# Chow rings of low-degree Hurwitz spaces

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

While there is much work and many conjectures surrounding the intersection
theory of the moduli space of curves, relatively little is known about the
intersection theory of the Hurwitz space $\mathcal{H}_{k,g}$ parametrizing
smooth degree $k$, genus $g$ covers of $\mathbb{P}^{1}$. Let $k=3,4,5$. We
prove that the rational Chow rings of $\mathcal{H}_{k,g}$ stabilize in a
suitable sense as $g$ tends to infinity. In the case $k=3$, we completely
determine the Chow rings for all $g$. We also prove that the rational Chow
groups of the simply branched Hurwitz space
$\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}$ are zero in codimension up to
roughly $g/k$. In [8], results developed in this paper are used to prove that
the Chow rings of $\mathcal{M}_{7},\mathcal{M}_{8},$ and $\mathcal{M}_{9}$ are
tautological.

###### 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

Intersection theory on the moduli space of curves $\mathcal{M}_{g}$ has
received much attention since Mumford’s famous paper [29], in which he
introduced the Chow ring of $\mathcal{M}_{g}$. Based on Harer’s result [22]
that the cohomology of the moduli space of curves is independent of the genus
$g$ in degrees small relative to $g$, Mumford conjectured that the stable
cohomology ring is isomorphic to
$\mathbb{Q}[\kappa_{1},\kappa_{2},\kappa_{3},\ldots]$. Madsen–Weiss [26] later
proved Mumford’s conjecture. It is unknown whether there is an analogous
stabilization result in the Chow ring of $\mathcal{M}_{g}$. Upon restricting
attention to the tautological ring, however, more is known.

The _tautological subring_ $R^{*}(\mathcal{M}_{g})\subseteq
A^{*}(\mathcal{M}_{g})$ is defined to be the subring of the rational Chow ring
generated by the kappa classes. There are many conjectures concerning the
relations and structure of the tautological ring. Prominent among them is
Faber’s conjecture [18, Conjecture 1], which states that the tautological ring
should be Gorenstein with socle in codimension $g-2$ and generated by the
first $\lfloor g/3\rfloor$ kappa classes with no relations in degree less than
$\lfloor g/3\rfloor$. The Gorenstein part of Faber’s conjecture is unknown,
although it has been shown to hold when $g\leq 23$ by a direct computer
calculation of Faber. The second portion of Faber’s conjecture has been
proved: Ionel [24] proved that the tautological ring is generated by
$\kappa_{1},\kappa_{2},\ldots,\kappa_{\lfloor g/3\rfloor}$, and Boldsen [6]
proved that there are no relations among the $\kappa$-classes in degrees less
than $\lfloor g/3\rfloor$. In other words, there is a surjection

(1.1) $\mathbb{Q}[\kappa_{1},\kappa_{2},\ldots,\kappa_{\lfloor
g/3\rfloor}]\twoheadrightarrow R^{*}(\mathcal{M}_{g}),$

which is an isomorphism in degrees less than $\lfloor g/3\rfloor$.

In this paper, we study the Chow rings of low-degree Hurwitz spaces. Our main
theorem is a stabilization result of a similar flavor to (1.1). Let
$\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$
covers of $\mathbb{P}^{1}$ up to automorphisms of the target. 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
$f_{*}(c_{1}(\omega_{f})^{i}\cdot\alpha^{*}c_{1}(\omega_{\pi})^{j})=\pi_{*}(\alpha_{*}(c_{1}(\omega_{f})^{i})\cdot
c_{1}(\omega_{\pi})^{j})$. Let $\mathcal{E}^{\vee}$ be the cokernel of the map
$\mathcal{O}_{\mathcal{P}}\to\alpha_{*}\mathcal{O}_{\mathcal{C}}$ (the
universal “Tschirnhausen bundle”). Set
$z=-\frac{1}{2}c_{1}(\omega_{\pi})``=c_{1}(\mathcal{O}_{\mathcal{P}}(1))"$.
Our theorem will be stated in terms of the tautological classes
$c_{2}=-\pi_{*}(z^{3})\in A^{2}(\mathcal{H}_{k,g})$ and

$a_{i}=\pi_{*}(c_{i}(\mathcal{E})\cdot z)\in
A^{i}(\mathcal{H}_{k,g})\qquad\text{and}\qquad
a_{i}^{\prime}=\pi_{*}(c_{i}(\mathcal{E}))\in A^{i-1}(\mathcal{H}_{k,g}).$

When $k=3,4,5$, our main theorem gives a minimal set of generators for
$R^{*}(\mathcal{H}_{k,g})$ and determines all relations among them in degrees
up to roughly $g/k$. In contrast with the case of $\mathcal{M}_{g}$ in (1.1),
the tautological ring of $\mathcal{H}_{k,g}$ does _not_ require a growing
number of generators as $g$ increases. In degree $3$, we determine the full
Chow ring of $\mathcal{H}_{3,g}$. When $k=3,5$, our results imply that the
dimensions of the Chow groups of $\mathcal{H}_{k,g}$ are independent of $g$
for $g$ sufficiently large. In degree $4$, _factoring covers_ — i.e. covers
$C\to\mathbb{P}^{1}$ that factor as a composition of two double covers $C\to
C^{\prime}\to\mathbb{P}^{1}$ — present a difficulty. We instead obtain
stabilization results for the Chow groups of
$\mathcal{H}_{4,g}^{\mathrm{nf}}\subseteq\mathcal{H}_{4,g}$, the open substack
parametrizing non-factoring covers, or equivalently covers whose monodromy
group is not contained in the dihedral group $D_{4}$.

######  Theorem 1.1.

Let $g\geq 2$ be an integer.

  1. (1)

The rational Chow ring of $\mathcal{H}_{3,g}$ is

$A^{*}(\mathcal{H}_{3,g})=R^{*}(\mathcal{H}_{3,g})=\begin{cases}\mathbb{Q}&\text{if
$g=2$}\\\ \mathbb{Q}[a_{1}]/(a_{1}^{2})&\text{if $g=3,4,5$}\\\
\mathbb{Q}[a_{1}]/(a_{1}^{3})&\text{if $g\geq 6$.}\end{cases}$

  2. (2)

Let $r_{i}=r_{i}(g)$ be defined as in Section 5.4. For each $g$ there is a map

$\frac{\mathbb{Q}[a_{1},a_{2}^{\prime},a_{3}^{\prime}]}{\langle
r_{1},r_{2},r_{3},r_{4}\rangle}\twoheadrightarrow
R^{*}(\mathcal{H}_{4,g})\subseteq A^{*}(\mathcal{H}_{4,g})\to
A^{*}(\mathcal{H}_{4,g}^{\mathrm{nf}}),$

such that the composition is an isomorphism in degrees up to
$\frac{g+3}{4}-4$. Furthermore, the dimension of the Chow group
$A^{i}(\mathcal{H}^{\mathrm{nf}}_{4,g})$ is independent of $g$ for $g>4i+12$.
When $g>4i+12$, the dimensions are given by

$\dim A^{i}(\mathcal{H}^{\mathrm{nf}}_{4,g})=\dim
R^{i}(\mathcal{H}_{4,g})=\begin{cases}2&i=1,4\\\ 4&i=2\\\ 3&i=3\\\ 1&i\geq
5.\end{cases}$

  3. (3)

Let $r_{i}=r_{i}(g)$ be as defined in Section 6.4. There is a map

$\frac{\mathbb{Q}[a_{1},a_{2}^{\prime},a_{2},c_{2}]}{\langle
r_{1},r_{2},r_{3},r_{4},r_{5}\rangle}\twoheadrightarrow
R^{*}(\mathcal{H}_{5,g})\subseteq A^{*}(\mathcal{H}_{5,g})$

such that the composition is an isomorphism in degrees $\leq\frac{g+4}{5}-16$.
Furthermore, the dimension of the Chow group $A^{i}(\mathcal{H}_{5,g})$ is
independent of $g$ for $g>5i+76$. When $g>5i+76$, the dimensions are given by

$\dim A^{i}(\mathcal{H}_{5,g})=\dim
R^{i}(\mathcal{H}_{5,g})=\begin{cases}2&i=1,i\geq 7\\\ 5&i=2\\\ 6&i=3\\\
7&i=4\\\ 4&i=5\\\ 3&i=6.\end{cases}$

######  Remark 1.2.

Angelina Zheng recently computed the rational cohomology of
$\mathcal{H}_{3,5}$ in [33], and, in subsequent work [34], finds the stable
rational cohomology of $\mathcal{H}_{3,g}$. Together, our results prove that
the cycle class map is injective. The corresponding statement for
$\mathcal{M}_{g}$ is unknown, but when $g\leq 6$ it follows from the fact that
the tautological ring is the entire Chow ring.

######  Remark 1.3.

Note that Theorem 1.1(2) implies that the restriction map
$R^{i}(\mathcal{H}_{4,g})\to R^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})$ is an
isomorphism for $i<\frac{g+3}{4}-4$. This implies an interesting vanishing
result: Any tautological class of codimension $i<\frac{g+3}{4}-4$ supported on
the locus of factoring covers is zero.

######  Remark 1.4.

In [1, 2, 4, 3, 5], Bhargava famously applied structure theorems for degree
$3,4$, and $5$ covers to counting number fields. As in Bhargava’s work, our
techniques rely on special aspects of structure theorems that do not seem to
extend to covers of degree $k\geq 6$. Our need to throw out factoring covers
in order to obtain asymptotic results for the full Chow ring seems to parallel
the fact that, when quartic covers are counted by discriminant, the $D_{4}$
covers constitute a positive proportion of all covers [3, Theorem 4].

######  Remark 1.5.

Ellenberg-Venkatesh-Westerland [16] have studied stability in the _homology_
of Hurwitz spaces of $G$ covers (which in particular separates out factoring
covers). Like the work of Harer and Madsen-Weiss, their techniques are
topological. On the other hand, ours are algebro-geometric: they are about the
Chow groups rather than (co)homology and they work in characteristic $p>5$
without the use of a comparison theorem.

######  Remark 1.6.

For $g$ suitably large, our proof of Theorem 1.1 (2) shows that $\dim
R^{i}(\mathcal{H}_{4,g})\leq 1$ for all $i\geq 5$, and similarly in (3) that
$\dim R^{i}(\mathcal{H}_{5,g})\leq 2$ for all $i\geq 7$. Hence,
$R^{*}(\mathcal{H}_{4,g})$ and $R^{*}(\mathcal{H}_{5,g})$ are _not_ Gorenstein
because there cannot be a perfect pairing for dimension reasons. On the other
hand, $A^{*}(\mathcal{H}_{3,g})=R^{*}(\mathcal{H}_{3,g})$ is Gorenstein.

Our method of proof is to study a large open substack
$\mathcal{H}_{k,g}^{\circ}\subset\mathcal{H}_{k,g}$, which can be represented
as an open substack of a vector bundle $\mathcal{X}_{k,g}^{\circ}$ over a
certain moduli stack of vector bundles on $\mathbb{P}^{1}$. The fact that the
moduli space admits such a description comes from the structure theorms of
degree $3,4,5$ covers and is precisely what is so special about these low-
degree cases. We then determine the Chow ring of $\mathcal{H}_{k,g}^{\circ}$
via excision on the complement of $\mathcal{H}_{k,g}^{\circ}$ inside
$\mathcal{X}_{k,g}^{\circ}$. This complement is a “discriminant locus”
parametrizing singular covers and maps that are not even finite. The stability
of the Chow groups we find fits in with the philosophy of Vakil–Wood [31]
about discriminants and suggests some possible variations on their theme. The
key point, which is reflected in the ampleness assumptions in some of the
conjectures from [31], is that the covers we parametrize correspond to
sections of a vector bundle that becomes “more positive” as the genus of the
curve grows. We compute generators for the Chow ring of the discriminant locus
by constructing a resolution whose Chow ring we can compute. See Figure 1 in
Section 5.3 for a picture summarizing our method.

We also give formulas in Section 7 that express other natural classes on
$\mathcal{H}_{k,g}$ — namely the $\kappa$-classes pulled back from
$\mathcal{M}_{g}$ and the classes corresponding to covers with certain
ramification profiles — in terms of the generators from Theorem 1.1. We give
two applications of these formulas. First, we show that for $k=4,5$, “the push
forward of tautological classes on $\mathcal{H}_{k,g}$ are tautological on
$\mathcal{M}_{g}$.” (The case $k=3$ already follows from Patel–Vakil’s result
that $A^{*}(\mathcal{H}_{3,g})=R^{*}(\mathcal{H}_{3,g})$ is generated by
$\kappa_{1}$ when $g>3$, and and all classes on $\mathcal{M}_{3}$ are
tautological.) Note that for $k>3$, there are tautological classes on
$\mathcal{H}_{k,g}$ that are _not_ pullbacks of tautological classes on
$\mathcal{M}_{g}$: Theorem 1.1 implies $\dim R^{1}(\mathcal{H}_{k,g})>1$, so
it cannot be spanned by the pullback of $\kappa_{1}$. Hence, our claim
regarding pushforwards is not a priori true. To set the stage for the theorem,
let $\beta:\mathcal{H}_{k,g}\to\mathcal{M}_{g}$ be the forgetful morphism.
Define $\mathcal{M}_{g}^{k}\subset\mathcal{M}_{g}$ to be the locus of curves
of gonality $\leq k$. There is a proper morphism
$\beta^{\prime}:\mathcal{H}_{k,g}\setminus\beta^{-1}(\mathcal{M}_{g}^{k-1})\to\mathcal{M}_{g}\smallsetminus\mathcal{M}_{g}^{k-1}$.
We define a class to be tautological on
$\mathcal{M}_{g}\smallsetminus\mathcal{M}_{g}^{k-1}$ if it is the restriction
of a tautological class on $\mathcal{M}_{g}$.

######  Theorem 1.7.

Let $g\geq 2$ be an integer and $k\in\\{3,4,5\\}$. The $\beta^{\prime}$ push
forward of classes in $R^{*}(\mathcal{H}_{k,g})$ are tautological on
$\mathcal{M}_{g}\smallsetminus\mathcal{M}_{g}^{k-1}$.

######  Remark 1.8.

Theorem 1.7 is a key tool in recent work of the authors [8], which proves that
the Chow rings of $\mathcal{M}_{7},\mathcal{M}_{8}$ and $\mathcal{M}_{9}$ are
tautological. Because the tautological ring has been computed in these cases
by Faber [18], this work settles the next open case in the program suggested
by Mumford [29] of determining the Chow ring of $\mathcal{M}_{g}$ for small
$g$.

######  Remark 1.9.

We emphasize that when $k=4$, there can be non-tautological classes in low
codimension supported on the locus of factoring covers. In particular, the
fundamental class of the bielliptic locus on $\mathcal{M}_{12}$ is not
tautological by a theorem of van Zelm [32], so Theorem 1.7 implies
$R^{*}(\mathcal{H}_{4,g})\neq A^{*}(\mathcal{H}_{4,g})$ for $g=12$.

The second application of our formulas is to vanishing results for the Chow
groups of the simply branched Hurwitz space
$\mathcal{H}^{s}_{k,g}\subseteq\mathcal{H}_{k,g}$. The Hurwitz space Picard
rank conjecture [23, Conjecture 2.49] says that
$A^{1}(\mathcal{H}^{s}_{k,g})=\operatorname{Pic}(\mathcal{H}^{s}_{k,g})\otimes\mathbb{Q}=0$
. This conjecture is known for $k\leq 5$ [13], and for $k>g-1$ [28]. In the
cases $k=2,3$, the stronger vanishing result $A^{i}(\mathcal{H}_{k,g}^{s})=0$
holds for all $i>0$. The following theorem provides further evidence for a
generalization of the Hurwitz space Picard rank conjecture to higher
codimension cycles.

######  Theorem 1.10.

Let $g\geq 2$ be an integer. The rational Chow groups of the simply-branched
Hurwitz space satisfy

$\displaystyle A^{i}(\mathcal{H}^{s}_{4,g})$ $\displaystyle=0\qquad\text{for
$1\leq i<\frac{g+3}{4}-4$}$ $\displaystyle A^{i}(\mathcal{H}^{s}_{5,g})$
$\displaystyle=0\qquad\text{for $1\leq i<\frac{g+4}{5}-16$.}$

The paper is structured as follows. In Section 2, we introduce some notational
conventions and some basic ideas from (equivariant) intersection theory that
we will use throughout the paper. We prove a lemma, the “Trapezoid Lemma”,
which establishes a useful set up where one can determine all relations coming
from certain excisions with an appropriate resolution. In Section 3, we
introduce certain bundles of principal parts, which will be used throughout
the remainder of the paper. Loosely speaking, these bundles help detect
singularities and ramification behavior. As we shall see in the later sections
of the paper, constructing a suitable principal parts bundle often requires
geometric insights and can be somewhat involved. In Sections 4, 5, and 6, we
use principal parts bundles and the Trapezoid Lemma to produce relations among
tautological classes in $A^{*}(\mathcal{H}_{3,g})$,
$A^{*}(\mathcal{H}_{4,g})$, and $A^{*}(\mathcal{H}_{5,g})$, respectively. From
these calculations, we obtain the proof Theorem 1.1. Finally, in Section 7, we
rewrite the $\kappa$-classes and classes that parametrize covers with certain
ramification behavior in terms of our preferred generators. These calculations
allow us to prove Theorems 1.7 and 1.10.

Several of the calculations in this paper were using the Macaulay2 [20]
package Schubert2 [21]. All of the code used in this paper is provided in a
Github repository [9]. Whenever there is a reference to a calculation done
with a computer, one can find the code to perform that calculation in the
Github repository.

### Acknowledgments

We are grateful to our advisors, Elham Izadi and Ravi Vakil, respectively, for
the many helpful conversations. In addition, we are grateful to Anand Patel,
who pointed out the need for special arguments for $\mathcal{H}_{3,g}$ when
$g$ is small. We thank Maxwell da Paixão de Jesus Santos for his
correspondence, which inspired us to formulate Theorem 1.7. We thank Aaron
Landesman and Andrea Di Lorenzo for their comments and insights.

##  2\. Conventions and some intersection theory

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 and Grassmann bundles

We follow the subspace convention for projective bundles: given a scheme (or
stack) $X$ and a vector bundle $E$ of rank $r$ on $X$, 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 $\mathbb{P}E$.

More generally, we define the Grassmann bundle $G(n,E)$ of $n$-dimensional
subspaces in $E$, which is equipped with a tautological sequence

$0\rightarrow S\rightarrow\gamma^{*}E\rightarrow Q\rightarrow 0$

where $\gamma:G(n,E)\rightarrow X$ is the structure map and $S$ has rank $n$.
The relative tangent bundle of $G(n,E)\to X$ is $\mathcal{H}om(S,Q)$. The Chow
ring $A^{*}(G(n,E))$ is generated as an $A^{*}(X)$-algebra by the classes
$\zeta_{i}=c_{i}(Q)$. Of particular interest to us will be Grassmann bundles
$A^{*}(G(2,E))$ when the rank of $E$ is either $4$ or $5$. If the rank of $E$
is $4$, $A^{*}(G(2,E))$ is generated as a $A^{*}(X)$-module by
$\zeta_{1}^{i}\zeta_{2}^{j}$ for $0\leq i\leq 2$, $0\leq j\leq 2$, $0\leq
i+j\leq 2$. If the rank of $E$ is $5$, $A^{*}(G(2,E))$ is generated as a
$A^{*}(X)$ module by $\zeta_{1}^{i}\zeta_{2}^{j}\zeta_{3}^{k}$ for $0\leq
i\leq 2$, $0\leq j\leq 2$, $0\leq k\leq 2$ and $0\leq i+j+k\leq 2$. See [19]
for a much more general discussion on the Chow rings of flag bundles. In
particular, these bases seem to be the preferred ones of the Macaulay2 [20]
package Schubert2 [21], which is what we use for calculations in this paper.

