j+s}+w_{j+s}+\ldots$
$\displaystyle g_{0,j}$ $\displaystyle=-(s+e)v_{j+s}+(e-j)w_{j+s}+\ldots.$

Hence, in
$A^{*}(\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e})$,
the classes $t_{n}$ for $r\leq n\leq d$ and $u_{m}$ for $r+1\leq m\leq d$ are
expressible as polynomials in $c_{2},t_{1},\ldots,t_{r-1},u_{1},\ldots,u_{r}$.
Moreover, after eliminating these higher degree generators, the $f_{i,j}$
produce no additional relations in degrees less than or equal to $d$ among the
restrictions to
$\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$ of
$c_{2},t_{1},\ldots,t_{r-1},u_{1},\ldots,u_{r},v_{1},\ldots,v_{e},w_{1},\ldots,w_{s+e}$.
With the analogous calculation for the $g_{i,j}$, equation (4.6) then implies
(4.5), and hence the statement of the theorem. ∎

###  4.2. Splitting loci

Every vector bundle $E$ on $\mathbb{P}^{1}$ splits as a direct sum of line
bundles, $E\cong\mathcal{O}(e_{1})\oplus\cdots\oplus\mathcal{O}(e_{r})$ for
integers $e_{1}\leq\cdots\leq e_{r}$. We call the non-decreasing sequence of
integers $\vec{e}=(e_{1},\ldots,e_{r})$ the _splitting type_ of $E$ and will
often abbreviate the corresponding sum of line bundles by
$\mathcal{O}(\vec{e}):=\mathcal{O}(e_{1})\oplus\cdots\oplus\mathcal{O}(e_{r})$.
Given a family of vector bundles $\mathcal{E}$ on a $\mathbb{P}^{1}$-bundle
$\pi:\mathcal{P}\to B$, the base $B$ is stratified by locally closed
subvarieties

$\\{b\in B:\mathcal{E}_{\pi^{-1}(b)}\cong\mathcal{O}(\vec{e})\\},$

which we call the _splitting locus for $\vec{e}$_. A subscheme structure on
splitting loci is defined in [18, Section 2], though it will not be necessary
here.

The splitting type $\vec{e}$ of $E$ is equivalent to the data of the ranks of
cohomology groups $h^{0}(\mathbb{P}^{1},E(m))$ for all $m\in\mathbb{Z}$.
Conversely, the locus of points $b\in B$ where the fibers of $\mathcal{E}$
satisfy some cohomological condition is a union of splitting loci. For
example, the locus in $B$ where $\mathcal{E}$ fails to be globally generated
on fibers is the union of splitting loci for splitting types $\vec{e}$ with
$e_{1}\leq-1$. Similarly, $\operatorname{Supp}R^{1}\pi_{*}\mathcal{E}(-2)$ is
the union of all splitting loci with $e_{1}\leq 0$.

Following the argument in [2, Lemma 5.1], the codimension in
$\mathcal{V}_{r,d}$ of the splitting locus where the universal $\mathcal{E}$
over $\mathcal{V}_{r,d}$ has splitting type $\vec{e}$ on fibers of
$\mathcal{P}\to\mathcal{V}_{r,d}$ is
$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))$. If we have a
$\mathbb{P}^{1}$-bundle equipped with two vector bundles, we can consider the
intersections of splitting loci for both bundles. The simultaneous splitting
locus in $\mathcal{V}_{r,d}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{s,e}$
where $\mathcal{E}$ has splitting type $\vec{e}$ and $\mathcal{F}$ has
splitting type $\vec{f}$ is equal to the product of the $\vec{e}$ splitting
locus in $\mathcal{V}_{r,d}$ with the $\vec{f}$ splitting locus in
$\mathcal{V}_{s,e}$, and therefore has codimension

(4.7)
$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f}))).$

##  5\. The good opens and codimension bounds

For each $k=3,4,5$ and genus $g$, we will define a stack $\mathcal{B}_{k,g}$
parametrizing the vector bundles associated to a degree $k$, genus $g$ cover
of $\mathbb{P}^{1}$. The stack $\mathcal{B}_{k,g}$ will come equipped with a
universal $\mathbb{P}^{1}$-bundle $\pi:\mathcal{P}\to\mathcal{B}_{k,g}$. Then,
we will define a vector bundle $\mathcal{U}_{k,g}$ on $\mathcal{P}$ whose
sections on a fiber of $\mathcal{P}\to\mathcal{B}_{k,g}$ is the relevant space
of sections in the linear algebraic data of covers appearing in Section 3.

Before treating the case for each $k$ in depth, we briefly outline our
construction of certain open substacks of the Hurwitz stack. For $k=3$, we
shall define $\mathcal{B}_{3,g}^{\prime}\subseteq\mathcal{B}_{3,g}$ to be the
open substack over which $\mathcal{U}_{3,g}$ is globally generated on the
fibers of $\pi:\mathcal{P}\to\mathcal{B}$. For $k=4,5$, let us define the open
substack

(5.1) $\displaystyle\mathcal{B}_{k,g}^{\prime}$
$\displaystyle:=\mathcal{B}_{k,g}\smallsetminus\operatorname{Supp}(R^{1}\pi_{*}\mathcal{U}_{k,g}).$

By the theorem on cohomology and base change, the restriction of
$\pi_{*}\mathcal{U}_{k,g}$ to $\mathcal{B}_{k,g}^{\prime}$ is locally free
with fibers given by the relevant space of sections in the linear algebraic
data of covers appearing in Section 3. We denote the total space of this
vector bundle on $\mathcal{B}_{k,g}^{\prime}$ by

$\mathcal{X}_{k,g}^{\prime}:=\pi_{*}\mathcal{U}_{k,g}|_{\mathcal{B}_{k,g}^{\prime}}.$

As discussed in Section 4.2, the complement of $\mathcal{B}_{k,g}^{\prime}$ is
a union of splitting loci. The splitting loci in the complement are determined
by the condition that $\mathcal{U}_{k,g}$ has a summand of degree $-2$ or
less. It will also be convenient to work with a slightly smaller open substack

(5.2)
$\mathcal{B}_{k,g}^{\circ}:=\mathcal{B}_{k,g}\smallsetminus\operatorname{Supp}(R^{1}\pi_{*}(\mathcal{U}_{k,g}\otimes\mathcal{O}_{\mathcal{P}}(-2))),$

whose complement consists of splitting loci where $\mathcal{U}_{k,g}$ has a
summand of degree $0$ or less. (Having a cut-off in terms of a degree $0$
summand will be cleaner than a cut-off in terms of a $-2$ summand for our next
step of bounding the codimension of the complement. The open
$\mathcal{B}_{k,g}^{\circ}$ also plays an important role in our sequel [3],
where the slightly stronger positivity condition on the fibers of
$\mathcal{U}_{k,g}$ will be needed.)

Pulling back these open substacks along the natural map
$\mathcal{H}_{k,g}\to\mathcal{B}_{k,g}$ defines open substacks of the Hurwitz
space as in the diagram below

(5.3)
${\mathcal{H}_{k,g}^{\circ}}$${\mathcal{H}_{k,g}^{\prime}}$${\mathcal{H}_{k,g}}$${\mathcal{B}_{k,g}^{\circ}}$${\mathcal{B}_{k,g}^{\prime}}$${\mathcal{B}_{k,g}.}$

In all cases, we shall see that $\mathcal{H}_{k,g}^{\prime}$ is an open
substack inside the vector bundle $\mathcal{X}_{k,g}^{\prime}$ over
$\mathcal{B}_{k,g}^{\prime}$. In particular, we will find that the Chow ring
of $\mathcal{H}_{k,g}^{\prime}$ is generated by tautological classes. To turn
this into meaningful results for the Chow ring of $\mathcal{H}_{k,g}$ we must
describe the complement of
$\mathcal{H}_{k,g}^{\prime}\subseteq\mathcal{H}_{k,g}$. When $k=3$, it turns
out $\mathcal{H}_{k,g}^{\prime}=\mathcal{H}_{k,g}$. For $k=4$, the complement
of $\mathcal{H}_{4,g}^{\prime}$ contains covers that factor through an
intermediate curve of low genus. Nevertheless, we show that the complement of
$\mathcal{H}_{k,g}^{\prime}$ intersects $\mathcal{H}_{4,g}^{\mathrm{nf}}$ in
high codimension. Finally, for $k=5$, we show that the complement of
$\mathcal{H}_{5,g}^{\prime}$ has high codimension.

Of course, the complement of $\mathcal{H}_{k,g}^{\prime}$ is contained in the
complement of $\mathcal{H}_{k,g}^{\circ}$, so it will suffice to bound the
codimension of the complement of $\mathcal{H}_{k,g}^{\circ}$ (which in turn
provides a lower bound on the codimension of the complement of
$\mathcal{H}_{k,g}^{\prime}$). For arbitrary $g$, the coefficient of $g$ in
the bounds we obtain will be sharp (and the same for
$\mathcal{H}_{k,g}^{\circ}$ and $\mathcal{H}_{k,g}^{\prime}$). In any
particular case, however, one may find slight improvements on our bounds by
enumerating the splitting loci in the complement $\mathcal{H}_{k,g}^{\prime}$,
and calculating their codimensions.

