# Effective divisors on projectivized Hodge
bundles and Modular Forms

Gerard van der Geer  and  Alexis Kouvidakis

###### Abstract.

We construct vector-valued modular forms on moduli spaces of curves and
abelian varieties using effective divisors in projectivized Hodge bundles over
moduli of curves. Cycle relations tell us the weight of these modular forms.
In particular we construct basic modular forms for genus $2$ and $3$. We also
discuss modular forms on the moduli of hyperelliptic curves. In that case the
relative canonical bundle is a pull back of a line bundle on a
${\mathbb{P}}^{1}$-bundle over the moduli of hyperelliptic curves and we
extend that line bundle to a compactification so that its push down is (close
to) the Hodge bundle and use this to construct modular forms. In an appendix
we use our method to calculate divisor classes in the dual projectivized
$k$-Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-
Zograf.

##  1\. Introduction

Moduli spaces of curves and of abelian varieties come with a natural vector
bundle, the Hodge bundle ${\mathbb{E}}$. Starting from this vector bundle one
can construct other natural vector bundles by applying Schur functors, like
${\rm Sym}^{n}({\mathbb{E}})$ or $\det({\mathbb{E}})^{\otimes m}$. Sections of
such bundles are called modular forms. For example, for the moduli space
$\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g$
these are Siegel modular forms, and for the moduli space $\mathcal{M}_{g}$ of
curves of genus $g$ these are Teichmüller modular forms. If the Schur functor
corresponds to an irreducible representation $\rho$ we say that a section of
${\mathbb{E}}_{\rho}$ is a modular form of weight $\rho$. The Hodge bundle
extends to appropriate compactifications of such moduli spaces and in many
cases the sections also extend automatically to the compactifications, e.g.
for $\mathcal{A}_{g}$ with $g\geq 2$ by the so-called Koecher principle.

In this paper we try to construct modular forms in a geometric way. It is
well-known that an effective divisor on $\mathcal{A}_{g}$ or on
$\overline{\mathcal{M}}_{g}$ with $g\geq 2$ representing the cycle class
$m\lambda$ with $\lambda=c_{1}(\det({\mathbb{E}}))$ and
$m\in{\mathbb{Z}}_{>0}$ yields a scalar-valued modular form of weight $m$,
that is, a section of $\det({\mathbb{E}})^{\otimes m}$. We will exploit
explicit effective divisors on projectivized vector bundles to construct
vector-valued modular forms. In particular, we will construct in this way
certain modular forms that play a pivotal role in low genera.

For example, in the case of $g=2$ there is the modular form $\chi_{6,8}$, a
section of ${\rm Sym}^{6}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$, that
appeared in [5] as follows. Recall that the Torelli morphism
$\mathcal{M}_{2}\hookrightarrow\mathcal{A}_{2}$ has dense image and we have an
equality of standard compactifications
$\overline{\mathcal{M}}_{2}=\tilde{\mathcal{A}}_{2}$. The moduli space
$\mathcal{M}_{2}$ has another description as a stack quotient. This derives
from the fact that a smooth complete curve of genus $2$ over a field $k$ of
characteristic not $2$ is a double cover of ${\mathbb{P}}^{1}$ ramified at six
points, so can be given as $y^{2}=f$ with $f$ a polynomial of degree $6$ with
non-vanishing discriminant. Writing $f$ as a homogeneous polynomial in two
variables, say $f\in{\rm Sym}^{6}(W)$ with $W$ the $k$-vector space generated
by $x_{1},x_{2}$, and observing that we may change the basis of $W$, we find a
presentation of $\mathcal{M}_{2}$ as a stack quotient

$\mathcal{M}_{2}\sim[W_{6,-2}^{0}/{\rm GL}(W)],$

where we write $W_{a,b}$ for the ${\rm GL}(W)$-representation ${\rm
Sym}^{a}(W)\otimes\det(W)^{b}$. Here the space $W_{6,-2}$ can be seen as the
vector space of binary sextics $f$ with an action of ${\rm GL}(W)$ by

$f(x_{1},x_{2})\mapsto(ad-bc)^{-2}f(ax_{1}+bx_{2},cx_{1}+dx_{2})$

for a matrix $\left(\begin{smallmatrix}a&b\\\ c&d\\\
\end{smallmatrix}\right)\in{\rm GL}(2)$. The subspace $W^{0}_{6,-2}$ of
$W_{6,-2}$ is the space of $f$ with non-vanishing discriminant. The twisting
by $\det(W)^{-2}$ is required to get the right stabilizer for the generic $f$,
namely $\pm{\rm Id}_{W}$.

This interpretation of $\mathcal{M}_{2}$ was used in [5] to construct vector-
valued Siegel modular forms of degree $2$ by using invariant theory of binary
sextics, thus extending and simplifying the description of scalar-valued
Siegel modular forms by invariants by Igusa [18, 19]. Covariants define
vector-valued modular forms and all Siegel modular forms of degree $2$ on
$\mathcal{A}_{2}$ can be constructed this way. In [5] it was shown that the
most basic covariant, the universal binary sextic, defines a meromorphic
Siegel modular form $\chi_{6,-2}$ of weight $(6,-2)$, that is, it defines a
meromorphic section of ${\rm
Sym}^{6}({\mathbb{E}})\otimes\det({\mathbb{E}})^{-2}$ on $\mathcal{A}_{2}$.
After multiplying $\chi_{6,-2}$ by Igusa’s cusp form $\chi_{10}$ one obtains
the holomorphic modular form $\chi_{6,8}$, the ‘first’ vector-valued Siegel
modular cusp form of degree $2$.

In the case of $g=3$, there is an analogous form $\chi_{4,0,8}$, a section of
${\rm Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$. Here it derives
from the description of the moduli space $\mathcal{M}_{3}^{nh}$ of non-
hyperelliptic curves of genus three as a stack quotient

$\mathcal{M}_{3}^{nh}\sim[W_{4,0,-1}^{0}/{\rm GL}(W)]\,,$

where $W$ is now of dimension $3$ and $W_{4,0,-1}^{0}\subset{\rm
Sym}^{4}(W)\otimes\det(W)^{-1}$ represents ternary quartics defining smooth
curves. In [6] this description led to the construction of a meromorphic
Teichmüller modular form $\chi_{4,0,-1}$ of weight $(4,0,-1)$ and a
(holomorphic) Siegel modular form $\chi_{4,0,8}$ of degree $3$ and weight
$(4,0,8)$. Also in this case all Teichmüller and Siegel modular forms of genus
$3$ on $\overline{\mathcal{M}}_{3}$ and $\mathcal{A}_{3}$ can be constructed
from these forms by invariant theory.

This paper arises from the desire to construct these basic forms and similar
forms in a geometric way. We use cycle relations for effective divisors (or
almost effective divisors) on the projectivized Hodge bundle to construct our
forms. It is based on the observation that an effective divisor $D$ on the
projectivized Hodge bundle ${\mathbb{P}}({\mathbb{E}})$ with cycle class

$[D]=[\mathcal{O}(j)]+k\,\lambda-\Delta$

with positive integers $j,k$ and $\Delta$ an effective boundary class gives
rise to a section of ${\rm
Sym}^{j}({\mathbb{E}})\otimes\det({\mathbb{E}})^{k}$ vanishing on boundary
divisors, that is, a modular form. This method produces the basic modular
forms $\chi_{6,8}$ and $\chi_{4,0,8}$ of degree $2$ and $3$ in an efficient
way.

This connection between divisors and modular forms can also be used in the
other direction, obtaining cycle classes for divisors on projectivized Hodge
bundles. We give some examples of this.

Another objective of this paper is to construct modular forms on moduli spaces
of hyperelliptic curves of genus $g$. For this we work with two descriptions
of the moduli, a description as a stack quotient and a description as a
Hurwitz space. The latter space $\mathcal{H}_{g,2}$ has as compactification
the space $\overline{\mathcal{H}}_{g,2}$ of admissible degree $2$ covers of
genus $g$. In the stack description modular forms pull back to covariants for
the action of ${\rm GL}(2)$ on the space of binary forms of degree $2g+2$.

In the Hurwitz space description the relative canonical bundle of the
universal curve over ${\mathcal{H}}_{g,2}$ can be viewed as the pull back of
$\mathcal{O}(g-1)$ from the trivial ${\mathbb{P}}^{1}$-bundle $P$ over
${\mathcal{H}}_{g,2}$ equipped with $2g+2$ non-intersecting sections. Using
the theory of admissible covers, $P$ is compactified to a space
$\overline{P}$, a fibration of rational stable curves with $2g+2$ marked
points over $\overline{\mathcal{H}}_{g,2}$, and we show that the line bundle
$\mathcal{O}(g-1)$ on $P$ extends to a line bundle on $\overline{P}$ with the
property that its push down to $\overline{\mathcal{H}}_{g,2}$ is close to the
Hodge bundle. This allows us to construct modular forms on
$\overline{\mathcal{H}}_{g,2}$.

When we consider projectivized bundles projectivization is meant in the
Grothendieck sense, so that for a vector space $V$ the projective space
${\mathbb{P}}(V)$ parametrizes hyperplanes in $V$.

In an appendix we apply a method used in this paper to calculate the classes
of certain divisors in the dual projectivized $k$-Hodge bundle that were
determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.

## Acknowledgements

We thank Fabien Cléry, Carel Faber and Gavril Farkas and the referee for
useful remarks.

##  2\. The Case of Genus Two

Let $k$ be a field of characteristic not $2$. We consider the moduli space
$\mathcal{M}_{2}$ of curves of genus $2$ over $k$. This is a Deligne-Mumford
stack and it carries a universal curve $\pi:\mathcal{C}\to\mathcal{M}_{2}$ of
genus $2$. The relative dualizing sheaf $\omega_{\pi}$ is base point free and
thus defines a morphism $\varphi:\mathcal{C}\to{\mathbb{P}}({\mathbb{E}})$.
For a curve $C$ the map $\varphi:C\to{\mathbb{P}}(\mathbb{E}_{C})$ associates
to a point the space of differentials vanishing in that point. We have a
commutative diagram with $u$ the natural morphism

$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\textstyle{{\mathbb{P}}({\mathbb{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\mathcal{M}_{2}}$

We let $\overline{\mathcal{M}}_{2}$ be the Deligne-Mumford compactification
and $\overline{\pi}:\overline{\mathcal{C}}\to\overline{\mathcal{M}}_{2}$ the
corresponding universal curve. However, the extension
$\omega_{\overline{\pi}}$ of $\omega_{\pi}$ does not define an extension of
the morphism $\varphi$ to ${\mathbb{P}}({\mathbb{E}})$ over the boundary
component $\Delta_{1}$ that parametrizes reducible curves.

We consider the branch divisor $D\subset{\mathbb{P}}({\mathbb{E}})$ of the
morphism $\varphi$. The divisor $D$ is of relative degree $6$ in the
${\mathbb{P}}^{1}$-bundle ${\mathbb{P}}({\mathbb{E}})$ over the base
$\mathcal{M}_{2}$. We define $\overline{D}$ to be the closure of $D$ in
${\mathbb{P}}({\mathbb{E}})$ over $\overline{\mathcal{M}}_{2}$. In the
rational Picard group of ${\mathbb{P}}({\mathbb{E}})$ we can write

$[\overline{D}]=[\mathcal{O}(6)]+u^{*}(A)$

with $A$ a class in the rational Picard group of $\overline{\mathcal{M}}_{2}$
and $u:{\mathbb{P}}({\mathbb{E}})\to\overline{\mathcal{M}}_{2}$ the natural
projection.

We want to determine $A$ in terms of the generators $\lambda$, $\delta_{0}$ of
the Picard group of $\overline{\mathcal{M}}_{2}$. We write $\lambda$ for the
first Chern class of ${\mathbb{E}}$ and $\delta_{1}$ (resp. $\delta_{0}$) for
the class of $\Delta_{1}$ (resp. $\Delta_{0}$) in the Picard group of the
stack $\overline{\mathcal{M}}_{2}$; here $\Delta_{0}$ is the boundary
component that parametrizes irreducible curves with a double point.

In order to do this we extend the morphism $\varphi$. It extends over a
Zariski open part of $\Delta_{0}$ since $\omega_{\pi}$ has no base points
there. However, over $\Delta_{i}$ with $i>0$ this system has base points. We
then use a base change as described in the appendix in Section 18. After a
base change we have in an open neighborhood $U_{i}$ of the generic point of
$\Delta_{i}$ a semi-stable family. If we take the base to be $1$-dimensional
we get a semi-stable family $\tilde{f}:\tilde{\mathcal{C}}\to\tilde{B}$ with
as central fibre a chain $C^{\prime}+R+C^{\prime\prime}$ with $R$ a
$(-2)$-curve and $C^{\prime}$ and $C^{\prime\prime}$ of genus $1$. The
extension $\varphi^{\prime}$ of $\varphi$ is given by $\omega_{\tilde{f}}(-R)$
with $\tilde{f}_{*}(\omega_{\tilde{f}}(-R))={\mathbb{E}}_{\tilde{B}}$ and the
morphism $\varphi^{\prime}:\tilde{C}\to{\mathbb{P}}({\mathbb{E}})$ contracts
$C^{\prime}$ and $C^{\prime\prime}$ and is of degree $2$ on $R$. We refer to
the appendix, Section 18 for the details. The morphism $\varphi^{\prime}$ has
the property that

${\varphi^{\prime}}^{*}(\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}_{\tilde{B}})}(1))=\omega_{\tilde{f}}(-R)\,.$

######  Proposition 2.1.

We have
$[\overline{D}]=6\,[\mathcal{O}(1)]+u^{*}(8\,\lambda-\delta_{0}-\delta_{1})$ .

###### Proof.

We write $[\overline{D}]=6\,[\mathcal{O}(1)]+u^{*}(A)$. We work with the above
two types of $1$-dimensional families $f:C\to B$. The morphism $\varphi$ is
ramified over $D$, and thus $\varphi^{\prime}$ is ramified over $\overline{D}$
and contracts $C^{\prime}$ and $C^{\prime\prime}$. We denote the ramification
divisor by $S$. We thus get (writing abusively line bundles and divisors for
the corresponding divisor classes)

$\omega_{\tilde{f}}={\varphi^{\prime}}^{*}\omega_{u}+S+2\,(C^{\prime}+C^{\prime\prime}),\quad{\varphi^{\prime}}^{*}\overline{D}/2=S+3\,(C^{\prime}+C^{\prime\prime})\,,$

where the first equation comes from adjunction
$\omega_{\tilde{f}}+C^{\prime}_{|C^{\prime}}=\mathcal{O}_{C^{\prime}}$ for
$C^{\prime}$ and similarly for $C^{\prime\prime}$, and the second one from
$C^{\prime}\cdot{\varphi^{\prime}}^{*}\overline{D}=0=C^{\prime\prime}\cdot{\varphi^{\prime}}^{*}\overline{D}$.
This gives

$\displaystyle\omega_{\tilde{f}}$
$\displaystyle={\varphi^{\prime}}^{*}(\omega_{u}+\overline{D}/2)-(C^{\prime}+C^{\prime\prime})$
$\displaystyle={\varphi^{\prime}}^{*}(\mathcal{O}(-2)+u^{*}(\lambda)+\mathcal{O}(3)+u^{*}(A/2))-(C^{\prime}+C^{\prime\prime})$
$\displaystyle={\varphi^{\prime}}^{*}(\mathcal{O}(1)+u^{*}(\lambda+A/2))-(C^{\prime}+C^{\prime\prime})$
$\displaystyle=\omega_{\tilde{f}}-R+\tilde{f}^{*}(\lambda+A/2)-(C^{\prime}+C^{\prime\prime})$
$\displaystyle=\omega_{\tilde{f}}+\tilde{f}^{*}(\lambda+A/2-b_{1})\,$

with $b_{1}$ the special point of $\tilde{B}$. This shows that
$A=-2\,\lambda+2\,b_{1}$. Because of the base change that we executed, we have
$2\,b_{1}=\delta_{1}$ and we obtain $A=-2\,\lambda+\delta_{1}$. Now we use the
well-known relation $10\,\lambda=\delta_{0}+2\,\delta_{1}$ (see [25]) and thus
get $A=-2\lambda+\delta_{1}=8\lambda-\delta_{0}-\delta_{1}$. ∎

######  Remark 2.2.

We indicate an alternative proof of this result in Remark 13.5.