###  2.2. The Trapezoid Lemma

Let $\tau:V\to B$ be a rank $r$ vector bundle. If $\sigma$ is a section of $V$
which vanishes in codimension $r$, then the vanishing locus of $\sigma$ has
fundamental class $c_{r}(V)\in A^{r}(B)$. The identity induces a section of
$\tau^{*}V$ on the total space of $V$ whose vanishing locus is the zero
section. Thus, a special case of this fact is that the zero section in the
total space of a vector bundle has class $c_{r}(\tau^{*}V)=\tau^{*}c_{r}(V)\in
A^{r}(V)\cong\tau^{*}A^{r}(B)$. More generally, suppose $\rho:X\to B$ is
another vector bundle on $B$ and we are given a map of vector bundles
$\phi:X\to V$ over $B$. Composing $\phi$ after the section induced by the
identity on the total space of $X$ defines a section of $\rho^{*}V$ on the
total space of $X$. We call the vanishing locus $K$ of this section the
_preimage under $\phi$ of the zero section in $V$_. If $\phi$ is a surjection
of vector bundles, then $K$ is simply the total space of the kernel subbundle.
If $K$ has codimension $r$ inside the total space of $W$, then its fundamental
class is $[K]=c_{r}(\rho^{*}V)=\rho^{*}c_{r}(V)\in
A^{r}(X)\cong\rho^{*}A^{r}(B)$.

A basic tool we shall use repeatedly is the following.

######  Lemma 2.1 (“Trapezoid push forwards”).

Suppose $\widetilde{B}\to B$ is proper (e.g. a tower of Grassmann bundles).
Let $X$ be a vector bundle on $B$ and let $V$ be a vector bundle of rank $r$
on $\widetilde{B}$. Suppose that we are given a map of vector bundles
$\phi:\sigma^{*}X\rightarrow V$ on $\widetilde{B}$. Let $K\subset\sigma^{*}X$
be the preimage under $\phi$ of the zero section in $V$, and suppose that $K$
has codimension $r$. We call this a trapezoid diagram:

${K}$${\sigma^{*}X}$${X}$${\widetilde{B}}$${B.}$$\scriptstyle{\rho^{\prime\prime}}$$\scriptstyle{\iota}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\sigma^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{\sigma}$

The image of $(\sigma^{\prime}\circ\iota)_{*}:A_{*}(K)\to A_{*}(X)$ contains
the ideal generated by $\rho^{*}(\sigma_{*}(c_{r}(V)\cdot\alpha_{i}))$ as
$\alpha_{i}\in A^{*}(\widetilde{B})$ ranges over generators for
$A^{*}(\widetilde{B})$ as a $A^{*}(B)$-module. Equality holds if $\phi$ is a
surjection. In other words, we have a surjective map of rings

$A^{*}(B)/\langle\sigma_{*}(c_{r}(V)\cdot\alpha_{i}))\rangle\rightarrow
A^{*}(X\smallsetminus\sigma^{\prime}(\iota(K))),$

which is an isomorphism when $\phi$ is a surjection of vector bundles.

###### Proof.

The pullback maps $(\rho^{\prime})^{*}$ and $\rho^{*}$ are isomorphisms on
Chow rings. The fundamental class of $K$ in $\sigma^{*}X$ is
$(\rho^{\prime})^{*}c_{r}(V)$, since it is defined by the vanishing of a
section of $(\rho^{\prime})^{*}V$. Consider classes in $A^{*}(K)$ of the form
$(\rho^{\prime\prime})^{*}\alpha$, where $\alpha\in A^{*}(\widetilde{B})$. The
effect of $(\sigma^{\prime}\circ\iota)_{*}$ on such classes is

(2.2)
$\sigma^{\prime}_{*}\iota_{*}(\rho^{\prime\prime})^{*}\alpha=\sigma^{\prime}_{*}\iota_{*}\iota^{*}(\rho^{\prime})^{*}\alpha=\sigma^{\prime}_{*}([K]\cdot(\rho^{\prime})^{*}\alpha)=\sigma^{\prime}_{*}(\rho^{\prime})^{*}(c_{r}(V)\cdot\alpha)=\rho^{*}\sigma_{*}(c_{r}(V)\cdot\alpha).$

The last step uses that flat pull back and proper push forward commute in
fiber diagram.

If $\alpha=\sum_{i}(\sigma^{*}\beta_{i})\cdot\alpha_{i}$, then the projection
formula gives

$\rho^{*}\sigma_{*}(c_{r}(V)\cdot\alpha)=\sum_{i}\rho^{*}(\beta_{i})\cdot\rho^{*}(\sigma_{*}(c_{r}(V)\cdot\alpha_{i})).$

If $K$ is a vector bundle, then _every_ class in $A^{*}(K)$ has the form
$(\rho^{\prime\prime})^{*}\alpha$ for some $\alpha\in A^{*}(\widetilde{B})$.
Thus, if $K$ is a vector bundle, the image of
$(\sigma^{\prime}\circ\iota)_{*}$ is generated over
$A^{*}(X)\cong\rho^{*}A^{*}(B)$ by the classes
$\rho^{*}(\sigma_{*}(c_{r}(V)\cdot\alpha_{i}))$, as $\alpha_{i}$ runs over
generators for $A^{*}(\widetilde{B})$ as a $A^{*}(B)$-module. ∎

###  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 [30, 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 denote by
$\mathcal{H}_{k,g}$ from now on.

Explicitly,
$\mathcal{H}_{k,g}=[\mathcal{H}_{k,g}^{\dagger}/\operatorname{SL}_{2}]$ 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.

We shall also work with $\mathcal{H}_{k,g}^{\mathrm{nf}}$, the open substack
of $\mathcal{H}_{k,g}$ parametrizing covers that do not factor through a lower
genus curve. When $k$ is prime,
$\mathcal{H}_{k,g}^{\mathrm{nf}}=\mathcal{H}_{k,g}$. In Section 7 of the
paper, we will consider the open substack
$\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}$, which parametrizes covers
that are simply branched. Note that
$\mathcal{H}^{s}_{k,g}\subseteq\mathcal{H}_{k,g}^{\mathrm{nf}}$.

##  3\. Relative bundles of principal parts

In this section, we collect some background on bundles of principal parts,
which will be used to produce relations among tautological classes in Sections
4, 5, 6, and to compute classes of certain ramification strata in Section 7.
For the basics, we follow the exposition in Eisenbud-Harris [15].

###  3.1. Basic properties

Let $b:Y\rightarrow Z$ be a smooth proper morphism. Let $\Delta_{Y/Z}\subset
Y\times_{Z}Y$ be the relative diagonal. With $p_{1}$ and $p_{2}$ the
projection maps, we obtain the following commutative diagram:

${\Delta_{Y/Z}}$${Y\times_{Z}Y}$${Y}$${Y}$${Z.}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{b}$$\scriptstyle{b}$

######  Definition 3.1.

Let $\mathcal{W}$ be a vector bundle on $Y$ and let
$\mathcal{I}_{\Delta_{Y/Z}}$ denote the ideal sheaf of the diagonal in
$Y\times_{Z}Y$. The bundle of relative $m^{\text{th}}$ order principal parts
$P^{m}_{Y/Z}(\mathcal{W})$ is defined as

$P^{m}_{Y/Z}(\mathcal{W})=p_{2*}(p_{1}^{*}\mathcal{W}\otimes\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}_{\Delta_{Y/Z}}^{m+1}).$

The following explains all the basic properties of bundles of principal parts
that we need. Parts (1) and (2) are Theorem 11.2 in [15]. Let $m\Delta_{Y/Z}$
be the closed subscheme of $Y\times_{Z}Y$ defined by the ideal sheaf
$\mathcal{I}_{\Delta_{Y/Z}}^{m}$. Part (3) below follows because the
restriction of $p_{2}$ to the thickened diagonal $m\Delta_{Y/Z}\to Y$ is
finite, so the push forward is exact.

######  Proposition 3.2.

With notation as above,

  1. (1)

The quotient map $p_{1}^{*}\mathcal{W}\rightarrow
p_{1}^{*}\mathcal{W}\otimes\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}_{\Delta_{Y/Z}}^{m+1}$
pushes forward to a map

$b^{*}b_{*}\mathcal{W}\cong p_{2*}p_{1}^{*}\mathcal{W}\rightarrow
P^{m}_{Y/Z}(\mathcal{W}),$

which, fiber by fiber, associates to a global section $\delta$ of
$\mathcal{W}$ a section $\delta^{\prime}$ whose value at $z\in Z$ is the
restriction of $\delta$ to an $m^{\text{th}}$ order neighborhood of $z$ in the
fiber $b^{-1}b(z)$.

  2. (2)

$P^{0}_{Y/Z}(\mathcal{W})=\mathcal{W}$. For $m>1$, the filtration of the
fibers $P^{m}_{Y/Z}(\mathcal{W})_{y}$ by order of vanishing at $y$ gives a
filtration of $P^{m}_{Y/Z}(\mathcal{W})$ by subbundles that are kernels of the
natural surjections $P^{m}_{Y/Z}(\mathcal{W})\rightarrow
P^{k}_{Y/Z}(\mathcal{W})$ for $k<m$. The graded pieces of the filtration are
identified by the exact sequences

$0\rightarrow\mathcal{W}\otimes\operatorname{Sym}^{m}(\Omega_{Y/Z})\rightarrow
P^{m}_{Y/Z}(\mathcal{W})\rightarrow P^{m-1}_{Y/Z}(\mathcal{W})\rightarrow 0.$

  3. (3)

A short exact sequence $0\rightarrow
K\to\mathcal{W}\to\mathcal{W}^{\prime}\rightarrow 0$ of vector bundles on $Y$
induces an exact sequence of principal parts bundles

$0\to P^{m}_{Y/Z}(K)\to P^{m}_{Y/Z}(\mathcal{W})\to
P^{m}_{Y/Z}(\mathcal{W}^{\prime})\to 0$

We will need to know when the map from part (1) is surjective.

######  Lemma 3.3.

Suppose $\mathcal{W}$ is a relatively very ample line bundle on $Y$. Then the
evaluation map $b^{*}b_{*}\mathcal{W}\rightarrow P^{1}_{Y/Z}(\mathcal{W})$ is
surjective.

###### Proof.

The statement can be checked fiber by fiber over $Z$. Then, it follows from
the fact that very ample line bundles separate points and tangent vectors. ∎

Together with the above lemma, the following two lemmas will help us establish
when evaluation maps are surjective in our particular setting.

######  Lemma 3.4.

Let $E=\mathcal{O}(e_{1})\oplus\cdots\oplus\mathcal{O}(e_{r})$ be a vector
bundle on $\mathbb{P}^{1}$ with $e_{1}\leq\cdots\leq e_{r}$ and let
$\gamma:\mathbb{P}E^{\vee}\to\mathbb{P}^{1}$ be the projectivization. The line
bundle
$L=\gamma^{*}\mathcal{O}_{\mathbb{P}^{1}}(a)\otimes\mathcal{O}_{\mathbb{P}E^{\vee}}(m)$
is very ample if and only if $m\geq 1$ and $a+me_{1}\geq 1$, equivalently if
and only if
$h^{1}(\mathbb{P}^{1},\gamma_{*}L\otimes\mathcal{O}_{\mathbb{P}^{1}}(-2))=0$.

###### Proof.

First, note that $L$ is the pullback of $\mathcal{O}(1)$ under a degree $m$
relative Veronese embedding
$\mathbb{P}E^{\vee}\hookrightarrow\mathbb{P}(\mathcal{O}(a)\otimes\operatorname{Sym}^{m}E)^{\vee}$.
The $\mathcal{O}(1)$ on the projective bundle
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(a)\otimes\operatorname{Sym}^{m}E)^{\vee}$
is very ample if and only if all summands of
$\mathcal{O}_{\mathbb{P}^{1}}(a)\otimes\operatorname{Sym}^{m}E=\gamma_{*}L$
have positive degree (see [15, Section 9.1.1]). These summands have degrees of
the form $a+e_{i_{1}}+\ldots+e_{i_{m}}$, all of which are at least $a+me_{1}$.
∎

######  Lemma 3.5.

Suppose $\mathcal{E}$ is a vector bundle on a $\mathbb{P}^{1}$-bundle
$\pi:\mathcal{P}\to B$ and let
$\gamma:\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ be the projectivization.
Suppose
$\mathcal{W}=(\gamma^{*}A)\otimes\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(m)$
for some $m\geq 1$ and vector bundle $A$ on $\mathcal{P}$. If
$R^{1}\pi_{*}[\gamma_{*}\mathcal{W}\otimes\mathcal{O}_{\mathcal{P}}(-2)]=0$,
then the evaluation map

$(\pi\circ\gamma)^{*}(\pi\circ\gamma)_{*}\mathcal{W}\to
P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/B}(\mathcal{W})$

is surjective.

###### Proof.

It suffices to check surjectivity in each of the fibers over $B$, so we are
reduced to the case that $B$ is a point. Now we may assume $A$ splits as a sum
of line bundles, say
$A\cong\mathcal{O}(a_{1})\oplus\cdots\oplus\mathcal{O}(a_{r})$. By cohomology
and base change, we have
$h^{1}(\mathbb{P}^{1},\gamma_{*}\mathcal{W}\otimes\mathcal{O}_{\mathbb{P}^{1}}(-2))=0$,
which implies
$h^{1}(\mathbb{P}^{1},\gamma_{*}(\gamma^{*}\mathcal{O}(a_{i})\otimes\mathcal{O}_{\mathbb{P}E^{\vee}}(m))\otimes\mathcal{O}(-2))=0$
for each $i$. By Lemma 3.4, we have that $\mathcal{W}$ is a sum of very ample
line bundles (over $B$). The bundle of principal parts respects direct sums,
so the evaluation map is surjective by Lemma 3.3. ∎

The following lemma should be thought of as saying “pulled back sections have
vanishing vertical derivatives.”

######  Lemma 3.6.

Let $X\xrightarrow{a}Y\xrightarrow{b}Z$ be a tower of schemes with $a$ and $b$
smooth, and let $\mathcal{W}$ be a vector bundle on $Y$. For each $m$ there is
a natural map $a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow
P^{m}_{X/Z}(a^{*}\mathcal{W})$. This map fits in an exact sequence

$0\rightarrow a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow
P^{m}_{X/Z}(a^{*}\mathcal{W})\rightarrow F_{m}\rightarrow 0,$

where $F_{1}\cong\Omega_{X/Y}\otimes a^{*}\mathcal{W}$ and $F_{m}$ for $m>1$
is filtered as

$0\rightarrow\operatorname{Sym}^{m-1}\Omega_{X/Z}\otimes\Omega_{X/Y}\otimes
a^{*}\mathcal{W}\rightarrow F_{m}\rightarrow F_{m-1}\rightarrow 0.$

In particular, the evaluation map

$b^{*}b_{*}\mathcal{W}\rightarrow P^{m}_{Y/Z}(\mathcal{W})$

gives rise to a composition

$a^{*}b^{*}b_{*}\mathcal{W}\rightarrow
a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow P^{m}_{X/Z}(a^{*}\mathcal{W}),$

which, fiber by fiber, gives the Taylor expansion of sections of $\mathcal{W}$
along the “horizontal” pulled back directions.

###### Proof.

We begin by constructing the map $a^{*}P^{m}_{X/Z}(\mathcal{W})\rightarrow
P^{m}_{Y/Z}(a^{*}\mathcal{W})$. Consider the following commutative diagram:

${X}$${X\times_{Z}X}$${X}$${Z}$${Y}$${Y\times_{Z}Y}$${Y}$${Z}$$\scriptstyle{a}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{a\times
a}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{2}}$$\scriptstyle{b}$

Let $\Delta_{Y}\subset Y\times_{Z}Y$ denote the relative diagonal, and
similarly for $\Delta_{X}\subset X\times_{Z}X$. By definition, we have

$a^{*}P^{m}_{Y/Z}(\mathcal{W})=a^{*}(q_{1*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}}\otimes
q_{2}^{*}\mathcal{W})).$

The natural transformation of functors $a^{*}q_{1*}\rightarrow p_{1*}(a\times
a)^{*}$ induces a map

$a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow p_{1*}((a\times
a)^{*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}})\otimes(a\times
a)^{*}q_{2}^{*}\mathcal{W})).$

The transform $(q_{2}\circ(a\times a))^{*}\rightarrow(a\circ p_{2})^{*}$
induces a map

$p_{1*}((a\times
a)^{*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}})\otimes(a\times
a)^{*}q_{2}^{*}\mathcal{W}))\rightarrow p_{1*}((a\times
a)^{*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}})\otimes
p_{2}^{*}a^{*}\mathcal{W}).$

The natural morphism of sheaves $\mathcal{O}_{Y\times_{Z}Y}\rightarrow(a\times
a)_{*}\mathcal{O}_{X\times_{Z}X}$ induces a map on quotients
$\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}}\rightarrow(a\times
a)_{*}(\mathcal{O}_{X\times_{Z}X}/\mathcal{I}^{m+1}_{\Delta_{X}}).$ By
adjunction, we obtain a map

$(a\times
a)^{*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}})\rightarrow\mathcal{O}_{X\times_{Z}X}/\mathcal{I}^{m+1}_{\Delta_{X}}.$

Then we have a morphism

$p_{1*}((a\times
a)^{*}(\mathcal{O}_{Y\times_{Z}Y}/\mathcal{I}^{m+1}_{\Delta_{Y}})\otimes
p_{2}^{*}a^{*}\mathcal{W})\rightarrow
p_{1*}(\mathcal{O}_{X\times_{Z}X}/\mathcal{I}_{\Delta_{X}}^{m+1}\otimes
p_{2}^{*}(a^{*}\mathcal{W}))=P^{m}_{X/Z}(a^{*}\mathcal{W}).$

By construction, the maps $a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow
P^{m}_{X/Z}(a^{*}\mathcal{W})$ are compatible with the filtrations on the
fibers by order of vanishing, so we obtain an induced map on the graded pieces
of the filtrations:

${0}$${\operatorname{Sym}^{m}(a^{*}\Omega_{Y/Z})\otimes
a^{*}\mathcal{W}}$${a^{*}P^{m}_{Y/Z}(\mathcal{W})}$${a^{*}P^{m-1}_{Y/Z}(\mathcal{W})}$${0}$${0}$${\operatorname{Sym}^{m}(\Omega_{X/Z})\otimes
a^{*}\mathcal{W}}$${P^{m}_{X/Z}(a^{*}\mathcal{W})}$${P^{m-1}_{X/Z}(a^{*}\mathcal{W})}$${0}$

When $m=1$, the right vertical map is the identity on $a^{*}\mathcal{W}$.
Hence, $a^{*}P^{1}_{Y/Z}(\mathcal{W})\rightarrow
P^{1}_{X/Z}(a^{*}\mathcal{W})$ is injective. By the snake lemma, the cokernel
is isomorphic to the cokernel of the left vertical map, which in turn is
$\Omega_{X/Y}\otimes a^{*}\mathcal{W}$ because $a$ and $b$ are smooth and
$\mathcal{W}$ is locally free. For $m>1$, we may assume by induction that the
right vertical map is injective, hence the center vertical map is injective.
The filtration of the cokernel $F_{m}$ of the center vertical map follows by
induction and the snake lemma. ∎

###  3.2. Directional refinements

Much of the exposition in this subsection is based on unpublished notes of
Ravi Vakil. Suppose we have a tower $X\xrightarrow{a}Y\xrightarrow{b}Z$ and
$a^{*}\Omega_{Y/Z}$ admits a filtration on $X$

(3.1) $0\rightarrow\Omega_{y}\rightarrow
a^{*}\Omega_{Y/Z}\rightarrow\Omega_{x}\rightarrow 0.$

For example, take $X=\mathbb{P}(\Omega_{Y/Z})$ or $G(n,\Omega_{Y/Z})$ with the
filtration given by the tautological sequence. First, suppose $\Omega_{x}$ and
$\Omega_{y}$ are rank $1$. The filtration (3.1) is the same as saying we can
choose local coordinates $x,y$ at each point of $Y$ where $y$ is well-defined
up to $(x,y)^{2}$, and $x$ is only defined modulo $y$. The vanishing of $y$
defines a distinguished “$x$-direction” on the tangent space $T_{Y/Z}$ at each
point, which is dual to the quotient $a^{*}\Omega_{Y/Z}\to\Omega_{x}$.