Along the way, we also bound the codimension of the complement of
$\mathcal{B}_{k,g}^{\circ}$. We point out some immediate corollaries regarding
the Chow rings of $\mathcal{B}_{k,g}^{\circ}$, which will be used in our
subsequent work [3].

###  5.1. Degree $3$

As a warm-up, we first explain our set-up in degree $3$, as it is simplest.
The results in this subsection are not new (they have already been established
by Bolognesi–Vistoli [2] and Patel–Vakil [23]) but spelling them out in our
language will be instructive; it will also be useful in our subsequent work
[3].

In Section 4, we gave a construction for $\mathcal{V}_{r,d}$ as the moduli
space of vector bundles on $\mathbb{P}^{1}$-bundles. As discussed in Section
3.1, the linear algebraic data of a degree $3$, genus $g$ cover involves a
rank $2$, degree $g+2$ vector bundle $E$ on $\mathbb{P}^{1}$ and section of
$\det E^{\vee}\otimes\operatorname{Sym}^{3}E$. We set

$\mathcal{B}_{3,g}:=\mathcal{V}_{2,g+2}\qquad\text{and}\qquad\mathcal{U}_{3,g}:=\det\mathcal{E}^{\vee}\otimes\operatorname{Sym}^{3}\mathcal{E},$

where $\mathcal{E}$ is the universal rank $2$ bundle on
$\pi:\mathcal{P}\to\mathcal{V}_{2,g+2}$. There is a natural map
$\mathcal{H}_{3,g}\to\mathcal{B}_{3,g}$ that sends a family of triple covers
$C\xrightarrow{\alpha}P\rightarrow S$ in $\mathcal{H}_{3,g}(S)$ to the
associated rank $2$ vector bundle $E_{\alpha}$ on $P\to S$ in
$\mathcal{B}_{3,g}(S)$. If $C\xrightarrow{\alpha}\mathbb{P}^{1}$ is an
integral triple cover and
$E_{\alpha}=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})$ is the associated rank
$2$ vector bundle on $\mathbb{P}^{1}$, then by [2, Proposition 2.2], we have
$e_{1},e_{2}\geq\frac{g+2}{3}$. Equivalently, every summand of $\det
E_{\alpha}^{\vee}\otimes\operatorname{Sym}^{3}E_{\alpha}$ is non-negative.
Hence, the map $\mathcal{H}_{3,g}\to\mathcal{B}_{3,g}$ factors through the
substack $\mathcal{B}_{3,g}^{\prime}\subseteq\mathcal{B}_{3,g}$ over which
$\mathcal{U}_{3,g}$ is globally generated on fibers of
$\mathcal{P}\to\mathcal{B}_{3,g}$. In particular,
$\mathcal{H}_{3,g}^{\prime}=\mathcal{H}_{3,g}$. We define
$\mathcal{X}_{3,g}^{\prime}:=\pi_{*}\mathcal{U}_{3,g}|_{\mathcal{B}_{3,g}^{\prime}}$,
which is a vector bundle on $\mathcal{B}_{3,g}^{\prime}$ by the theorem on
cohomology and base change.

######  Lemma 5.1.

There is an open inclusion
$\mathcal{H}_{3,g}=\mathcal{H}_{3,g}^{\prime}\to\mathcal{X}_{3,g}^{\prime}$.
In particular, $A^{*}(\mathcal{H}_{3,g})$ is generated by the CE classes
$c_{2},a_{1},a_{2},a_{2}^{\prime}$, and therefore
$A^{*}(\mathcal{H}_{3,g})=R^{*}(\mathcal{H}_{3,g})$.

###### Proof.

The first sentence was essentially observed in [2, p. 12]. We include an
explanation using our notation. Given a scheme $S$, the objects of
$\mathcal{X}_{3,g}(S)$ are tuples $(P\to S,E,\eta)$ where $(P\to S,E)$ is an
object of $\mathcal{B}_{3,g}^{\prime}(S)$ and $\eta\in
H^{0}(P,\operatorname{Sym}^{3}E\otimes\det E^{\vee})$. We define an open
substack $\mathcal{Y}_{3,g}^{\prime}\subset\mathcal{X}_{3,g}^{\prime}$ by the
condition that $V(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$ is a family of
smooth curves, where $\Phi$ is as in (3.6). Considering the Hilbert polynomial
of $V(\Phi(\eta))$, one sees that the fibers have arithmetic genus $g$.
Theorem 3.4 now shows that there is an equivalence
$\mathcal{H}_{3,g}^{\prime}\cong\mathcal{Y}_{3,g}^{\prime}$.

By excision, the Chow ring of $\mathcal{H}_{3,g}=\mathcal{H}_{3,g}^{\prime}$
is generated by restrictions of classes on $\mathcal{X}_{3,g}^{\prime}$. Since
$\mathcal{X}_{3,g}^{\prime}$ is a vector bundle over
$\mathcal{B}_{3,g}^{\prime}$, their Chow rings are isomorphic, so the
statement about generators follows from Theorem 4.4. ∎

###  5.2. Degree $4$

By Casnati–Ekedahl’s characterization of quadruple covers (Theorem 3.6), the
linear algebraic data of a quadruple cover of $\mathbb{P}^{1}$ is equivalent
to the data of: a rank $3$ vector bundle $E$; a rank $2$ vector bundle $F$; an
isomorphism $\det F\cong\det E$; and a global section of
$F^{\vee}\otimes\operatorname{Sym}^{2}E$ on $\mathbb{P}^{1}$ having the right
codimension. By Example 3.1, $\deg(E)=\deg(F)=g+3$. The stacks
$\mathcal{V}_{2,g+3}$ and $\mathcal{V}_{3,g+3}$ both admit natural morphisms
to $\operatorname{BSL}_{2}$, and the fiber product
$\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$ is the
stack whose objects are quadruples $(S,V,E,F)$ where $S$ is a $k$-scheme, $V$
is a rank $2$-vector bundle with trivial determinant, $E$ is a rank $3$ vector
bundle on $\mathbb{P}V$ whose restriction to the fibers of
$\mathbb{P}V\rightarrow S$ is globally generated of degree $g+3$, and $F$ is a
rank $2$ vector bundle on $\mathbb{P}V$ whose restriction to the fibers of
$\mathbb{P}V\rightarrow S$ is globally generated of degree $g+3$.

The additional data of an isomorphism $\det F\cong\det E$ is captured by a
$\mathbb{G}_{m}$ torsor over
$\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$
defined as follows. Let $\mathcal{E}$ be the universal rank $3$ bundle and
$\mathcal{F}$ be the universal rank $2$ bundle on the universal
$\mathbb{P}^{1}$-bundle
$\pi:\mathcal{P}\rightarrow\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$.
Since $\det\mathcal{E}^{\vee}\otimes\det\mathcal{F}$ has degree $0$ on each
fiber of
$\pi:\mathcal{P}\rightarrow\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$,
the theorem on cohomology and base change shows that
$\mathcal{L}:=\pi_{*}(\det\mathcal{E}^{\vee}\otimes\det\mathcal{F})$ is a line
bundle with
$\pi^{*}\mathcal{L}\cong\det\mathcal{E}^{\vee}\otimes\det\mathcal{F}$.

######  Definition 5.2.

With notation as above, define the stack $\mathcal{B}_{4,g}$ to be the
$\mathbb{G}_{m}$-torsor over
$\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$ given
by the complement of the zero section of the line bundle $\mathcal{L}$.

The objects of $\mathcal{B}_{4,g}$ are tuples $(S,V,E,F,\phi)$ where
$(S,V,E,F)$ is an object of
$\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3}$ and
$\phi$ is an isomorphism $\det F\cong\det E$. Recalling the notation of
Section 3.2, given an object $C\xrightarrow{\alpha}P\to S$ of
$\mathcal{H}_{4,g}(S)$, the restriction of $E_{\alpha}$ and $F_{\alpha}$ to
fibers of $P\to S$ are both known to be globally generated (see Proposition
5.6). Hence, there is a natural map $\mathcal{H}_{4,g}\to\mathcal{B}_{4,g}$
that sends the family $C\xrightarrow{\alpha}P\xrightarrow{\pi}S$ to the tuple
$(S,\pi_{*}\mathcal{O}_{P}(1)^{\vee},E_{\alpha},F_{\alpha},\phi_{\alpha})$.