An important remark is now that $u_{*}(\mathcal{O}(1))={\mathbb{E}}$ and
$u_{*}(\mathcal{O}(m))={\rm Sym}^{m}({\mathbb{E}})$ for $m\geq 1$. The divisor
$\overline{D}$ with

$[\overline{D}]=[\mathcal{O}(6)]+u^{*}(8\lambda-\delta_{0}-\delta_{1})$

is an effective divisor on ${\mathbb{P}}({\mathbb{E}})$. We apply $u_{*}$ to
the corresponding section $1$ of $\mathcal{O}(\overline{D})$. By Proposition
2.1 we see that we get a regular section $\chi_{6,8}$ of the vector bundle
${\rm Sym}^{6}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ over
$\overline{\mathcal{M}}_{2}$. Moreover, this section vanishes on the divisors
$\Delta_{0}$ and $\Delta_{1}$. Note that the Torelli map extends to an
isomorphism $\overline{\mathcal{M}}_{2}\cong\tilde{\mathcal{A}}_{2}$ with
$\tilde{\mathcal{A}}_{2}$ the standard smooth compactification of
$\mathcal{A}_{2}$. Therefore our section defines a Siegel modular form
$\chi_{6,8}$ of weight $(6,8)$ that is a cusp form.

######  Corollary 2.3.

Let $\overline{D}$ be the closure in ${\mathbb{P}}({\mathbb{E}})$ of the
branch divisor of the canonical map for the universal curve over
$\mathcal{M}_{2}$. The push forward $u_{*}(s)$, with $s$ the natural section
$1$ of $\mathcal{O}(\overline{D})$ on ${\mathbb{P}}({\mathbb{E}})$, defines a
Siegel modular cusp form $\chi_{6,8}$ of degree $2$ and weight $(6,8)$.

The relation $10\,\lambda=\delta_{0}+2\,\delta_{1}$ quoted above implies that
there exists a Siegel modular cusp form $\chi_{10}$ of degree $2$ and of
weight $10$ with divisor $\delta_{0}+2\,\delta_{1}$. The quotient
$\chi_{6,-2}:=\chi_{6,8}/\chi_{10}$ defines a meromorphic section of ${\rm
Sym}^{6}({\mathbb{E}})\otimes\det{\mathbb{E}}^{-2}$ that is regular outside
$\Delta_{1}$.

We now analyze the orders of vanishing along $\delta_{1}$ of $\chi_{6,8}$ and
$\chi_{6,-2}$. When identifying $\overline{\mathcal{M}}_{2}$ with
$\tilde{A}_{2}$ we also write $\mathcal{A}_{1,1}$ for $\delta_{1}$; it is the
locus of products of elliptic curves.

We analyze the orders by working locally on a family over a local base $B$
with central fibre a general point $b_{1}$ of the boundary divisor
$\Delta_{1}$. As we mentioned before, the map
$\varphi:{\mathcal{C}}\to{\mathbb{P}}({\mathbb{E}})$ defined over
${\mathcal{M}}_{2}$ does not extend to the whole $\overline{\mathcal{C}}$ over
$\overline{\mathcal{M}}_{2}$ due to the fact that the canonical system has
base points at the nodes of the curves over the boundary divisor $\Delta_{1}$.
On the other hand, by the theory of admissible covers, the ramification
divisor of the above map $\varphi$ extends to a divisor $S$ on
$\overline{\mathcal{C}}$ in a way that avoids the above nodal locus. Namely,
over $b_{1}\in\Delta_{1}$ the fibre is a nodal curve $C$ which is the union of
two elliptic curves $C_{1}$ and $C_{2}$ meeting at a point $p$. The
restriction of the ramification divisor on each component is the union of the
three –additional to $p$– ramification points of the system
$|\mathcal{O}(2p)|$. Therefore the extension of the map $\varphi$ is defined
on the ramification divisor $S$. The map $\varphi$ maps
$C_{1}\backslash{\\{p\\}}$ and $C_{2}\backslash{\\{p\\}}$ to two distinct
points $p_{1}$ and $p_{2}$ respectively which are defined as follows. The
fibre of ${\mathbb{P}}({\mathbb{E}})$ over $b_{1}$ can be identified with
${\mathbb{P}}(H^{0}(C,\omega_{C}))$. The points of $H^{0}(C,\omega_{C})$ have
the form $(s_{1},s_{2})$, with $s_{i}$ a section of
$H^{0}(C_{i},\mathcal{O}(p))$. Then the point $p_{1}$ corresponds to the
hyperplane $\\{(0,s_{2}),s_{2}\in H^{0}(C_{2},\mathcal{O}(p))\\}$ and the
point $p_{2}$ corresponds to the hyperplane $\\{(s_{1},0),s_{1}\in
H^{0}(C_{1},\mathcal{O}(p))\\}$.

The divisor $\overline{D}$, the image of the ramification divisor under the
extended map $\varphi$, splits then into six irreducible components denoted by
$D_{1},\ldots,D_{6}$. Over our local base $B$ we thus have the six local
sections $D_{i}$ ($i=1,\ldots,6$) of the family ${\mathbb{P}}({\mathbb{E}})\to
B$. By the above description of the extension of the map $\varphi$, we may
conclude that $D_{1},D_{2},D_{3}$ pass through $p_{1}$ and $D_{4},D_{5},D_{6}$
through $p_{2}$.

Lifting the sections $D_{i}$ locally to sections $\sigma_{i}$ of
${\mathbb{E}}$ and choosing a basis $e_{1}$, $e_{2}$ of ${\mathbb{E}}$ over
$B$ such that $e_{1}$ and $e_{2}$ determine $p_{1}$ and $p_{2}$ in the fibre
of ${\mathbb{P}}({\mathbb{E}})$ over $z=0$, we can write
$\sigma_{i}=a_{i}e_{1}+b_{i}e_{2}$ for $i=1,\ldots,6$. Then at $z=0$ the
functions $b_{1},b_{2},b_{3}$ and $a_{4},a_{5},a_{6}$ vanish, while
$a_{1},a_{2},a_{3},b_{4},b_{5},b_{6}$ do not vanish. Since by blowing up once
we can separate, we may assume that these sections vanish with order $1$ at
$z=0$. By construction the section $\chi$ of ${\rm
Sym}^{6}({\mathbb{E}})\otimes\det({\mathbb{E}})^{-2}$ is locally given by

$\frac{\sigma_{1}\cdots\sigma_{6}}{z}\,.$

We may write $\sigma_{1}\cdots\sigma_{6}$ as

$\displaystyle a_{1}\cdots
a_{6}\,e_{1}^{6}+(a_{1}a_{2}a_{3}a_{4}a_{5}b_{6}+\ldots+b_{1}a_{2}a_{3}a_{4}a_{5}a_{6})\,e_{1}^{5}e_{2}+$
$\displaystyle(a_{1}a_{2}a_{3}a_{4}b_{5}b_{6}+\ldots+b_{1}b_{2}a_{3}a_{4}a_{5}a_{6})\,e_{1}^{4}e_{2}^{2}+$
$\displaystyle(a_{1}a_{2}a_{3}b_{4}b_{5}b_{6}+\ldots+b_{1}b_{2}b_{3}a_{4}a_{5}a_{6})\,e_{1}^{3}e_{2}^{3}+\ldots+b_{1}\cdots
b_{6}\,e_{2}^{6}\,.$

The order at $z=0$ of the coefficient of $e_{1}^{i}e_{2}^{6-i}$ equals

$\min_{\\#\Lambda=i}(\\#\Lambda^{c}\cap\\{1,2,3\\}+\\#\Lambda\cap\\{4,5,6\\})$

with $\Lambda$ running though the subsets of $\\{1,\ldots,6\\}$ of cardinality
$i$ and $\Lambda^{c}$ denoting the complement. We find for these orders
$(3,2,1,0,1,2,3)$ for $i=0,\ldots,6$, hence for the section $\chi$ given by
$\sigma_{1}\cdots\sigma_{6}/z$ we find the orders $(2,1,0,-1,0,1,2)$.

######  Corollary 2.4.

The section $1$ of the line bundle $\mathcal{O}(\overline{D})$ on
${\mathbb{P}}({\mathbb{E}})$ over $\overline{\mathcal{M}}_{2}$ pushes down via
${\mathbb{P}}({\mathbb{E}})\to\overline{\mathcal{M}}_{2}$ to the meromorphic
modular form $\chi_{6,-2}$ on
$\overline{\mathcal{M}}_{2}=\tilde{\mathcal{A}}_{2}$. The orders of the seven
coordinates of $\chi_{6,-2}$ along $\mathcal{A}_{1,1}$ in $\mathcal{A}_{2}$
are $(2,1,0,-1,0,1,2)$.

These orders are in agreement with the result of [5] where $\chi_{6,-2}$ was
constructed by invariant theory and properties of modular forms were used to
determine these orders.

A different way to construct the form $\chi_{6,8}$ uses the so-called
Weierstrass divisor $W$ in the dual bundle:

$W:=\\{(C,\eta)\in{\mathbb{P}}({\mathbb{E}}^{\vee}):\text{\rm${\rm div}(\eta)$
contains a Weierstrass point}\\}$

over $\mathcal{M}_{2}$. Here $C$ denotes a curve of genus $2$ and $\eta$ a
differential form on $C$. We let $\overline{W}$ be the closure of $W$ over
$\overline{\mathcal{M}}_{2}$. We then have an identity due to Gheorghita [13,
Thm 1]

$[\overline{W}]=6\,[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)]+34\,\lambda-3\,\delta_{0}-5\,\delta_{1}\,,$

where we write $\lambda$ and $\delta_{i}$ for the pullback of $\lambda$ and
$\delta_{i}$ to ${\mathbb{P}}({\mathbb{E}}^{\vee})$. Now $\overline{W}$ is an
effective divisor and the push forward of the section $1$ of
$\mathcal{O}(\overline{W})$ is a section of ${\rm
Sym}^{6}({\mathbb{E}}^{\vee})\otimes\det({\mathbb{E}})^{34}\otimes\mathcal{O}(-3\delta_{0}-5\delta_{1})$.
For $g=2$ we have
${\mathbb{E}}^{\vee}\cong{\mathbb{E}}\otimes\det({\mathbb{E}})^{-1}$, hence
${\rm Sym}^{6}({\mathbb{E}}^{\vee})\cong{\rm
Sym}^{6}({\mathbb{E}})\otimes\det({\mathbb{E}})^{-6}$. This implies that under
the isomorphism of ${\mathbb{P}}^{1}$-bundles
${\mathbb{P}}({\mathbb{E}})\cong{\mathbb{P}}({\mathbb{E}}^{\vee})$ the
isomorphism identifies $[\overline{W}]$ with $[\overline{D}]$, and we get in
the dual bundle

$[\overline{W}]=6\,[\mathcal{O}(1)]+28\,\lambda-3\,\delta_{0}-5\,\delta_{1}=6\,[\mathcal{O}(1)]+8\,\lambda-\delta_{0}-\delta_{1}\,.$

Using push forward we find again a form of weight $(6,8)$ vanishing on
$\delta_{1}$ and $\delta_{0}$. Up to a multiplicative non-zero constant this
is $\chi_{6,8}$.

######  Remark 2.5.

The identity
$[\overline{W}]=6\,[\mathcal{O}(1)]+28\,\lambda-3\,\delta_{0}-5\,\delta_{1}$
implies that there exists a modular form of weight $(6,28)$ vanishing with
multiplicity $3$ on $\Delta_{0}$ and multiplicity $5$ on $\Delta_{1}$, but
this is (up to a multiplicative constant) the form $\chi_{10}^{2}\,\chi_{6,8}$
with $\chi_{10}$ the form of weight $10$ with divisor $\delta_{0}+2\delta_{1}$
that displays the relation $10\,\lambda=\delta_{0}+2\delta_{1}$.

##  3\. The class of the $k$-canonically embedded curve

For the calculation of the classes of effective divisors in
${\mathbb{P}}({\mathbb{E}})$ related to the canonical image of the universal
curve it is helpful to have (part of) the class of the closure of the
canonical image in ${\mathbb{P}}({\mathbb{E}})$ over
$\overline{\mathcal{M}}_{g}$. Without extra effort we can and will extend the
calculation to the case of the $k$-canonically embedded curve for $k\geq 1$.

We consider the universal family
$\pi:\overline{\mathcal{C}}_{g}\to\overline{\mathcal{M}}_{g}$. This comes with
a natural vector bundle ${\mathbb{E}}_{k}=\pi_{*}(\omega_{\pi}^{\otimes k})$
for $k\in{\mathbb{Z}}_{\geq 1}$ and for $k=1$ this is the Hodge bundle
${\mathbb{E}}_{1}={\mathbb{E}}$. We write
$u:{\mathbb{P}}({\mathbb{E}}_{k})\to\overline{\mathcal{M}}_{g}$ for the
natural map. For $k\geq 2$ the sheaf $\omega_{\pi}^{\otimes k}$ is base point
free for stable curves and the surjection
$\pi^{*}{\mathbb{E}}_{k}\to\omega_{\pi}^{\otimes k}$ defines a morphism
$\varphi_{k}:\overline{\mathcal{C}}_{g}\to{\mathbb{P}}({\mathbb{E}}_{k})$. For
$k=1$ the sheaf $\omega_{\pi}$ is base point free on
$\mathcal{M}_{g}\cup\Delta_{0}^{0}$, with $\Delta_{0}^{0}\subset\Delta_{0}$
the open locus with only disconnecting nodes, but it has base points on the
nodes lying over the generic points of the boundary components $\Delta_{i}$
for $i>0$. The appendix Section 18 describes the closure of the image over an
open neighborhood of the generic point of $\Delta_{i}$.

We denote by $\Gamma_{k}$ the image of $\varphi_{k}(\mathcal{C}_{g})$ over
$\mathcal{M}_{g}\cup\Delta_{0}^{0}$ and by $\overline{\Gamma}_{k}$ the closure
of $\Gamma_{k}$ in ${\mathbb{P}}({\mathbb{E}}_{k})$ over
$\overline{\mathcal{M}}_{g}$. We can write the class of $\Gamma_{k}$ as a
cycle on ${\mathbb{P}}({\mathbb{E}}_{k})$ as

$None$
$[\overline{\Gamma}_{k}]=\sum_{i=0}^{r-2}h_{k}^{i}u^{*}(\beta_{r-2-i}),$

with $r={\rm rank}({\mathbb{E}}_{k})$,
$h_{k}=c_{1}(\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}_{k})}(1))$ and $\beta_{j}$
a codimension $j$ cycle on $\overline{\mathcal{M}}_{g}$.