The goal of this section is to define principal parts bundles that measure
certain parts of a Taylor expansion with respect to these local coordinates.
These principal parts bundles will be indexed by admissible sets $S$ of
monomials in $x$ and $y$ (defined below). If $x^{i}y^{j}\in S$, then
$P^{S}_{Y/Z}(\mathcal{W})$ will keep track of the coefficient of $x^{i}y^{j}$
in the Taylor expansion of a section of $\mathcal{W}$. For example,
$S=\\{1,x\\}$ will correspond to a quotient of $a^{*}P^{1}_{Y/Z}(\mathcal{W})$
that measures only derivatives in the $x$-direction. The set
$S=\\{1,x,y,x^{2},xy,y^{2}\\}$ corresponds to the pullback of the usual second
order principal parts. It is helpful to visualize these sets with diagrams as
below, where we place a dot at coordinate $(i,j)$ if $x^{i}y^{j}\in S$.

$i$$j$$\\{1,x\\}$$\\{1,x,y,x^{2},xy,y^{2}\\}$

More generally, if $\Omega_{x}$ and $\Omega_{y}$ have any ranks, the quotient
$\Omega_{x}$ is dual to a distinguished subspace of $T_{Y/Z}$. The
construction below will build bundles $P^{S}_{Y/Z}(\mathcal{W})$ such that if
$x^{i}y^{j}\in S$, then $P^{S}_{Y/Z}(\mathcal{W})$ tracks the coefficients of
all monomials corresponding to
$\operatorname{Sym}^{i}\Omega_{x}\otimes\operatorname{Sym}^{j}\Omega_{y}$. In
other words, $P^{S}_{Y/Z}(\mathcal{W})$ will admit a filtration with quotients
$\operatorname{Sym}^{i}\Omega_{x}\otimes\operatorname{Sym}^{j}\Omega_{y}\otimes\mathcal{W}$
for each $(i,j)$ such that $x^{i}y^{j}\in S$. Each dot in the diagram
corresponds to a piece of this filtration. Only diagrams of certain shapes are
allowed.

######  Definition 3.7.

A set $S$ is admissible if the following hold

  * •

If $x^{i}y^{j}\in S$, then $x^{i-1}y^{j}\in S$ (if $i-1\geq 0$). That is, for
each dot in the diagram, the dot to its left is also in the diagram if
possible.

  * •

If $x^{i}y^{j}\in S$, then $x^{i-2}y^{j+1}$ (if $i-2\geq 0$). That is, for
each dot in the diagram, the dot two to the left and one down is also in the
diagram if possible.

Equivalently, the diagram associated to $S$ is built, via intersections and
unions, from triangular collections of lattice points bounded by the axes and
a line of slope $1$ or slope $\frac{1}{2}$.

To build the principal parts bundles $P^{S}_{Y/Z}(W)$, let us consider the
diagram

${\widetilde{\Delta}}$${\Delta}$${X\times_{Z}Y}$${Y\times_{Z}Y}$${Y}$${X}$${Y}$${Z,}$$\scriptstyle{\iota}$$\scriptstyle{\widetilde{p}_{2}}$$\scriptstyle{\widetilde{a}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{b}$$\scriptstyle{a}$$\scriptstyle{b}$

where $\Delta=\Delta_{Y/Z}\subset Y\times_{Z}Y$ is the diagonal and all
squares are fibered squares. The composition of vertical maps give
isomorphisms $\Delta\cong Y$ and $\widetilde{\Delta}\cong X$. There is an
identification
$\iota_{*}\Omega_{\widetilde{\Delta}/Z}\cong\mathcal{I}_{\widetilde{\Delta}}/\mathcal{I}_{\widetilde{\Delta}}^{2}$.
Using (3.1) and the isomorphism $\widetilde{\Delta}\cong X$, we obtain an
injection

$\iota_{*}\Omega_{y}\to\iota_{*}a^{*}\Omega_{Y/Z}\to\iota_{*}\Omega_{X/Z}\cong\mathcal{I}_{\widetilde{\Delta}}/\mathcal{I}_{\widetilde{\Delta}}^{2},$

which determines a subsheaf
$\mathcal{J}\subset\mathcal{I}_{\widetilde{\Delta}}=:\mathcal{I}$. The sheaf
$\mathcal{I}$ corresponds to the monomials $\\{x^{i}y^{j}:i+j\geq 1\\}$ (see
(3.3) below). The subsheaf $\mathcal{J}$ corresponds to the monomials
$\\{x^{i}y^{j}:i+j\geq 2\text{ or }j\geq 1\\}$ (see (3.4) below). The
condition $i+j\geq 2$ says $\mathcal{I}^{2}\subset\mathcal{J}$. The condition
$j\geq 1$ says $\mathcal{J}\subset\mathcal{I}$ and it “picks out our
$y$-coordinate(s) to first order.”

In the next paragraph, we will explain how to construct an ideal
$\mathcal{I}_{S}$, via intersections and unions of $\mathcal{I}$ and
$\mathcal{J}$, corresponding to monomials not in $S$. Our refined principal
parts bundles will then be defined as

$P^{S}_{Y/Z}(\mathcal{W}):=\widetilde{p}_{2*}\left(\widetilde{a}^{*}p_{1}^{*}\mathcal{W}\otimes\mathcal{O}_{X\times_{Z}Y}/\mathcal{I}_{S}\right),$

The bundle $P^{S}_{Y/Z}(\mathcal{W})$ is defined on $X$ and will be a quotient
of $a^{*}P^{m}_{Y/Z}(\mathcal{W})$ for $m=\max\\{i+j:x^{i}y^{j}\in S\\}$. In
particular, there are restricted evaluation maps

(3.2) $a^{*}b^{*}b_{*}\mathcal{W}\to a^{*}P^{m}_{Y/Z}(\mathcal{W})\to
P^{S}_{Y/Z}(\mathcal{W}),$

which we think of as Taylor expansions only along certain directions specified
by $S$.

To start, we shall have $\mathcal{I}_{\\{1,x,y\\}}:=\mathcal{I}$ and
$\mathcal{I}_{\\{1,x\\}}:=\mathcal{J}$. Powers of these ideals correspond to
regions below lines of slope $1$ and $\frac{1}{2}$ respectively.

(3.3) $\ddots$$\mathcal{I}$ $\ddots$$\mathcal{I}^{2}$
$\mathcal{I}^{3}$$\ddots$

(3.4) $\ddots$$\mathcal{J}$ $\ddots$$\mathcal{J}^{2}$
$\mathcal{J}^{3}$$\ddots$

To say that $S$ is admissible is to say that $\mathcal{I}_{S}$ is built by
taking unions and intersections such half planes, which corresponds to
intersections and unions of $\mathcal{I}$ and $\mathcal{J}$. We list below the
principal parts bundles we require in the remainder of the paper and their
associated ideal $\mathcal{I}_{S}$.

  1. (1)

$S=\\{1,x\\}$ with $\mathcal{I}_{S}=\mathcal{J}$, which we call the bundle of
_restricted principal parts_.

  2. (2)

$S=\\{1,x,y,x^{2}\\}$ with $\mathcal{I}_{S}=\mathcal{J}^{2}$ will arise in
triple point calculations.

  3. (3)

$S=\\{1,x,y,x^{2},xy\\}$ with
$\mathcal{I}_{S}=\mathcal{I}^{3}+\mathcal{J}^{3}$ arises when finding
quadruple points in a pencil of conics.

  4. (4)

$S=\\{1,x,y,x^{2},xy,x^{3}\\}$ with $\mathcal{I}_{S}=\mathcal{J}^{3}$ will
arise in finding quadruple points in pentagonal covers.

Diagrams corresponding to these sets appear at the end of the next subsection.
Given two admissible sets $S\subset S^{\prime}$, there is a natural surjection
$P^{S^{\prime}}_{Y/Z}(\mathcal{W})\to P^{S}_{Y/Z}(\mathcal{W})$, which
corresponds to truncating Taylor series. This determines the order(s) that the
terms
$\operatorname{Sym}^{i}\Omega_{x}\otimes\operatorname{Sym}^{j}\Omega_{y}\otimes\mathcal{W}$
corresponding to $x^{i}y^{j}\in S^{\prime}$ may appear as quotients in a
filtration: a term corresponding to $x^{i}y^{j}\in S^{\prime}$ is a well-
defined subbundle of $P^{S^{\prime}}_{Y/Z}(\mathcal{W})$ if
$S^{\prime}\smallsetminus x^{i}y^{j}$ is an admissible set.

###  3.3. Bundle-induced refinements

Now suppose that $a^{*}\mathcal{W}$ admits a filtration on $X$ by

(3.5) $0\rightarrow K\rightarrow
a^{*}\mathcal{W}\rightarrow\mathcal{W}^{\prime}\rightarrow 0,$

where $\mathcal{W}^{\prime}$ is a vector bundle, and hence so is $K$.
Exactness of principal parts for $X$ over $Z$ gives an exact sequence

$0\rightarrow P^{m}_{X/Z}(K)\rightarrow
P^{m}_{X/Z}(a^{*}\mathcal{W})\rightarrow
P^{m}_{X/Z}(\mathcal{W}^{\prime})\rightarrow 0.$

We are interested in the restriction of this filtration to
$a^{*}P^{m}_{Y/Z}(\mathcal{W})\subset P^{m}_{X/Z}(a^{*}\mathcal{W})$. First,
we need the following fact.

######  Lemma 3.8.

The intersection of the two subbundles

(3.6) $P^{m}_{X/Z}(K)\subset
P^{m}_{X/Z}(a^{*}\mathcal{W})\qquad\text{and}\qquad
a^{*}P^{m}_{Y/Z}(\mathcal{W})\subset P^{m}_{X/Z}(a^{*}\mathcal{W})$

is a subbundle.

###### Proof.

We proceed by induction. For $m=0$, the claim is just that $K$ is a subbundle
of $a^{*}\mathcal{W}$. The question is local, so we can assume that the
vanishing order filtration exact sequences

$0\rightarrow\operatorname{Sym}^{m}\Omega_{X/Z}\otimes
a^{*}\mathcal{W}\rightarrow P^{m}_{X/Z}(a^{*}\mathcal{W})\rightarrow
P^{m-1}_{X/Z}(a^{*}\mathcal{W})\rightarrow 0,$

are split. By induction and the (locally split) exact sequences,

$0\rightarrow\operatorname{Sym}^{m}\Omega_{X/Z}\otimes K\rightarrow
P^{m}_{X/Z}(K)\rightarrow P^{m-1}_{X/Z}(K)\rightarrow 0$

and

$0\rightarrow a^{*}\operatorname{Sym}^{m}\Omega_{Y/Z}\otimes
a^{*}\mathcal{W}\rightarrow a^{*}P^{m}_{Y/Z}(\mathcal{W})\rightarrow
a^{*}P^{m-1}_{Y/Z}(\mathcal{W})\rightarrow 0$

it suffices to show that the intersection of
$\operatorname{Sym}^{m}\Omega_{X/Z}\otimes K$ and
$a^{*}\operatorname{Sym}^{m}\Omega_{Y/Z}\otimes a^{*}\mathcal{W}$ is a
subbundle of $\operatorname{Sym}^{m}\Omega_{X/Z}\otimes a^{*}\mathcal{W}$. But
this intersection is given by $a^{*}\operatorname{Sym}^{m}\Omega_{Y/Z}\otimes
K$, which is a subbundle. ∎

######  Definition 3.9.

We define $\underline{P}^{m}_{Y/Z}(K)$ to be the intersection of the two
subbundles in (3.6). This subbundle tracks principal parts of $K$ in the
directions of $Y/Z$. We include the underline to remind ourselves that this
bundle is defined on $X$ since $K$ is defined on $X$. We define
$Q^{m}_{Y/Z}(\mathcal{W}^{\prime})$ to be the cokernel of
$\underline{P}^{m}_{Y/Z}(K)\to a^{*}P^{m}_{Y/Z}(\mathcal{W})$.

When $K=a^{*}K^{\prime}$ for a bundle $K^{\prime}$ on $Y$, the bundle
$\underline{P}^{m}_{Y/Z}(K)$ is just the bundle
$a^{*}P^{m}_{Y/Z}(K^{\prime})$.

The vanishing order filtrations from Proposition 3.2 of $P^{m}_{X/Z}(K)$ and
$a^{*}P^{m}_{Y/Z}(\mathcal{W})$ restrict to a vanishing order filtration on
$\underline{P}^{m}_{Y/Z}(K)$, which in turn induces a vanishing order
filtration on $Q^{m}_{Y/Z}(\mathcal{W}^{\prime})$. We describe this for $m=1$
below for future use.

######  Lemma 3.10.

The bundle $Q^{1}_{Y/Z}(\mathcal{W}^{\prime})$ is equipped with a surjection
$a^{*}P^{1}_{Y/Z}(\mathcal{W})\to Q^{1}_{Y/Z}(\mathcal{W}^{\prime})$ and a
filtration

$0\rightarrow a^{*}\Omega_{Y/Z}\otimes\mathcal{W}^{\prime}\to
Q^{1}_{Y/Z}(\mathcal{W}^{\prime})\rightarrow\mathcal{W}^{\prime}\rightarrow 0$

A section $\mathcal{O}_{Y}\xrightarrow{\delta}\mathcal{W}$ on $X$ induces a
section
$\mathcal{O}_{X}\xrightarrow{\delta^{\prime}}a^{*}P^{1}_{Y/Z}(\mathcal{W})\to
Q^{1}_{Y/Z}(\mathcal{W}^{\prime})$ that records the values and “horizontal
derivatives” of $\delta$ in the quotient $\mathcal{W}^{\prime}$.

###  3.4. Directional and bundle-induced refinements

The principal parts bundles constructed in this subsection will not be needed
until Section 7. Here, we suppose that we have filtrations as in (3.1) and
(3.5). We have an inclusion $\underline{P}^{m}_{Y/Z}(K)\hookrightarrow
a^{*}P^{m}_{Y/Z}(\mathcal{W})$ as well as a quotient
$a^{*}P^{m}_{Y/Z}(\mathcal{W})\to P^{S}_{Y/Z}(\mathcal{W})$. We define
$\underline{P}^{S}_{Y/Z}(K)$ to be image of the composition

$\underline{P}^{m}_{Y/Z}(K)\hookrightarrow a^{*}P^{m}_{Y/Z}(\mathcal{W})\to
P^{S}_{Y/Z}(\mathcal{W}),$

which tracks the principal parts of $K$ in the $Y/Z$ directions specified by
$S$.

Given two admissible sets $S\subset S^{\prime}$, there is a quotient
$\underline{P}^{S^{\prime}}_{Y/Z}(K)\to\underline{P}^{S}_{Y/Z}(K)$. Let
$V\subset\underline{P}^{S^{\prime}}_{Y/Z}(K)$ be the kernel. We define
$P^{S\subset S^{\prime}}_{Y/Z}(\mathcal{W}\to\mathcal{W}^{\prime})$ to be the
cokernel of the composition

$V\hookrightarrow\underline{P}^{S^{\prime}}_{Y/Z}(K)\hookrightarrow
P^{S^{\prime}}_{Y/Z}(\mathcal{W}).$

The bundle $P^{S\subset S^{\prime}}_{Y/Z}(\mathcal{W}\to\mathcal{W}^{\prime})$
tracks the principal parts associated to $S$ on $\mathcal{W}$ and then the
principal parts associated to the rest of $S^{\prime}$ but just in the
$\mathcal{W}^{\prime}$ quotient. We visualize $P^{S\subset
S^{\prime}}_{Y/Z}(\mathcal{W}\to\mathcal{W}^{\prime})$ by a decorated diagram
of shape $S^{\prime}$, where the dots are filled in the subshape $S$ and half
filled (representing values just in $\mathcal{W}^{\prime}$) in the remainder
$S^{\prime}\smallsetminus S$ (colored in blue below). A preview of the cases
we shall need later are pictured below.

  1. (6.4A)

$S=\\{1,x\\}$ and $S^{\prime}=\\{1,x,y,x^{2}\\}$, for triple points in Section
7.3.

  2. (6.4B)

$S=\\{1,x\\}$ and $S^{\prime}=\\{1,x,y,x^{2},xy\\}$, for quadruple points in
Lemma 7.7.

  3. (6.4C)

$S=\\{1,x\\}$ and $S^{\prime}=\\{1,x,y,x^{2},xy,x^{3}\\}$, for quadruple
points in Lemma 7.13.

Revisiting Definition 3.9,
$Q^{1}_{Y/Z}(\mathcal{W}^{\prime})=P^{\varnothing\subset\\{1,x,y\\}}(\mathcal{W}\to\mathcal{W}^{\prime})$
would be represented by

##  4\. The Chow ring in degree $3$

###  4.1. Set up

We begin by recalling the linear algebraic data associated to a degree $3$
cover as developed by Miranda and Casnsati–Ekedahl [27, 11]. For more details
in our context, see [7] and [10, Section 3.1]. Given a degree $3$, genus $g$
cover, $\alpha:C\to\mathbb{P}^{1}$, define
$E_{\alpha}:=(\alpha_{*}\mathcal{O}_{C}/\mathcal{O}_{\mathbb{P}^{1}})^{\vee}$,
which is a rank $2$, degree $g+2$ vector bundle on $\mathbb{P}^{1}$. There is
a natural embedding $C\subset\mathbb{P}E_{\alpha}^{\vee}$ and $C$ is the zero
locus of a section of

$H^{0}(\mathbb{P}E^{\vee}_{\alpha},\gamma^{*}\det
E_{\alpha}^{\vee}\otimes\mathcal{O}_{\mathbb{P}E_{\alpha}^{\vee}}(3))\cong
H^{0}(\mathbb{P}^{1},\det
E_{\alpha}^{\vee}\otimes\operatorname{Sym}^{3}E_{\alpha}),$

where $\gamma:\mathbb{P}E^{\vee}_{\alpha}\to\mathbb{P}^{1}$ is the structure
map. Conversely, given a globally generated, rank $2$, degree $g+2$ vector
bundle $E$ on $\mathbb{P}^{1}$ with $\operatorname{Sym}^{3}E\otimes\det
E^{\vee}$ globally generated, the vanishing of a general section $\delta\in
H^{0}(\mathbb{P}E^{\vee},\gamma^{*}\det
E^{\vee}\otimes\mathcal{O}_{\mathbb{P}E^{\vee}}(3))$ defines a smooth, genus
$g$ triple cover
$\alpha:C=V(\delta)\subset\mathbb{P}E^{\vee}\to\mathbb{P}^{1}$ such that
$E_{\alpha}\cong E$. First, let us give a characterization of which sections
do not yield covers parametrized by $\mathcal{H}_{3,g}$.

######  Lemma 4.1.

Let $E$ be a rank $2$, degree $g+2$ vector bundle on $\mathbb{P}^{1}$ such
that $\operatorname{Sym}^{3}E\otimes\det E^{\vee}$ is globally generated. Let
$\delta\in H^{0}(\mathbb{P}E,\gamma^{*}\det
E^{\vee}\otimes\mathcal{O}_{\mathbb{P}E_{\alpha}}(3))$. Suppose that the zero
locus $C=V(\delta)\subseteq\mathbb{P}E^{\vee}$ is not a smooth, irreducible
genus $g$ triple cover of $\mathbb{P}^{1}$. Then there exists a point $p\in C$
such that $\dim T_{p}C=2$.

###### Proof.

If $\delta=0$, then $C$ is $2$-dimensional and the claim follows. Now suppose
$\delta\neq 0$. We will show that $C$ is connected, which implies that if $C$
fails to be an irreducible triple cover, it must have a point with $2$
dimensional tangent space. If $\operatorname{Sym}^{3}E\otimes\det E^{\vee}$ is
globally generated, then both summands of
$E=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})$ have degree at least
$\frac{g+2}{3}$. Hence, $h^{0}(\mathbb{P}^{1},E^{\vee})=0$. If
$C\to\mathbb{P}^{1}$ is finite we have
$h^{0}(C,\mathcal{O}_{C})=h^{0}(\mathbb{P}^{1},\alpha_{*}\mathcal{O}_{C})=h^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}})+h^{0}(\mathbb{P}^{1},E_{\alpha}^{\vee})=1$,
so $C$ is connected. Now suppose $C$ has a positive dimensional fiber over
$\mathbb{P}^{1}$. Any curve in the class
$\mathcal{O}_{\mathbb{P}E^{\vee}}(3)\otimes\gamma^{*}\det E^{\vee}$ has a
component that meets every fiber, thus $C$ is again connected.