By slight abuse of notation, let us denote the pullback to $\mathcal{B}_{4,g}$
of the universal $\mathbb{P}^{1}$-bundle by
$\pi:\mathcal{P}\rightarrow\mathcal{B}_{4,g}$, and the universal rank $3$ and
$2$ vector bundles on it by $\mathcal{E}$ and $\mathcal{F}$. Let
$z=\mathcal{O}_{\mathcal{P}}(1)$ and write

$c_{i}(\mathcal{E})=\pi^{*}a_{i}+(\pi^{*}a_{i}^{\prime})z\qquad\text{and}\qquad
c_{i}(\mathcal{F})=\pi^{*}b_{i}+(\pi^{*}b_{i}^{\prime})z.$

for $a_{i},b_{i}\in A^{i}(\mathcal{B}_{4,g})$ and
$a_{i}^{\prime},b_{i}^{\prime}\in A^{i-1}(\mathcal{B}_{4,g})$. Note that
$a_{1}^{\prime}=b_{1}^{\prime}=g+3$. Moreover, by definition of
$\mathcal{B}_{4,g}$, we have
$c_{1}(\det\mathcal{E}^{\vee}\otimes\det\mathcal{F})=0$, so $a_{1}=b_{1}$.
Further, by Lemma 2.1, we have

(5.4) $\displaystyle
A^{*}(\mathcal{B}_{4,g})=A^{*}(\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3})/\langle
c_{1}(\mathcal{L})\rangle=A^{*}(\mathcal{V}_{3,g+3}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{2,g+3})/\langle
a_{1}-b_{1}\rangle.$

Thus, Theorem 4.4 shows that
$c_{2},a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2}^{\prime},b_{2}$
generate $A^{*}(\mathcal{B}_{4,g})$ and

(5.5)
$\operatorname{Trun}^{g+4}A^{*}(\mathcal{B}_{4,g})=\operatorname{Trun}^{g+4}\mathbb{Q}[c_{2},a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2}^{\prime},b_{2}].$

Next, we define
$\mathcal{U}_{4,g}:=\mathcal{F}^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{E}$
on $\mathcal{P}$. We then define $\mathcal{B}_{4,g}^{\prime}$ and
$\mathcal{B}_{4,g}^{\circ}$ by (5.1) and (5.2) respectively. Correspondingly,
the open substacks
$\mathcal{H}_{4,g}^{\circ}\subseteq\mathcal{H}_{4,g}^{\prime}\subseteq\mathcal{H}_{4,g}$
are described by

$\displaystyle\\{S\rightarrow\mathcal{H}_{4,g}^{\circ}\\}$
$\displaystyle=\\{S\rightarrow\mathcal{H}_{4,g}:R^{1}(\pi_{S})_{*}(\mathcal{F}_{S}^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{E}_{S}\otimes\mathcal{O}_{\mathcal{P}_{S}}(-2))=0\\}$
$\displaystyle\\{S\rightarrow\mathcal{H}_{4,g}^{\prime}\\}$
$\displaystyle=\\{S\rightarrow\mathcal{H}_{4,g}:R^{1}(\pi_{S})_{*}(\mathcal{F}_{S}^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{E}_{S})=0\\}.$

The key property of $\mathcal{H}_{4,g}^{\prime}$ is that the map
$\mathcal{H}_{4,g}^{\prime}\to\mathcal{B}_{4,g}^{\prime}$ factors through an
open inclusion in the total space of a vector bundle
$\mathcal{X}_{4,g}^{\prime}:=\pi_{*}\mathcal{U}_{4,g}|_{\mathcal{B}_{4,g}^{\prime}}$.

######  Lemma 5.3.

There is an open inclusion
$\mathcal{H}_{4,g}^{\prime}\to\mathcal{X}_{4,g}^{\prime}$. In particular,
$A^{*}(\mathcal{H}_{4,g}^{\prime})=R^{*}(\mathcal{H}_{4,g}^{\prime})$ is
generated by the CE classes
$c_{2},a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2}^{\prime},b_{2}$.

###### Proof.

The objects of $\mathcal{X}_{4,g}^{\prime}$ are tuples $(S,V,E,F,\phi,\eta)$
where $(S,V,E,F,\phi)\in\mathcal{B}_{4,g}^{\prime}$ and $\eta\in
H^{0}(\mathbb{P}V,F^{\vee}\otimes\operatorname{Sym}^{2}E)$. Letting $\Phi$ be
as in (3.9), we define $\mathcal{Y}_{4,g}\subset\mathcal{X}_{4,g}^{\prime}$ to
be the open substack defined by the condition that
$V(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$ is a family of smooth curves.
Considering the Hilbert polynomial of $V(\Phi(\eta))$, using (3.7), we see
that the fibers have arithmetic genus $g$. Using Theorem 3.6, we see that
$\mathcal{H}_{4,g}^{\prime}$ is equivalent to $\mathcal{Y}_{4,g}^{\prime}$

By excision, the Chow ring of $\mathcal{H}_{4,g}^{\prime}$ is generated by
restriction of classes from $\mathcal{X}_{4,g}^{\prime}$. Since
$\mathcal{X}_{4,g}^{\prime}$ is a vector bundle over
$\mathcal{B}_{4,g}^{\prime}$, their Chow rings are isomorphic, so the
statement about generators follows from (5.4). ∎

######  Lemma 5.4.

The codimension of
$\operatorname{Supp}(R^{1}\pi_{*}(\mathcal{U}_{4,g}\otimes\mathcal{O}_{\mathcal{P}}(-2)))$
is at least $\frac{g+3}{4}-4$. That is, the codimension of the complement of
$\mathcal{B}_{4,g}^{\circ}\subseteq\mathcal{B}_{4,g}$ has codimension at least
$\frac{g+3}{4}-4$.

###### Proof.

By equation (4.7), the codimension of the support of
$R^{1}\pi_{*}(\mathcal{F}^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{E}\otimes\mathcal{O}_{\mathcal{P}}(-2))$
is the minimum value of
$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f}))$
as we range over splitting types $\vec{e}=(e_{1},e_{2},e_{3})$ with $e_{1}\leq
e_{2}\leq e_{3}$ and $\vec{f}=(f_{1},f_{2})$ with $f_{1}\leq f_{2}$ and

$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes(\operatorname{Sym}^{2}\mathcal{O}(\vec{e}))\otimes\mathcal{O}_{\mathbb{P}^{1}}(-2))>0\qquad\Leftrightarrow\qquad
2e_{1}\leq f_{2}.$

We have

$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))\geq
2e_{3}-2e_{1}-3+f_{2}-f_{1}-1.$

To find the minimum, we consider the function of $5$ real variables

$f(x_{1},x_{2},x_{3},y_{1},y_{2}):=2x_{3}-2x_{1}+y_{2}-y_{1}$

on the compact region $D$ defined by

$\displaystyle 0\leq x_{1}\leq x_{2}\leq x_{3},\quad x_{1}+x_{2}+x_{3}=1,\quad
y_{1}\leq y_{2},\quad y_{1}+y_{2}=1,\quad 2x_{1}\leq y_{2}.$

Since $f$ is piecewise linear, its extreme values are attained where multiple
boundary conditions intersect at a point. Code provided at [5] performs the
linear algebra to locate such points and evaluates $f$ at the them to
determine its minimum. The minimum is $\frac{1}{4}$, attained at
$(\frac{1}{4},\frac{3}{8},\frac{3}{8},\frac{1}{2},\frac{1}{2})$. Thus,

$\dim\operatorname{Supp}R^{1}\pi_{*}(\mathcal{U}_{4,g}\otimes\mathcal{O}_{\mathcal{P}}(-2))\geq(g+3)\cdot\min_{D}(f)-4=\frac{g+3}{4}-4.\qed$

For later use, let us note an immediate consequence of the previous lemma:
Using excision and (5.4), we see

(5.6)
$\operatorname{Trun}^{(g+3)/4-4}A^{*}(\mathcal{B}_{4,g}^{\circ})=\operatorname{Trun}^{(g+3)/4-4}\mathbb{Q}[c_{2},a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2}^{\prime},b_{2}].$

Just because the complement of $\mathcal{B}_{4,g}^{\circ}$ has high
codimension inside $\mathcal{B}_{4,g}$ does not mean that the complement of
$\mathcal{H}_{4,g}^{\circ}$ will have high codimension in $\mathcal{H}_{4,g}$.
The condition for $\alpha:C\to\mathbb{P}^{1}$ to be in
$\mathcal{H}_{4,g}^{\circ}$ is that
$h^{1}(\mathbb{P}^{1},F_{\alpha}^{\vee}\otimes\operatorname{Sym}^{2}E_{\alpha})=0$.
We shall refer to this as “our cohomological condition.” Our cohomological
condition fails for factoring covers, as we explain now. Suppose
$\alpha:C\rightarrow\mathbb{P}^{1}$ factors as
$C\xrightarrow{\beta}C^{\prime}\xrightarrow{h}\mathbb{P}^{1}$ where
$C^{\prime}$ has genus $g^{\prime}$. We claim
$E_{\alpha}=\mathcal{O}(g^{\prime}+1)\oplus E^{\prime}$ for some rank $2$
bundle $E^{\prime}$. Indeed, because $\beta$ is a double cover, we have

$\beta_{*}\mathcal{O}_{C}\cong\mathcal{O}_{C^{\prime}}\oplus L,$

where $L$ is a line bundle on $C^{\prime}$. Pushing forward again by $h$,

$\alpha_{*}\mathcal{O}_{C}\cong\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-g^{\prime}-1)\oplus
h_{*}L.$

This establishes that $E_{\alpha}$ has an $\mathcal{O}(g^{\prime}+1)$ summand.
In particular, since some summand of $F$ has degree at least $\frac{g+3}{2}$,

$h^{1}(\mathbb{P}^{1},F^{\vee}\otimes\operatorname{Sym}^{2}E)\geq\frac{g+3}{2}-2(g^{\prime}+1)-1.$

Thus, covers that factor with $g^{\prime}$ small are never in
$\mathcal{H}^{\circ}_{4,g}$. More precisely, if a factoring cover does satisfy
our cohomological condition, then the genus of the intermediate curve must
satisfy $2(g^{\prime}+1)\geq\frac{g+3}{2}$.