######  Proposition 3.1.

We have $\beta_{0}=2\,k(g-1)$ and
$\beta_{1}=k^{2}\kappa_{1}-2k(g-1)c_{1}({\mathbb{E}}_{k})-\epsilon$ with
$\epsilon=\sum_{i=1}^{[g/2]}\delta_{i}$ for $k=1$ and $\epsilon=0$ else.

Using $c_{1}({\mathbb{E}}_{k})=\frac{k(k-1)}{2}\kappa_{1}+\lambda$ and
$\kappa_{1}=12\lambda-\sum_{i=0}^{[g/2]}\delta_{i}$ (by [24]) we can write
$\beta_{1}$ as

$\beta_{1}=\left((1-g)(12k^{3}-12k^{2}+2k)+12\,k^{2}\right)\,\lambda+\left((g-1)k-g\right)\,k^{2}\sum_{i=0}^{[g/2]}\delta_{i}-\epsilon\,.$

###### Proof.

We start with the case $k\geq 2$. In this case $\overline{\Gamma}_{k}$ is the
image of $\overline{\mathcal{C}}_{g}$ under $\varphi_{k}$ and
$\varphi_{k}^{*}(h_{k})=k\,\omega_{\pi}$. Since the image of the generic fibre
has degree $2k(g-1)$ we find $\beta_{0}=2k(g-1)$. The hyperplane class $h_{k}$
satisfies $\sum_{i=0}^{r}(-1)^{i}h_{k}^{i}c_{r-i}({\mathbb{E}}_{k})=0$. For
dimension reasons we have $u_{*}(h_{k}^{i})=0$ for $i\leq r-2$ and
$u_{*}(h_{k}^{r-1})=1$. Thus we get

$u_{*}({\varphi_{k}}_{*}[1]h_{k}^{2})=u_{*}{\varphi_{k}}_{*}(k^{2}\omega_{\pi}^{2})=k^{2}\pi_{*}(\omega_{\pi}^{2})=k^{2}\,\kappa_{1}\,,$

with $\pi_{*}(\omega_{\pi}^{2})=\kappa_{1}$, while on the other hand by (1)

$u_{*}(({\varphi_{k}}_{*}[1])h_{k}^{2})=(u_{*}h_{k}^{r})\beta_{0}+(u_{*}h_{k}^{r-1})\beta_{1}=2k(g-1)\,c_{1}({\mathbb{E}}_{k})+\beta_{1}\,,$

giving $\beta_{1}=k^{2}\kappa_{1}-2k(g-1)c_{1}({\mathbb{E}}_{k})$ and this
settles the case $k\geq 2$. The same argument works for $k=1$ as long as we
work on $\mathcal{M}_{g}\cup\Delta_{0}^{0}$. To get the coefficients of the
$\delta_{i}$ for $i>0$ we work over a $1$-dimensional base $B$ in an open
neighborhood $U_{i}$ of the generic point of $\Delta_{i}$ with special fibre
$C^{\prime}+R+C^{\prime\prime}$ as in the appendix Section 18 where the
extension $\varphi^{\prime}$ of $\varphi$ is defined by
$\omega_{\pi^{\prime}}(-R)$. The contribution of $\delta_{i}$ to $2\beta_{1}$
is now $\pi^{\prime}_{*}(\omega(-R)^{2})$, where the coefficient $2$ of
$\beta_{1}$ comes from the fact that $\varphi^{\prime}$ is of degree $2$ on
$R$. We get
$\pi^{\prime}_{*}(\omega_{\pi^{\prime}}(-R)^{2})=\pi^{\prime}_{*}(\omega_{\pi^{\prime}}^{2})+\pi^{\prime}_{*}(R^{2})=-2\,\delta_{i}-2\,\delta_{i}=-4\,\delta_{i}$,
as the fibre has two singular points and $R^{2}=-2$. Putting everything
together results in the given formula. ∎

##  4\. The Case of Genus Three

Here there is no restriction on the characteristic. We consider the moduli
stack $\mathcal{M}_{3}$ of curves of genus $3$ over our field $k$ and the
universal curve $\pi:\mathcal{C}\to\mathcal{M}_{3}$. The canonical map defines
a morphism $\varphi:\mathcal{C}\to{\mathbb{P}}({\mathbb{E}})$ and we thus
obtain the image divisor $D$ in ${\mathbb{P}}({\mathbb{E}})$ over
$\mathcal{M}_{3}$. We have a commutative diagram

$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\textstyle{{\mathbb{P}}({\mathbb{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\mathcal{M}_{3}}$

We consider the closure $\overline{D}$ of $D$ in ${\mathbb{P}}({\mathbb{E}})$
over $\overline{\mathcal{M}}_{3}$. The canonical image of the generic curve is
a quartic curve. Thus we have a relation
$[\overline{D}]=[\mathcal{O}(4)]+u^{*}(A)$ in the rational Picard group of
${\mathbb{P}}({\mathbb{E}})$ with $A$ a divisor class on
$\overline{\mathcal{M}}_{3}$ given in Proposition 3.1.

######  Corollary 4.1.

We have
$[\overline{D}]=[\mathcal{O}(4)]+u^{*}(8\,\lambda-\delta_{0}-2\,\delta_{1})$.

The divisor $\overline{D}$ is effective over
$\overline{\mathcal{M}}_{3}-\Delta_{1}$. Because of the relation
$[\overline{D}]=[\mathcal{O}(4)]+u^{*}(8\,\lambda-\delta_{0}-2\,\delta_{1})$
the corresponding section $1$ of $\mathcal{O}(\overline{D})$ maps under
$u_{*}$ to a section $\psi$ of ${\rm
Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ that is regular outside
$\Delta_{1}$ and vanishes on $\Delta_{0}$. In view of the even powers ${\rm
Sym}^{4}$ and $8$, this section $\psi$ is invariant under the action of $-1$
on the fibres of ${\mathbb{E}}$. As the action of $-1$ defines the involution
of the double covering of stacks $\mathcal{M}_{3}\to\mathcal{A}_{3}$, the
section $\psi$ descends to a section $\chi_{4,0,8}$ of ${\rm
Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ on the image of
$\overline{\mathcal{M}}_{3}-\Delta_{1}$ under the Torelli morphism
$\overline{\mathcal{M}}_{3}\to\tilde{\mathcal{A}}_{3}$, with
$\tilde{\mathcal{A}}_{3}$ the standard second Voronoi compactification of
$\mathcal{A}_{3}$. Since the image of $\Delta_{1}$ in
$\tilde{\mathcal{A}}_{3}$ is of codimension $2$, the section $\chi_{4,0,8}$
extends to a regular section of ${\rm
Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ on all of
$\mathcal{A}_{3}$, and then by the Koecher Principle it extends to
$\tilde{\mathcal{A}}_{3}$. Thus it defines a regular Siegel modular cusp form
$\chi_{4,0,8}$ of degree $3$ and weight $(4,0,8)$.

######  Corollary 4.2.

Let $\overline{D}$ be the closure of the canonical curve over
$\mathcal{M}_{3}$ in ${\mathbb{P}}({\mathbb{E}})$ and $s$ the natural section
$1$ of $\mathcal{O}(\overline{D})$. Then $\chi_{4,0,8}=u_{*}(s)$ is a
Teichmüller modular form and it descends to a Siegel modular cusp form of
degree $3$ and weight $(4,0,8)$.

The class $\mathfrak{H}$ of the hyperelliptic locus
$\overline{\mathcal{H}}_{3}$ in $\overline{\mathcal{M}}_{3}$ satisfies ([15,
p. 140])

$None$ $\mathfrak{h}=9\,\lambda-\delta_{0}-3\,\delta_{1}\,.$

The relation (2) shows that there exists a scalar-valued Teichmüller modular
form $\chi_{9}$ of weight $9$ on $\overline{\mathcal{M}}_{3}$. Its square is
invariant under the action of $-1$ on the fibres of ${\mathbb{E}}$, hence
descends to a Siegel modular form of weight $18$. Up to a multiplicative
scalar this is Igusa’s modular form $\chi_{18}$.

If we divide $\chi_{4,0,8}$ by $\chi_{9}$ we obtain a meromorphic section of
${\rm Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{-1}$ on
$\overline{\mathcal{M}}_{3}$ that is regular on $\mathcal{M}_{3}$ outside the
hyperelliptic locus. This form was used in [6] to construct Teichmüller
modular forms and Siegel modular forms by invariant theory.

##  5\. Moduli of hyperelliptic curves as a stack quotient

In this section we discuss the stack quotient description of the moduli of
hyperelliptic curves. We consider hyperelliptic curves in characteristic not
$2$. A hyperelliptic curve of genus $g$ is a morphism
$\alpha:C\to{\mathbb{P}}^{1}$ of degree $2$ where $C$ is a smooth curve of
genus $g$. A morphism $a:\alpha\to\alpha^{\prime}$ between two hyperelliptic
curves is a commutative diagram

$\textstyle{{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}}$$\textstyle{{\mathbb{P}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{1}}$

A hyperelliptic curve $C$ of genus $g$ can be written as $y^{2}=f(x)$ with
$f\in{\mathbb{C}}[x]$ of degree $2g+2$. In fact, choosing a basis
$(x_{1},x_{2})$ of the $g^{1}_{2}$ defines the morphism $\alpha$. Let
$W=\langle x_{1},x_{2}\rangle$, a vector space (over our algebraically closed
base field) of dimension $2$, and $L=\alpha^{*}(O_{\mathbb{P}^{1}}(1))$. By
Riemann-Roch we have $\dim H^{0}(C,L^{g+1})=g+3$, while $\dim{\rm
Sym}^{g+1}(W)=g+2$, so we have a non-zero element $y\in H^{0}(C,L^{g+1})$
which is anti-invariant under the involution corresponding to $\alpha$. The
anti-invariant subspace of $H^{0}(C,L^{g+1})$ has dimension $1$. Then $y^{2}$
is invariant and lies in ${\rm Sym}^{2g+2}(W)$. Thus we find the equation
$y^{2}=f(x_{1},x_{2})$ with $f$ homogeneous of degree $2g+2$ and with non-zero
discriminant.

We have made two choices here: a generator $y$ of $H^{0}(C,L^{g+1})^{(-1)}$, a
space of dimension $1$, and a basis of $W$. We can change the choice of $y$
(by a non-zero scalar) and the choice of a basis of $W$ by
$\gamma=(a,b;c,d)\in{\rm GL}(W)$. The action of ${\rm GL}(W)$ is on the right
via

$f(x_{1},x_{2})\mapsto f(ax_{1}+bx_{2},cx_{1}+dx_{2})\,.$

If we let ${\rm GL}(W)$ act on $y$ by a power of the determinant, then this
action preserves the type of equation. In inhomogeneous form the action by
${\rm GL}(W)$ is by

$f\mapsto f(\frac{ax+b}{cx+d}),\quad y\mapsto y/(cx+d)^{g+1}\,,$

with the following effect on the equation:

$y^{2}=f(x)\mapsto y^{2}=(cx+d)^{2g+2}f(\frac{ax+b}{cx+d})\,.$

The last expression on the right-hand-side can be written as binary form of
degree $2g+2$.

The stabilizer of a generic $f\in{\rm Sym}^{2g+2}(W)$ is $\mu_{2g+2}$, the
roots of unity of order dividing $2g+2$. Since we want a stabilizer of order
$2$ for the generic element, we consider a twisted action: define the ${\rm
GL}(W)$-representation

$W_{a,b}={\rm Sym}^{a}(W)\otimes{\det}(W)^{\otimes b}\,.$

This can be identified with ${\rm Sym}^{a}(W)$ as a vector space, but the
action by ${\rm GL}(W)$ is different. Inside this space $W_{a,b}$ we have the
open subspace $W_{a,b}^{0}$ of homogeneous polynomials of degree $a$ with non-
zero discriminant. We now distinguish two cases.

Case 1. $g$ even. Here we consider the stack quotient

$[W_{2g+2,-g}^{0}/{\rm GL}(W)]\,.$

This stack quotient can be identified with the moduli stack $\mathcal{H}_{g}$
of hyperelliptic curves of genus $g$ for $g$ even. Indeed, the action of
$t\cdot{\rm Id}_{W}$ is (on inhomogeneous equations) by

$f\mapsto t^{-2g}f,\quad y\mapsto y/t^{g+1}\,,$

hence $y^{2}=f$ maps to $y^{2}=t^{2}\,f$, so that the stabilizer is $\mu_{2}$,
as required. Note also that the action of $-1\in{\rm GL}(W)$ is by
$y\mapsto-y$, so $y$ is an odd element. A basis of $H^{0}(C,\Omega^{1}_{C})$
is given by

$x^{i}dx/y,\quad(i=0,\ldots,g-1)\,.$

The action on $dx$ is by $(ad-bc)dx/(cx+d)^{2}$ resulting in the action on the
space of differentials by

$x^{i}dx/y\mapsto(ad-bc)(cx+d)^{g-1-i}(ax+b)^{i}\,dx/y\,.$

If we forget the twisted action on $y$, we can identify
$H^{0}(C,\Omega^{1}_{C})$ with $W_{g-1,1}$. But $y^{2}$ must be viewed as an
element of $W_{2g+2,-g}$, so the action of $t\,1_{W}$ on $y$ should be twisted
by $t^{-g}={\det}^{-g/2}$. We get

$H^{0}(C,\Omega_{C}^{1})\cong W_{g-1,(2-g)/2}\quad\text{\rm for $g$ even.}$

We see that under the identification $h:[W_{g+2,-g}^{0}/{\rm
GL}(W)]\buildrel{\sim}\over{\longrightarrow}\mathcal{H}_{g}$ the pullback
$h^{*}({\mathbb{E}})$ of the Hodge bundle ${\mathbb{E}}$ is the equivariant
bundle $W_{g-1,(2-g)/2}$. The action of $-1_{W}$ is by $-1$ on
$W_{g-1,(2-g)/2}$. We also observe
$h^{*}(\det({\mathbb{E}}))={\det}(W)^{g/2}$.

Case 2. $g$ odd. Here we take $W_{2g+2,-g+1}$.

######  Remark 5.1.

If we consider $W_{2g+2,r}$ then $r$ has to be even, since as above we later
view $y^{2}$ as an element of $W_{2g+2,r}$ and we need an action by
${\det}^{r/2}$ on $y$.