∎

We now recall some notation and constructions from [10]. The association of
$\alpha:C\to\mathbb{P}^{1}$ with $E_{\alpha}$ gives rise to a map of
$\mathcal{H}_{3,g}$ to the moduli stack $\mathcal{B}_{3,g}$ of rank $2$,
degree $g+3$, globally generated vector bundles on $\mathbb{P}^{1}$-bundles.
Let $\pi:\mathcal{P}\to\mathcal{B}_{3,g}$ be the universal
$\mathbb{P}^{1}$-bundle and let $\mathcal{E}$ be the universal rank $2$ vector
bundle on $\mathcal{P}$. Continuing the notation of [10], let
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$ and define classes $a_{i}\in
A^{i}(\mathcal{B}_{3,g})$ and $a_{i}^{\prime}\in A^{i-1}(\mathcal{B}_{3,g})$
by the formula

$c_{i}(\mathcal{E})=\pi^{*}a_{i}+\pi^{*}a_{i}^{\prime}z.$

We also define $c_{2}=-\pi_{*}(z^{3})\in A^{2}(\mathcal{B}_{3,g})$, which is
the pullback of the universal second Chern class on $\operatorname{BSL}_{2}$.
By [10, Theorem 4.4],

(4.1) $\text{$a_{1},a_{2},a_{2}^{\prime},c_{2}$ generate
$A^{*}(\mathcal{B}_{3,g})$ and satisfy no relations in degrees up to $g+2$}.$

Now, let $\gamma:\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ and define
$\mathcal{W}:=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(3)\otimes\gamma^{*}\det\mathcal{E}^{\vee}$,
which is a line bundle on $\mathbb{P}\mathcal{E}^{\vee}$. Consider the bundle
$\mathcal{U}_{3,g}:=\gamma_{*}\mathcal{W}=\operatorname{Sym}^{3}\mathcal{E}\otimes\det\mathcal{E}^{\vee}$.
We define

(4.2)
$\mathcal{B}_{3,g}^{\prime}:=\mathcal{B}_{3,g}\smallsetminus\operatorname{Supp}R^{1}\pi_{*}\mathcal{U}_{3,g}(-1).$

Equivalently, $\mathcal{B}_{3,g}^{\prime}$ is the open locus where
$\mathcal{U}_{3,g}$ is globally generated on the fibers of $\pi$. By the
theorem on cohomology and base change

$\mathcal{X}^{\prime}_{3,g}:=\pi_{*}\mathcal{U}_{3,g}|_{\mathcal{B}_{3,g}^{\prime}}$

is a vector bundle. In [10, Lemma 5.1], we showed that the map
$\mathcal{H}_{3,g}\to\mathcal{B}_{3,g}$ factors through an open embedding
$\mathcal{H}_{3,g}\to\mathcal{X}^{\prime}_{3,g}$. Hence, the Chow ring of
$\mathcal{H}_{3,g}$ is generated by the pullbacks of the classes
$a_{1},a_{2}^{\prime},a_{2},c_{2}$ from $\mathcal{B}^{\prime}_{3,g}$. We must
determine the relations among these classes that come from excising
$\operatorname{Supp}R^{1}\pi_{*}\mathcal{U}_{3,g}(-1)$ from
$\mathcal{B}_{3,g}$ and from excising

$\Delta_{3,g}:=\mathcal{X}^{\prime}_{3,g}\smallsetminus\mathcal{H}_{3,g}.$

In other words, we shall compute the Chow ring $A^{*}(\mathcal{H}_{3,g})$ by
computing the image of the left-hand map in the excision sequence

$A^{*-1}(\Delta_{3,g})\rightarrow A^{*}(\mathcal{X}_{3,g}^{\prime})\rightarrow
A^{*}(\mathcal{H}_{3,g})\rightarrow 0.$

###  4.2. Resolution and excision

We begin by constructing a space $\widetilde{\Delta}_{3,g}$, which corresponds
to triple covers (or worse) with a marked singular point. By forgetting the
marked point, we will obtain a proper surjective morphism
$\widetilde{\Delta}_{3,g}\rightarrow\Delta_{3,g}$ by Lemma 4.1. Because our
Chow rings are taken with rational coefficients, pushforward induces a
surjection on Chow groups $A^{*}(\widetilde{\Delta}_{3,g})\rightarrow
A^{*}(\Delta_{3,g})$. Thus, it will suffice to describe the image of
$A^{*-1}(\widetilde{\Delta}_{3,g})\to A^{*}(\mathcal{X}_{3,g}^{\prime})$.

To build $\widetilde{\Delta}_{3,g}$, we use the machinery of bundles of
relative principal parts. By Proposition 3.2 part (1), there is an evaluation
map

(4.3)
$\gamma^{*}\pi^{*}\mathcal{X}_{3,g}^{\prime}=(\pi\circ\gamma)^{*}(\pi\circ\gamma)_{*}\mathcal{W}\rightarrow
P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}^{\prime}_{3,g}}(\mathcal{W}).$

A geometric point of $\gamma^{*}\pi^{*}\mathcal{X}_{3,g}^{\prime}$ is the data
of $(E,\delta,p)$ where $E$ is a geometric point of
$\mathcal{B}_{3,g}^{\prime}$, $\delta$ a section of
$\mathcal{O}_{\mathbb{P}E}(3)\otimes\gamma^{*}\det E^{\vee}$, and $p$ a point
of $\mathbb{P}E^{\vee}$. Such a point lies in the kernel of the evaluation map
(4.3) precisely when $\delta(p)=0$ and the first order derivatives of $\delta$
also vanish at $p$, which is to say $V(\delta)\subset\mathbb{P}E^{\vee}$ has
$2$-dimensional tangent space at $p$. A similar description works in arbitrary
families. We define $\widetilde{\Delta}_{3,g}$ to be the preimage of the zero
section of (4.3) so we obtain a “trapezoid” diagram:

(4.4)
${{\widetilde{\Delta}_{3,g}}}$${\gamma^{*}\pi^{*}\mathcal{X}_{3,g}^{\prime}}$${\pi^{*}\mathcal{X}_{3,g}^{\prime}}$${\mathcal{X}_{3,g}^{\prime}}$${\mathbb{P}\mathcal{E}^{\vee}}$${\mathcal{P}}$${{\mathcal{B}_{3,g}^{\prime}.}}$$\scriptstyle{i}$$\scriptstyle{\rho^{\prime\prime}}$$\scriptstyle{\gamma^{\prime}}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

We can thus determine information about the Chow ring of
$\mathcal{H}_{3,g}=\mathcal{X}_{3,g}^{\prime}\smallsetminus(\pi^{\prime}\circ\gamma^{\prime}\circ
i)(\widetilde{\Delta}_{3,g})$ using the Trapezoid Lemma 2.1.

######  Lemma 4.2.

The rational Chow ring of $\mathcal{H}_{3,g}$ is a quotient of
$\mathbb{Q}[a_{1}]/(a_{1}^{3})$. Moreover,

  1. (1)

For all $g\geq 3$, we have $A^{1}(\mathcal{H}_{3,g})=\mathbb{Q}a_{1}$.

  2. (2)

For all $g\geq 6$, we have $A^{2}(\mathcal{H}_{3,g})=\mathbb{Q}a_{1}^{2}$.

###### Proof.

Let $z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$ and
$\zeta=c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1))$, so
$z^{i}\zeta^{j}$ for $0\leq i,j\leq 1$ form a basis for
$A^{*}(\mathbb{P}\mathcal{E}^{\vee})$ as a $A^{*}(\mathcal{B}_{3,g}^{\prime})$
module. Let $I$ be the ideal generated by
$(\pi\circ\gamma)_{*}(c_{3}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{3,g}^{\prime}}(\mathcal{W}))\cdot
z^{i}\zeta^{j})$ for $0\leq i,j\leq 1$. We compute expressions for these push
forwards in terms of $a_{1},a_{2},a_{2}^{\prime},c_{2}$, and we find
$\mathbb{Q}[a_{1},a_{2}^{\prime},a_{2},c_{2}]/I\cong\mathbb{Q}[a_{1}]/(a_{1}^{3})$.
The code to do the above computations is provided at [9]. For example, when
$i=j=0$, because $\widetilde{\Delta}_{3,g}\to\Delta_{3,g}$ is generically one-
to-one, this allows us to find

(4.5)
$[\Delta_{3,g}]=\pi^{\prime}_{*}\gamma^{\prime}_{*}[\widetilde{\Delta}_{3,g}]=\rho^{*}(\pi\circ\gamma)_{*}(c_{3}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{3,g}^{\prime}}(\mathcal{W}))=(8g+12)a_{1}-9a_{2}^{\prime}.$

By the Trapezoid Lemma 2.1, we have that $A^{*}(\mathcal{H}_{3,g})$ is a
quotient of $A^{*}(\mathcal{B}_{3,g}^{\prime})/I$. Since
$A^{*}(\mathcal{B}_{3,g}^{\prime})$ is a quotient of
$\mathbb{Q}[a_{1},a_{2},a_{2}^{\prime},c_{2}]$, it follows that
$A^{*}(\mathcal{H}_{3,g})$ is a quotient of $\mathbb{Q}[a_{1}]/(a_{1}^{3})$.

First, note that the complement of $\mathcal{B}_{3,g}^{\prime}$ inside
$\mathcal{B}_{3,g}$ is the union of splitting loci where
$E=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})$ for $3e_{1}<g+2$ (see [10,
Section 4.2] for a review of splitting loci in our context). The codimension
of the $(e_{1},e_{2})$ splitting locus with $e_{1}\leq e_{2}$ is
$\max\\{0,e_{2}-e_{1}-1\\}$. Using this, one readily checks that the
complement of $\mathcal{B}_{3,g}^{\prime}$ has codimension at least $2$ for
$g\geq 3$ and at least $3$ for $g\geq 6$. Thus, by (4.1), for $g\geq 3$, the
only relations in codimension $1$ come from the push forwards of classes on
$\widetilde{\Delta}_{3,g}$. Further, for $g\geq 6$, the only relations in
codimension $2$ come from the push forwards of classes supported on
$\widetilde{\Delta}_{3,g}$.

To prove (1) and (2), it suffices to show that $I$ already accounts for all
such relations in codimension $1$ when $g\geq 3$ and for all such relations in
codimension $2$ when $g\geq 6$. Precisely, let
$\mathcal{Z}\subset\mathbb{P}\mathcal{E}^{\vee}$ be the locus where the map
(4.3) fails to be surjective on fibers. We will show that

(4.6)
$A^{0}(\widetilde{\Delta}_{3,g})=A^{0}(\widetilde{\Delta}_{3,g}\smallsetminus\rho^{\prime\prime-1}(\mathcal{Z}))\cong\rho^{\prime\prime*}A^{0}(\mathbb{P}\mathcal{E}^{\vee}\smallsetminus\mathcal{Z})=\rho^{\prime\prime*}A^{0}(\mathbb{P}\mathcal{E}^{\vee})$

and when $g\neq 4$, that

(4.7)
$A^{1}(\widetilde{\Delta}_{3,g})=A^{1}(\widetilde{\Delta}_{3,g}\smallsetminus\rho^{\prime\prime-1}(\mathcal{Z}))\cong\rho^{\prime\prime*}A^{1}(\mathbb{P}\mathcal{E}^{\vee}\smallsetminus\mathcal{Z})=\rho^{\prime\prime*}A^{1}(\mathbb{P}\mathcal{E}^{\vee}).$

The middle isomorphism follows in both cases from the fact that
$\widetilde{\Delta}_{3,g}\smallsetminus\rho^{\prime\prime-1}(\mathcal{Z})$ is
a vector bundle over $\mathbb{P}\mathcal{E}^{\vee}\smallsetminus\mathcal{Z}$.
To show the other equalities we use excision.

We claim that the map (4.3) always has rank at least $2$. To see this,
consider the diagram

(4.8)
${\gamma^{*}\pi^{*}\pi_{*}\gamma_{*}\mathcal{W}}$${P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{3,g}^{\prime}}(\mathcal{W})}$${\gamma^{*}\gamma_{*}\mathcal{W}}$${P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{P}}(\mathcal{W})}$

The left vertical map is a surjection because $\gamma_{*}\mathcal{W}$ is
relatively globally generated along $\mathcal{P}\to\mathcal{B}_{3,g}^{\prime}$
(by definition of $\mathcal{B}_{3,g}^{\prime}$, see (4.2)); the bottom
horizontal map is surjective by Lemma 3.3 because $\mathcal{W}$ is relatively
very ample on $\mathbb{P}\mathcal{E}^{\vee}$ over $\mathcal{P}$. Thus, the top
horizontal map must have rank at least
$2=\mathrm{rank}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{P}}(\mathcal{W}))$.
It follows that

(4.9)
$\operatorname{codim}(\rho^{\prime\prime-1}(\mathcal{Z})\subset\widetilde{\Delta}_{3,g})=\operatorname{codim}(\mathcal{Z}\subset\mathbb{P}\mathcal{E}^{\vee})-1.$

By the argument in Lemma 3.5, $\mathcal{Z}$ is the locus where $\mathcal{W}$
fails to induce a relative embedding on $\mathbb{P}\mathcal{E}^{\vee}$ over
$\mathcal{B}_{3,g}^{\prime}$. By Lemma 3.4, the restriction to a fiber over
$\mathcal{B}_{3,g}^{\prime}$, say
$\mathcal{W}|_{\mathbb{P}E^{\vee}}\cong\mathcal{O}_{\mathbb{P}E^{\vee}}(3)\otimes\gamma^{*}\mathcal{O}_{\mathbb{P}^{1}}(-g-2)$
fails to be very ample if and only if
$E\cong\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})$ with $3e_{1}\leq g+2$.
Moreover, in this case, the linear system fails to induce an embedding
precisely along the directrix of $\mathbb{P}E^{\vee}$. By definition of
$\mathcal{B}_{3,g}^{\prime}$, we always have $3e_{1}\geq g+2$. Thus,
$\gamma(\mathcal{Z})$ is contained in at most one splitting locus, which is
nonempty if and only if $g\equiv 1\pmod{3}$. In particular:

  1. (1)

if $g=4$, then $\gamma(\mathcal{Z})$ is the splitting locus
$(e_{1},e_{2})=(2,4)$, which has codimension $1$

  2. (2)

if $g=7$, then $\gamma(\mathcal{Z})$ is the splitting locus
$(e_{1},e_{2})=(3,6)$, which has codimension $2$

  3. (3)

if $g\neq 4,7$, then $\gamma(\mathcal{Z})$ has codimension at least $3$

Since the directrix has codimension $1$, it follows that

$\operatorname{codim}(\mathcal{Z}\subset\mathbb{P}\mathcal{E}^{\vee})=\begin{cases}2&\text{if
$g=4$}\\\ 3&\text{if $g=7$}\\\ \geq 4&\text{otherwise.}\end{cases}$

By (4.9), we see then that $\rho^{\prime\prime-1}(\mathcal{Z})$ has suitably
high codimension so that (4.6) is satisfied for all $g$ and (4.7) is satisfied
for $g\neq 4$. ∎

This completes the proof of Theorem 1.1(1) when $g\geq 6$.

###  4.3. Low genus calculations

The lemmas in this section show that the remaining Chow groups not already
determined by Lemma 4.2 vanish. This is due to certain geometric phenomena
that occur in low codimension when the genus is small.

######  Lemma 4.3.

When $g=2$, we have $A^{*}(\mathcal{H}_{3,2})=0$.

###### Proof.

When $g=2$, the complement of
$\mathcal{B}_{3,2}^{\prime}\subset\mathcal{B}_{3,2}$ is the $(1,3)$ splitting
locus, which has codimension $1$. As a consequence, $a_{1}$ and
$a_{2}^{\prime}$ satisfy a relation on $\mathcal{B}_{3,2}^{\prime}$. Using
[25, Lemma 5.1], we calculate the class of the $(1,3)$ splitting locus as the
degree $1$ piece of a ratio of total Chern classes below, which we compute
with the code [9]:

$0=s_{1,3}=\left[\frac{c((\pi_{*}\mathcal{E}(-2)\otimes\pi_{*}\mathcal{O}_{\mathcal{P}}(1))^{\vee})}{c((\pi_{*}\mathcal{E}(-1))^{\vee})}\right]_{1}=a_{2}^{\prime}-2a_{1}$

on $\mathcal{B}_{3,2}^{\prime}$. Specializing (4.5) to $g=2$, we also have the
additional relation $0=[\Delta_{3,2}]=28a_{1}-9a_{2}^{\prime}$ in
$A^{1}(\mathcal{H}_{3,g})$, so we conclude $a_{1}=a_{2}^{\prime}=0$ and hence,
the Chow ring is trivial. ∎

######  Lemma 4.4.

For $g=3,4,5$, we have $A^{2}(\mathcal{H}_{3,g})=0$.

###### Proof.

We first explain the case $g=3$. Here, the complement of
$\mathcal{B}_{3,3}^{\prime}$ inside $\mathcal{B}_{3,3}$ is the closure of the
splitting locus $(e_{1},e_{2})=(1,4)$, which has codimension $2$. The
universal formulas for classes of splitting loci [25] say that the class of
this unbalanced splitting locus is the degree $2$ piece of a ratio of total
Chern classes, which we computed in the code [9],

$s_{1,4}=\left[\frac{c((\pi_{*}\mathcal{E}(-2)\otimes\pi_{*}\mathcal{O}_{\mathcal{P}}(1))^{\vee})}{c((\pi_{*}\mathcal{E}(-1))^{\vee})}\right]_{2}=3a_{1}^{2}+\frac{1}{2}a_{2}-\frac{5}{2}a_{1}a_{2}^{\prime}+\frac{1}{2}a_{2}^{\prime
2}+3c_{2}.$

It follows that $A^{*}(\mathcal{H}_{3,3})$ is a quotient of
$\mathbb{Q}[a_{1},a_{2},a_{2}^{\prime},c_{2}]/(I+\langle s_{1,4}\rangle)$. We
checked in the code [9] that the codimension $2$ piece of this ring is zero.

The case $g=5$ is very similar so we explain it next. The complement of
$\mathcal{B}_{3,5}^{\prime}$ inside $\mathcal{B}_{3,5}$ is the closure of the
splitting locus $(e_{1},e_{2})=(2,5)$, which has codimension $2$. The class of
this splitting locus is computed similarly:

$s_{2,5}=\left[\frac{c((\pi_{*}\mathcal{E}(-3)\otimes\pi_{*}\mathcal{O}_{\mathcal{P}}(1))^{\vee})}{c((\pi_{*}\mathcal{E}(-2))^{\vee})}\right]_{2}=6a_{1}^{2}+\frac{1}{2}a_{2}-\frac{7}{2}a_{1}a_{2}^{\prime}+\frac{1}{2}a_{2}^{\prime
2}+6c_{2}.$

Therefore, $A^{*}(\mathcal{H}_{5,3})$ is a quotient
$\mathbb{Q}[a_{1},a_{2},a_{2}^{\prime},c_{2}]/(I+\langle s_{2,5}\rangle)$,
whose codimension $2$ piece we also checked is zero [9].