######  Lemma 5.5.

The locus of degree $4$ covers $C\to\mathbb{P}^{1}$ that factor $C\rightarrow
C^{\prime}\rightarrow\mathbb{P}^{1}$ where $C^{\prime}$ has genus $g^{\prime}$
has codimension $2(g^{\prime}+1)$ in $\mathcal{H}_{4,g}.$ Hence, the
complement of
$\mathcal{H}^{\circ}_{4,g}\cap\mathcal{H}^{\mathrm{nf}}_{4,g}\subset\mathcal{H}^{\circ}_{4,g}$
has codimension at least $\frac{g+3}{2}$.

###### Proof.

The dimension of the Hurwitz stack is the degree of the branch locus minus
$3=\dim\operatorname{Aut}(\mathbb{P}^{1})$, giving
$\dim\mathcal{H}_{4,g}=2g+3$. Meanwhile, by Riemann–Hurwitz, the dimension of
the space of genus $g$ double covers of a fixed curve $C^{\prime}$ of genus
$g^{\prime}$ is $2g-2-2(2g^{\prime}-2)$. The dimension of the stack of genus
$g^{\prime}$ double covers of $\mathbb{P}^{1}$ modulo
$\operatorname{Aut}(\mathbb{P}^{1})$ is $2g^{\prime}-1$. Therefore, the
dimension of the space of degree $4$ covers that factor through a curve of
genus $g^{\prime}$ is

$2g-2-2(2g^{\prime}-2)+2g^{\prime}-1=2g+1-2g^{\prime}=\dim\mathcal{H}_{4,g}-2(g^{\prime}+1).\qed$

Covers that factor through a curve of low $g^{\prime}$ are therefore loci of
fixed codimension that fail our cohomological condition. For this reason, in
degree $4$, our techniques will only prove that certain Chow groups of the
locus of non-factoring covers are generated by tautological classes. Below, we
collect some results about the splitting types of the vector bundles
associated to a degree $4$ cover. These facts were known to Schreyer [25]
(though Schreyer’s notation differs from ours). We include proofs here as they
demonstrate the geometric meaning of splitting types.

######  Proposition 5.6.

Suppose $\alpha:C\rightarrow\mathbb{P}^{1}$ is a degree $4$ cover and
$E_{\alpha}=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})\oplus\mathcal{O}(e_{3})$
with $e_{1}\leq e_{2}\leq e_{3}$, and
$F=\mathcal{O}(f_{1})\oplus\mathcal{O}(f_{2})$ with $f_{1}\leq f_{2}$. The
following are true:

  1. (1)

$e_{1}+e_{2}+e_{3}=f_{1}+f_{2}=g+3$ and with $e_{1}\geq 1$ if $C$ irreducible.

  2. (2)

If $C$ is irreducible, $2e_{1}\geq f_{1}$, and $2e_{2}\geq f_{2}$. Hence $F$
is globally generated.

  3. (3)

If $\alpha$ does not factor then $e_{1}+e_{3}-f_{2}\geq 0$.

###### Proof.

(1) follows from Example 3.1 and fact that $\det E_{\alpha}\cong\det
F_{\alpha}$. If $C$ is irreducible, we have
$h^{0}(\mathbb{P}^{1},E_{\alpha}^{\vee})=h^{0}(\mathbb{P}^{1},\alpha_{*}\mathcal{O}_{C})-1=0$,
so $e_{1}\geq 1$.

The remaining conditions can be seen from the description of $C$ as the
intersection of two relative quadrics on $\mathbb{P}E_{\alpha}^{\vee}$. Let us
choose a splitting
$E=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})\oplus\mathcal{O}(e_{3})$ and
corresponding coordinates $X,Y,Z$ on $\mathbb{P}E^{\vee}$. The two quadrics
that define $C$ are of the form

(5.7) $\displaystyle p$
$\displaystyle=p_{1,1}X^{2}+p_{1,2}XY+p_{2,2}Y^{2}+p_{1,3}XZ+p_{2,3}YZ+p_{3,3}Z^{2}$
(5.8) $\displaystyle q$
$\displaystyle=q_{1,1}X^{2}+q_{1,2}XY+q_{2,2}Y^{2}+q_{1,3}XZ+q_{2,3}YZ+q_{3,3},Z^{2}$

where $p_{i,j}$ is a polynomial on $\mathbb{P}^{1}$ of degree
$e_{i}+e_{j}-f_{1}$ and $q_{i,j}$ is a polynomial on $\mathbb{P}^{1}$ of
degree $e_{i}+e_{j}-f_{2}$. If this degree is negative, then we mean this
coefficient is zero.

(2) If $2e_{1}<f_{1}$, then $p_{1,1}=q_{1,1}=0$ and $C=V(p,q)$ would contain
the curve $Y=Z=0$, forcing $C$ to be reducible. If $2e_{2}<f_{2}$, then
$q_{1,1}=q_{1,2}=q_{2,2}=0$ so $Z$ divides $q$. If $C$ were irreducible, it
would be contained in one of the linear components of $V(q)$ but this is
impossible. The global generation of $F$ follows because the inequalities
imply $f_{1}=g+3-f_{2}\geq e_{1}\geq 1$.

(3) If $e_{1}+e_{3}-f_{2}\leq-1$, then we show $\alpha$ factors. This
inequality implies

$2e_{1}-f_{2}\leq e_{1}+e_{2}-f_{2}\leq e_{1}+e_{3}-f_{2}\leq-1,$

so the coefficients $p_{1,1},p_{1,2}$, and $p_{1,3}$ vanish. Therefore, $p$ is
a combination of $Y^{2},YZ,$ and $Z^{2}$. Hence, $V(p)$ is reducible in every
fiber and contains the point $[1,0,0]$ in each fiber.

In other words, each fiber of $C\to\mathbb{P}^{1}$ consists of two pairs of
points colinear with $[1,0,0]$. Projection away from the line $Y=Z=0$ defines
a double cover $C\to C^{\prime}$ that factors $\alpha$. ∎

The simultaneous splitting loci of the universal $\mathcal{E}$ and
$\mathcal{F}$ over $\mathcal{H}_{4,g}$ give rise to a stratification of
$\mathcal{H}_{4,g}$. In [10, Remark 4.2], Deopurkar–Patel show that the
codimension of the splitting locus where $\mathcal{E}$ has splitting type
$\vec{e}$ and $\mathcal{F}$ has splitting type $\vec{f}$ is

(5.9)
$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))-h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e})).$

Note that this differs from (4.7) by
$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e}))$!

######  Example 5.7 ($g=6$).

We have $\dim\mathcal{H}_{4,6}=\dim\mathcal{M}_{6}=15$. Using Proposition 5.6
(2), we see that the non-empty strata are

  1. (1)

$\vec{e}=(3,3,3),\vec{f}=(4,5)$, (codimension $0$): The generic stratum.

  2. (2)

$\vec{e}=(2,3,4),\vec{f}=(4,5)$, (codimension $1$): By Casnati-Del Centina
[8], the bielliptic locus is contained in this stratum as the locus where
$p_{1,2}=0$ and $p_{1,3}=0$. Note that $\deg(p_{1,2})=0$ and
$\deg(p_{1,3})=1$, so this represents $3$ conditions, making the bielliptic
locus codimension $4$ inside $\mathcal{H}_{4,6}$.

  3. (3)

$\vec{e}=(3,3,3),\vec{f}=(3,6)$, (codimension $2$): This stratum consists of
trigonal curves. We have
$\mathbb{P}E^{\vee}\cong\mathbb{P}^{1}\times\mathbb{P}^{2}$. Since
$\deg(q_{i,j})=0$ and $\deg(p_{i,j})=3$ for all $i,j$, the projection onto the
$\mathbb{P}^{2}$ factor realizes $C$ as a degree $3$ cover of a conic in
$\mathbb{P}^{2}$.

  4. (4)

$\vec{e}=(2,3,4),\vec{f}=(3,6)$, (codimension 2): Curves with a $g^{2}_{5}$.
We have $p_{1,1}=0$ and $\deg(q_{1,1})=1$, so the curve meets the line $Y=Z=0$
in $\mathbb{P}E^{\vee}$ in one point, say $\nu\in C$. The canonical line
bundle on $C$ is the restriction of
$\mathcal{O}_{\mathbb{P}E^{\vee}}(1)\otimes\omega_{\mathbb{P}^{1}}$, which
contracts the line $Y=Z=0$ in the map $\mathbb{P}E^{\vee}\to\mathbb{P}^{5}$.
Thus, $\nu$ is contained in each of the planes spanned by the image of a fiber
of $\alpha$ under the canoncial. Hence, the $g^{1}_{4}$ plus $\nu$ is a
$g^{2}_{5}$. The locus of genus $6$ curves possessing a $g^{2}_{5}$ is
codimension $3$ in $\mathcal{M}_{6}$, but this stratum has codimension $2$ in
$\mathcal{H}_{4,6}$ because projection from any point on a plane quintic gives
a $g^{1}_{4}$.