Here the stabilizer of a generic element is $\mu_{4}$. Now on inhomogeneous
equations the action is by

$f\mapsto t^{-2g+2}\,f,\quad y\mapsto y/t^{g+1}\,,$

hence $y^{2}=f$ maps to $y^{2}=t^{4}\,f$. Note that here $-1_{W}$ acts by
$f\mapsto f$ and $y\mapsto y$. But $\sqrt{-1}_{W}$ acts by $f\mapsto f$ and
$y\mapsto-y$. To get the right stack quotient with stabilizer of the generic
element of order $2$, we take

$[W^{0}_{2g+2,1-g}/({\rm GL}(W)/(\pm 1_{W}))]\,.$

The action on the differentials $x^{i}dx/y$ with $i=0,\ldots,g-1$ is by

$x^{i}dx/y\mapsto(ad-bc)^{(1-g)/2}(cx+d)^{g-1-i}(ax+b)^{i}\,dx/y\,,$

hence without twisting we get $H^{0}(C,\Omega_{C}^{1})=W_{g,1}$. Since we view
$y^{2}$ as element of $W_{g+3,1-g}$ we find under $h:[W^{0}_{2g+2,1-g}/({\rm
GL}(W)/(\pm 1_{W}))]\buildrel{\sim}\over{\longrightarrow}\mathcal{H}_{g}$ that
$h^{*}({\mathbb{E}})=W_{g-1,(3-g)/2}$. The element $\sqrt{-1}\,1_{W}$ acts on
the differentials as $(-1)^{(3-g)/2}(\sqrt{-1})^{g-1}=-1$.

We summarize.

######  Proposition 5.2.

Writing $W_{a,b}={\rm Sym}^{a}(W)\otimes\det(W)^{b}$ we have the
identification of stacks

$h^{-1}:\mathcal{H}_{g}\buildrel{\sim}\over{\longrightarrow}\begin{cases}[W_{2g+2,-g}^{0}/{\rm
GL}(W)]&g\quad\text{\rm even}\\\ [W^{0}_{2g+2,1-g}/({\rm GL}(W)/(\pm
1_{W}))]&g\quad\text{\rm odd,}\\\ \end{cases}$

and

$h^{*}({\mathbb{E}})\cong\begin{cases}W_{g-1,(2-g)/2}&g\quad\text{\rm even}\\\
W_{g-1,(3-g)/2}&g\quad\text{\rm odd,}\\\ \end{cases}\qquad
h^{*}(\det({\mathbb{E}}))=\begin{cases}\det(W)^{g/2}&g\quad\text{\rm even}\\\
\det(W)^{g}&g\quad\text{\rm odd.}\\\ \end{cases}$

For a somewhat different description see [3, Cor. 4.7, p. 654].

Recall that the moduli stack $\mathcal{H}_{g}$ has as compactification the
closure $\overline{\mathcal{H}}_{g}$ of $\mathcal{H}_{g}$ inside the moduli
stack $\overline{\mathcal{M}}_{g}$. The Picard group of $\mathcal{H}_{g}$ is
known by [3] to be finite cyclic for $g\geq 2$ of order $4g+2$ if $g$ is even
and $8g+4$ else. The rational Picard group of $\overline{\mathcal{H}}_{g}$ is
known by Cornalba (see [8]) to be free abelian of rank $g$ generated by
classes $\delta_{i}$ and $\zeta_{j}$ for $i=0,\ldots,\lfloor g/2\rfloor$ and
$j=1,\ldots\lfloor(g-1)/2\rfloor$. Cornalba gives also the first Chern class
$\lambda$ of the Hodge bundle ${\mathbb{E}}$ on $\overline{\mathcal{H}}_{g}$

$(8g+4)\,\lambda=g\,\delta_{0}+4\sum_{i=1}^{\lfloor
g/2\rfloor}i(g-i)\,\delta_{i}+2\,\sum_{i=1}^{\lfloor(g-1)/2\rfloor}(i+1)(g-i)\,\zeta_{i}\,,$

where the generic point of the divisor $\zeta_{i}$ has an admissible model
$C^{\prime}\cup C^{\prime\prime}$ with two nodes $C^{\prime}\cap
C^{\prime\prime}=\\{p,q\\}$ mapping to a union of two ${\mathbb{P}}^{1}$, with
$2i+2$ marked points on $C^{\prime}$, see Figure 1 in Section 7.

##  6\. Modular forms on the hyperelliptic locus of genus three

Let ${\mathbb{E}}$ be the Hodge bundle on $\overline{\mathcal{H}}_{3}$. By a
modular form of weight $k$ on $\overline{\mathcal{H}}_{3}$ we mean a section
of $\det({\mathbb{E}})^{\otimes k}$. The construction in the preceding section
shows that a modular form of weight $k$ on $\overline{\mathcal{H}}_{3}$ when
pulled back to the stack $[W^{0}_{2,0}/({\rm GL}(W)/\pm{\rm id}_{W})]$ gives
rise to an invariant of degree $3k/2$. Indeed, it defines a section of the
equivariant bundle $\det(W)^{3k}$ invariant under ${\rm SL}(W)$, but in view
of the fact that we divide by the action of ${\rm GL}(W)/(\pm{\rm id}_{W})$
this yields an invariant of degree $3k/2$.

Let $M_{k}(\Gamma_{3})=H^{0}(\mathcal{A}_{3},\det({\mathbb{E}})^{k})$ be the
space of Siegel modular forms of degree $3$ on $\Gamma_{3}={\rm
Sp}(6,{\mathbb{Z}})$. In [20] Igusa considered an exact sequence

$0\to
M_{k-18}(\Gamma_{3})\buildrel{\cdot\chi_{18}}\over{\longrightarrow}M_{k}(\Gamma_{3})\to
I_{3k/2}(2,8)$

with $I_{d}(2,8)$ the vector space of invariants of degree $d$ of binary
octics. We can interpret Igusa’s sequence in the following way. A Siegel
modular form of weight $k$ defines by restriction to the hyperelliptic locus a
modular form of weight $k$ on $\overline{\mathcal{H}}_{3}$ and it thus defines
an invariant of degree $3k/2$.

For each irreducible representation $\rho$ of ${\rm GL}(3)$ we have a vector
bundle ${\mathbb{E}}_{\rho}$ made from ${\mathbb{E}}$ by a Schur functor. By a
modular form of weight $\rho$ on $\overline{\mathcal{H}}_{3}$ we mean a
section of a vector bundle ${\mathbb{E}}_{\rho}$. We can pull back to the
stack $[W^{0}_{8,-2}/({\rm GL}(W)/\pm{\rm id}_{W})]$, but the situation is
more involved as ${\rm Sym}^{n}({\rm Sym}^{2}(W))$ decomposes as a
representation of ${\rm GL}(W)$. For example, we have with $W_{a,b}={\rm
Sym}^{a}(W)\otimes\det(W)^{b}$

$h^{*}({\rm Sym}^{4}({\mathbb{E}}))={\rm Sym}^{4}({\rm
Sym}^{2}(W))=W_{8,0}\oplus W_{4,2}\oplus W_{0,4}\,.$

Here and in the rest of this section we assume that the characteristic is $0$,
or not $2$, and high enough for the representation theory (plethysm) to
work111Alternatively one could use divided powers as in [2, 3.1]. In this case
we can consider the restriction of the Siegel modular form $\chi_{4,0,8}$ to
the hyperelliptic locus and we know that it does not vanish identically by [6,
Lemma 7.7]. On the other hand we have the basic covariant $f_{8,-2}$, the
diagonal section of $W_{8,-2}$ over the stack $[W_{8,-2}^{0}/({\rm
GL}(W)/\pm{\rm id}_{W})]$.

The discriminant form $\mathfrak{d}$ of binary octics, an invariant of degree
$14$, does not define a modular form, but its third power $\mathfrak{d}^{3}$
does. It defines a modular form of weight $28$, see [28, p. 811] and also
Remark 13.1.

Via the projection $p_{8,0}:{\rm Sym}^{4}({\rm Sym}^{2}(W))\to W_{8,0}$ a
section of ${\rm Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ defines
a covariant of bi-degree $(8,24/2)=(8,12)$ for the action of ${\rm GL}(W)$.

######  Proposition 6.1.

The restriction to the hyperelliptic locus of the section $\chi_{4,0,8}$
corresponds via the projection $p_{8,0}$ to a multiple of the covariant
$f_{8,-2}\cdot\mathfrak{d}$ with $\mathfrak{d}$ the discriminant of binary
octics.

###### Proof.

By restricting and projecting we obtain a covariant of bi-degree $(8,12)$.
This covariant is divisible by the discriminant and does not vanish on the
locus of smooth hyperelliptic curves. Therefore, division by $\mathfrak{d}$
gives a non-vanishing covariant of bidegree $(8,-2)$. Taking into account the
‘twisting’ by $\det(W)^{-2}$, this must be a multiple of the universal binary
octic. ∎

We will discuss the other two projections later in Lemma 14.3. Note that the
divisor $D$, the canonical image of the universal curve in
${\mathbb{P}}({\mathbb{E}})$ that defines $\chi_{4,0,8}$, has a restriction to
the locus of smooth hyperelliptic curves which is divisible by $2$. Indeed,
the canonical image of a hyperelliptic curve is a double conic. This suggests
that we can take the ‘square root’ of the restriction of $\chi_{4,0,8}$ to the
hyperelliptic locus. However, the boundary divisors prevent this. If we take a
level cover of the moduli space we can construct a modular form of weight
$(2,0,4)$. We will carry this out later (in Corollary 13.4), working on a
Hurwitz space that we shall introduce in the next section.

##  7\. The Hurwitz space of admissible covers of degree two

In this and the following sections will use the other description of the
moduli of hyperelliptic curves, namely the moduli space
$\overline{\mathcal{H}}_{g,2}$ of admissible covers of degree $2$ and genus
$g$ in the sense of [17], see [16]. Thus we are looking at covers $f:C\to P$
of degree $2$ with $C$ nodal of genus $g$ and $P$ a stable $b$-pointed curve
of genus $0$. Here the $b=2g+2$ branch points are ordered and
$\mathcal{H}_{g,2}\to\mathcal{H}_{g}$ is a Galois cover with Galois group the
symmetric group $\mathfrak{S}_{2g+2}$.

The boundary $\overline{\mathcal{H}}_{g,2}-\mathcal{H}_{g,2}$ consists of
finitely many divisors that we shall denote by
$\Delta_{b}^{\Lambda}=\Delta^{\Lambda}$, where we omit the index $b$ if $g$ is
clear. Here the index $\Lambda$ defines a partition
$\\{1,2,\ldots,b\\}=\Lambda\sqcup\Lambda^{c}$, and the generic point of
$\Delta^{\Lambda}$ corresponds to an admissible cover that maps to a stable
curve of genus $0$ that is the union of two copies of ${\mathbb{P}}^{1}$, one
containing the points with mark in $\Lambda$, the other one those with mark in
$\Lambda^{c}$. Here we will assume that $\\#\Lambda=j$ with $2\leq j\leq g+1$.

The parity of $\\#\Lambda$ plays an important role here. If $\\#\Lambda=2i+2$
is even, then the generic admissible cover corresponding to a point of
$\Delta^{\Lambda}$ is a union $C_{i}\cup C_{g-i-1}$ that is a double cover of
a union of two rational curves ${\mathbb{P}}_{1}$ and ${\mathbb{P}}_{2}$ with
$C_{i}$ lying over ${\mathbb{P}}_{1}$ and $C_{g-i-1}$ over ${\mathbb{P}}_{2}$.
Here $C_{i}$ (resp. $C_{g-i-1}$) has genus $i$ (resp. $g-i-1$) with $0\leq
i\leq(g-1)/2$ and is ramified over the points of $\Lambda$ (resp.
$\Lambda^{c}$).

$C_{i}$$C_{g-i-1}$${\mathbb{P}}_{1}$${\mathbb{P}}_{2}$$p$$q$$s$Fig. 1:
$\Lambda$ even

If $\\#\Lambda=2i+1$ is odd with $1\leq i\leq g/2$, then we have a union
$C_{i}\cup C_{g-i}$ lying over ${\mathbb{P}}_{1}\cup{\mathbb{P}}_{2}$, where
$C_{i}$ (resp. $C_{g-i}$) of genus $i$ (resp. $g-i$) is ramified over
$\Lambda$ and in $p$, the intersection of $C_{i}$ and $C_{g-i}$, (resp. over
$\Lambda^{c}$ and in $p$). Note that $p$ is a simple node.

$C_{i}$$C_{g-i}$${\mathbb{P}}_{1}$${\mathbb{P}}_{2}$$p$$s$Fig. 2: $\Lambda$
odd

Assuming that $g$ and $b=2g+2$ are fixed we will write

$\Delta_{j}=\sum_{\\#\Lambda=j}\Delta^{\Lambda}\qquad\text{\rm with $2\leq
j<b/2$}$

and provide the symmetric case with a factor $1/2$, that is,
$\Delta_{b/2}=\frac{1}{2}\sum_{\\#\Lambda=b/2}\Delta^{\Lambda}$.

##  8\. Divisors on the moduli of stable curves of genus zero

For later use we recall some notation and facts concerning divisors on the
moduli spaces $\overline{\mathcal{M}}_{0,n}$. We refer to [21]. The boundary
divisors on $\overline{\mathcal{M}}_{0,n}$ are denoted by $S_{n}^{\Lambda}$
and are indexed by partitions $\\{1,\ldots,n\\}=\Lambda\sqcup\Lambda^{c}$ into
two disjoint sets with $2\leq\\#\Lambda\leq n-2$ and we have
$S_{n}^{\Lambda}=S_{n}^{\Lambda^{c}}$. Via the natural map
$\pi_{n+1}:\overline{\mathcal{M}}_{0,n+1}\to\overline{\mathcal{M}}_{0,n}$ we
may view $\overline{\mathcal{M}}_{0,n+1}$ as the universal curve and
$\pi_{n+1}$ has $n$ sections. The generic point of $S_{n}^{\Lambda}$
corresponds to a stable curve with two rational components, one of which
contains the points marked by $\Lambda$. For pullback by $\pi_{n+1}$ we have
the relation

$\pi_{n+1}^{*}(S_{n}^{\Lambda})=S_{n+1}^{\\{\Lambda,n+1\\}}\cup
S_{n+1}^{\\{\Lambda^{c},n+1\\}}\,.$

The $n$ sections of $\pi_{n+1}$ have images $S_{n+1}^{\\{i,n+1\\}}$ with
$i=1,\ldots,n$.

We can collect these boundary divisors on $\overline{\mathcal{M}}_{0,n+1}$ via

$T_{n+1,j}=\sum_{\\#\Lambda=j}S_{n+1}^{\\{\Lambda,n+1\\}},\quad
T_{n+1,j}^{c}=\sum_{\\#\Lambda=j}S_{n+1}^{\\{\Lambda^{c},n+1\\}}\,,$

with the convention that in view of the symmetry we add a factor $1/2$ for
even $n$ and $j=n/2$

$T_{n+1,n/2}=\frac{1}{2}\sum_{\\#\Lambda=n/2}S_{n+1}^{\\{\Lambda,n+1\\}},\quad
T_{n+1,n/2}^{c}=\frac{1}{2}\sum_{\\#\Lambda=n/2}S_{n+1}^{\\{\Lambda^{c},n+1\\}}\,.$

Later, when a fixed index $k$ is given we will split these divisors as
$T=T(k^{+})+T(k^{-})$ where $(k^{+})$ (resp. $(k^{-})$) indicates that the sum
is taken over $\Lambda$ containing $k$ (resp. not containing $k$). So
$T_{n+1,j}(k^{+})=\sum_{\\#\Lambda=j,k\in\Lambda}S_{n+1}^{\\{\Lambda,n+1\\}}$
(and with a factor $1/2$ if $j=n/2$).