In the case $g=4$, our additional relation will come from
$\rho^{\prime\prime-1}(\mathcal{Z})\subset\widetilde{\Delta}_{3,4}$, which has
codimension $1$, and whose push forward therefore determines a class that is
zero in $A^{2}(\mathcal{H}_{3,4})$. By (4.8), we have that
$\rho^{\prime\prime-1}(\mathcal{Z})$ is the transverse intersection of
$\rho^{\prime-1}(\mathcal{Z})$ with the kernel subbundle of
$\gamma^{*}\pi^{*}\pi_{*}\gamma_{*}\mathcal{W}\to
P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{P}}(\mathcal{W})$. That is, our
possible additional relation is given by

(4.10)
$s:=\pi^{\prime}_{*}\gamma^{\prime}_{*}i_{*}[\rho^{\prime\prime-1}(\mathcal{Z})]=\gamma^{\prime}_{*}\pi^{\prime}_{*}(\rho^{\prime*}[\mathcal{Z}]\cdot\rho^{\prime*}c_{2}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{P}}(\mathcal{W})))=\rho^{*}\gamma_{*}\pi_{*}([\mathcal{Z}]\cdot
c_{2}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{P}}(\mathcal{W}))).$

It remains to compute $[\mathcal{Z}]$, which we do now. Let
$\Sigma=\gamma(\mathcal{Z})\subset\mathcal{B}_{3,4}^{\prime}$ be the $(2,4)$
splitting locus. Using the formulas for classes of splitting loci [25], we
compute

$[\Sigma]=s_{1,4}=\left[\frac{c(((\pi_{*}\mathcal{E}(-3)\otimes\pi_{*}\mathcal{O}_{\mathcal{P}}(1))^{\vee})}{c(((\pi_{*}\mathcal{E}(-2))^{\vee})}\right]_{1}=a_{2}^{\prime}-3a_{1}.$

Over $\Sigma$, there is a sequence

(4.11)
$0\rightarrow\pi^{*}\mathcal{M}(-2)\rightarrow\mathcal{E}^{\vee}|_{\Sigma}\rightarrow\pi^{*}\mathcal{N}(-4)\rightarrow
0$

for line bundles $\mathcal{M}$ and $\mathcal{N}$ on $\Sigma$. Let
$m=c_{1}(\mathcal{M})$ and $n=c_{1}(\mathcal{N})$. The directrix over $\Sigma$
is
$\mathcal{Z}=\mathbb{P}(\pi^{*}\mathcal{M}(-2))\subset\mathbb{P}\mathcal{E}^{\vee}|_{\Sigma}$.
By [15, Proposition 9.13], the fundamental class of $\mathcal{Z}$ inside
$\mathbb{P}\mathcal{E}^{\vee}|_{\Sigma}$ is
$\zeta+c_{1}(\pi^{*}\mathcal{N}(-4))=\zeta+n-4z$. Considering Chern classes in
the exact sequence (4.11), we learn (recall $a_{1}^{\prime}=g+2=6$)

$\displaystyle-a_{1}|_{\Sigma}-6z=c_{1}(\mathcal{E}^{\vee}|_{\Sigma})$
$\displaystyle=m-4z+n-2z$ $\displaystyle\Rightarrow$ $\displaystyle\quad m+n$
$\displaystyle=-a_{1}|_{\Sigma}$ and $\displaystyle
a_{2}|_{\Sigma}+(a_{2}^{\prime}|_{\Sigma})\cdot
z=c_{2}(\mathcal{E}^{\vee}|_{\Sigma})$ $\displaystyle=(m-4z)(n-2z)$
$\displaystyle=mn-c_{2}-(2m+4n)z$ $\displaystyle\Rightarrow$
$\displaystyle\quad 2m+4n$ $\displaystyle=-a_{2}^{\prime}|_{\Sigma}.$

In particular,
$n=\left.\left(a_{1}-\frac{a_{2}^{\prime}}{2}\right)\right|_{\Sigma}$. Hence,
the fundamental class of $\mathcal{Z}$ inside all of
$\mathbb{P}\mathcal{E}^{\vee}$ is

$[\mathcal{Z}]=(\zeta+a_{1}-\tfrac{a_{2}^{\prime}}{2}-4z)\cdot[\Sigma]=(\zeta+a_{1}-\tfrac{a_{2}^{\prime}}{2}-4z)(a_{2}^{\prime}-3a_{1}).$

This allows us to compute $s$ in (4.10), and our code confirms that the
codimension $2$ piece of
$\mathbb{Q}[a_{1},a_{2},a_{2}^{\prime},c_{2}]/(I+\langle s\rangle)$ is zero
[9]. ∎

Together, Lemmas 4.2, 4.3 and 4.4 determine the rational Chow ring of
$\mathcal{H}_{3,g}$ for all $g$:

$A^{*}(\mathcal{H}_{3,g})=\begin{cases}\mathbb{Q}&\text{if $g=2$}\\\
\mathbb{Q}[a_{1}]/(a_{1}^{2})&\text{if $g=3,4,5$}\\\
\mathbb{Q}[a_{1}]/(a_{1}^{3})&\text{if $g\geq 6$.}\end{cases}$

This completes the proof of Theorem 1.1(1).

##  5\. The Chow ring in degree $4$

###  5.1. Set up

We begin by briefly recalling the linear algebraic data associated to a degree
$4$ cover, as developed by Casnati–Ekedahl [11]. For more details in our
context, see [10, Section 3.2]. Given a degree $4$ cover
$\alpha:C\to\mathbb{P}^{1}$, we associate two vector bundles on
$\mathbb{P}^{1}$:

$E_{\alpha}:=(\alpha_{*}\mathcal{O}_{C}/\mathcal{O}_{\mathbb{P}^{1}})^{\vee}=\ker(\alpha_{*}\omega_{\alpha}\to\mathcal{O}_{\mathbb{P}^{1}})\qquad\text{and}\qquad
F_{\alpha}:=\ker(\operatorname{Sym}^{2}E_{\alpha}\to\alpha_{*}\omega_{\alpha}^{\otimes
2}).$

The first is rank $3$ and the second is rank $2$. If $C$ has genus $g$, then
both bundles have degree $g+3$. Geometrically, the curve $C$ is embedded in
$\gamma:\mathbb{P}E^{\vee}_{\alpha}\rightarrow\mathbb{P}^{1}$ as the zero
locus of a section

$\delta_{\alpha}\in
H^{0}(\mathbb{P}E_{\alpha}^{\vee},\mathcal{O}_{\mathbb{P}E^{\vee}_{\alpha}}(2)\otimes\gamma^{*}F_{\alpha}^{\vee}).$

In each fiber of $\gamma$, the four points are the base locus of a pencil of
conics parametrized by $F_{\alpha}$.

Conversely, given vector bundles $E,F$ of ranks $3$ and $2$, both of degree
$g+3$, we wish to characterize when a section

$\delta\in
H^{0}(\mathbb{P}E^{\vee},\mathcal{O}_{\mathbb{P}E^{\vee}}(2)\otimes\gamma^{*}F^{\vee})$

fails to produce a smooth degree $4$, genus $g$ cover.

######  Lemma 5.1.

Suppose $E,F,\delta$ are as above with
$F^{\vee}\otimes\operatorname{Sym}^{2}E$ globally generated. If the zero locus
$C=V(\delta)$ is not an irreducible, smooth quadruple cover of
$\mathbb{P}^{1}$, then there is a point $p\in C$ such that $\dim T_{p}C\geq
2$.

###### Proof.

If $C$ is connected or has a component of dimension at least $2$ then the
lemma is immediate. Suppose $C$ is $1$-dimensional and disconnected. We first
rule out the case in which $C$ has at least $2$ connected components, both
mapping finitely onto $\mathbb{P}^{1}$. In this case,
$\alpha_{*}\mathcal{O}_{C}$ has more than one $\mathcal{O}$ factor; then $E$
has a degree $0$ summand, so $\operatorname{Sym}^{2}E\otimes F^{\vee}$ would
have a negative degree summand, which we are assuming is not the case.

Next suppose $C$ has a component $C_{0}$ which does not map finitely onto
$\mathbb{P}^{1}$. Then $C_{0}$ must be contained in a fiber of
$\gamma:\mathbb{P}E^{\vee}\rightarrow\mathbb{P}^{1}$. The restriction of the
zero locus of $\delta$ to a fiber is the intersection of two (possibly
singular) conics in $\mathbb{P}^{2}$. The only way for such an intersection to
have a $1$-dimensional component is for the conics to have a common component
$C_{0}$. Hence, some fiber of $C$ is equal to $C_{0}$ union a finite subscheme
of length less than $4$ (length $1$ if $C_{0}$ is a line, empty if $C_{0}$ is
a conic).

Since the generic fiber consists of $4$ points, some of those $4$ points must
specialize to $C_{0}$, which means $C$ is singular at those points on $C_{0}$
(and $C$ is connected). ∎

The association of $\alpha:C\to\mathbb{P}^{1}$ with the pair
$(E_{\alpha},F_{\alpha})$ gives rise to map $\mathcal{H}_{4,g}$ to the moduli
stack $\mathcal{B}_{4,g}$ of pairs of vector bundles on
$\mathbb{P}^{1}$-bundles, as defined in [10, Definition 5.2]. Let
$\pi:\mathcal{P}\to\mathcal{B}_{4,g}$ be the universal
$\mathbb{P}^{1}$-bundle. Let $\mathcal{E}$ be the universal rank $3$ vector
bundle on $\mathcal{P}$, and $\mathcal{F}$ the universal rank $2$ bundle on
$\mathcal{P}$. Continuing the notation of [10], let
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$ and define classes $a_{i},b_{i}\in
A^{i}(\mathcal{B}_{4,g})$ and $a_{i}^{\prime},b_{i}^{\prime}\in
A^{i-1}(\mathcal{B}_{4,g})$ by the formula

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

(Note that there is a “determinant compatibility” condition which implies
$a_{1}=b_{1}$, see [10, Equation 5.4].) We also define
$c_{2}=-\pi_{*}(z^{3})\in A^{2}(\mathcal{B}_{4,g})$, which is the pullback of
the universal second Chern class on $\operatorname{BSL}_{2}$. By [10, Equation
5.5],

(5.1)
$a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2},b_{2}^{\prime},c_{2}$
generate $A^{*}(\mathcal{B}_{4,g})$ and satisfy $\displaystyle\text{no
relations in degrees up to $g+3$}.$

We call the pullbacks of $\mathcal{E}$ and $\mathcal{F}$ to
$\mathcal{H}_{4,g}$ the _CE bundles_ (these are the bundles appearing in the
Casnati–Ekedahl resolution for the universal curve). We call the pullbacks to
$\mathcal{H}_{4,g}$ of the associated classes in (5.1) the _CE classes_. By
[10, Theorem 3.10], the CE classes are tautological and generate the
tautological ring.

Up to this point, the set up has been quite similar to degree $3$. However,
unlike in degree $3$, the full Hurwitz stack $\mathcal{H}_{4,g}$ _cannot_ be
realized as an open substack of a vector bundle over an open substack of
$\mathcal{B}_{4,g}$. This is why we were unable to determine the full Chow
ring of $\mathcal{H}_{4,g}$ with our techniques. We now proceed in two steps.
First, in Section 5.2 we shall produce several relations among CE classes on
$\mathcal{H}_{4,g}$ using principal parts bundles. Then, in Section 5.3, we
shall define an open substack
$\mathcal{H}_{4,g}^{\circ}\subset\mathcal{H}_{4,g}$, which does lie inside a
vector bundle over an open substack
$\mathcal{B}_{4,g}^{\circ}\subset\mathcal{B}_{4,g}$, and use it to demonstrate
that we have found all relations in degrees up to roughly $g/4$. It may help
to think of $\mathcal{H}_{4,g}^{\circ}$ as an open substack that is “large
enough to witness the independence of many CE classes.”

###  5.2. Relations among CE classes

Let $\mathcal{E}$ and $\mathcal{F}$ be the CE bundles on the universal
$\mathbb{P}^{1}$-bundle $\pi:\mathcal{P}\to\mathcal{H}_{4,g}$. Let
$\gamma:\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ be the structure map. We
define a rank $2$ vector bundle on $\mathbb{P}\mathcal{E}^{\vee}$ by
$\mathcal{W}:=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(2)\otimes\mathcal{F}^{\vee}$.
By the Casnati–Ekedahl theorem in degree 4, see [10, Equation 3.7] or [11],
the universal curve $\mathcal{C}\subset\mathbb{P}\mathcal{E}^{\vee}$
determines a global section $\delta^{\mathrm{\mathrm{univ}}}$ of
$\mathcal{W}$, whose vanishing locus is
$V(\delta^{\mathrm{\mathrm{univ}}})=\mathcal{C}\subset\mathbb{P}\mathcal{E}^{\vee}$.

The global section $\delta^{\mathrm{\mathrm{univ}}}$ induces a global section
$\delta^{\mathrm{\mathrm{univ}}}{{}^{\prime}}$ of the principal parts bundle
$P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{1}(\mathcal{W})$ on
$\mathbb{P}\mathcal{E}^{\vee}$, which records the value and derivatives of
$\delta^{\mathrm{\mathrm{univ}}}$. Now consider the tower

${{G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})}}$${\mathbb{P}\mathcal{E}^{\vee}}$${\mathcal{P}}$${{\mathcal{H}_{4,g}},}$$\scriptstyle{a}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

where $G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})$ parametrizes
$2$ dimensional subspaces of the vertical tangent space of
$\mathbb{P}\mathcal{E}^{\vee}$ over $\mathcal{H}_{4,g}$. Dualizing the
tautological sequence on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}^{\circ}})$ we obtain a
filtration

$0\rightarrow\Omega_{y}\rightarrow
a^{*}\Omega_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}\to\Omega_{x}\rightarrow
0,$

where $\Omega_{y}$ is rank $1$ and $\Omega_{x}$ is rank $2$. Let
$P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{\\{1,x\\}}(\mathcal{W})$
be the bundle of restricted principal parts as defined in Section 3.2.

On $G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})$, we obtain a
global section, call it $\delta^{\mathrm{\mathrm{univ}}}{{}^{\prime\prime}}$,
of
$P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{\\{1,x\\}}(\mathcal{W})$
by composing the section $a^{*}\delta^{\mathrm{\mathrm{univ}}}{{}^{\prime}}$
with the quotient
$a^{*}P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{1}(\mathcal{W})\to
P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{\\{1,x\\}}(\mathcal{W})$.
The vanishing locus of $\delta^{\mathrm{\mathrm{univ}}}{{}^{\prime\prime}}$ is
the space of pairs $(p,S)$ where $p\in
V(\delta^{\mathrm{\mathrm{univ}}})\subset\mathbb{P}\mathcal{E}^{\vee}$ and $S$
is a two-dimensional subspace of the tangent space of the fiber of
$V(\delta^{\mathrm{univ}})\to\mathcal{H}_{4,g}$ through $p$. But
$V(\delta^{\mathrm{univ}})=\mathcal{C}\to\mathcal{H}_{4,g}$ is smooth of
relative dimension $1$. Thus,
$\delta^{\mathrm{\mathrm{univ}}}{{}^{\prime\prime}}$ must be non-vanishing on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})$.

Since
$P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{\\{1,x\\}}(\mathcal{W})$
has a non-vanishing global section, its top Chern class,
$c_{6}(P_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}^{\\{1,x\\}}(\mathcal{W}))$,
must be $0$. Moreover, the push forward of this class times any class on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})$ is also zero. Such
relations are generated by the following classes.

######  Lemma 5.2.

Let $\tau=c_{1}(\Omega_{y}^{\vee})$ where $\Omega_{y}^{\vee}$ is the
tautological quotient bundle on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}})$. Let
$\zeta=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1)$ and
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$. Then all classes of the form

(5.2)
$\pi_{*}\gamma_{*}a_{*}(c_{6}(P^{\\{1,x\\}}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{H}_{4,g}}(\mathcal{W}))\cdot\tau^{i}\zeta^{j}z^{k})$

are zero in $R^{*}(\mathcal{H}_{4,g})\subseteq A^{*}(\mathcal{H}_{4,g})$.

It is straightforward for a computer to compute such push forwards as
polynomials in the CE classes. We describe the ideal these push forwards
generate in Section 5.4

###  5.3. All relations in low codimension

We now recall the construction of our “large open” substack
$\mathcal{H}_{4,g}^{\circ}\subset\mathcal{H}_{4,g}$. We start with
$\mathcal{B}_{4,g}$, the moduli space of pairs of vector bundles $E$ of rank
$3$, degree $g+3$ and $F$ of rank $2$, degree $g+3$ on
$\mathbb{P}^{1}$-bundles together with an isomorphism of their determinants
(see [10, Section 5.2]). Now, working over $\mathcal{B}_{4,g}$, let
$\mathcal{E}$ and $\mathcal{F}$ be the universal bundles on
$\pi:\mathcal{P}\rightarrow\mathcal{B}_{4,g}$ and let
$\gamma:\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ be the structure map.
Define
$\mathcal{W}:=\gamma^{*}\mathcal{F}^{\vee}\otimes\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(2)$,
and let
$\mathcal{U}_{4,g}:=\gamma_{*}\mathcal{W}=\mathcal{F}^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{E}$.
We consider an open substack
$\mathcal{B}^{\circ}_{4,g}\subset\mathcal{B}_{4,g}$, defined by a certain
positivity condition for the bundle $\mathcal{U}_{4,g}$

(5.3)
$\mathcal{B}^{\circ}_{4,g}:=\mathcal{B}_{4,g}\smallsetminus\operatorname{Supp}R^{1}\pi_{*}(\mathcal{U}_{4,g}(-2)).$

Let $\mathcal{H}^{\circ}_{4,g}$ denote the base change of
$\mathcal{H}_{4,g}\rightarrow\mathcal{B}_{4,g}$ along the open embedding
$\mathcal{B}_{4,g}^{\circ}\hookrightarrow\mathcal{B}_{4,g}$.

Figure 1. Summary of the method

######  Remark 5.3.

We note that the complement of
$\mathcal{H}^{\circ}_{4,g}\subset\mathcal{H}_{4,g}$ (represented in blue in
the right of Figure 1) contains covers that factor through a curve of low
genus (see [10, p. 21-22]). Thus, the codimension of the complement of
$\mathcal{H}^{\circ}_{4,g}\subset\mathcal{H}_{4,g}$ is $2$. However, upon
restricting to non-factoring covers, the codimension of the complement of
$\mathcal{H}^{\circ}_{4,g}\cap\mathcal{H}_{4,g}^{\mathrm{nf}}\subset\mathcal{H}_{4,g}^{\mathrm{nf}}$
has codimension at least $\frac{g+3}{4}-4$ [10, Lemma 5.5]. In this sense,
$\mathcal{H}_{4,g}^{\mathrm{nf}}$ and $\mathcal{H}_{4,g}^{\circ}$ are “good
approximations” to each other. This is what allows us to find stabilization
results for $\dim A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})$.

Over $\mathcal{B}^{\circ}_{4,g}$, we see that
$\mathcal{X}_{4,g}^{\circ}:=\pi_{*}\mathcal{U}_{4,g}|_{\mathcal{B}^{\circ}_{4,g}}$
is a vector bundle whose fibers correspond to sections of $\mathcal{U}_{4,g}$.
The open $\mathcal{H}_{4,g}^{\circ}$ is contained in the open
$\mathcal{H}_{4,g}^{\prime}$ of [10, Lemma 5.3], so that lemma implies
$\mathcal{H}^{\circ}_{4,g}\to\mathcal{B}_{4,g}^{\circ}$ factors through an
open embedding in $\mathcal{X}^{\circ}_{4,g}$. We define

$\Delta_{4,g}:=\mathcal{X}^{\circ}_{4,g}\smallsetminus\mathcal{H}^{\circ}_{4,g},$

represented in red in the middle column of Figure 1. Now we wish to use the
excision to determine the Chow ring of $\mathcal{H}_{4,g}^{\circ}$ in degrees
up to $\frac{g+3}{4}-4$. Note that $A^{*}(\mathcal{X}^{\circ}_{4,g})\cong
A^{*}(\mathcal{B}_{4,g}^{\circ})$, and we have already determined the latter
in degrees up to $\frac{g+3}{4}-4$ by [10, Equation 5.6].