  5. (5)

$\vec{e}=(1,4,4),\vec{f}=(2,7)$, (codimension $2$): Hyperelliptic curves

The open $\mathcal{H}_{4,6}^{\prime}$ is the union of strata (1), (2), and
(3), while $\mathcal{H}_{4,6}^{\circ}$ contains only the generic stratum (1).
The image in $\mathcal{M}_{g}$ of $\mathcal{H}_{4,6}^{\prime}$ under the
forgetful map is what Penev–Vakil [24] call the “Mukai general locus” of genus
$6$ curves.

Using the numerical results of Lemma 5.6, we show that the codimension of non-
factoring covers that fail our cohomological condition grows as a positive
fraction of the genus.

######  Lemma 5.8.

The locus of non-factoring degree $4$ covers
$\alpha:C\rightarrow\mathbb{P}^{1}$ such that

$h^{1}(\mathbb{P}^{1},F_{\alpha}^{\vee}\otimes\operatorname{Sym}^{2}E_{\alpha}\otimes\mathcal{O}(-2))>0$

has codimension at least $\frac{g+3}{4}-4$. That is, the codimension the
complement of
$\mathcal{H}^{\circ}_{4,g}\cap\mathcal{H}^{\mathrm{nf}}_{4,g}\subset\mathcal{H}^{\mathrm{nf}}_{4,g}$
is at least $\frac{g+3}{4}-4$.

###### Proof.

By equation (5.9), the codimension of the locus of covers $\alpha$ with
$E_{\alpha}=\mathcal{O}(\vec{e})$ and $F_{\alpha}=\mathcal{O}(\vec{f})$ is

$\displaystyle u(\vec{e},\vec{f})$
$\displaystyle:=h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))-h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e}))$
$\displaystyle\geq
2e_{3}-2e_{1}+f_{2}-f_{1}-4-h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e})).$

Assuming $\alpha$ does not factor, Proposition 5.6 (2) and (3) show that the
only summands of
$\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e})$
that can contribute to
$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e}))$
are $\mathcal{O}(2e_{1}-f_{2})$ and $\mathcal{O}(e_{1}+e_{2}-f_{2})$. Thus,
our task is to bound the function

$\displaystyle
2e_{3}-2e_{1}+f_{2}-f_{1}-4-\max\\{0,f_{2}-2e_{1}-1\\}-\max\\{0,f_{2}-e_{1}-e_{2}-1\\}$

from below on the region where the conditions of Proposition 5.6 hold and
$2e_{1}-f_{2}\leq 0$, which is equivalent to
$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e})\otimes\mathcal{O}(-2))>0$.

Let us introduce a function of $5$ real variables

$\displaystyle f(x_{1},x_{2},x_{3},y_{1},y_{2})$
$\displaystyle:=2x_{3}-2x_{1}+y_{2}-y_{1}-\max\\{0,y_{2}-2x_{1}\\}-\max\\{0,y_{2}-x_{1}-x_{2}\\}$

so that

$u(\vec{e},\vec{f})\geq(g+3)f\left(\frac{e_{1}}{g+3},\frac{e_{2}}{g+3},\frac{e_{3}}{g+3},\frac{f_{1}}{g+3},\frac{f_{2}}{g+3}\right)-4.$

We wish to minimize $f$ on the compact region defined by

$\displaystyle x_{1}+x_{2}+x_{3}=1,\quad y_{1}+y_{2}=1,\quad 0\leq x_{1}\leq
x_{2}\leq x_{3},\quad 0\leq y_{1}\leq 2x_{1}\leq y_{2}\leq
2x_{2},x_{1}+x_{3}.$

These correspond to the conditions from Proposition 5.6, together with the
condition that $2e_{1}\leq f_{2}$, which must be satisfied if the
cohomological condition is failed. Since $f$ is piecewise linear, its extreme
values are attained where multiple boundary conditions (including those where
the function changes) intersect at a point. A program provied in [5] performs
the linear algebra to locate such points and evaluates $f$ at them to
determine its minimum. The minimum is $\frac{1}{4}$, attained at
$(x_{1},x_{2},x_{3},y_{1},y_{2})=(\frac{1}{4},\frac{3}{8},\frac{3}{8},\frac{1}{2},\frac{1}{2})$.
It follows that, if
$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e})\otimes\mathcal{O}(-2))>0$,
then $u(\vec{e},\vec{f})\geq\frac{g+3}{4}-4$. ∎

######  Remark 5.9.

Aaron Landesman points out that our above Lemma 5.8 parallels [1, Lemma 11] of
Bhargava. Bhargava’s two cases $a_{11}=0$ or $a_{11}=a_{12}=0$ correspond to
the fact that either $\mathcal{O}(2e_{1}-f_{2})$ or
$\mathcal{O}(2e_{1}-f_{2})$ and $\mathcal{O}(e_{1}+e_{2}-f_{2})$ are the only
possible negative summands of
$F_{\alpha}^{\vee}\otimes\operatorname{Sym}^{2}E_{\alpha}$ for a non-factoring
cover $\alpha$.

Lemmas 5.5 and 5.8 together should be thought of as saying that
$\mathcal{H}_{4,g}^{\circ}$ and $\mathcal{H}_{4,g}^{\mathrm{nf}}$ are “good
approximations” of each other. We can now complete the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

Suppose $i<\frac{g+3}{4}-4$. Consider the restriction maps

$A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})\xrightarrow{\sim}A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}}\cap\mathcal{H}_{4,g}^{\circ})\xleftarrow{\sim}A^{i}(\mathcal{H}_{4,g}^{\circ}).$

Lemma 5.8 says the arrow on the left is an isomorphism; Lemma 5.5 says the
arrow on the right is an isomorphism. In turn then, we also have

$R^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})\xrightarrow{\sim}R^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}}\cap\mathcal{H}_{4,g}^{\circ})\xleftarrow{\sim}R^{i}(\mathcal{H}_{4,g}^{\circ}),$

where $R^{i}$ of an open substack of $\mathcal{H}_{4,g}$ means the image of
tautological classes under the restriction to that open. Since
$\mathcal{H}_{4,g}^{\circ}\subset\mathcal{H}_{4,g}^{\prime}$, Lemma 5.3
implies $A^{i}(\mathcal{H}_{4,g}^{\circ})=R^{i}(\mathcal{H}_{4,g}^{\circ})$.
Thus, we conclude

$A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})=A^{i}(\mathcal{H}_{4,g}^{\circ})=R^{i}(\mathcal{H}_{4,g}^{\circ})=R^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}}).\qed$

###  5.3. Degree $5$

Using Casnati’s characterization of regular degree $5$ covers (Theorem 3.8), a
regular degree $5$ cover of is equivalent to the data of a rank $4$ vector
bundle $E$; a rank $5$ vector bundle $F$; an isomorphism $(\det E)^{\otimes
2}\cong\det F$; and a global section of $\mathcal{H}om(E^{\vee}\otimes\det
E,\wedge^{2}F)$ satisfying certain conditions. By Example 3.1, if a cover
$\alpha:C\to\mathbb{P}^{1}$ has genus $g$, then $\deg(E_{\alpha})=g+4$. In
turn, $\deg(F_{\alpha})=2\deg(E_{\alpha})=2g+8$. To build the appropriate base
stack, we start with
$\mathcal{V}_{4,g+4}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{5,2g+8}$ which
parametrizes tuples $(S,V,E,F)$ where $V$ is a rank $2$ vector bundle on $S$
with trivial determinant, and $E$ and $F$ are vector bundles of the
appropriate ranks and degrees on $\mathbb{P}V$. We let $\mathcal{E}$ denote
the universal rank $4$ vector bundle and $\mathcal{F}$ the universal rank $5$
bundle on the universal $\mathbb{P}^{1}$-bundle
$\pi:\mathcal{P}\to\mathcal{V}_{4,g+4}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{5,2g+8}$.
Since $\det\mathcal{E}^{\otimes 2}\otimes\det\mathcal{F}^{\vee}$ is a line
bundle of degree $0$ on each fiber of $\pi$, we have $\det\mathcal{E}^{\otimes
2}\otimes\det\mathcal{F}^{\vee}\cong\pi^{*}\mathcal{L}$ where
$\mathcal{L}:=\pi_{*}(\det\mathcal{E}^{\otimes
2}\otimes\det\mathcal{F}^{\vee}),$ which is a line bundle by cohomology and
base change.

######  Definition 5.10.

With notation as above, we define the stack $\mathcal{B}_{5,g}$ as the
$\mathbb{G}_{m}$-torsor over
$\mathcal{V}_{5,g+4}\times_{\operatorname{BSL}_{2}}\mathcal{V}_{5,2g+8}$ given
by the complement of the zero section of the line bundle $\mathcal{L}$.