##  9\. A good model

We now will work with a ‘good model’ of the universal admissible cover over
$\overline{\mathcal{H}}_{g,2}$. Such a model was constructed in [12, Section
4]. We start with the observation that the space
$\overline{\mathcal{H}}_{g,2}$ is not normal, and we therefore normalize it.
The result $\widetilde{\mathcal{H}}_{g,2}$ is now a smooth stack over which we
have a universal curve $\tilde{\mathcal{C}}\to\widetilde{\mathcal{H}}_{g,2}$.

We have a natural map
$h:\widetilde{\mathcal{H}}_{g,2}\to\overline{\mathcal{M}}_{0,b}$ with $b=2g+2$
and the universal curve now fits into a commutative diagram

$\textstyle{{\widetilde{\mathcal{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{\varpi}$$\textstyle{{\overline{{\mathcal{M}}}}_{0,b+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{b+1}}$$\textstyle{\overline{\mathcal{H}}_{g,2}}$$\textstyle{{\widetilde{\mathcal{H}}}_{g,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{h}$$\textstyle{{\overline{{\mathcal{M}}}}_{0,b}}$

We can construct a proper flat map that extends the relative canonical
morphism ${\mathcal{C}}\to{\mathbb{P}}^{1}_{{{\mathcal{H}}}_{g,2}}$ by taking
the fibre product ${\mathbb{P}}$ of ${\overline{{\mathcal{M}}}}_{0,b+1}$ and
${\widetilde{\mathcal{H}}}_{g,2}$ over ${\overline{{\mathcal{M}}}}_{0,b}$ and
thus obtain a commutative diagram

$\textstyle{{\widetilde{\mathcal{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\varpi}$$\textstyle{{\mathbb{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi^{\prime}}$$\scriptstyle{c^{\prime}}$$\textstyle{{\overline{{\mathcal{M}}}}_{0,b+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{b+1}}$$\textstyle{{\widetilde{\mathcal{H}}}_{g,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\overline{{\mathcal{M}}}}_{0,b}}$

The resulting space ${\mathbb{P}}$ is not smooth, but has rational
singularities. Resolving these in a minimal way gives a model
$\widetilde{\mathbb{P}}$; taking the resolution $\widetilde{Y}$ of the
normalization $Y$ of the fibre product of ${\widetilde{\mathbb{P}}}$ and
$\widetilde{\mathcal{C}}$ over ${\mathbb{P}}$ gives us finally a commutative
diagram

$None$
$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{t}$$\textstyle{\widetilde{\mathbb{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\widetilde{\mathcal{H}}_{g,2}}$

where $B$ is our base $\widetilde{\mathcal{H}}_{g,2}$ or any other base
mapping to it. We write $\pi$ for the resulting morphism
$\widetilde{\mathbb{P}}\to B$, $h$ for the natural map
$B\to\overline{\mathcal{M}}_{0,b}$ and $\nu$ for
$B\to\overline{\mathcal{H}}_{g,2}$. We refer to [12, Section 4] for additional
details.

In the following we will assume that we have a physical family over a base
$B$. We will abuse the notation $\Delta^{\Lambda}$ for the pull back of the
divisor $\Delta^{\Lambda}$ under $\nu:B\to\overline{\mathcal{H}}_{g,2}$.

In the case that $\\#\Lambda$ is even, say $\\#\Lambda=2i+2$ with $0\leq
i\leq(g-1)/2$, the pull back of $\Delta^{\Lambda}$ decomposes as

$\pi^{*}(\Delta^{\Lambda})=\Pi^{\Lambda}+\Pi^{\Lambda^{c}}\,,$

corresponding to the two components of a general fibre of $\pi$, with
$\Pi^{\Lambda}$ mapping to $S_{b+1}^{\\{\Lambda,b+1\\}}$ under
$\widetilde{\mathbb{P}}\to\overline{\mathcal{M}}_{0,b+1}$, and similarly
$\Pi^{\Lambda^{c}}$ mapping to $S_{b+1}^{\\{\Lambda^{c},b+1\\}}$. Note that we
restrict $\\#\Lambda$ by $\leq g+1$, hence the notation $\Pi^{\Lambda}$ should
not lead to confusion.

In the case $\\#\Lambda$ is odd we find a similar decomposition

$\pi^{*}(\Delta^{\Lambda})=\Pi^{\Lambda}+R^{\Lambda}+\Pi^{\Lambda^{c}}\,,$

corresponding now to the fact that the general fibre of $\pi$ has three
components, one coming from the blowing up.

We notice

$h^{*}(S_{b}^{\Lambda})=\begin{cases}\Delta^{\Lambda}&\\#\Lambda\equiv
0\,(\bmod\,2)\\\ 2\,\Delta^{\Lambda}&\\#\Lambda\equiv 1\,(\bmod\,2)\\\
\end{cases}$

If we use the notation $\Delta_{j}=\sum_{\\#\Lambda=j}\Delta^{\Lambda}$, we
find for the tautological classes $\lambda=c_{1}({\mathbb{E}})$ and
$h^{*}(\psi_{k})$, simply denoted by $\psi_{k}$ and defined as the first Chern
class of the line bundle given by the cotangent space at the $k$th point of
our pointed curve ($k=1,\ldots,b$), the following formulas on our base $B$
(see [11])

$None$
$\lambda=\sum_{i=0}^{(g-1)/2}\frac{(i+1)(g-i)}{2(2g+1)}\Delta_{2i+2}+\sum_{i=1}^{g/2}\frac{i(g-i)}{2g+1}\Delta_{2i+1}$

and

$None$
$\displaystyle\psi_{k}=\sum_{i=0}^{(g-1)/2}\left(\frac{(g-i)(2g-2i-1)}{g(2g+1)}\Delta_{2i+2}(k^{+})+\frac{(i+1)(2i+1)}{g(2g+1)}\Delta_{2i+2}(k^{-})\right)$
$\displaystyle+2\,\sum_{i=1}^{g/2}\left(\frac{(g-i)(2g-2i+1)}{g(2g+1)}\Delta_{2i+1}(k^{+})+\frac{i(2i+1)}{g(2g+1)}\Delta_{2i+1}(k^{-})\right)$

where we use the notation $(k^{+})$ (resp. $(k^{-})$) to denote the condition
$k\in\Lambda$ (resp. $k\not\in\Lambda$) as above. The relation (4) implies the
following.

######  Corollary 9.1.

There exists a scalar-valued modular form of weight $2(2g+1)$ on the moduli
space $\widetilde{\mathcal{H}}_{g,2}$ whose divisor is a union of boundary
divisors. It descends to the hyperelliptic locus $\overline{\mathcal{H}}_{g}$
and corresponds to a power of the discriminant of the binary form of degree
$2g+2$.

##  10\. Extending the linear system

The canonical system on a hyperelliptic curve is defined by the pull back of
the sections of the line bundle of $\mathcal{O}(g-1)$ of degree $g-1$ on the
projective line. We now try to extend this line bundle over our
compactification.

A first attempt would be to consider the divisor $(g-1)\,\tilde{S}_{k}$ with
$\tilde{S}_{k}$ the pullback to $\widetilde{\mathbb{P}}$ of the section
$S_{k}$ of
$\pi_{b+1}:\overline{\mathcal{M}}_{0,b+1}\to\overline{\mathcal{M}}_{0,b}$.
Recall the morphism $t=\pi f\colon\widetilde{Y}\to B$. We can add a boundary
divisor $\Xi_{k}$ to it such that
$f^{*}\mathcal{O}_{\widetilde{\mathbb{P}}}(D_{k})$ with
$D_{k}=(g-1)\tilde{S}_{k}+\Xi_{k}$ coincides with $\omega_{t}$ on the fibres
of $t$, namely in view of the intersection numbers take $\Xi_{k}$ equal to

$\displaystyle\sum_{i=0}^{(g-1)/2}\left((g-1-i)\,\Pi_{2i+2}(k^{+})+i\,\Pi_{2i+2}^{c}(k^{-})\right)+$
$\displaystyle\sum_{i=1}^{g/2}\left((g-i-1)\,\Pi_{2i+1}(k^{+})-(g-i)\,\Pi^{c}_{2i+1}(k^{+})\right)+\sum_{i=1}^{g/2}\left((i-1)\,\Pi_{2i+1}^{c}(k^{-})-i\,\Pi_{2i+1}(k^{-})\right)$

Here $\Pi_{j}=\sum_{\\#\Lambda=j}\Pi^{\Lambda}$ and
$\Pi_{j}^{c}=\sum_{\\#\Lambda=j}\Pi^{\Lambda^{c}}$ and $(k^{+})$ (resp.
$(k^{-})$) indicates the condition that $k\in\Lambda$ (resp.
$k\not\in\Lambda$); moreover, we add a factor $1/2$ in case $j=b/2$.

Now $f^{*}\mathcal{O}(D_{k})$ and $\omega_{t}$ agree on the fibres of $t$, so
they differ by a pull back under $t=\pi\circ f$, see diagram (2).

To see the above, when e.g. $\\#\Lambda=2i+2$ is even: in that case the fibre
of $\tilde{Y}$ over $t$ is as in Fig. 1 and
$\omega_{C}=(\omega_{C_{i}}+p+q,\omega_{C_{g-i-1}}+p+q)=(i(p+q),(g-i-1)(p+q))=f^{*}(i,g-i-1)$,
where we indicate by $i$ the line bundle of degree $i$ on ${\mathbb{P}}^{1}$.
One then checks that with the above choice of $\Xi_{k}$ the restriction of
$D_{k}$ on the corresponding fibre of $\pi$ is of type $(i,g-1-i)$. Indeed, in
case $k\in\Lambda$ then $(g-1)\tilde{S}_{k}+(g-1-i)\Pi_{2i+2}(k^{+})$
restricts to $((g-1)-(g-i-1),g-i-1)=(i,g-i-1)$ and in case $k\notin\Lambda$
then $(g-1)\tilde{S}_{k}+i\Pi^{c}_{2i+2}(k^{-})$ restricts to
$(i,(g-1)-i)=(i,g-i-1)$. The case where $\\#\Lambda=2i+1$ is odd, although a
little more complicated, is treated similarly.

We therefore will change $D_{k}$ by a pull back under $\pi$. Define a divisor
class on $B$ by

$\displaystyle E_{k}=\frac{2g-1}{2}\psi_{k}$
$\displaystyle-\sum_{i=0}^{(g-1)/2}\left((g-i-1)\,\Delta_{2i+2}(k^{+})+i\,\Delta_{2i+2}(k^{-})\right)$
$\displaystyle-\sum_{i=1}^{g/2}\left((g-i-1)\,\Delta_{2i+1}(k^{+})+(i-1)\,\Delta_{2i+1}(k^{-})\right)$

and define a line bundle on $\tilde{\mathbb{P}}$ by

$None$ $M=\mathcal{O}(D_{k}+\pi^{*}E_{k})\,.$

######  Lemma 10.1.

The line bundle $M$ does not depend on $k$, satisfies $f^{*}(M)=\omega_{t}$
and restricts to the general fibre ${\mathbb{P}}^{1}$ of $\pi$ as
$\mathcal{O}(g-1)$. For $\\#\Lambda=2i+2$ its restriction to the general fibre
${\mathbb{P}}_{1}\cup{\mathbb{P}}_{2}$ over $\Delta^{\Lambda}$ is of degree
$(i,g-i-1)$, while for $\\#\Lambda=2i+1$ its restriction to the general fibre
${\mathbb{P}}_{1}\cup R\cup{\mathbb{P}}_{2}$ is of degrees $(i,-1,g-i)$.

###### Proof.

We use the section $\tau_{k}$ of $t:\widetilde{Y}\to B$ with
$f\tau_{k}=\tilde{s}_{k}$ with $\tilde{s}_{k}$ the natural sections of the map
$\pi$ with image $\tilde{S}_{k}$. Then we have
$\tau_{k}^{*}\omega_{t}=\psi_{k}/2$ and
$\tau_{k}^{*}f^{*}D_{k}=\tilde{s}_{k}^{*}D_{k}$ for which we have

$\displaystyle\tilde{s}_{k}^{*}D_{k}=-(g-1)\psi_{k}+\sum_{i=0}^{(g-1)/2}\left((g-i-1)\,\Delta_{2i+2}(k^{+})+i\,\Delta_{2i+2}(k^{-})\right)$
$\displaystyle+\sum_{i=1}^{g/2}\left((g-1-i)\,\Delta_{2i+1}(k^{+})+(i-1)\,\Delta_{2i+1}(k^{-})\right)$
$\displaystyle\,.$

From this we obtain
$\tau_{k}^{*}(\omega_{t})-\tau_{k}^{*}f^{*}D_{k}=\pi^{*}(E_{k})$, so that
$\omega_{t}=f^{*}(\mathcal{O}(D_{k}+E_{k}))$. We also see that the restriction
of $M$ on the fibres of $\pi$ does not depend on $k$. Moreover, we have

$\tilde{s}_{j}^{*}(D_{k}+\pi^{*}E_{k})=\tau_{j}^{*}f^{*}(D_{k}+E_{k})=\tau_{j}^{*}(\omega_{t})=\psi_{j}/2=\tilde{s}_{j}^{*}(D_{j}+\pi^{*}E_{j})\,,$

showing that the restrictions of $\mathcal{O}(D_{k}+\pi^{*}E_{k})$ and
$\mathcal{O}(D_{j}+\pi^{*}E_{j})$ agree on $\tilde{S}_{j}$. The restrictions
of the fibres of $\pi$ over the general points of $\Delta_{b}^{\Lambda}$ are
easily checked. ∎

We now want to compare $\pi_{*}(M)$ with the Hodge bundle
${\mathbb{E}}=t_{*}(\omega_{t})$ on $B$. The next proposition shows that these
agree up to codimension $2$.

######  Proposition 10.2.

We have an exact sequence $0\to\pi_{*}(M)\to{\mathbb{E}}\to\mathcal{T}\to 0$,
where $\mathcal{T}$ is a coherent sheaf that is a torsion sheaf supported on
the boundary. Moreover, we have $c_{1}(\pi_{*}(M))=\lambda$.

###### Proof.

By Lemma 10.1 we have $\omega_{t}=f^{*}(M)$. But $R^{1}\pi_{*}(M)=(0)$, so we
have

$\pi_{*}(M\otimes
f_{*}\mathcal{O}_{\tilde{Y}})=\pi_{*}f_{*}(f^{*}(M))=\pi_{*}f_{*}(\omega_{t})=t_{*}(\omega_{t})\,.$

We have an exact sequence $0\to\mathcal{O}_{\tilde{\mathbb{P}}}\to
f_{*}\mathcal{O}_{\tilde{Y}}\to\mathcal{F}\to 0$ with $\mathcal{F}$ a coherent
sheaf of rank $1$ that restricted to the smooth fibers of $\pi$ has degree
$-(g+1)$, as one sees by applying Riemann-Roch to $f$ and
$\mathcal{O}_{\tilde{Y}}$. Tensoring the sequence with $M$ and applying
$\pi_{*}$ gives the exact sequence

$0\to\pi_{*}(M)\to\pi_{*}(M\otimes
f_{*}\mathcal{O}_{\tilde{Y}})\to\pi_{*}(M\otimes\mathcal{F})\to 0\,.$

On the smooth fibers of $\pi$ the sheaf $M\otimes\mathcal{F}$ restricts to a
line bundle of degree $(g-1)-(g+1)=-2$, hence $\pi_{*}(M\otimes\mathcal{F})$
is a torsion sheaf.