The next step is to construct a space $\widetilde{\Delta}_{4,g}$ (pictured in
red on the far left of Figure 1), which surjects properly onto $\Delta_{4,g}$.
With rational coefficients, the push forward
$\widetilde{\Delta}_{4,g}\to\Delta_{4,g}$ will be surjective on Chow groups.
Thus, pushing forward all classes from $\widetilde{\Delta}_{4,g}$ will produce
all relations needed to describe $\mathcal{H}_{4,g}^{\circ}$ as a quotient of
$A^{*}(\mathcal{X}^{\circ}_{4,g})\cong A^{*}(\mathcal{B}_{4,g}^{\circ})$.

Each geometric point of $\mathcal{X}_{4,g}^{\circ}$ corresponds to a tuple
$(E,F,\delta)$ where $E,F$ are vector bundles on $\mathbb{P}^{1}$ and
$\delta\in
H^{0}(\mathbb{P}E^{\vee},F^{\vee}\otimes\mathcal{O}_{\mathbb{P}E^{\vee}}(2))$.
We now use restricted bundles of relative principal parts for
$\mathbb{P}\mathcal{E}^{\vee}\rightarrow\mathcal{B}_{4,g}^{\circ}$ to define a
space parametrizing triples

$((E,F,\delta)\in\mathcal{X}_{4,g}^{\circ},\ p\in V(\delta),\ S\subset
T_{p}V(\delta)\text{ of dimension 2}).$

Let
$a:G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}})\rightarrow\mathbb{P}\mathcal{E}^{\vee}$
be the Grassmann bundle of $2$-planes in the relative tangent bundle.
Dualizing the tautological sequence on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}})$ we obtain a
filtration

$0\rightarrow\Omega_{y}\rightarrow
a^{*}\Omega_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}}\to\Omega_{x}\rightarrow
0,$

where $\Omega_{y}$ is rank $1$ and $\Omega_{x}$ is rank $2$. Using the bundle
of restricted principal parts constructed in Section 3.2, we obtain an
evaluation map

(5.4) $a^{*}\gamma^{*}\pi^{*}\pi_{*}\gamma_{*}\mathcal{W}\cong
a^{*}\gamma^{*}\pi^{*}\mathcal{X}_{4,g}^{\circ}\rightarrow
P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}}(\mathcal{W})\rightarrow
P^{\\{1,x\\}}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}^{\circ}_{4,g}}(\mathcal{W}),$

which we claim is surjective. The rightmost map from principal parts to
restricted principal parts is always a surjection. Thus, it suffices to show
that the map $\gamma^{*}\pi^{*}\mathcal{X}_{4,g}^{\circ}\rightarrow
P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}^{\circ}_{4,g}}(\mathcal{W})$
is surjective. By definition of $\mathcal{B}_{4,g}^{\circ}$ (see (5.3)), we
have
$R^{1}\pi_{*}[(\gamma_{*}\mathcal{W})\otimes\mathcal{O}_{\mathcal{P}}(-2)]=0$,
so the surjectivity follows from Lemma 3.5.

We define $\widetilde{\Delta}_{4,g}$ to be the kernel bundle of (5.4). We have
the following “trapezoid” diagram:

(5.5)
${{\widetilde{\Delta}_{4,g}}}$${a^{*}\gamma^{*}\pi^{*}\mathcal{X}_{4,g}^{\circ}}$${\gamma^{*}\pi^{*}\mathcal{X}_{4,g}^{\circ}}$${\pi^{*}\mathcal{X}_{4,g}^{\circ}}$${\mathcal{X}_{4,g}^{\circ}}$${{G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}^{\circ}_{4,g}})}}$${\mathbb{P}\mathcal{E}^{\vee}}$${\mathcal{P}}$${{\mathcal{B}_{4,g}^{\circ}}}$$\scriptstyle{\rho^{\prime\prime}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\gamma^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{a}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

######  Proposition 5.4.

Let $\tau=c_{1}(\Omega_{y}^{\vee})$ where $\Omega_{y}^{\vee}$ is the
tautological quotient line bundle on
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}})$. Let
$\zeta=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1)$ and
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$. Let $I$ be the ideal generated by

(5.6)
$\pi_{*}\gamma_{*}a_{*}(c_{6}(P^{\\{1,x\\}}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}^{\circ}_{4,g}}(\mathcal{W}))\cdot\tau^{i}\zeta^{j}z^{k})\qquad\text{for
}0\leq i,j\leq 2,\ 0\leq k\leq 1.$

Then $A^{*}(\mathcal{H}_{4,g}^{\circ})\cong
A^{*}(\mathcal{B}_{4,g}^{\circ})/I$. Together with $a_{1}=b_{1}$, the classes
in (5.2) therefore generate all relations among the CE classes on
$\mathcal{H}_{4,g}$ in degrees less than $\frac{g+3}{4}-4$.

###### Proof.

By Lemma 5.1, $\widetilde{\Delta}_{4,g}$ surjects onto $\Delta_{4,g}$, so we
may apply the Trapezoid Lemma 2.1. Since
$T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}}$ has rank $3$, the
Grassmann bundle
$G(2,T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}})$ is just the
projectivization of
$T_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}}^{\vee}$; hence its
Chow ring is generated as a module over $A^{*}(\mathbb{P}\mathcal{E}^{\vee})$
by $\tau^{i}$ for $0\leq i\leq 2$. Similarly
$A^{*}(\mathbb{P}\mathcal{E}^{\vee})$ is generated as a module over
$A^{*}(\mathcal{P})$ by $\zeta^{j}$ for $0\leq j\leq 2$ and
$A^{*}(\mathcal{P})$ is generated as a module over
$A^{*}(\mathcal{B}_{4,g}^{\circ})$ by $z^{k}$ for $0\leq k\leq 1$. Thus, the
Trapezoid Lemma 2.1 implies that the classes in (5.6) generate all relations
among the pullbacks of classes on $\mathcal{B}_{4,g}^{\circ}$. In particular,
setting $i=j=k=0$, we obtain

(5.7)
$[\Delta_{4,g}]=\pi^{\prime}_{*}\gamma^{\prime}_{*}a^{\prime}_{*}[\widetilde{\Delta}_{4,g}]=\rho^{*}(\pi\circ\gamma\circ
a)_{*}(c_{6}(P^{1}_{\mathbb{P}\mathcal{E}^{\vee}/\mathcal{B}_{4,g}^{\circ}}(\mathcal{W})))=(8g+20)a_{1}-8a_{2}^{\prime}-b_{2}^{\prime}.$

To see the second claim, note that the classes in (5.6) pullback to the
classes in (5.2). By [10, Equation 5.6], the generators
$a_{1}=b_{1},a_{2},a_{2}^{\prime},a_{3},a_{3}^{\prime},b_{2},b_{2}^{\prime},c_{2}$
of $A^{*}(\mathcal{B}_{4,g}^{\circ})$ satisfy no relations in codimension less
than $\frac{g+3}{4}-4$ (besides $a_{1}=b_{1}$). Since one can only obtain more
relations under restriction $A^{*}(\mathcal{H}_{4,g})\to
A^{*}(\mathcal{H}_{4,g}^{\circ})$, we have found all relations among CE
classes in degrees less than $\frac{g+3}{4}-4$. ∎

###  5.4. Presentation of the ring and stabilization

We use the code [9] compute the classes in (5.2). Let $I$ be the ideal they
generate in the $\mathbb{Q}$-algebra on the CE classes. It turns out that
modulo $I$, all CE classes are expressible in terms of
$a_{1},a_{2}^{\prime},a_{3}^{\prime}$. In particular,

(5.8)
$\mathbb{Q}[c_{2},a_{1},a_{2},a_{3},a_{2}^{\prime},a_{3}^{\prime},b_{2}^{\prime},b_{2}]/I\cong\mathbb{Q}[a_{1},a_{2}^{\prime},a_{3}^{\prime}]/\langle
r_{1},r_{2},r_{3},r_{4}\rangle,$

where

$\displaystyle r_{1}$
$\displaystyle=(2g^{3}+9g^{2}+10g)a_{1}^{3}-(8g^{2}+24g+8)a_{1}a_{3}^{\prime}$
$\displaystyle r_{2}$
$\displaystyle=(12g^{3}+42g^{2}+36g)a_{1}^{2}a_{2}^{\prime}-(22g^{3}+121g^{2}+187g+66)a_{1}a_{3}^{\prime}-(24g^{2}+24g)a_{2}^{\prime}a_{3}^{\prime}$
$\displaystyle r_{3}$
$\displaystyle=(432g^{3}+1512g^{2}+1296g)a_{1}a_{2}^{\prime
2}-(1450g^{3}+8001g^{2}+13115g+5442)a_{1}a_{3}^{\prime}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \
-(1584g^{3}+5544g^{2}+3936g)a_{2}^{\prime}a_{3}^{\prime}$ $\displaystyle
r_{4}$
$\displaystyle=(14344g^{6}+165692g^{5}+747682g^{4}+1636869g^{3}+1719009g^{2}+677844g-540)a_{1}^{2}a_{3}^{\prime}$
$\displaystyle-(17280g^{4}+112320g^{3}+224640g^{2}+129600g)a_{2}^{\prime
2}a_{3}^{\prime}+(352g^{5}+1440g^{4}+1448g^{3}+120g^{2})a_{3}^{\prime 2}.$

######  Remark 5.5.

In contrast with the degree $3$ case, brute force computations show that there
is no presentation of the Chow ring whose relations do not involve $g$.

######  Corollary 5.6.

Suppose $g\geq 2$.

  1. (1)

$R^{1}(\mathcal{H}_{4,g})$ is spanned by $\\{a_{1},a_{2}^{\prime}\\}$.

  2. (2)

$R^{2}(\mathcal{H}_{4,g})$ is spanned by
$\\{a_{1}^{2},a_{1}a_{2}^{\prime},a_{2}^{\prime 2},a_{3}^{\prime}\\}$.

  3. (3)

$R^{3}(\mathcal{H}_{4,g})$ is spanned by $\\{a_{1}a_{3}^{\prime},a_{2}^{\prime
3},a_{2}^{\prime}a_{3}^{\prime}\\}$.

  4. (4)

$R^{4}(\mathcal{H}_{4,g})$ is spanned by $\\{a_{2}^{\prime 4},a_{3}^{\prime
2}\\}$.

  5. (5)

For $i\geq 5$, $R^{i}(\mathcal{H}_{4,g})$ is spanned by $\\{a_{2}^{\prime
i}\\}$.

For $g>4i+12$, the spanning set of $R^{i}(\mathcal{H}_{4,g})$ given above is a
basis.

###### Proof.

Our code [9] verifies that the lists above are bases for
$\mathbb{Q}[a_{1},a_{2}^{\prime},a_{3}^{\prime}]/\langle
r_{1},r_{2},r_{3},r_{4}\rangle$ in degrees $i\leq 9$. In particular, for
$5\leq i\leq 10$, every monomial in $a_{1},a_{2}^{\prime},a_{3}^{\prime}$ of
degree $i$ is a multiple of $a_{2}^{\prime i}$. By inspection, $a_{2}^{\prime
i}$ is not in the ideal $\langle r_{1},r_{2},r_{3},r_{4}\rangle$ for any $i$,
so $a_{2}^{\prime i}$ is non-zero for all $i$. For $i\geq 11$, every monomial
of degree $i$ in $a_{1},a_{2}^{\prime},a_{3}^{\prime}$ is expressible as a
product of monomials having degrees between $5$ and $10$. It follows that
every monomial of degree $i\geq 11$ is a multiple of $a_{2}^{\prime i}$.

Proposition 5.4 states that $I$ provides all relations among the CE classes in
degrees less than $\frac{g+3}{4}-4$. That is, the left-hand side of (5.8) maps
to $R^{*}(\mathcal{H}_{4,g})$ isomorphically in degrees $i<\frac{g+3}{4}-4$.
Hence, a basis for the degree $i$ piece of
$\mathbb{Q}[a_{1},a_{2}^{\prime},a_{3}^{\prime}]/\langle
r_{1},r_{2},r_{3},r_{4}\rangle$ is a basis for $R^{i}(\mathcal{H}_{4,g})$ when
$i<\frac{g+3}{4}-4$, equivalently when $g>4i+12$. ∎

###### Proof of Theorem 1.1(2).

Consider the equation

$\frac{\mathbb{Q}[a_{1},a_{2}^{\prime},a_{3}^{\prime}]}{\langle
r_{1},r_{2},r_{3},r_{4}\rangle}\rightarrow R^{*}(\mathcal{H}_{4,g})\rightarrow
R^{*}(\mathcal{H}_{4,g}^{\circ})=A^{*}(\mathcal{H}_{4,g}^{\circ}).$

The first map exists and is surjective by Proposition 5.2 and the presentation
(5.8). Meanwhile, Lemma 5.4 establishes that the composition is an isomorphism
in degrees less than $\frac{g+3}{4}-4$. Therefore, the first map can have no
kernel in codimension less than $\frac{g+3}{4}-4$.

Finally, in [10, Lemmas 5.5 and 5.8], we showed that
$\mathcal{H}_{4,g}^{\circ}$ and $\mathcal{H}_{4,g}^{\mathrm{nf}}$ are “good
approximations of each other” in the sense that the codimension the complement
of
$\mathcal{H}^{\circ}_{4,g}\cap\mathcal{H}^{\mathrm{nf}}_{4,g}\subset\mathcal{H}^{\mathrm{nf}}_{4,g}$
and of
$\mathcal{H}^{\circ}_{4,g}\cap\mathcal{H}^{\mathrm{nf}}_{4,g}\subset\mathcal{H}^{\circ}_{4,g}$
are both at least $\frac{g+3}{4}-4$. Therefore, by excision there is an
isomorphism $A^{i}(\mathcal{H}_{4,g}^{\circ})\cong
A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})$. In particular, we have $\dim
A^{i}(\mathcal{H}_{4,g}^{\mathrm{nf}})=\dim
A^{i}(\mathcal{H}_{4,g}^{\circ})=\dim R^{i}(\mathcal{H}_{4,g})$. The
calculation of $\dim R^{i}(\mathcal{H}_{4,g})$ follows from Corollary 5.6. ∎

##  6\. The Chow ring in degree $5$

###  6.1. Set up

We begin by recalling the linear algebraic data associated to degree $5$
covers, as developed by Casnati [12]. For more details in our context, see
[10, Section 3.3]. To a degree $5$, cover $\alpha:C\to\mathbb{P}^{1}$, we
again associate two vector bundles on $\mathbb{P}^{1}$:

$E_{\alpha}:=(\alpha_{*}\mathcal{O}_{C}/\mathcal{O}_{\mathbb{P}^{1}})^{\vee}=\ker(\alpha_{*}\omega_{\alpha}\to\mathcal{O}_{\mathbb{P}^{1}})\qquad\text{and}\qquad
F_{\alpha}:=\ker(\operatorname{Sym}^{2}E_{\alpha}\to\alpha_{*}\omega_{\alpha}^{\otimes
2}).$

If $C$ has genus $g$, then $E_{\alpha}$ has degree $g+4$, and rank $4$, while
$F_{\alpha}$ has degree $2g+8$ and rank $5$. Geometrically, the curve $C$ is
embedded in $\gamma:\mathbb{P}(E^{\vee}_{\alpha}\otimes\det
E_{\alpha})\rightarrow\mathbb{P}^{1}$, which further maps to
$\mathbb{P}(\wedge^{2}F_{\alpha})$ via an associated section

$\eta\in H^{0}(\mathbb{P}^{1},\mathcal{H}om(E_{\alpha}^{\vee}\otimes\det
E_{\alpha},\wedge^{2}F_{\alpha})).$

The curve $C$ is obtained as the intersection of the image of
$\mathbb{P}(E^{\vee}_{\alpha}\otimes\det E_{\alpha})$ with the Grassmann
bundle $G(2,F_{\alpha})\subset\mathbb{P}(\wedge^{2}F_{\alpha})$.

Conversely, suppose we are given a rank $4$, degree $g+4$ vector bundle $E$
and a rank $5$, degree $2g+8$ vector bundle $F$ on $\mathbb{P}^{1}$. We write
$E^{\prime}:=E^{\vee}\otimes\det E$ and
$\gamma:\mathbb{P}E^{\prime}\to\mathbb{P}^{1}$. We characterize which sections
$\eta$ fail to produce a smooth degree $5$, genus $g$ cover. Let

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

######  Lemma 6.1.

Let $E$ and $F$ be as above, with $\mathcal{H}om(E^{\prime},\wedge^{2}F)$
globally generated. Suppose we have a map $\eta:E^{\prime}\to\wedge^{2}F$.

  1. (1)

If $\eta$ is not injective on fibers then the subscheme
$D(\Phi(\eta))\subset\mathbb{P}E^{\prime}$ cut by the $4\times 4$ Pfaffians of
$\Phi(\eta)$ is not smooth of dimension $1$.

  2. (2)

If $\eta:E^{\prime}\rightarrow\wedge^{2}F$ is injective on fibers, the
intersection $C=\eta(\mathbb{P}E^{\prime})\cap G(2,F)$ fails to be a smooth,
irreducible genus $g$, degree $5$ cover of $\mathbb{P}^{1}$ if and only if
there exists $p\in C$ so that $\dim T_{p}C\geq 2$.

###### Proof.

(1) Suppose $\eta(e_{1})=0$ for $e_{1}$ a vector in the fiber of $E^{\prime}$
over $0\in\mathbb{P}^{1}$, where $\mathbb{P}^{1}$ has coordinate $t$. We can
choose coordinates $X_{1},X_{2},X_{3},X_{4}$ on $\mathbb{P}E^{\prime}$ so that
$\mathrm{span}(e_{1})\in\mathbb{P}E^{\prime}|_{0}\subset\mathbb{P}E^{\prime}$
is defined by vanishing of $t$ and $X_{2},X_{3},X_{4}$. Since $\eta(e_{1})$
vanishes at $t=0$, all entries of a matrix representative $M_{\eta}$ for
$\Phi(\eta)$ as in [10, Equation 5.13] would have coefficient of $X_{1}$
divisible by $t$. In particular, the quadrics $Q_{i}$ that define the Pfaffian
locus $C=D(\Phi(\eta))$ of $\eta$ lie in the ideal
$(t)+(X_{2},X_{3},X_{4})^{2}$. Hence, $T_{p}C$ contains the entire vertical
tangent space of $\mathbb{P}E^{\prime}\to\mathbb{P}^{1}$, and therefore has
dimension at least $3$.

(2) If $\eta(\mathbb{P}E^{\prime})\cap G(2,F)\subset\mathbb{P}(\wedge^{2}F)$
is connected, or has a component of dimension $\geq 2$, then we are done, so
we suppose $\dim C=1$. The general fiber of $C$ over $\mathbb{P}^{1}$ consists
of $5$ points. The global generation of
$\mathcal{H}om(E^{\prime},\wedge^{2}F)$ implies all summands of $E$ have
positive degree, so $h^{0}(\mathbb{P}^{1},E^{\vee})=0$. Hence, if $C$ has the
right codimension in each fiber, then
$h^{0}(C,\mathcal{O}_{C})=h^{0}(\mathbb{P}^{1},E^{\vee})+1=1$ so $C$ is
connected.

Now suppose that $C$ has a component $C_{0}$ that is contained in a fiber. We
claim $C$ is connected (and thus has a two dimension tangent space at some
point on $C_{0}$). Suppose that the fiber over $x\in\mathbb{P}^{1}$ is the
union of a one dimensional component $C_{0}$ together with a finite scheme
$\Gamma$. The image $\eta(\mathbb{P}E^{\prime}|_{x})$ is the intersection of
six hyperplanes $H_{i}$ in the fiber
$\mathbb{P}(\wedge^{2}\mathcal{F})|_{x}\cong\mathbb{P}^{9}$. Thus the fiber of
$C$ over $x$ is the intersection of six hyperplanes $H_{i}$ and the
Grassmannian $G(2,F|_{x})$ in its Plücker embedding. Because the Plücker
embedding is nondegenerate, we can arrange it so that $H_{1}\cap\cdots\cap
H_{5}\cap G(2,F|_{x})$ has pure dimension $1$, i.e. the excess dimension
appears only after intersecting with $H_{6}$; see [15, Section 13.3.6] for a
similar argument due to Vogel.