By slight abuse of notation, we continue to denote the universal
$\pi:\mathbb{P}^{1}$-bundle by $\mathcal{P}\to\mathcal{B}_{5,g}$ and the
universal rank $4$ and $5$ vector bundles on it by $\mathcal{E}$ and
$\mathcal{F}$. Let $z=\mathcal{O}_{\mathcal{P}}(1)$ and write

$c_{i}(\mathcal{E})=\pi^{*}a_{i}+(\pi^{*}a_{i}^{\prime})z\qquad\text{and}\qquad
c_{i}(\mathcal{F})=\pi^{*}b_{i}+(\pi^{*}b_{i}^{\prime})z.$

for $a_{i},b_{i}\in A^{i}(\mathcal{B}_{5,g})$ and
$a_{i}^{\prime},b_{i}^{\prime}\in A^{i-1}(\mathcal{B}_{5,g})$. Note that
$2a_{1}^{\prime}=b_{1}^{\prime}=2(g+4)$. Moreover, by definition of
$\mathcal{B}_{5,g}$, we have $c_{1}(\det\mathcal{E}^{\otimes
2}\otimes\det\mathcal{F}^{\vee})=0$, so $b_{1}=2a_{1}$. Using Lemma 2.1 and
Theorem 4.4 as in the previous subsection, we have

(5.10)
$\operatorname{Trun}^{g+5}A^{*}(\mathcal{B}_{5,g})=\operatorname{Trun}^{g+5}\mathbb{Q}[c_{2},a_{1},\ldots,a_{4},a_{2}^{\prime},\ldots,a_{4}^{\prime},b_{2},\ldots,b_{5},b_{2}^{\prime},\ldots,b_{5}^{\prime}].$

We define
$\mathcal{U}_{5,g}:=\mathcal{H}om(\mathcal{E}^{\vee}\otimes\det\mathcal{E},\wedge^{2}\mathcal{F})$,
and $\mathcal{B}_{5,g}^{\prime}$ and $\mathcal{B}_{5,g}^{\circ}$ as in (5.1)
and (5.2), respectively. Given a map $S\rightarrow\mathcal{H}_{5,g}$, let
$\pi_{S}:\mathcal{P}_{S}\rightarrow S$ denote the $\mathbb{P}^{1}$-bundle and
let $\mathcal{E}_{S}$ (resp. $\mathcal{F}_{S})$ be the rank $4$ (resp. rank
$5$) vector bundle on $\mathcal{P}_{S}$ associated to the family in the sense
of Casnati–Ekedahl. The open substacks
$\mathcal{H}^{\circ}_{5,g}\subseteq\mathcal{H}^{\prime}_{5,g}\subseteq\mathcal{H}_{5,g}$
are defined by

$\displaystyle\\{S\rightarrow\mathcal{H}^{\circ}_{5,g}\\}$
$\displaystyle=\\{S\rightarrow\mathcal{H}_{5,g}:R^{1}(\pi_{S})_{*}(\mathcal{H}om(\mathcal{E}^{\vee}_{S}\otimes\det\mathcal{E}_{S},\wedge^{2}\mathcal{F}_{S})\otimes\mathcal{O}_{\mathcal{P}_{S}}(-2))=0$
$\displaystyle\qquad\qquad\qquad\qquad\text{and $\mathcal{F}_{S}$ globally
generated on fibers of $\pi_{S}$}\\}.$
$\displaystyle\\{S\rightarrow\mathcal{H}^{\prime}_{5,g}\\}$
$\displaystyle=\\{S\rightarrow\mathcal{H}_{5,g}:R^{1}(\pi_{S})_{*}(\mathcal{H}om(\mathcal{E}^{\vee}_{S}\otimes\det\mathcal{E}_{S},\wedge^{2}\mathcal{F}_{S}))=0$
$\displaystyle\qquad\qquad\qquad\qquad\text{and $\mathcal{F}_{S}$ globally
generated on fibers of $\pi_{S}$}\\}.$

The important feature of the open $\mathcal{H}_{5,g}^{\prime}$ is that it can
be realized as an open inside the vector bundle
$\mathcal{X}_{5,g}^{\prime}:=\pi_{*}\mathcal{U}_{5,g}|_{\mathcal{B}_{5,g}^{\prime}}$
over $\mathcal{B}_{5,g}^{\prime}$.

######  Lemma 5.11.

There is an open inclusion
$\mathcal{H}_{5,g}^{\prime}\to\mathcal{X}_{5,g}^{\prime}$. In particular,
$A^{*}(\mathcal{H}_{5,g}^{\prime})=R^{*}(\mathcal{H}_{5,g}^{\prime})$ is
generated by the CE classes
$c_{2},a_{1},\ldots,a_{4},a_{2}^{\prime},\ldots,a_{4}^{\prime},b_{2},\ldots,b_{5},b_{2}^{\prime},\ldots,b_{5}^{\prime}$.

###### Proof.

The objects of $\mathcal{X}_{5,g}^{\prime}$ are tuples $(S,V,E,F,\phi,\eta)$
where $(S,V,E,F,\phi)\in\mathcal{B}_{5,g}^{\prime}$ and $\eta\in
H^{0}(\mathbb{P}V,\mathcal{H}om(E^{\vee}\otimes\det E,\wedge^{2}F))$. Using
the notation of Section 3.3, we define
$\mathcal{Y}_{5,g}^{\prime}\subset\mathcal{X}_{5,g}^{\prime}$ to be the open
substack defined by the condition that
$D(\Phi(\eta))\subset\mathbb{P}E^{\vee}\to S$ is a family of smooth curves.
Considering their Hilbert polynomials as determined by the resolution (3.14),
we see that the fibers of $D(\Phi(\eta))\to S$ have arithmetic genus $g$.
Applying Theorem 3.8, we see that $\mathcal{H}_{5,g}^{\prime}$ is equivalent
to $\mathcal{Y}_{5,g}^{\prime}$

By excision, the Chow ring of $\mathcal{H}_{5,g}^{\prime}$ is generated by
restriction of classes from $\mathcal{X}_{5,g}^{\prime}$. Since
$\mathcal{X}_{5,g}^{\prime}$ is a vector bundle over
$\mathcal{B}_{5,g}^{\prime}$, their Chow rings are isomorphic, so the
statement about generators follows from Theorem 4.4. ∎

Now we show that the complements of the opens we have defined have high
codimension.

######  Lemma 5.12.

The support of
$R^{1}\pi_{*}(\mathcal{U}_{5,g}\otimes\mathcal{O}_{\mathcal{P}}(-2))$ has
codimension at least $\frac{g+4}{5}-16$. That is, the codimension of the
complement of $\mathcal{B}_{5,g}^{\circ}\subset\mathcal{B}_{5,g}$ is at least
$\frac{g+4}{5}-16$.

###### Proof.

By (4.7), the codimension of the support of
$R^{1}\pi_{*}(\mathcal{H}om(\mathcal{E}^{\vee}\otimes\det\mathcal{E},\wedge^{2}\mathcal{F})\otimes\mathcal{O}_{\mathcal{P}}(-2))$
is the minimum value of

$h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))$

as we range over splitting types $\vec{e}$ of degree $g+4$ and $\vec{f}$ of
degree $2g+8$ so that

$h^{1}(\mathbb{P}^{1},\mathcal{H}om(\mathcal{O}(\vec{e})^{\vee}\otimes\det\mathcal{O}(\vec{e}),\wedge^{2}\mathcal{O}(\vec{f}))\otimes\mathcal{O}_{\mathbb{P}^{1}}(-2))>0\quad\Leftrightarrow\quad
e_{1}+f_{1}+f_{2}-(g+4)\leq 0.$

Similar to the proof of Lemma 5.4, we may find this minimum by finding the
minimum of the function

$f(x_{1},\ldots,x_{4},y_{1},\ldots,y_{5})=3x_{4}+x_{3}-x_{2}-3x_{1}+4y_{5}+2y_{4}-2y_{2}-4y_{1}$

on the compact region $D$ defined by

$\displaystyle 0\leq x_{1}\leq\cdots\leq x_{4},\quad
x_{1}+\ldots+x_{4}=1,\quad 0\leq y_{1}\leq\cdots\leq y_{5},\quad
y_{1}+\ldots+y_{5}=2$ $\displaystyle x_{1}+y_{1}+y_{2}-1\leq 0.$

Using our code [5], we find that the minimum of the linear function $f$ over
$D$ is $\frac{1}{5}$ attained at
$(\frac{1}{5},\frac{4}{15},\frac{4}{15},\frac{4}{15},\frac{2}{5},\frac{2}{5},\frac{2}{5},\frac{2}{5},\frac{2}{5})$.
Thus,

$\displaystyle\dim\operatorname{Supp}R^{1}\pi_{*}(\mathcal{U}_{5,g}\otimes\mathcal{O}_{\mathcal{P}}(-2))$
$\displaystyle\geq(g+4)\cdot\min_{D}(f)-16=\frac{g+4}{5}-16.\qed$

For later use, let us note an immediate consequence of the previous lemma:
Using excision and (5.10), we see

(5.11)
$\operatorname{Trun}^{(g+4)/5-16}A^{*}(\mathcal{B}_{5,g}^{\circ})=\operatorname{Trun}^{(g+4)/5-16}\mathbb{Q}[c_{2},a_{1},\ldots,a_{4},a_{2}^{\prime},\ldots,a_{4}^{\prime},b_{2},\ldots,b_{5},b_{2}^{\prime},\ldots,b_{5}^{\prime}].$

######  Lemma 5.13.

The codimension of the locus of smooth degree $5$ covers $\alpha$ such that

$h^{1}(\mathcal{H}om(E_{\alpha}^{\vee}\otimes\det
E_{\alpha},\wedge^{2}F_{\alpha})\otimes\mathcal{O}_{\mathbb{P}^{1}}(-2))>0$

has codimension at least $\frac{g+4}{5}-16$. That is, the codimension of the
complement of $\mathcal{H}^{\circ}_{5,g}$ inside $\mathcal{H}_{5,g}$ is at
least $\frac{g+4}{5}-16$.