We now calculate $c_{1}(\pi_{*}(M))$. We apply Grothendieck-Riemann-Roch to
$\pi$ and $\mathcal{O}(D_{k})$. It says

${\rm ch}(\pi_{!}(\mathcal{O}(D_{k})))=\pi_{*}({\rm
ch}(\mathcal{O}(D_{k}))){\rm Td}^{\vee}(\omega_{\pi})\,,$

which by $\pi_{*}({\rm Td}^{\vee}_{2}(\omega_{\pi}))=0$ gives

$c_{1}(\pi_{*}(\mathcal{O}(D_{k})))=\frac{1}{2}\pi_{*}(-D_{k}\,\omega_{\pi}+D_{k}^{2})\,.$

We calculate

$None$
$\pi_{*}(D_{k}\omega_{\pi})=(g-1)\psi_{k}-\sum_{i=0}^{(g-1)/2}\left((g-i-1)\,\Delta_{2i+2}(k^{+})+i\,\Delta_{2i+2}(k^{-})\right)+\sum_{i=1}^{g/2}\Delta_{2i+1}\,,$

and

$None$ $\displaystyle\pi_{*}(D_{k}^{2})=$
$\displaystyle-(g-1)^{2}\psi_{k}+\sum_{i=0}^{(g-1)/2}(g-i-1)(g+i-1)\,\Delta_{2i+2}(k^{+})$
$\displaystyle+\sum_{i=0}^{(g-1)/2}(2g-2-i)i\,\Delta_{2i+2}(k^{-})+\sum_{i=1}^{g/2}((2g-i-1)(i-1)-i^{2})\,\Delta_{2i+1}\,.$

Adding $\pi_{*}(\pi^{*}E_{k})$ gives

$\displaystyle
c_{1}(\pi_{*}(M))=\frac{g^{2}}{2}\,\psi_{k}-\frac{1}{2}\sum_{i=0}^{(g-1)/2}\left((g-i-1)(g-i)\,\Delta_{2i+2}(k^{+})+i(i+1)\,\Delta_{2i+2}(k^{-})\right)$
$\displaystyle-\sum_{i=1}^{g/2}\left((g-i)^{2}\,\Delta_{2i+1}(k^{+})+i^{2}\,\Delta_{2i+1}(k^{-})\right)\,.$

Substituting the formula for $\psi_{k}$ we find

$c_{1}(\pi_{*}(M))=\sum_{i=0}^{(g-1)/2}\frac{(g-i)(i+1)}{2(2g+1)}\Delta_{2i+2}+\sum_{i=1}^{g/2}\frac{i(g-i)}{2g+1}\Delta_{2i+1}=\lambda\,.$

∎

The line bundle $M$ on $\tilde{\mathbb{P}}$ is not base point free as
Proposition 10.2 shows; the restriction to the $R$-part has negative degree.
We can make it base point free by defining

$None$ $N=M(-R)=\mathcal{O}(D_{k}+\pi^{*}E_{k}-R)\,.$

Now the restriction of $N$ to a general fibre over $\Delta_{2i+1}$, which is a
chain of three rational curves ${\mathbb{P}}_{1},R,{\mathbb{P}}_{2}$, has
degrees $(i-1,1,g-i-1)$ and one checks that $N$ is base point free.

######  Lemma 10.3.

Up to codimension $2$ we have on $B$ that $\pi_{*}(N)={\mathbb{E}}$.

###### Proof.

We have $R^{1}\pi_{*}(N)=0$. Therefore the exact sequence $0\to N\to M\to
M_{|R}\to 0$ yields the exact sequence

$0\to\pi_{*}(N)\to\pi_{*}(M)\to\pi_{*}(M_{|R})\to 0\,.$

We now show that $c_{1}(\pi_{*}(N))=c_{1}(\pi_{*}(M))$. Since
$R^{1}\pi_{*}(M)=0=R^{1}\pi_{*}(N)$ we find by Grothendieck-Riemann-Roch that

$c_{1}(\pi_{*}(M))=\frac{1}{2}\pi_{*}(c_{1}(M)^{2}-c_{1}(M)\omega_{\pi}),\quad
c_{1}(\pi_{*}(N))=\frac{1}{2}\pi_{*}(c_{1}(N)^{2}-c_{1}(N)\omega_{\pi})\,.$

By the definition of $N$ and the fact that $R$ is a $(-2)$-curve if we take a
base $B$ of dimension $1$, and thus has intersection number $0$ with a fibre,
we have

$\pi_{*}(c_{1}(N))^{2}=\pi_{*}(c_{1}(M))^{2}$

and $c_{1}(N)\,\omega_{\pi}=c_{1}(M)\,\omega_{\pi}$ since the restriction of
$\omega_{\pi}$ to $R$ is trivial. ∎

##  11\. The rational normal curve

The image of a hyperelliptic curve by the canonical map is a rational normal
curve, that is, ${\mathbb{P}}^{1}$ embedded in ${\mathbb{P}}^{g-1}$ via the
linear system of degree $g-1$.

In our setting we can see the rational normal curve and its degenerations
using the extension $N$ of the line bundle of degree $g-1$, as defined in (9),
to the compactification as constructed in the preceding section.

We let $u:{\mathbb{P}}({\mathbb{E}})\to B$ be the natural projection. Now $N$
is base point free and up to codimension $2$ we have
$\pi_{*}(N)={\mathbb{E}}$, so the global-to-local map $\pi^{*}\pi_{*}(N)\to N$
induces a surjective map $\nu:\pi^{*}({\mathbb{E}})\to N$ over
$\tilde{\mathbb{P}}$. This induces a morphism
$\phi:\tilde{\mathbb{P}}\to{\mathbb{P}}({\mathbb{E}})$ by associating to a
point of $\tilde{\mathbb{P}}$ the kernel of $\nu$. It fits into a diagram

$\textstyle{\tilde{\mathbb{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}({\mathbb{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{B}$

######  Proposition 11.1.

For a point of $B$ with smooth fibre under $\pi$ the image of $\phi$ is a
rational normal curve of degree $g-1$. For a general point
$\beta\in\Delta_{2i+2}$ with fibre ${\mathbb{P}}_{1}\cup{\mathbb{P}}_{2}$ the
image is a union of two rational normal curves of degree $i$ and $g-i-1$. For
a general point $\beta\in\Delta_{2i+1}$ with fibre
${\mathbb{P}}_{1},R,{\mathbb{P}}_{2}$ the image is a union of three rational
normal curves of degree $i-1$, $1$ and $g-i-1$. Here we interpret the case of
degree $0$ as a contracted curve.

###### Proof.

The proposition follows almost immediately from Lemma 10.1. ∎

######  Remark 11.2.

If $i=1$ then ${\mathbb{P}}_{1}$ is contracted. If also $g=2$ then both
${\mathbb{P}}_{1}$ and ${\mathbb{P}}_{2}$ are contracted and the image of $R$
coincides with the fibre of ${\mathbb{P}}({\mathbb{E}})$.

######  Remark 11.3.

The sections $\tilde{s}_{i}:B\to\tilde{\mathbb{P}}$ for $i=1,\ldots,b$ induce
sections $\sigma_{i}=\phi\circ s_{i}:B\to{\mathbb{P}}({\mathbb{E}})$ by
sending $\beta$ to the kernel of
${\mathbb{E}}=\tilde{s}_{i}^{*}\pi^{*}({\mathbb{E}})\to s_{i}^{*}(N)$.

######  Remark 11.4.

In the case $g=2$ the map $\phi$ is a birational map
$\tilde{\mathbb{P}}\to{\mathbb{P}}({\mathbb{E}})$ that blows down boundary
components. More precisely, over $\Delta_{2}$ it blows down ${\Pi}_{2}$ and
over $\Delta_{3}$ the components supported at ${\Pi}_{3}={\Pi}^{c}_{3}$.

##  12\. Symmetrization

We have been working with the moduli space ${\mathcal{H}}_{g,2}$ and
$\mathcal{M}_{0,b}$ and their compactifications. Here the symmetric group
$\mathfrak{S}_{b}$ acts. We therefore make our construction symmetric.

We put $D=\sum_{k=1}^{b}D_{k}$ and $E=\sum_{k=1}^{b}E_{k}$ and set

$\tilde{M}=\mathcal{O}(D+E)\,,\quad\psi=\sum_{k=1}^{b}\psi_{k}\,,\quad\text{\rm
and}\quad\tilde{S}=\sum_{k=1}^{b}\tilde{S}_{k}\,.$

We find

$\psi=4\,\sum_{i=0}^{(g-1)/2}\frac{(g-i)(i+1)}{2g+1}\,\Delta_{2i+2}+2\,\sum_{i=1}^{g/2}\frac{(2g-2i+1)(2i+1)}{2g+1}\,\Delta_{2i+1}$

and

$\displaystyle
D=(g-1)\tilde{S}+2\sum_{i=0}^{(g-1)/2}\left((g-i-1)(i+1)\Pi_{2i+2}+i\,(g-i)\,\Pi^{c}_{2i+2}\right)+$
$\displaystyle\sum_{i=1}^{g/2}\left((g-4i-1)\Pi_{2i+1}-(3g-4i+1)\,\Pi^{c}_{2i+1}\right)$

and

$E=\frac{2g-1}{2}\psi-2\sum_{i=0}^{(g-1)/2}\left((g-i)(2i+1)-(i+1)\right)\Delta_{2i+2}-\sum_{i=1}^{g/2}\left(4i\,(g-i)-(g+2)\right)\Delta_{2i+1}$

##  13\. The Case of Hyperelliptic Genus Three

The Hurwitz space $\mathcal{H}_{3,2}$ admits a compactification
$\overline{\mathcal{H}}_{3,2}$ with boundary components $\Delta^{\Lambda}$
with $\\#\Lambda\in\\{2,3,4\\}$. Taking the components with $\Lambda$ of fixed
cardinality together gives boundary components $\Delta_{2},\Delta_{3}$ and
$\Delta_{4}$. Under the morphism
$\overline{\mathcal{H}}_{3,2}\to\overline{\mathcal{M}}_{3}$ the components
$\Delta_{2}$ and $\Delta_{4}$ are mapped to $\delta_{0}$, while $\Delta_{3}$
goes to $\delta_{1}$. The formulas (4) and (5) specialize to

$None$
$\lambda=\frac{3}{14}\Delta_{2}+\frac{2}{7}\Delta_{3}+\frac{2}{7}\Delta_{4}$

and

$\psi_{k}=\frac{5}{7}\,\Delta_{2}(k^{+})+\frac{1}{21}\,\Delta_{2}(k^{-})+\frac{20}{21}\,\Delta_{3}(k^{+})+\frac{2}{7}\,\Delta_{3}(k^{-})+\frac{2}{7}\,\Delta_{4}\,.$

######  Remark 13.1.

The equation (10) shows that on $\overline{\mathcal{H}}_{3,2}$ there exists a
scalar-valued modular form of weight $14$ whose square equals $\chi_{28}$, a
form mentioned in Section 6. Since on $\overline{\mathcal{H}}_{3}$ we have
$28\,\lambda=3\,\delta_{0}+8\,\delta_{1}+8\,\zeta_{1}$, an integral class not
divisible by $2$, there is not a modular form of weight $14$ on
$\overline{\mathcal{H}}_{3}$ with square $\chi_{28}$. Compare with Cornalba’s
formula at the end of Section 5.

We have the line bundle $M$ on $\tilde{\mathbb{P}}$ defined in (6)
corresponding to the divisor class $D_{k}+E_{k}$ given by

$D_{k}=2\,\tilde{S}_{k}+2\,\Pi_{2}(k^{+})+2\,\Pi_{4}(k^{+})+\Pi_{3}(k^{+})-2\,\Pi_{3}^{c}(k^{+})-\Pi_{3}(k^{-})$

and

$E_{k}=\frac{5}{2}\psi_{k}-(2\,\Delta_{2}(k^{+})+\Delta_{3}(k^{+})+\Delta_{4})\,,$

where $\psi_{k}$ is given in (6). Define the rational divisor class

$U:=\frac{1}{14}\Delta_{2}+\frac{3}{7}(\Delta_{3}+\Delta_{4})=\frac{3}{2}\psi_{k}-(\Delta_{2}(k^{+})+\Delta_{3}(k^{+}))\,.$

The divisor class of $D_{k}+E_{k}$ is independent of $k$ as observed in Lemma
10.1, but this can be seen also directly from the next lemma.

######  Lemma 13.2.

We have the linear equivalence
$D_{k}+E_{k}\sim-\omega_{\pi}+\Pi_{2}+\Pi_{3}+\pi^{*}(U)$.

###### Proof.

One checks that $-\omega_{\pi}+\Pi_{2}+\Pi_{3}$ and $D_{k}+E_{k}$ have the
same restriction to fibres of $\pi$. We have
$s_{k}^{*}(-\omega_{\pi}+\Pi_{2}+\Pi_{3})=-\psi_{k}+\Delta_{2}(k^{+})+\Delta_{3}(k^{+})$
and $s_{k}^{*}(D_{k}+E_{k})=\psi_{k}/2$. ∎

Let $Q$ be the image of
$\phi:\tilde{\mathbb{P}}\to{\mathbb{P}}({\mathbb{E}})$, see Proposition 11.1.
The map $\phi$ is the composition of a map
$\phi^{\prime}:\tilde{\mathbb{P}}\to Q$ with the inclusion map
$\iota:Q\hookrightarrow{\mathbb{P}}(\mathbb{E})$. The generic fibre of $Q\to
B$ is a conic, hence $\mathcal{O}(Q)=\mathcal{O}(2)\otimes\mathcal{O}(u^{*}A)$
for some divisor $A$ on $B$. We determine $A$.