To obtain the excess component $C_{0}$ in the final intersection, we must have
that

$H_{1}\cap\cdots\cap H_{5}\cap G(2,F|_{x})=C_{0}\cup\Phi$

with $C_{0}\subset H_{6}$. Note that the reducible curve $C_{0}\cup\Phi$ must
have degree $5=\deg G(2,F|_{x})$, so each component has degree at most $4$.
Therefore, the finite scheme $\Gamma=\Phi\cap H_{6}$ has degree at most $4$.
Because the general fiber of $C$ over $\mathbb{P}^{1}$ consists of a degree
five zero dimensional subscheme, it follows that some of the five points in
the general fiber must specialize into $C_{0}$, and the intersection
$\eta(\mathbb{P}E^{\prime})\cap G(2,F)$ is singular there. ∎

The association of $\alpha:C\rightarrow\mathbb{P}^{1}$ with the pair
$(E_{\alpha},F_{\alpha})$ gives rise to a map
$\mathcal{H}_{5,g}\rightarrow\mathcal{B}_{5,g}$, where $\mathcal{B}_{5,g}$ is
the moduli stack of pairs of vector bundles on $\mathbb{P}^{1}$-bundles, as
defined in [10, Definition 5.10]. Let $\pi:\mathcal{P}\to\mathcal{B}_{5,g}$ be
the universal $\mathbb{P}^{1}$-bundle and let $\mathcal{E}$ be the universal
rank $4$ vector bundle on $\mathcal{P}$. Continuing the notation of [10], let
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$ and define classes $a_{i},b_{i}\in
A^{i}(\mathcal{B}_{5,g})$ and $a_{i}^{\prime},b_{i}^{\prime}\in
A^{i-1}(\mathcal{B}_{5,g})$ by the formula

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

(Note that there is a “determinant compatibility condition” which implies
$2a_{1}=b_{1}$, see [10, p. 25].) We also define $c_{2}=-\pi_{*}(z^{3})\in
A^{2}(\mathcal{B}_{5,g})$, which is the pullback of the universal second Chern
class on $\operatorname{BSL}_{2}$.

By [10, Equation 5.10]

(6.1)
$a_{1},a_{2},a_{3},a_{4},a_{2}^{\prime},a_{3}^{\prime},a_{4}^{\prime},b_{2},b_{3},b_{4},b_{5},b_{2}^{\prime},b_{3}^{\prime},b_{4}^{\prime},b_{5}^{\prime},c_{2}$
generate $A^{*}(\mathcal{B}_{5,g})$ and satisfy $\displaystyle\text{no
relations in degrees up to $g+4$}.$

We call the pullbacks of $\mathcal{E}$ and $\mathcal{F}$ to
$\mathcal{H}_{5,g}$ the _CE bundles_ , just like in the degree $4$ case.
Similarly, the pullbacks of the classes appearing in 6.1 to
$\mathcal{H}_{5,g}$ are called the _CE classes_. By [10, Theorem 3.10], the CE
classes are tautological and generate the tautological ring.

In order to prove Theorem 1.1(3), we proceed in two steps, just like we did in
degree $4$. First, in Section 6.2, we construct a certain bundle of principal
parts, and it to find relations among the CE classes. In Section 6.3, we
define an open substack $\mathcal{H}^{\circ}_{5,g}\subset\mathcal{H}_{5,g}$,
which is an open substack of a vector bundle over
$\mathcal{B}^{\circ}_{5,g}\subset\mathcal{B}_{5,g}$, and use it to demonstrate
that we have found all relations in degrees up to roughly $g/5$. Just like in
degree $4$, the method is summarized by Figure 1, but this time there are no
factoring covers, so one can ignore the top row.

###  6.2. The construction of the bundle of principal parts and relations

In this section, we will perform a construction that starts with the data
$(\mathcal{P}\rightarrow B,\mathcal{E},\mathcal{F},\eta)$ associated to degree
$5$ covers and produces a vector bundle called
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$ whose
sections help us detect when the associated subscheme
$D(\Phi(\eta))\subset\mathbb{P}\mathcal{E}^{\prime}$ defined by the vanishing
of Pfaffians fails to be smooth of relative dimension $1$ over $B$. The
formation of this bundle commutes with base change. We will use this
construction to produce relations among CE classes in the Chow ring of
$\mathcal{H}_{5,g}$.

Suppose we are given the data $(\mathcal{P}\rightarrow
B,\mathcal{E},\mathcal{F},\eta)$ where $\mathcal{P}\rightarrow B$ is a
$\mathbb{P}^{1}$-bundle, $\mathcal{E}$ is a rank $4$ vector bundle on
$\mathcal{P}$, $\mathcal{F}$ is a rank $5$ vector bundle on $\mathcal{P}$, and
$\eta\in
H^{0}(\mathcal{P},\mathcal{H}om(\mathcal{E}^{\vee}\otimes\det\mathcal{E},\wedge^{2}\mathcal{F}))$.
Set $\mathcal{E}^{\prime}=\mathcal{E}^{\vee}\otimes\det\mathcal{E}$.
Furthermore, we will assume that
$\eta:\mathcal{E}^{\prime}\rightarrow\wedge^{2}\mathcal{F}$ is injective with
locally free cokernel. It thus induces an inclusion
$\mathbb{P}\eta:\mathbb{P}\mathcal{E}^{\prime}\rightarrow\mathbb{P}(\wedge^{2}\mathcal{F})$.

To set up this construction, let
$\mathcal{Y}:=G(2,\mathcal{F})\times_{\mathcal{P}}\mathbb{P}\mathcal{E}^{\prime}$
and let $p_{1}:\mathcal{Y}\to G(2,\mathcal{F})$ and
$p_{2}:\mathcal{Y}\to\mathbb{P}\mathcal{E}^{\prime}$ be the projection maps,
so we have the diagram below.

${\mathcal{Y}}$${{G(2,\mathcal{F})}}$${{\small\mathbb{P}\mathcal{E}^{\prime}\times_{\mathcal{P}}\mathbb{P}(\wedge^{2}\mathcal{F})}}$${\mathbb{P}(\wedge^{2}\mathcal{F})}$${\mathbb{P}\mathcal{E}^{\prime}}$${\mathcal{P}}$${{B}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{i}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{\epsilon}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

These spaces come equipped with tautological sequences, which we label as
follows. On $G(2,\mathcal{F})$, we have an exact sequence

$0\rightarrow\mathcal{T}\rightarrow
i^{*}\epsilon^{*}\mathcal{F}\rightarrow\mathcal{R}\rightarrow 0,$

where $\mathcal{T}$ is rank $2$ and $\mathcal{R}$ is rank $3$. Meanwhile, on
$\mathbb{P}(\wedge^{2}\mathcal{F})$, we have an exact sequence

(6.2)
$0\rightarrow\mathcal{O}_{\mathbb{P}(\wedge^{2}\mathcal{F})}(-1)\rightarrow\epsilon^{*}(\wedge^{2}\mathcal{F})\rightarrow\mathcal{U}_{9}\rightarrow
0$

where $\mathcal{U}_{9}$ is the tautological rank $9$ quotient bundle. Noting
that the Plücker embedding satisfies
$i^{*}\mathcal{O}_{\mathbb{P}(\wedge^{2}\mathcal{F})}(-1)=\wedge^{2}\mathcal{T}$,
the restriction of (6.2) to $G(2,\mathcal{F})$ takes the form

(6.3) $0\rightarrow\wedge^{2}\mathcal{T}\rightarrow
i^{*}\epsilon^{*}(\wedge^{2}\mathcal{F})\rightarrow
i^{*}\mathcal{U}_{9}\rightarrow 0.$

It follows that the map
$i^{*}\epsilon^{*}(\wedge^{2}\mathcal{F})\rightarrow\wedge^{2}\mathcal{R}$
descends to a map

(6.4) $i^{*}\mathcal{U}_{9}\to\wedge^{2}\mathcal{R}.$

######  Remark 6.2.

The tensor product of (6.4) with
$i^{*}\mathcal{O}_{\mathbb{P}(\wedge^{2}\mathcal{F})}(1)$ is the natural map
from the restriction of the tangent bundle to the normal bundle,
$i^{*}T_{\mathbb{P}(\wedge^{2}\mathcal{F})}\to
N_{G(2,\mathcal{F})/\mathbb{P}(\wedge^{2}\mathcal{F})}$.

We define

$\mathcal{W}:=\mathcal{H}om(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(-1),\gamma^{*}(\wedge^{2}\mathcal{F}))=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1)\otimes\gamma^{*}(\wedge^{2}\mathcal{F})\otimes\det\mathcal{E},$

which is a rank $10$ vector bundle on $\mathbb{P}\mathcal{E}^{\prime}$. The
composition

$\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(-1)\to\gamma^{*}\mathcal{E}^{\prime}\xrightarrow{\gamma^{*}\eta}\gamma^{*}(\wedge^{2}\mathcal{F})$

defines a section $\delta$ of $\mathcal{W}$. Pulling back to
$\mathbb{P}\mathcal{E}^{\prime}\times_{\mathcal{P}}\mathbb{P}(\wedge^{2}\mathcal{F})$,
consider the further composition

(6.5) $q_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(-1)\to
q_{2}^{*}\gamma^{*}\mathcal{E}^{\prime}\xrightarrow{q_{1}^{*}\epsilon^{*}\eta}q_{1}^{*}\epsilon^{*}(\wedge^{2}\mathcal{F})\rightarrow
q_{1}^{*}\mathcal{U}_{9}.$

The vanishing locus of this composition is precisely the graph of
$\mathbb{P}\eta$ inside
$\mathbb{P}\mathcal{E}^{\prime}\times_{\mathcal{P}}\mathbb{P}(\wedge^{2}\mathcal{F})$.
Restricting (6.5) to $\mathcal{Y}$, we obtain a section, which we call
$\overline{\delta}$, of the rank $9$ vector bundle

$\mathcal{W}^{\prime}:=\mathcal{H}om(p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(-1),p_{1}^{*}i^{*}\mathcal{U}_{9}).$

The vanishing $V(\overline{\delta})\subset\mathcal{Y}$ is the intersection of
the graph of $\mathbb{P}\eta$ with $\mathcal{Y}$ and is therefore identified
with the intersection
$G(2,\mathcal{F})\cap\mathbb{P}\eta(\mathbb{P}\mathcal{E}^{\prime})$. Viewed
inside $\mathbb{P}\mathcal{E}^{\prime}$, this intersection is equal to the
desired associated subscheme
$D(\Phi(\eta))\subset\mathbb{P}\mathcal{E}^{\prime}$.

######  Remark 6.3.

The subscheme $D(\Phi(\eta))\subseteq\mathbb{P}\mathcal{E}^{\prime}$ is not in
general the zero locus of a section of a vector bundle. However, we have found
how to realize this scheme as the zero locus of a section of a vector bundle
on $\mathcal{Y}$, basically by using the fact that the graph of
$\mathbb{P}\eta$ is defined by the zero locus of a section of a vector bundle.

Next, we are going to construct a certain restricted principal parts bundle
from $\mathcal{W}^{\prime}$ that will detect when fibers of
$\mathcal{C}=V(\overline{\delta})\to B$ have vertical tangent space of
dimension $2$ or more. Before giving the construction, let us describe the
geometric picture on a single fiber $\mathbb{P}^{1}$ of $\mathcal{P}\to B$.
Let $E$ and $F$ be vector bundles on $\mathbb{P}^{1}$ of ranks $4$ and $5$
respectively and suppose $\eta:E^{\prime}\to\wedge^{2}F$ is an injection of
vector bundles with locally free cokernel. Let $p\in\mathbb{P}E^{\prime}$. The
intersection $G(2,F)\cap\eta(\mathbb{P}E^{\prime})$ has a two dimensional
tangent space at $\eta(p)\in G(2,F)$ if and only if there exists a two
dimensional subspace $S\subset T_{p}\mathbb{P}E^{\prime}$ such that the
differential of the projectivization of $\eta$ sends $S$ into
$T_{q}G(2,F)\subset T_{q}\mathbb{P}(\wedge^{2}F)$. Equivalently, the
composition $S\subset
T_{p}\mathbb{P}E^{\prime}\xrightarrow{\mathrm{d}\mathbb{P}\eta}T_{\eta(p)}\mathbb{P}(\wedge^{2}F)\to
N_{G(2,F)/\mathbb{P}(\wedge^{2}F)}|_{\eta(p)}$ is zero (see Figure 2).

Figure 2. Does $\mathrm{d}\mathbb{P}\eta$ send $S$ into $T_{\eta(p)}G$?

First consider
$Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$ (see
Definition 3.9), which comes equipped with a filtration

(6.6) $0\rightarrow
p_{2}^{*}\Omega_{\mathbb{P}\mathcal{E}^{\prime}/B}\otimes\mathcal{W}^{\prime}\to
Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})\rightarrow\mathcal{W}^{\prime}\rightarrow
0.$

Given any section $\delta$ of $\mathcal{W}$, there is an induced section of
$Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$, which
records the values and first order changes of the induced section
$\overline{\delta}$ of $\mathcal{W}^{\prime}$ as we move across
$\mathbb{P}\mathcal{E}^{\prime}$. Now let
$\widetilde{\mathcal{Y}}:=G(2,p_{2}^{*}T_{\mathbb{P}\mathcal{E}^{\prime}/B})\xrightarrow{a}\mathcal{Y}$,
which comes equipped with a tautological sequence

$0\rightarrow\Omega_{x}^{\vee}\rightarrow
a^{*}p_{2}^{*}T_{\mathbb{P}\mathcal{E}^{\prime}/B}\rightarrow\Omega_{y}^{\vee}\rightarrow
0,$

where $\Omega_{x}$ and $\Omega_{y}$ are both rank $2$. Dualizing the left map
gives

(6.7) $a^{*}p_{2}^{*}\Omega_{\mathbb{P}\mathcal{E}^{\prime}/B}\to\Omega_{x}.$

Meanwhile, tensoring the $p_{1}^{*}$ of (6.4) with
$p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)$, we have a quotient

(6.8) $\mathcal{W}^{\prime}\to
p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)\otimes
p_{1}^{*}(\wedge^{2}\mathcal{R}).$

######  Remark 6.4.

If one has an injection $\eta:\mathcal{E}^{\prime}\to\wedge^{2}\mathcal{F}$,
then one has an isomorphism of
$p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)$ with
$p_{1}^{*}i^{*}\mathcal{O}_{\mathbb{P}(\wedge^{2}\mathcal{F})}(1)$ on
$V(\overline{\delta})$ (coming from (6.5)). By Remark 6.2, the restriction of
(6.8) to $V(\overline{\delta})$ then agrees with the restriction of
$p_{1}^{*}i^{*}T_{\mathbb{P}(\wedge^{2}\mathcal{F})}\to
p_{1}^{*}N_{G(2,\mathcal{F})/\mathbb{P}(\wedge^{2}\mathcal{F})}$ to
$V(\overline{\delta})$. This was the geometric intuition behind the definition
we are about to make.

Pulling back (6.8) to $\widetilde{\mathcal{Y}}$ and tensoring with (6.7), we
obtain a quotient

(6.9)
$a^{*}(p_{2}^{*}\Omega_{\mathbb{P}\mathcal{E}^{\prime}/B}\otimes\mathcal{W}^{\prime})\to\Omega_{x}\otimes
a^{*}(p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)\otimes
p_{1}^{*}(\wedge^{2}\mathcal{R})).$

Note that the term on the left of (6.9) is the $a^{*}$ of the term on the left
of (6.6) (the “derivatives part” of the principal parts bundle). Let
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$ be the
quotient of
$a^{*}Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$ by the
kernel of (6.9). This bundle comes equipped with a filtration

(6.10) $0\rightarrow\Omega_{x}\otimes
a^{*}(p_{2}^{*}\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)\otimes
p_{1}^{*}(\wedge^{2}\mathcal{R}))\to
RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})\to\mathcal{W}^{\prime}\rightarrow
0$

and has rank $15$. The bundle
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$ remembers
derivatives just in the “$x$-directions” (i.e. along a distinguished
$2$-plane) and remembers their values under the quotient (6.8). Considering
Remark 6.4 and Figure 2, this is telling us to what extent vectors in the
subspace $S$ corresponding to “$x$-directions” leave $T_{\eta(p)}G(2,F)$. This
will be spelled out in local coordinates in the lemma below.

The global section $\overline{\delta}$ of $\mathcal{W}^{\prime}$ induces a
global section $\overline{\delta}^{\prime}$ of
$Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$, which in
turn gives rise to a global section $\overline{\delta}^{\prime\prime}$ of
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/B}(\mathcal{W}^{\prime})$. The
following lemma describes the geometric condition for such an induced section
to vanish at a geometric point of $\widetilde{\mathcal{Y}}$.

######  Lemma 6.5.

Let $E$ and $F$ be vector bundles on $\mathbb{P}^{1}$ of ranks $4$ and $5$
respectively. Let $Y=\mathbb{P}E^{\prime}\times_{\mathbb{P}^{1}}G(2,F)$ and
let $W$, $W^{\prime}$, $R$, $Q^{1}_{\mathbb{P}E^{\prime}}(W^{\prime})$ and
$RQ^{1}_{\mathbb{P}E^{\prime}}(W^{\prime})$ be defined analogously to the
constructions above (working over a point instead of $B$). Suppose
$\eta:E^{\prime}\to\wedge^{2}F$ is an injection of vector bundles. Then the
following are true:

  1. (1)

The induced section $\overline{\delta}$ of $W^{\prime}$ corresponding to
$\eta$ vanishes at $(p,q)\in Y$ if and only if the projectivization of $\eta$
sends $p$ to $q$.

  2. (2)

The induced section $\overline{\delta}^{\prime\prime}$ of
$RQ^{1}_{\mathbb{P}E^{\prime}}(W^{\prime})$ corresponding to $\eta$ vanishes
at $(p,q,S)\in\widetilde{Y}$ if and only if the differential of the
projectivization of $\eta$ sends the subspace $S\subset
T_{p}\mathbb{P}E^{\prime}$ into the subspace $T_{q}G(2,F)\subset
T_{q}\mathbb{P}(\wedge^{2}F)$.

Hence, given any family $(\mathcal{P}\to B,\mathcal{E},\mathcal{F},\eta)$, the
image of the vanishing of the induced section
$\overline{\delta}^{\prime\prime}$ of
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}}(\mathcal{W}^{\prime})$ is the locus
in $B$ over which fibers of $D(\Phi(\eta))\to B$ fail to be smooth of relative
dimension $1$.

###### Proof.

(1) Let $t$ be a coordinate on $\mathbb{P}^{1}$, and let
$p\in\mathbb{P}E^{\prime}$ and $q\in G(2,F)$ be points in the fiber over
$0\in\mathbb{P}^{1}$. To say $\eta$ sends $p$ to $q$ is to say that $\eta$
sends the subspace of $E^{\prime}|_{0}$ corresponding to $p$ into the subspace
of $\wedge^{2}F|_{0}$ corresponding to $q$. Hence, by the definition of the
tautological sequences, $\eta$ sends $p$ to $q$ if and only if the composition

$p_{2}^{*}\mathcal{O}_{\mathbb{P}E^{\prime}}(-1)\to
p_{2}^{*}\gamma^{*}E^{\prime}\to p_{1}^{*}i^{*}\epsilon^{*}(\wedge^{2}F)\to
p_{1}^{*}i^{*}U_{9}$

vanishes at $(p,q)$, which is to say $\overline{\delta}$ vanishes.

(2) Trivializing $E$ and $F$ over an open $0\in U\subset\mathbb{P}^{1}$, we
may choose a basis $e_{1},\ldots,e_{4}$ for $E$ so that
$p=\mathrm{span}(e_{1})$ and a basis $f_{1},\ldots,f_{5}$ for $F$ so that
$q=\mathrm{span}(f_{1}\wedge f_{2})$. Let $\eta_{k,ij}$ be the coefficient of
$f_{i}\wedge f_{j}$ in $\eta(e_{k})$, so $\eta_{k,ij}$ is a polynomial in $t$.
In these local coordinates, to say $\eta$ sends $p$ to $q$ is to say that
$\eta_{1,ij}|_{t=0}=0$ for $ij\neq 12$.