###### Proof.

The cohomological statement depends only on the splitting type of $E_{\alpha}$
and $F_{\alpha}$. In the proof of [10, Proposition 5.2], Deopurkar–Patel show
that the codimension of the locus of covers such that
$E_{\alpha}\cong\mathcal{O}(\vec{e})$ and
$F_{\alpha}\cong\mathcal{O}(\vec{f})$ has codimension

(5.12) $\displaystyle u(\vec{e},\vec{f})$
$\displaystyle:=h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))$
$\displaystyle\qquad-h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{e})\otimes\wedge^{2}\mathcal{O}(\vec{f})\otimes\mathcal{O}_{\mathbb{P}^{1}}(-g-4)).$

A cover with these discrete invariants corresponds to a global section $\eta$
of

$\mathcal{H}om(\mathcal{O}(\vec{e})^{\vee}\otimes\det\mathcal{O}(\vec{e}),\wedge^{2}\mathcal{O}(\vec{f}))=\mathcal{O}(\vec{e})\otimes\wedge^{2}\mathcal{O}(\vec{f})\otimes\mathcal{O}_{\mathbb{P}^{1}}(-g-4).$

Such a global section can be represented by a skew-symmetric matrix

(5.13) $M_{\eta}=\left(\begin{matrix}0&L_{1,2}&L_{1,3}&L_{1,4}&L_{1,5}\\\
-L_{1,2}&0&L_{2,3}&L_{2,4}&L_{2,5}\\\ -L_{1,3}&-L_{2,3}&0&L_{3,4}&L_{3,5}\\\
-L_{1,4}&-L_{2,4}&-L_{3,4}&0&L_{4,5}\\\
-L_{1,5}&-L_{2,5}&-L_{3,5}&-L_{4,5}&0\end{matrix}\right),$

where $L_{i,j}\in
H^{0}(\mathcal{O}(f_{i}+f_{j})\otimes\mathcal{O}(\vec{e})\otimes\mathcal{O}(-g-4))$.
The corresponding curve $C\subset\mathbb{P}E^{\vee}$ is cut out by the
$4\times 4$ Pfaffians of the main minors of $M_{\eta}$. The Pfaffian of the
submatrix obtained by deleting the last row and column is

$Q_{5}=L_{1,2}L_{3,4}-L_{1,3}L_{2,4}+L_{2,3}L_{1,4}.$

If $Q_{5}$ is reducible, then $C$ is reducible. Indeed, if $C$ were
irreducible, it would be contained in one component of $Q_{5}$, forcing every
fiber to be contained in a hyperplane, violating the Geometric-Riemann-Roch
theorem. Therefore, as observed in [10, p. 21], $L_{1,2}$ and $L_{1,3}$ cannot
both be identically zero, and so

(5.14) $f_{1}+f_{3}+e_{4}-(g+4)\geq 0.$

Let $X_{1},\ldots,X_{4}$ be coordinates on $\mathbb{P}E^{\vee}$ corresponding
to a choice of splitting $E\cong\mathcal{O}(\vec{e})$, so we think of
$L_{i,j}$ as a linear form in $X_{1},\ldots,X_{4}$ where the coefficient of
$X_{k}$ is a section of
$\mathcal{O}(f_{i}+f_{j})\otimes\mathcal{O}(e_{k})\otimes\mathcal{O}(-g-4)$,
i.e. a homogeneous polynomial of degree $f_{i}+f_{j}+e_{k}-(g+4)$ on
$\mathbb{P}^{1}$. If $Q_{5}$ is irreducible, it cannot be divisible by
$X_{4}$. Observe that if $f_{i}+f_{j}+e_{3}-(g+4)<0$, then the coefficients of
$X_{k}$ for $k\leq 3$ vanish, so $X_{4}$ divides $L_{i,j}$. If $X_{4}$ divides
$L_{1,2}$, $L_{1,3}$ and $L_{1,4}$, then $X_{4}$ divides $Q_{5}$ and $Q_{5}$
is reducible. To prevent this, we must have

(5.15) $f_{1}+f_{4}+e_{3}-(g+4)\geq 0.$

Similarly, if $X_{4}$ divides $L_{1,2}$, $L_{1,3}$ and $L_{2,3}$, then $X_{4}$
divides $Q_{5}$ and $Q_{5}$ is reducible. To prevent this, we must have

(5.16) $f_{2}+f_{3}+e_{3}-(g+4)\geq 0.$

For splitting types satisfying (5.14), (5.15), and (5.16), at most $11$ of the
$40$ summands of the form $\mathcal{O}(e_{i}+f_{j}+f_{k}-(g+4))$ in
$\mathcal{O}(\vec{e})\otimes\wedge^{2}\mathcal{O}(\vec{f})\otimes\mathcal{O}_{\mathbb{P}^{1}}(-g-4)$
can be negative. For these allowed splitting types, we have

$\displaystyle u(\vec{e},\vec{f})$
$\displaystyle=h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e})))+h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f})))$
$\displaystyle\qquad-\sum_{i=1}^{4}\max\\{0,g+3-f_{1}-f_{2}-e_{i}\\}-\sum_{i=1}^{3}\max\\{0,g+3-f_{1}-f_{3}-e_{i}\\}$
$\displaystyle\qquad-\sum_{i=1}^{2}\max\\{0,g+3-f_{1}-f_{4}-e_{i}\\}-\sum_{i=1}^{2}\max\\{0,g+3-f_{2}-f_{3}-e_{i}\\}.$

We seek a lower bound on $u(\vec{e},\vec{f})$ given that
$\mathcal{O}(\vec{e})\otimes\wedge^{2}\mathcal{O}(\vec{f})\otimes\mathcal{O}(-g-4)$
has a non-positive summand, i.e. in the region where
$e_{1}+f_{1}+f_{2}-(g+4)\leq 0$. Note that

$\displaystyle h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{e}))$
$\displaystyle\geq 3e_{4}+e_{3}-e_{2}-3e_{1}-6$ $\displaystyle
h^{1}(\mathbb{P}^{1},\mathcal{E}nd(\mathcal{O}(\vec{f}))$ $\displaystyle\geq
4f_{5}+2f_{4}-2f_{2}-4f_{1}-10.$

Let us define a function of $9$ real variables

$\displaystyle f(x_{1},\ldots,x_{4},y_{1},\ldots,y_{5})$
$\displaystyle:=3x_{4}+x_{3}-x_{2}-3x_{1}+4y_{5}+2y_{4}-2y_{2}-4y_{1}$
$\displaystyle\quad-\sum_{i=1}^{4}\max\\{0,1-y_{1}-y_{2}-x_{i}\\}-\sum_{i=1}^{3}\max\\{0,1-y_{1}-y_{3}-x_{i}\\}$
$\displaystyle\quad-\sum_{i=1}^{2}\max\\{0,1-y_{1}-y_{4}-x_{i}\\}-\sum_{i=1}^{2}\max\\{0,1-y_{2}-y_{3}-x_{i}\\}$

so that

$u(\vec{e},\vec{f})\geq(g+4)f\left(\frac{e_{1}}{g+4},\ldots,\frac{e_{4}}{g+4},\frac{f_{1}}{g+4},\ldots,\frac{f_{5}}{g+4}\right)-16.$

Now we wish to find the minimum of $f$ on the compact region defined by

$\displaystyle 0\leq x_{1}\leq\cdots\leq x_{4},\quad
x_{1}+\ldots+x_{4}=1,\quad 0\leq y_{1}\leq\cdots\leq y_{5},\quad
y_{1}+\ldots+y_{5}=2$ $\displaystyle y_{1}+y_{3}+x_{4}-1\geq 0,\quad
y_{1}+y_{4}+x_{3}-1\geq 0,\qquad y_{2}+y_{3}+x_{3}-1\geq 0$ $\displaystyle
x_{1}+y_{1}+y_{2}-1\leq 0.$

Since $f$ is piecewise linear, its extreme values are attained at points where
multiple boundary conditions (including those where the linear function
changes) intersect to give a single point. Our code [5] performs the linear
algebra to locate such points and determines that the minimum is
$\frac{1}{5}$, which is attained at
$(\frac{1}{5},\frac{4}{15},\frac{4}{15},\frac{4}{15},\frac{2}{5},\frac{2}{5},\frac{2}{5},\frac{2}{5},\frac{2}{5})$.
It follows that if $\vec{e}$ and $\vec{f}$ satisfy
$h^{1}(\mathbb{P}^{1},\mathcal{O}(\vec{e})\otimes\wedge^{2}\mathcal{O}(\vec{f})\otimes\mathcal{O}(-g-4)\otimes\mathcal{O}(-2))>0$
then $u(\vec{e},\vec{f})\geq\frac{g+4}{5}-16$. ∎

We now prove Theorem 1.4 to complete the $k=5$ case.