######  Lemma 13.3.

On ${\mathbb{P}}({\mathbb{E}})$ we have the linear equivalence

$[Q]\sim[\mathcal{O}(2)]+u^{*}(4\,\lambda-(\Delta_{2}+\Delta_{3}+\Delta_{4}))\,.$

###### Proof.

We have $\omega_{\mathbb{P}({\mathbb{E}})}\otimes
u^{*}(\omega_{B}^{-1})=\mathcal{O}(-3)\otimes u^{*}(\det{\mathbb{E}})$ and by
adjunction
$\omega_{Q}=\iota^{*}(\mathcal{O}(Q)\otimes\omega_{\mathbb{P}({\mathbb{E}})})$.
Since $\phi^{\prime}$ is a blow down we have
$\omega_{\tilde{\mathbb{P}}}=(\phi^{\prime})^{*}\omega_{Q}\otimes\mathcal{O}(\Pi_{2}+\Pi_{3})$.
We get

$\displaystyle\phi^{*}(\mathcal{O}(Q))=$
$\displaystyle{\phi^{\prime}}^{*}\omega_{Q}\otimes\phi^{*}\omega^{-1}_{\mathbb{P}({\mathbb{E}})}$
$\displaystyle=$
$\displaystyle\omega_{\tilde{\mathbb{P}}}\otimes\mathcal{O}(-\Pi_{2}-\Pi_{3})\otimes\phi^{*}\mathcal{O}(3)\otimes\pi^{*}\det({\mathbb{E}})^{-1}\otimes\pi^{*}\omega_{B}^{-1}$
$\displaystyle=$
$\displaystyle\omega_{\pi}\otimes\mathcal{O}(-\Pi_{2}-\Pi_{3})\otimes\phi^{*}\mathcal{O}(3)\otimes\pi^{*}\det({\mathbb{E}})^{-1}\,.$

On the other hand we have
$\phi^{*}\mathcal{O}(Q)=\phi^{*}\mathcal{O}(2)\otimes\mathcal{O}(\pi^{*}A)$
and $\phi^{*}\mathcal{O}(1)=N$, hence we get

$None$
$\mathcal{O}(u^{*}A)=N\otimes\omega_{\pi}\otimes\mathcal{O}(-\Pi_{2}-\Pi_{3})\otimes\pi^{*}\det({\mathbb{E}})^{-1}\,.$

By Lemma 13.2 we have
$N=\omega_{\pi}^{-1}\otimes\mathcal{O}(\Pi_{2}+\Pi_{3})\otimes{O}(U)$.
Substituting this in (11) we get the desired result. ∎

The effective divisor $Q$ yields a modular form and Lemma 13.3 gives its
weight.

######  Corollary 13.4.

The effective divisor $Q$ on ${\mathbb{P}}({\mathbb{E}})$ defines a modular
form $\chi_{2,0,4}$ on $\tilde{\mathcal{H}}_{3,2}$ of weight $(2,0,4)$, that
is, a non-zero section of ${\rm
Sym}^{2}({\mathbb{E}})\otimes\det({\mathbb{E}})^{4}$.

Since the divisor $\Delta_{2}+\Delta_{3}+\Delta_{4}$ is not a pull back from
the moduli space $\overline{\mathcal{H}}_{3}$, the modular form does not
descend to $\overline{\mathcal{H}}_{3}$. Recall that the modular form
$\chi_{4,0,8}$ restricted to the hyperelliptic locus was associated to a
divisor $D$ that equals $2\,Q$.

######  Remark 13.5.

In the same vein as above we can determine in an alternative way the result of
Proposition 2.1 on class of the closure $\overline{D}$ of the ramification
divisor $D$ of the universal genus $2$ curve. By the theory of admissible
covers there is a natural map
$\overline{\mathcal{H}}_{2,2}\to\overline{\mathcal{M}}_{2}$ with the property
that the pull back of the Hodge bundle on $\overline{\mathcal{M}}_{2}$ is the
Hodge bundle on $\overline{\mathcal{H}}_{2,2}$ associated to the corresponding
family of admissible covers. Hence the pull back of the $\mathcal{O}(1)$ of
the bundle ${\mathbb{P}}({\mathbb{E}})\to\overline{\mathcal{M}}_{2}$ equals
the $\mathcal{O}(1)$ of the bundle
${\mathbb{P}}({\mathbb{E}})\to\overline{\mathcal{H}}_{2,2}$. Let
$\Sigma=\phi_{*}(\sum_{k=1}^{6}\tilde{S}_{k})$, with
$\phi:{\mathbb{P}}\to{\mathbb{P}}({\mathbb{E}})$ the map defined in Section
11. By geometry, the pull back of $\overline{D}$ to the bundle
${\mathbb{P}}({\mathbb{E}})$ over $\overline{\mathcal{H}}_{2,2}$ equals
$\Sigma$. By Remark 11.4 we have
$\phi^{*}\Sigma=\tilde{S}+2\,\Pi_{2}+6\,\Pi_{3}$. By using the formulas of
Section 12, we have for $g=2$:

$\phi^{*}\mathcal{O}(6)=\tilde{M}-6\,R=\tilde{S}+\frac{12}{5}\Pi_{2}+\frac{2}{5}\Pi_{2}^{c}+\frac{24}{5}\Pi_{3}-\frac{3}{5}R\,.$

We now write
$[\overline{D}]=\mathcal{O}(6)+u^{*}(a\,\delta_{0}+b\,\delta_{1})$. By pulling
back to $\tilde{\mathbb{P}}$ and using the above formulas, we get (we refer to
the diagram in Section 11 for notation)

$\tilde{S}+\frac{12}{5}\Pi_{2}+\frac{2}{5}\Pi_{2}^{c}+\frac{24}{5}\Pi_{3}-\frac{3}{5}R+\pi^{*}(2a\,\Delta_{0}+b\,\Delta_{3})=\tilde{S}+2\,\Pi_{2}+6\,\Pi_{3}\,.$

This implies

$\pi^{*}(2a\,\Delta_{0}+b\,\Delta_{3})=-\frac{2}{5}(\Pi_{2}+\Pi_{2}^{c})+\frac{3}{5}(2\,\Pi_{3}+R)=\pi^{*}(-\frac{2}{5}\Delta_{2}+\frac{3}{5}\Delta_{3})\,,$

hence $a=-1/5$ and $b=3/5$ and the result follows by using the formula
$10\,\lambda=\delta_{0}+2\,\delta_{1}$.

##  14\. Comparison with the Hodge Bundle

We know by Lemma 10.3 that the line bundle
$N=\mathcal{O}_{\widetilde{\mathbb{P}}}(D_{k}+E_{k}-R)$ on
$\tilde{\mathbb{P}}$ over $\widetilde{\mathcal{H}}_{3,2}$ has the property
that $\pi_{*}(N)\cong{\mathbb{E}}$ up to codimension $2$. We now deal with the
push forward of the tensor powers of $N$.

######  Lemma 14.1.

We have for $m\in{\mathbb{Z}}_{\geq 1}$

$c_{1}(\pi_{*}(N^{\otimes
m}))=\frac{2\,m^{2}+m}{14}\,\Delta_{2}+\frac{5\,m^{2}-m}{14}(\Delta_{3}+\Delta_{4})\,.$

###### Proof.

We apply Grothendieck-Riemann-Roch to $\pi$ and $N^{\otimes m}$ as in (10) in
the proof of Proposition 10.2. Recall that $N$ corresponds to the
divisor(class) $D_{k}+E_{k}-R$. We use that $R^{1}\pi_{*}N^{\otimes m}=0$ for
all $m$ and find

$c_{1}(\pi_{*}(N^{\otimes
m}))=\frac{1}{2}\pi_{*}\left(m^{2}(D_{k}+E_{k}-R)^{2}-m\,\omega_{\pi}\cdot(D_{k}+E_{k}-R)\right)$

and using the relations (8) and (9) of the proof of Proposition 10.2 we get

$c_{1}(\pi_{*}(N^{\otimes
m}))=\frac{2\,m^{2}+m}{14}\,\Delta_{2}+\frac{5\,m^{2}-m}{14}(\Delta_{3}+\Delta_{4})$

as required. ∎

######  Proposition 14.2.

On $B$ we have the exact sequence

$0\to{\rm Sym}^{m-2}({\mathbb{E}})\otimes\mathcal{O}(-A)\to{\rm
Sym}^{m}({\mathbb{E}})\to\pi_{*}(N^{\otimes m})\to 0\,,$

with $A=4\,\lambda-(\Delta_{2}+\Delta_{3}+\Delta_{4})$.

###### Proof.

By Lemma 13.3 we have on ${\mathbb{P}}({\mathbb{E}})$ the exact sequence

$0\to\mathcal{O}(m-2)\otimes
u^{*}\mathcal{O}(-A)\to\mathcal{O}(m)\to\mathcal{O}(m)_{|Q}\to 0$

Applying $u_{*}$ and observing that $R^{1}u_{*}\mathcal{O}(m-2)$ vanishes
gives the result. ∎

A section of ${\rm Sym}^{j}({\mathbb{E}})\otimes\det({\mathbb{E}})^{k}$ over
$\mathcal{H}_{3}$ pulls back to the stack $[W^{0}_{8,-2}/({\rm GL}(W)/(\pm
1_{W}))]$ as a section of ${\rm Sym}^{j}({\rm
Sym}^{2}(W))\otimes\det(W)^{k/2}$ for even $k$. We have an isotypical
decomposition

${\rm Sym}^{j}({\rm Sym}^{2}(W))=\oplus_{n=0}^{\lfloor j/2\rfloor}{\rm
Sym}^{2j-4n}(W)\otimes\det(W)^{2n}\,,$

where we assume here and in the rest of this section that the characteristic
is $0$ or not $2$ and high enough for this identity to hold (or use divided
powers as in [2, 3.1]). A section of ${\rm
Sym}^{j}({\mathbb{E}})\otimes\det({\mathbb{E}})^{k}$ over
$\mathcal{M}_{3}^{nh}$ pulls back to $[V_{4,0,-1}/{\rm GL}(V)]$, where we now
write $V$ for the standard space of dimension $3$. An identification
$V\cong{\rm Sym}^{2}(W)$ corresponds to an embedding
${\mathbb{P}}^{1}\hookrightarrow{\mathbb{P}}^{2}$ with image a smooth quadric.
If we view $V$ with basis $x,y,z$, the kernel of the projection

${\rm Sym}^{j}(V)={\rm Sym}^{j}({\rm Sym}^{2}(W))\to{\rm Sym}^{2j}(W)$

consists of the polynomials of degree $j$ in $x,y,z$ that vanish on the
quadric. Thus in view of the isotypical decomposition above the exact sequence

$0\to{\rm Sym}^{m-2}({\mathbb{E}})\otimes\mathcal{O}(-A)\to{\rm
Sym}^{m}({\mathbb{E}})\to\pi_{*}(N^{\otimes m})\to 0\,.$

corresponds to (the pullback to $W_{8,-2}^{0}$ of) an exact sequence

$0\to\left({\rm Sym}^{m-2}({\rm Sym}^{2}W)\right)\otimes\det(W)^{2}\to{\rm
Sym}^{m}({\rm Sym}^{2}W)\to{\rm Sym}^{2m}(W)\to 0\,.$

The section $\chi_{4,0,8}$ of ${\rm
Sym}^{4}({\mathbb{E}})\otimes\det({\mathbb{E}})^{8}$ restricted to the
hyperelliptic locus allows three projections according to the decomposition

$None$ ${\rm Sym}^{4}({\rm Sym}^{2}W)\otimes\det(W)^{24}=W_{8,24}\oplus
W_{4,26}\oplus W_{0,28}\,.$

######  Lemma 14.3.

The projections to the three summands in (12) of the pull back of
$\chi_{4,0,8}$ to $\mathcal{H}_{3,2}$ define modular forms on
$\overline{\mathcal{H}}_{3,2}$ of weights $(4,0,8)$, $(2,0,4)$ and $(0,0,14)$
and these are up to a scalar given by the covariants $f_{8,-2}\,\mathfrak{d}$,
$f_{4,-1}\,\mathfrak{d}$ and the discriminant $\mathfrak{d}$.

###### Proof.

The identification of ${\mathbb{E}}$ with ${\rm Sym}^{2}(W)$ corresponds to
the embedding of ${\mathbb{P}}^{1}$ as a conic $C$ in ${\mathbb{P}}^{2}$. A
ternary quartic $Q$ contains $C$ either $0$, $1$ or $2$ times, say $Q=mC+R$
with $0\leq m\leq 2$. The three projections correspond to $R\cap C$ and give
the universal binary octic, the universal binary quartic and $1$ up to
twisting. The first projection was identified in Proposition 6.1. The argument
for the second is similar, while the third descends to
$\overline{\mathcal{H}}_{3}$ and does not vanish on $\mathcal{H}_{3}$.
Therefore it must be a multiple of the disciminant. Taking into account the
action of ${\rm GL}_{2}/{\pm 1_{W}}$ we get the indicated weights (namely
$2(14+\epsilon)$ with $\epsilon=-2,-1,0$). ∎

##  15\. More Modular Forms for Genus Three

We will use more effective divisors on projectivized Hodge bundles to produce
more modular forms. Note that the connection between divisors on projectivized
Hodge bundles and modular forms can also be used in the other direction:
obtaining results on cycle classes using modular forms. We give a few
examples. To a canonical quartic plane curve $C$ we can associate a curve
$\check{S}$ in the dual plane of lines intersecting $C$ equianharmonically. It
corresponds to a contravariant (concomitant) $\sigma$ of the ternary quartic
given by Salmon in [26, p. 264] and it is defined by an equivariant ${\rm
GL}(3)$ embedding $W[4,4,0]\hookrightarrow{\rm Sym}^{2}({\rm Sym}^{4}(W))$. It
gives rise to a divisor in ${\mathbb{P}}({\mathbb{E}}^{\vee})$ and a modular
form $\chi_{0,4,16}$ of weight $(0,4,16)$. We refer to [6, p. 54] for the
relation between invariant theory of ternary quartics and modular forms. The
Siegel modular form $\chi_{0,4,16}$ vanishes with order $2$ at infinity and
order $4$ along the locus $\mathcal{A}_{2,1}$ of decomposable abelian
threefolds. With
$\check{u}:{\mathbb{P}}({\mathbb{E}}^{\vee})\to\overline{\mathcal{M}}_{3}$ the
projection we have
$\check{u}_{*}(\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1))={\mathbb{E}}^{\vee}\cong\wedge^{2}{\mathbb{E}}\otimes\det({\mathbb{E}})^{-1}$
and we thus find an effective divisor on ${\mathbb{P}}({\mathbb{E}}^{\vee})$
over $\tilde{\mathcal{A}}_{3}$ with class
$[\check{S}]=[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(4)]+20\,\lambda-2\,\delta$
and it vanishes with multiplicity $4$ along $\mathcal{A}_{2,1}$. We thus find
on ${\mathbb{P}}({\mathbb{E}}^{\vee})$ over $\overline{\mathcal{M}}_{3}$ a
relation

$[\check{S}]=[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(4)]+20\,\lambda-2\,\delta_{0}-4\,\delta_{1}\,,$

where we identify $\lambda$ and $\delta_{i}$ with their pullbacks to
${\mathbb{P}}({\mathbb{E}}^{\vee})$. Similarly, in the dual plane we have the
sextic $\check{T}$ of lines intersecting the quartic curve in a quadruple of
points with $j$-invariant $1728$. The corresponding concomitant $\tau$
corresponds to $W[6,6,0]\hookrightarrow{\rm Sym}^{3}({\rm Sym}^{4}(W))$ and
defines a modular form of weight $(0,6,24)$ vanishing with multiplicity $3$ at
infinity and multiplicity $6$ along $\mathcal{A}_{2,1}$. We thus get a cycle
relation

$[\check{T}]=[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(6)]+30\,\lambda-3\,\delta_{0}-6\,\delta_{1}\,.$

The concomitant $\sigma^{3}-27\,\tau^{2}$ vanishes on the locus of double
conics and the corresponding modular form of weight $(0,12,48)$ is divisible
by $\chi_{18}^{2}$ as can be checked using the methods of [6]. Dividing by
$\chi_{18}^{2}$ gives a cusp form of weight $(0,12,12)$ vanishing with
multiplicity $2$ at infinity and multiplicity $3$ along $\mathcal{A}_{2,1}$.
It is classically known (see e.g. [4, p. 43]) that this concomitant defines
the dual curve $\check{C}$ to the canonical image $C$ in
${\mathbb{P}}({\mathbb{E}})$. We thus find an effective divisor in
${\mathbb{P}}({\mathbb{E}}^{\vee})$ containing the closure of the dual curve
with class

$12\,[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)]+24\,\lambda-2\,\delta_{0}-3\,\delta_{1}\,.$

This effective divisor class can also be given by the cycle

$B=\\{(C,\eta)\in{\mathbb{P}}({\mathbb{E}}^{\vee}):\,\text{${\rm div}(\eta)$
has a point of multiplicity $2$}\\}$

over $\mathcal{M}_{3}$ and Korotkin and Zograf in [23, Thm. 1] determined the
class of its closure $\overline{B}$

$[\overline{B}]=12\,[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)]+24\,\lambda-2\,\delta_{0}-3\,\delta_{1}\,.$

Another example of an effective divisor for genus $3$ is provided by the
Weierstrass divisor $W$ with class

$[\overline{W}]=24\,[\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)]+68\,\lambda-6\,\delta_{0}-12\,\delta_{1}$

as given by Gheorghita in [13]. Here we get a section of

${\rm
Sym}^{24}(\wedge^{2}{\mathbb{E}})\otimes\det({\mathbb{E}})^{44}(-6\,\delta_{0}-12\,\delta_{1})$

This gives a Teichmüller modular form of weight $(0,24,44)$ vanishing with
multiplicity $6$ at the cusp. It descends to a Siegel modular form.