The map $p_{1}^{*}\wedge^{2}F\rightarrow\wedge^{2}R$ corresponds to projection
onto the span of $f_{3}\wedge f_{4},f_{3}\wedge f_{5},$ and $f_{4}\wedge
f_{5}$. If $\eta$ sends $p$ to $q$, then the induced section
$\overline{\delta}$ of $W^{\prime}$ already vanishes. Therefore, the value of
$\overline{\delta}^{\prime\prime}$ at $(p,q)$ lands in the subbundle
$p_{2}^{*}\Omega_{\mathbb{P}E^{\prime}/B}\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}E^{\prime}}(1)\otimes
p_{1}^{*}(\wedge^{2}R)\subset Q^{1}_{\mathbb{P}E^{\prime}}(W^{\prime})$. This
“value” of $\overline{\delta}^{\prime\prime}$ at $(p,q)$ records the first
order information of $\eta_{1,ij}$ for $ij=34,35,45$ as $p$ deforms.

First order deformations of $p$ are of the form
$\mathrm{span}(e_{1})\mapsto\mathrm{span}(e_{1}+\epsilon(ae_{2}+be_{3}+ce_{4}))|_{t=\epsilon
d}$, where $\epsilon^{2}=0$. Here, $a,b,c,d$ are coordinates on the tangent
space at $p$ ($a,b,c$ are vertical coordinates and $d$ is the horizontal
coordinate). The coefficient of $f_{i}\wedge f_{j}$ in
$\eta(e_{1}+\epsilon(ae_{2}+be_{3}+ce_{4}))|_{t=\epsilon d}$ is

(6.11)
$\eta_{1,ij}+\left(d\left.\left(\frac{d}{dt}\eta_{1,ij}\right)\right|_{t=0}+a\eta_{2,ij}|_{t=0}+b\eta_{3,ij}|_{t=0}+c\eta_{4,ij}|_{t=0}\right)\epsilon\qquad\text{for
$ij=34,35,45$}.$

Locally, $a\epsilon,b\epsilon,c\epsilon,d\epsilon$ are our basis for
$\Omega_{\mathbb{P}E}$ and $f_{i}\wedge f_{j}$ for $ij=34,35,45$ is our basis
for $\wedge^{2}R$. The “value” we wish to extract in the fiber of
$p_{2}^{*}\Omega_{\mathbb{P}E^{\prime}/B}\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}E^{\prime}}(1)\otimes p_{1}^{*}(\wedge^{2}R)$
over $(p,q)$ is the coefficients of $a\epsilon,b\epsilon,c\epsilon,$ and
$d\epsilon$ in (6.11) for $ij=34,35,45$.

Now suppose $\eta$ is injective on fibers, so $\mathbb{P}\eta$ is well-
defined. In particular, $\eta_{1,12}|_{t=0}\neq 0$. With respect to $a,b,c,d$
the differential of $\mathbb{P}\eta$, from
$T_{p}\mathbb{P}E^{\prime}\rightarrow T_{q}\mathbb{P}(\wedge^{2}F)$, is
represented by a $9\times 4$ matrix

(6.12)
$\frac{1}{\eta_{1,12}|_{t=0}}\left(\begin{matrix}\frac{d}{dt}\eta_{1,13}|_{t=0}&\eta_{2,13}|_{t=0}&\eta_{3,13}|_{t=0}&\eta_{4,13}|_{t=0}\\\
\frac{d}{dt}\eta_{1,14}|_{t=0}&\eta_{2,14}|_{t=0}&\eta_{3,14}|_{t=0}&\eta_{4,14}|_{t=0}\\\
\vdots&&&\vdots\\\
\frac{d}{dt}\eta_{1,45}|_{t=0}&\eta_{2,45}|_{t=0}&\eta_{3,45}|_{t=0}&\eta_{4,45}|_{t=0}\end{matrix}\right).$

The subspace $T_{q}G(2,F)\subset T_{q}\mathbb{P}(\wedge^{2}F)$ corresponds to
the first $6$ coordinates. (A first order deformation of $f_{1}\wedge f_{2}$
remains a pure wedge to first order if and only if the $f_{i}\wedge f_{j}$
with non-zero coefficient in the deformation have one of $i,j$ is equal to $1$
or $2$. See also Remark 6.4.) Thus, $\mathbb{P}\eta$ sends
$T_{p}\mathbb{P}E^{\prime}$ into $T_{q}G(2,F)$ if and only if the bottom three
rows of (6.12) vanish, which occurs if and only if the coefficients of
$a,b,c,d$ in (6.11) vanish. More generally, a tangent vector in
$T_{p}\mathbb{P}E^{\prime}$ is sent into $T_{q}G(2,F)$ if and only if (6.11)
vanishes (for $ij=34,35,45$) when the corresponding values of $a,b,c,d$ are
plugged in. Plugging in values for $a,b,c,d$ in a given two dimensional
subspace $S$ of $T_{p}\mathbb{P}E^{\prime}$ then corresponds to the “value” of
$\eta$ in $S^{\vee}\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}E^{\prime}}(1)\otimes p_{1}^{*}(\wedge^{2}R)$
over $(p,q)$. By the filtration (6.10), this “value” is zero if and only if
$\overline{\delta}^{\prime\prime}$ vanishes at $(p,q,S)\in\widetilde{X}$.

Since the formation of these (refined) principal parts bundles commutes with
base change, the claim regarding families follows. ∎

We now apply the above construction in the case $B=\mathcal{H}_{5,g}$ and
$\eta=\eta^{\mathrm{univ}}$, the section associated to the universal cover
$\mathcal{C}\to\mathcal{P}$. By Lemma 6.5 and the fact that the universal
curve $\mathcal{C}=V(\overline{\delta}^{\prime\prime})$ is smooth of relative
dimension $1$ over $\mathcal{H}_{5,g}$, the global section
$\overline{\delta}^{\prime\prime}$ of
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{H}_{5,g}}(\mathcal{W}^{\prime})$
is nowhere vanishing. We therefore have the following lemma, which gives a
source of relations among the CE classes on $\mathcal{H}_{5,g}$.

######  Lemma 6.6.

Let
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1)),\zeta=c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)),\sigma_{i}=c_{i}(\mathcal{R}),$
and $s_{i}=c_{i}(\Omega_{y}^{\vee})$. All classes of the form (some pullbacks
omitted for ease of notation):

$a_{*}p_{2*}\gamma_{*}\pi_{*}(c_{15}(RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{H}_{5,g}}(\mathcal{W}^{\prime}))\cdot
s_{1}^{l_{1}}s_{2}^{l_{2}}\sigma_{1}^{k_{1}}\sigma_{2}^{k_{2}}\sigma_{3}^{k_{3}}\zeta^{j}z^{i})$

are zero in $R^{*}(\mathcal{H}_{5,g})\subseteq A^{*}(\mathcal{H}_{5,g})$.

###  6.3. All relations in low codimension

We recall the construction of an open substack
$\mathcal{H}_{5,g}^{\circ}\subset\mathcal{H}_{5,g}$ and what we already know
about its Chow ring from [10]. We start with $\mathcal{B}_{5,g}$, the moduli
space of pairs of vector bundles $E$ of rank $4$, degree $g+4$ and $F$ of rank
$5$, degree $g+5$ on $\mathbb{P}^{1}$-bundles together with an isomorphism of
$\det E^{\otimes 2}$ and $\det F$ (see [10, Section 5.3]). Let $\mathcal{E}$
and $\mathcal{F}$ be the universal bundles on
$\pi:\mathcal{P}\rightarrow\mathcal{B}_{5,g}$ and let
$\gamma:\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ be the structure map.
Define
$\mathcal{U}_{5,g}:=\mathcal{H}om(\mathcal{E}^{\vee}\otimes\det\mathcal{E},\wedge^{2}\mathcal{F})$.
We consider an open substack
$\mathcal{B}^{\circ}_{5,g}\subset\mathcal{B}_{5,g}$, defined by a certain
positivity condition for the bundle $\mathcal{U}_{5,g}$

(6.13)
$\mathcal{B}^{\circ}_{5,g}:=\mathcal{B}_{5,g}\smallsetminus\operatorname{Supp}R^{1}\pi_{*}(\mathcal{U}_{5,g}(-2)).$

Let $\mathcal{H}^{\circ}_{5,g}$ denote the base change of
$\mathcal{H}_{5,g}\rightarrow\mathcal{B}_{5,g}$ along the open embedding
$\mathcal{B}_{5,g}^{\circ}\hookrightarrow\mathcal{B}_{5,g}$.

Over $\mathcal{B}^{\circ}_{5,g}$, we see that
$\mathcal{X}_{5,g}^{\circ}:=\pi_{*}\mathcal{U}_{5,g}|_{\mathcal{B}^{\circ}_{5,g}}$
is a vector bundle whose fibers correspond to sections of $\mathcal{U}_{5,g}$.
The open $\mathcal{H}_{5,g}^{\circ}$ is contained in the open
$\mathcal{H}_{5,g}^{\prime}$ of [10, Lemma 5.11], so that lemma implies
$\mathcal{H}^{\circ}_{5,g}\to\mathcal{B}_{5,g}^{\circ}$ factors through an
open embedding in $\mathcal{X}^{\circ}_{5,g}$. We define

$\Delta_{5,g}:=\mathcal{X}^{\circ}_{5,g}\smallsetminus\mathcal{H}^{\circ}_{5,g},$

represented in red in the middle column of Figure 1. Now we wish to use
excision to determine the Chow ring of $\mathcal{H}_{5,g}^{\circ}$ in degrees
up to $\frac{g+4}{5}-16$. We already understand
$A^{*}(\mathcal{X}^{\circ}_{5,g})\cong A^{*}(\mathcal{B}_{5,g}^{\circ})$ in
degrees up to $\frac{g+4}{5}-16$ by [10, Equation 5.11]. Lemma 6.1 says we
need to remove the locus of non-injective maps and the locus of injective maps
such that the induced intersection of $\mathbb{P}\mathcal{E}^{\prime}$ and
$G(2,\mathcal{F})$ has a singular point.

We begin by computing the relations obtained from removing the locus of non-
injective maps $\mathcal{E}^{\prime}\rightarrow\wedge^{2}\mathcal{F}$, i.e.
maps that drop rank along some point on $\mathcal{P}$. Consider the projective
bundle
$\gamma:\mathbb{P}\mathcal{E}^{\prime}\rightarrow\mathcal{P}\to\mathcal{B}_{5,g}^{\circ}$,
and let
$\mathcal{W}:=\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)\otimes\gamma^{*}(\wedge^{2}\mathcal{F})$.
We have that
$\gamma_{*}\mathcal{W}=\mathcal{H}om(\mathcal{E}^{\prime},\wedge^{2}\mathcal{F})=\mathcal{U}_{5,g}$,
so by the definition of $\mathcal{B}_{5,g}^{\circ}$ (see (6.13)) and Lemma
3.5, the map

(6.14)
$\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}=\gamma^{*}\pi^{*}\pi_{*}\gamma_{*}\mathcal{W}\to
P^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W})\qquad\text{is
surjective.}$

Composing with the surjection
$P^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W})\to\mathcal{W}$,
we obtain a surjection
$\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}\to\mathcal{W}$, whose kernel we
define to be $\widetilde{\mathcal{X}}^{\mathrm{ni}}$. The fiber of
$\widetilde{\mathcal{X}}^{\mathrm{ni}}$ at a point
$p\in\mathbb{P}\mathcal{E}^{\prime}$ corresponds to maps of
$\mathcal{E}^{\prime}\to\wedge^{2}\mathcal{F}$ (on the fiber over
$\pi(\gamma(p))$) whose kernel contains the subspace referred to by $p$.

We then have the following trapezoid diagram:

${\widetilde{\mathcal{X}}^{\mathrm{ni}}}$${{\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{\mathcal{X}_{5,g}^{\circ}}}$${\mathbb{P}\mathcal{E}^{\prime}}$${\mathcal{P}}$${{\mathcal{B}^{\circ}_{5,g}}}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

Thus, Lemma 2.1 yields:

######  Proposition 6.7.

The image of the pushforward map
$A_{*}(\widetilde{\mathcal{X}}^{\mathrm{ni}})\rightarrow
A_{*}(\mathcal{X}_{5,g})$ is equal to the ideal generated by

(6.15)
$\rho^{*}\pi_{*}\gamma_{*}(c_{10}(\mathcal{W}))\cdot\zeta^{j}z^{i}),\qquad
0\leq j\leq 3,\ 0\leq i\leq 1.$

where $\zeta=c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1))$ and
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1))$.

Next, we excise the locus of injective maps such that the induced intersection
of $\mathbb{P}\mathcal{E}^{\prime}$ and $G(2,\mathcal{F})$ has a singular
point. From the construction in Section 6.2 applied to the case
$B=\mathcal{B}^{\circ}_{5,g}$, we have a rank $15$ vector bundle
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}^{\circ}_{5,g}}(\mathcal{W}^{\prime})$
on $\widetilde{\mathcal{Y}}$, which comes equipped with a series of
surjections (see Lemma 3.10 for the first map; the second map comes from the
construction of
$RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}^{\circ}_{5,g}}(\mathcal{W}^{\prime})$,
which was made just after (6.9)):

(6.16)
$a^{*}p_{1}^{*}P^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W})\to
a^{*}Q^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W}^{\prime})\to
RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W}^{\prime}).$

Applying $a^{*}p_{2}^{*}$ to (6.14) and composing the result with (6.16), we
obtain a surjection

(6.17) $a^{*}p_{2}^{*}\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}\to
RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W}^{\prime}).$

Define $\widetilde{\Delta}_{5,g}$ to be the kernel of (6.17), so that we
obtain a trapezoid diagram:

${\widetilde{\Delta}_{5,g}}$${{\sigma^{*}p_{2}^{*}\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{p_{2}^{*}\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{\gamma^{*}\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{\pi^{*}\mathcal{X}_{5,g}^{\circ}}}$${{\mathcal{X}_{5,g}^{\circ}}}$${\widetilde{\mathcal{Y}}}$${\mathcal{Y}}$${\mathbb{P}\mathcal{E}^{\prime}}$${\mathcal{P}}$${{\mathcal{B}^{\circ}_{5,g}}}$$\scriptstyle{i}$$\scriptstyle{\rho^{\prime\prime}}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{a}$$\scriptstyle{p_{2}}$$\scriptstyle{\gamma}$$\scriptstyle{\pi}$

######  Lemma 6.8.

Let
$z=c_{1}(\mathcal{O}_{\mathcal{P}}(1)),\zeta=c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\prime}}(1)),\sigma_{i}=c_{i}(\mathcal{R}),$
and $s_{i}=c_{i}(\Omega_{y}^{\vee})$. The image of the push forward
$A_{*}(\widetilde{\Delta}_{5,g})\to A_{*}(\mathcal{X}_{5,g}^{\circ})$ is the
ideal generated by

(6.18)
$\rho^{*}a_{*}p_{2*}\gamma_{*}\pi_{*}(c_{15}(RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}^{\circ}_{5,g}}(\mathcal{W}^{\prime}))\cdot
s_{1}^{l_{1}}s_{2}^{l_{2}}\sigma_{1}^{k_{1}}\sigma_{2}^{k_{2}}\sigma_{3}^{k_{3}}\zeta^{j}z^{i})$

for $0\leq j\leq 3$, $0\leq i\leq 1$, $0\leq l_{1},l_{2}\leq 2$ with
$l_{1}+l_{2}\leq 2$, and $0\leq k_{1},k_{2},k_{3}\leq 2$ with
$k_{1}+k_{2}+k_{3}\leq 2$.

###### Proof.

The monomials
$s_{1}^{l_{1}}s_{2}^{l_{2}}\sigma_{1}^{k_{1}}\sigma_{2}^{k_{2}}\sigma_{3}^{k_{3}}\zeta^{j}z^{i}$
with exponents satisfying the inequalities in the statement of the lemma
generate $A^{*}(\widetilde{\mathcal{Y}})$ as an
$A^{*}(\mathcal{B}_{5,g}^{\circ})$ module (see the last paragraph of Section
2.1). The result now follows from the Trapezoid Lemma 2.1. In codimension $1$,
for example, since $\widetilde{\Delta}_{5,g}\to\Delta_{5,g}$ is generically
one-to-one, we see

(6.19) $[\Delta_{5,g}]=\rho^{*}(\pi\circ\gamma\circ p_{2}\circ
a)_{*}(c_{15}(RQ^{1}_{\mathbb{P}\mathcal{E}^{\prime}/\mathcal{B}_{5,g}^{\circ}}(\mathcal{W}^{\prime})))=(10g+36)a_{1}-7a_{2}^{\prime}-b_{2}^{\prime}.\qed$

######  Lemma 6.9.

Let $I$ be the ideal generated by the classes in (6.15) and (6.18). Then
$A^{*}(\mathcal{H}_{5,g}^{\circ})=A^{*}(\mathcal{B}_{5,g}^{\circ})/I$. In
fact, $I$ is generated by the classes in (6.18), so Lemma 6.6 determines all
relations among $CE$ classes in codimension up to $\frac{g+4}{5}-16$.

###### Proof.

By Lemmas 6.5 and 6.1, we have that $\Delta_{5,g}$ is the union of the image
of $\widetilde{\Delta}_{5,g}$ in $\mathcal{X}_{5,g}^{\circ}$ with the image of
$\widetilde{\mathcal{X}}^{\mathrm{ni}}$ in $\mathcal{X}_{5,g}^{\circ}$. The
first claim now follows from excision, the fact that push forward is
surjective with rational coefficients, and Lemmas 6.7 and 6.8.

Meanwhile, direct computation [9] shows that $I$ is generated by the classes
in (6.18). Since $\rho$ is flat, the classes in (6.18) equal the classes of
Lemma 6.6. Next, [10, Equation 5.11] says that our generators on
$\mathcal{B}_{5,g}^{\circ}$ satisfy no relations in codimension less than
$\frac{g+4}{5}-16$. Thus, we have determined all relations among CE classes in
codimension up to $\frac{g+4}{5}-16$ ∎

###  6.4. Presentation of the ring and stabilization

Modulo the relations in Lemma 6.9, it turns out $R^{*}(\mathcal{H}_{5,g})$ is
generated by $a_{1},a_{2}^{\prime}\in R^{1}(\mathcal{H}_{5,g})$ and
$a_{2},c_{2}\in R^{2}(\mathcal{H}_{5,g})$, as we now explain. Let $I$ be the
ideal generated by the classes in (6.15) and (6.18) in the
$\mathbb{Q}$-algebra on the CE classes. Using Macaulay, we determined a
simplified presentation

(6.20)
$\mathbb{Q}[c_{2},a_{1},\ldots,a_{4},a_{2}^{\prime},\ldots,a_{4}^{\prime},b_{2},\ldots,b_{5},b_{2}^{\prime},\ldots,b_{5}^{\prime}]/I\cong\mathbb{Q}[a_{1},a_{2}^{\prime},a_{2},c_{2}]/\langle
r_{1},r_{2},r_{3},r_{4},r_{5}\rangle,$

where

$\displaystyle r_{1}$
$\displaystyle=(1064g+3610)a_{1}^{3}-1074a_{1}^{2}a_{2}^{\prime}+(-2148g-7272)a_{1}a_{2}+2160a_{2}a_{2}^{\prime}+$
$\displaystyle\qquad+(-1064g^{3}-10830g^{2}-36680g-41360)a_{1}c_{2}+(1074g^{2}+7272g+12288)a_{2}^{\prime}c_{2}$
$\displaystyle r_{2}$
$\displaystyle=(-6412g-21255)a_{1}^{3}+6207a_{1}^{2}a_{2}^{\prime}+(12414g+40896)a_{1}a_{2}+(-11880)a_{2}a_{2}^{\prime}+$
$\displaystyle\qquad+(6412g^{3}+63765g^{2}+211540g+234480)a_{1}c_{2}+(-6207g^{2}-40896g-68184)a_{2}^{\prime}c_{2}$
$\displaystyle r_{3}$