###### Proof of Theorem 1.4.

Suppose $i<\frac{g+4}{5}-16$. Then, by excision and Lemma 5.13, the
restriction map $A^{i}(\mathcal{H}_{5,g})\to A^{i}(\mathcal{H}_{5,g}^{\circ})$
is an isomorphism. Hence, $R^{i}(\mathcal{H}_{5,g})\to
R^{i}(\mathcal{H}_{5,g}^{\circ})$ is also an isomorphism. Since
$\mathcal{H}_{5,g}^{\circ}\subseteq\mathcal{H}_{5,g}^{\prime}$, Lemma 5.11
tells us that
$A^{i}(\mathcal{H}_{5,g}^{\circ})=R^{i}(\mathcal{H}_{5,g}^{\circ})$. Hence, we
have shown

$A^{i}(\mathcal{H}_{5,g})=A^{i}(\mathcal{H}_{5,g}^{\circ})=R^{i}(\mathcal{H}_{5,g}^{\circ})=R^{i}(\mathcal{H}_{5,g}).\qed$

##  6\. Conclusion and preview of subsequent work

At this point, we have established that, for $k\leq 5$, a large portion of the
Chow ring of the (non-factoring) Hurwitz space
$\mathcal{H}_{k,g}^{\mathrm{nf}}$ is tautological. We did so by showing that
$\mathcal{H}_{k,g}^{\mathrm{nf}}$ is closely approximated by an open substack
$\mathcal{H}_{k,g}^{\circ}$ which, in turn, can be realized as an open
substack of a vector bundle
$\mathcal{X}_{r,d}^{\circ}=\mathcal{X}_{r,d}^{\prime}|_{\mathcal{B}_{r,d}^{\circ}}$
over the stack $\mathcal{B}_{r,d}^{\circ}$. By (5.6) and (5.11), we understand
$A^{*}(\mathcal{B}_{r,d}^{\circ})\cong A^{*}(\mathcal{X}_{r,d}^{\circ})$ well.
First of all, we know $A^{*}(\mathcal{B}_{r,d}^{\circ})$ is generated by
classes which pullback to the CE classes on $\mathcal{H}_{k,g}^{\circ}$; this
is how we saw
$A^{*}(\mathcal{H}_{k,g}^{\circ})=R^{*}(\mathcal{H}_{k,g}^{\circ})$. However,
we actually know a bit more: the generators we list for
$A^{*}(\mathcal{B}_{r,d}^{\circ})\cong A^{*}(\mathcal{X}_{r,d}^{\circ})$
satisfy no relations in low degrees. In other words, all relations that their
pullbacks to $\mathcal{H}_{k,g}^{\circ}$ satisfy come from performing excision
on the complement of
$\mathcal{H}_{k,g}^{\circ}\subset\mathcal{X}_{r,d}^{\circ}$.

Determining these relations will be the focus of our subsequent work [3]. The
central innovation there is to find an appropriate resolution of the
complement of $\mathcal{H}_{k,g}^{\circ}\subset\mathcal{X}_{r,d}^{\circ}$,
which allows us to determine _all_ relations in degrees up to roughly $g/k$.
Furthermore, we will prove that the relations we find among the restrictions
of CE classes to $\mathcal{H}_{k,g}^{\circ}$ actually hold on _all_ of
$\mathcal{H}_{k,g}$. Using the codimension bounds we established in Section 5
here, results about the structure of
$A^{*}(\mathcal{H}_{k,g}^{\circ})=R^{*}(\mathcal{H}_{k,g}^{\circ})$ will then
translate into results about the structure of
$A^{*}(\mathcal{H}_{k,g}^{\mathrm{nf}})$ and $R^{*}(\mathcal{H}_{k,g})$ in
degrees up to roughly $g/k$.

## References

  * [1] Manjul Bhargava, _The density of discriminants of quartic rings and fields_ , Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288
  * [2] Michele Bolognesi and Angelo Vistoli, _Stacks of trigonal curves_ , Trans. Amer. Math. Soc. 364 (2012), no. 7, 3365–3393. MR 2901217
  * [3] Samir Canning and Hannah Larson, _Chow rings of low-degree Hurwitz spaces_ , (2021).
  * [4] by same author, _The Chow rings of the moduli spaces of curves of genus $7,8$ and $9$_, arXiv:2104.05820 (2021).
  * [5] by same author, _Low-degree-hurwitz_ , https://github.com/src2165/Low-Degree-Hurwitz/, 2021.
  * [6] G. Casnati and T. Ekedahl, _Covers of algebraic varieties. I. A general structure theorem, covers of degree $3,4$ and Enriques surfaces_, J. Algebraic Geom. 5 (1996), no. 3, 439–460. MR 1382731
  * [7] Gianfranco Casnati, _Covers of algebraic varieties. II. Covers of degree $5$ and construction of surfaces_, J. Algebraic Geom. 5 (1996), no. 3, 461–477. MR 1382732
  * [8] Gianfranco Casnati and Andrea Del Centina, _A characterization of bielliptic curves and applications to their moduli spaces_ , Ann. Mat. Pura Appl. (4) 181 (2002), no. 2, 213–221. MR 1911936
  * [9] Gianfranco Casnati and Roberto Notari, _On some gorenstein loci in $hilb^{6}(\mathbb{P}_{k}^{4})$_, Journal of Algebra 308 (2007), no. 2, 493 – 523.
  * [10] Anand Deopurkar and Anand Patel, _The Picard rank conjecture for the Hurwitz spaces of degree up to five_ , Algebra Number Theory 9 (2015), no. 2, 459–492. MR 3320849
  * [11] Dan Edidin, _Equivariant geometry and the cohomology of the moduli space of curves_ , Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 259–292. MR 3184166
  * [12] Dan Edidin and William Graham, _Equivariant intersection theory_ , Invent. Math. 131 (1998), no. 3, 595–634. MR 1614555
  * [13] Carel Faber, _Chow rings of moduli spaces of curves. I. The Chow ring of $\overline{\mathscr{M}}_{3}$_, Ann. of Math. (2) 132 (1990), no. 2, 331–419. MR 1070600
  * [14] by same author, _Chow rings of moduli spaces of curves. II. Some results on the Chow ring of $\overline{\mathscr{M}}_{4}$_, Ann. of Math. (2) 132 (1990), no. 3, 421–449. MR 1078265
  * [15] John Harer, _The second homology group of the mapping class group of an orientable surface_ , Invent. Math. 72 (1983), no. 2, 221–239. MR 700769
  * [16] E. Izadi, _The Chow ring of the moduli space of curves of genus $5$_, The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 267–304. MR 1363060
  * [17] Hannah Larson, _The intersection theory of the moduli stack of vector bundles on $\mathbb{P}^{1}$_, arXiv:2104.14642 (2021).
  * [18] Hannah K. Larson, _Universal degeneracy classes for vector bundles on $\mathbb{P}^{1}$ bundles_, Adv. Math. 380 (2021), 107563, 20. MR 4200467
  * [19] Ib Madsen and Michael Weiss, _The stable moduli space of Riemann surfaces: Mumford’s conjecture_ , Ann. of Math. (2) 165 (2007), no. 3, 843–941. MR 2335797
  * [20] Rick Miranda, _Triple covers in algebraic geometry_ , Amer. J. Math. 107 (1985), no. 5, 1123–1158. MR 805807
  * [21] Scott Mullane, _The Hurwitz space Picard rank conjecture for $d>g-1$_, arXiv:2009.10063 (2020).
  * [22] David Mumford, _Towards an enumerative geometry of the moduli space of curves_ , Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR 717614
  * [23] Anand Patel and Ravi Vakil, _On the Chow ring of the Hurwitz space of degree three covers of $\mathbb{P}^{1}$_, arXiv:1505.04323 (2015).
  * [24] Nikola Penev and Ravi Vakil, _The Chow ring of the moduli space of curves of genus six_ , Algebr. Geom. 2 (2015), no. 1, 123–136. MR 3322200
  * [25] Frank-Olaf Schreyer, _Syzygies of canonical curves and special linear series_ , Math. Ann. 275 (1986), no. 1, 105–137. MR 849058
  * [26] Jason van Zelm, _Nontautological bielliptic cycles_ , Pacific J. Math. 294 (2018), no. 2, 495–504. MR 3770123
  * [27] Angelo Vistoli, _Chow groups of quotient varieties_ , J. Algebra 107 (1987), no. 2, 410–424. MR 885804
  * [28] by same author, _Alexander duality in intersection theory_ , Compositio Math. 70 (1989), no. 3, 199–225. MR 1002043