######  Corollary 15.1.

The dual of the canonical curve defines a Siegel modular cusp form of degree
$3$ of weight $(0,12,12)$ vanishing with multiplicity $2$ at infinity. The
Weierstrass divisor defines a cusp form of weight $(0,24,44)$ vanishing with
multiplicity $6$ at infinity.

##  16\. The hypertangent divisor

A generic canonically embedded curve $C$ of genus $3$ has $24$ (Weierstrass)
points where the tangent line intersects $C$ with multiplicity $3$. The union
of these $24$ lines forms a divisor in ${\mathbb{P}}^{2}$. Taking the closure
of this divisor for the universal curve over $\mathcal{M}_{3}$ defines a
divisor $H$ in ${\mathbb{P}}({\mathbb{E}})$ over $\overline{\mathcal{M}}_{3}$
which we call the hypertangent line divisor. We calculate the class of this
divisor over $\overline{\mathcal{M}}_{3}$ and also calculate the class of a
corresponding divisor over $\overline{\mathcal{H}}_{3,2}$.

The calculation over $\overline{\mathcal{M}}_{3}$ uses the divisors
$\check{S}$ and $\check{T}$ in ${\mathbb{P}}({\mathbb{E}}^{\vee})$ over
$\overline{\mathcal{M}}_{3}$ as defined in the preceding section. It is a
classical result that the intersection $\check{S}\cdot\check{T}$ in the
generic fibre is the $0$-cycle consisting of the $24$ points defining the $24$
hyperflexes of the generic curve $C$, see [26]. We consider the incidence
variety

$I=\\{(p,\ell)\in{\mathbb{P}}({\mathbb{E}})\times_{\overline{\mathcal{M}}_{3}}{\mathbb{P}}({\mathbb{E}}^{\vee}):p\in\ell\\}$

Let $\rho:I\to{\mathbb{P}}({\mathbb{E}})$ and
$\check{\rho}:I\to{\mathbb{P}}({\mathbb{E}}^{\vee})$ be the two projections
fitting in the commutative diagram

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\check{\rho}}$$\textstyle{{\mathbb{P}}({\mathbb{E}}^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{u}}$$\textstyle{{\mathbb{P}}({\mathbb{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\overline{\mathcal{M}}_{3}}$

We have the tautological sequence on ${\mathbb{P}}({\mathbb{E}})$

$0\to F\to
u^{*}({\mathbb{E}})\to\mathcal{O}_{{\mathbb{P}}({\mathbb{E}})}(1)\to 0$

and a similar one on ${\mathbb{P}}({\mathbb{E}}^{\vee})$

$0\to\check{F}\to\check{u}^{*}({\mathbb{E}}^{\vee})\to\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)\to
0\,.$

Now note that $I$ can be identified with the ${\mathbb{P}}^{1}$-bundle
${\mathbb{P}}({{F}^{\vee}})$ on ${\mathbb{P}}({\mathbb{E}})$, but also with
the ${\mathbb{P}}^{1}$-bundle ${\mathbb{P}}(\check{F}^{\vee})$ on
${\mathbb{P}}({\mathbb{E}}^{\vee})$.

The tautological inclusion $F\to u^{*}{\mathbb{E}}$ induces a surjection
$u^{*}{\mathbb{E}}^{\vee}\to F^{\vee}$ and this gives an inclusion
${\mathbb{P}}(F^{\vee})\to{\mathbb{P}}(u^{*}{\mathbb{E}}^{\vee})$ of
projective bundles over ${\mathbb{P}}({\mathbb{E}})$ which composed with
natural map
${\mathbb{P}}(u^{*}{\mathbb{E}}^{\vee})\to{\mathbb{P}}({\mathbb{E}}^{\vee})$
gives the map
$\check{\rho}:I={\mathbb{P}}(F^{\vee})\to{\mathbb{P}}({\mathbb{E}}^{\vee})$.
This implies

$None$
$\mathcal{O}_{\mathbb{P}(F^{\vee})}(1)=\check{\rho}^{*}\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1)\quad\text{\rm
and
similarly}\quad\mathcal{O}_{\mathbb{P}(\check{F}^{\vee})}(1)={\rho}^{*}\mathcal{O}_{{\mathbb{P}}({\mathbb{E}})}(1)\,.$

With

$f=c_{1}(F),\quad\check{f}=c_{1}(\check{F}),\quad
h=c_{1}(\mathcal{O}_{{\mathbb{P}}({\mathbb{E}})}(1)),\quad\check{h}=c_{1}(\mathcal{O}_{{\mathbb{P}}({\mathbb{E}}^{\vee})}(1))\,,$

this gives the identities of pullbacks of the first Chern classes
$c_{1}({\mathbb{E}})=-c_{1}({\mathbb{E}}^{\vee})=\lambda$

$\rho^{*}(f)+\rho^{*}(h)=\rho^{*}u^{*}(\lambda)=\check{\rho}^{*}\check{u}^{*}(\lambda)=-\check{\rho}^{*}(\check{f})-\check{\rho}^{*}(\check{h})\,.$

Since $I={\mathbb{P}}(F^{\vee})$ over ${\mathbb{P}}({\mathbb{E}})$ and
$\check{\rho}^{*}\check{h}=c_{1}({\mathcal{O}}_{{\mathbb{P}}(F^{\vee})}(1))$,
the Chern classes of $F^{\vee}$ and the first Chern class of the tautological
line bundle satisfy the relation

$None$
$\check{\rho}^{*}\check{h}^{2}+\rho^{*}(f)\,\check{\rho}^{*}(\check{h})+\rho^{*}(c_{2}(F))=0\,.$

######  Corollary 16.1.

Under the map $\rho_{*}\check{\rho}^{*}$ we have

$\check{h}^{2}\mapsto
h-u^{*}(\lambda),\quad\check{h}\,\check{u}^{*}(\xi)\mapsto
u^{*}(\xi),\quad\check{u}^{*}(\eta)\mapsto 0$

for $\xi\in{\rm CH}^{1}(\overline{\mathcal{M}}_{3})$ and $\eta\in{\rm
CH}^{2}(\overline{\mathcal{M}}_{3})$.

###### Proof.

Using relation (14) gives

$\rho_{*}(\check{\rho}^{*}(\check{h})^{2})=-\rho_{*}(\rho^{*}(f)\check{\rho}^{*}(\check{h})-\rho^{*}(c_{2}(F))=-f=h-u^{*}(\lambda)\,.$

The other properties follow from general intersection theory. ∎

Let now $\psi$ be the class of the codimension $2$ cocycle
$\check{S}\cdot\check{T}$.

######  Lemma 16.2.

We have
$\rho_{*}\check{\rho}^{*}\psi=24\,h+216\,\lambda-24\,\delta_{0}-48\,\delta_{1}$.

###### Proof.

By the results of the preceding section we have

$\psi=24\,\check{h}^{2}+240\,\check{h}\lambda-24\,\check{h}\delta_{0}-48\,\check{h}\delta_{1}+r$

with $r\in\check{\rho}^{*}{\rm CH}^{2}(\overline{\mathcal{M}}_{3})$. Corollary
16.1 implies the result. ∎

We now claim that the codimension $2$ cycle $\check{S}\cdot\check{T}$ when
restricted to the hyperelliptic locus is of the form $12\,\check{h}$, in other
words, by (2) it contains an effective codimension $2$ cycle with class

$12\,(9\,\lambda-\delta_{0}-3\,\delta_{1})\,\check{h}+\check{u}^{*}(\xi)$

with $\xi$ a codimension $2$ class on $\overline{\mathcal{M}}_{3}$. We check
this using the explicit form of the two concomitants $\sigma$ and $\tau$
defining $\check{S}$ and $\check{T}$. Here $\sigma$ is a polynomial of degree
$4$ in $a_{0},\ldots,a_{14}$ and degree $4$ in the coordinates
$u_{0},u_{1},u_{2}$ where $a_{0},\ldots,a_{14}$ are the coefficients of the
general ternary quartic. A calculation shows that $\sigma$ restricted to the
locus of double conics becomes a square $q^{2}$ with $q$ of degree $2$ in the
$u_{i}$, while $\tau$ becomes a cube $q^{3}$. Hence the cycle $S^{\vee}\cdot
T^{\vee}$ restricted to the hyperelliptic locus is represented by an effective
cycle representing $6\,q\sim 12\check{h}$. By Corollary 16.1 under
$\rho_{*}\hat{\rho}^{*}$ this is sent to an effective cycle with class
$12(9\,\lambda-\delta_{0}-3\,\delta_{1})$. Since $H$ is defined as the closure
of the hypertangent divisor in the generic fibre, the class of $H$ equals
$\rho_{*}\check{\rho}^{*}\psi$ minus $12$ times the class of the hyperelliptic
locus; by Lemma 16.2 we get

$24\,h+216\,\lambda-24\,\delta_{0}-48\,\delta_{1}-12(9\lambda-\delta_{0}-3\delta_{1})=24\,h+108\,\lambda-12\,\delta_{0}-12\,\delta_{1}\,.$

We summarize.

######  Proposition 16.3.

The class $[H]$ of the hypertangent divisor $H$ in
${\mathbb{P}}_{\overline{\mathcal{M}}_{3}}({\mathbb{E}})$ equals
$[\mathcal{O}_{\mathbb{P}({\mathbb{E}})}(24)]+108\,\lambda-12\,\delta_{0}-12\,\delta_{1}$.
It gives rise to a Siegel modular form of degree $3$ and weight $(24,0,108)$
vanishing with multiplicity $12$ along the boundary.

We now work on the Hurwitz space and define and calculate the class of a
hypertangent $H_{h}$ divisor there. It is defined by taking the eight tangent
lines at the ramification points of the canonical image. More precisely, on
$\tilde{\mathbb{P}}$ we have the line bundle $N$ defined in (9). Recall that
$\tilde{S}_{k}$ for $1\leq k\leq 8$ is the pullback of the section $S_{k}$ of
$\pi_{9}:\overline{\mathcal{M}}_{0,9}\to\overline{\mathcal{M}}_{0,8}$. Under
restriction to the hyperelliptic locus the Weierstrass points degenerate to
the ramification points. We define the corresponding hypertangent divisor
$H_{h}$ in ${\mathbb{P}}({\mathbb{E}})$ over $\overline{\mathcal{H}}_{3,2}$ by
taking the tangents to the canonical image of the generic curve at the points
of the sections $\tilde{S_{k}}$, $k=1,\ldots,8$ over $\mathcal{H}_{3,2}$ and
then taking the closure over $\overline{\mathcal{H}}_{3,2}$.

We now consider the bundle $N(-2\tilde{S}_{k})$ on $\tilde{\mathbb{P}}$. This
line bundle is trivial on the generic fibre of $\pi:\tilde{\mathbb{P}}\to B$,
so $\pi_{*}(N(-2\tilde{S}_{k}))$ is a line bundle on $B$.

######  Lemma 16.4.

We have

$c_{1}(R^{1}\pi_{*}N(-2\tilde{S}_{k}))=\Delta_{2}(k^{+})+\Delta_{3}(k^{+}),\quad\text{\rm
and}\quad
c_{1}(\pi_{*}N(-2\tilde{S}_{k}))=-\Delta_{3}(k^{+})-\Delta_{3}+E_{k}\,.$

###### Proof.

Recall that $N=\mathcal{O}(D_{k}+\pi^{*}(E_{k})-R)$. The first statement
follows by analyzing the restrictions over the boundary components. For the
second we apply Grothendieck-Riemann-Roch as in the proof of Proposition 10.2.
By (7) and (8) we have

$\displaystyle c_{1}(\pi_{*}N(-2\tilde{S}_{k}))$
$\displaystyle=-\Delta_{2}(k^{+})-2\,\Delta_{3}(k^{+})-\Delta_{3}+c_{1}(R^{1}\pi_{*}N(-2\tilde{S}_{k}))$
$\displaystyle=-\Delta_{3}(k^{+})-\Delta_{3}+E_{k}\,.$

∎

Put $\mathcal{F}_{k}=\pi_{*}(N(-2\tilde{S}_{k}))$. The injection
$N(-2\tilde{S}_{k})\hookrightarrow N$ induces an injection
$\mathcal{F}_{k}\to{\mathbb{E}}$. Pulling back to $\mathbb{P}({\mathbb{E}})$
via $u^{*}$ and composing with the canonical surjection
$u^{*}({\mathbb{E}})\to\mathcal{O}_{\mathbb{P}({\mathbb{E}})}(1)$ we get an
induced map

$q:u^{*}\mathcal{F}_{k}\to\mathcal{O}_{\mathbb{P}({\mathbb{E}})}(1)\,.$

The degeneracy locus of $q$ is an effective divisor $F_{k}$ that is the
vanishing divisor of a section of
$\mathcal{O}_{\mathbb{P}({\mathbb{E}})}(1)\otimes u^{*}\mathcal{F}_{k}^{-1}$.
The interpretation is as follows. The map $\phi$ defines an embedding of the
generic fibre of $\tilde{\mathbb{P}}$ into the generic fibre
${\mathbb{P}}({\mathbb{E}})$. If we identify
$H^{0}({\mathbb{P}}^{1},\mathcal{O}(2))$ with the fibre of ${\mathbb{E}}$ and
projectivize, the divisor $p_{1}+p_{2}\in|\mathcal{O}(2)|$ is mapped to the
line through through the points $\phi(p_{1})$, $\phi(p_{2})$. We now sum these
divisors $F_{k}$ and get an effective divisor $H_{h}$ with class

$\displaystyle{}[H_{h}]=8\,[\mathcal{O}(1)]-\sum_{k=1}^{8}[u^{*}\mathcal{F}_{k}]$
$\displaystyle=[\mathcal{O}(8)]+u^{*}(3\,\Delta_{3}+8\,\Delta_{3}-\sum_{k=1}^{8}E_{k})$
$\displaystyle=[\mathcal{O}(8)]+u^{*}(8\,\lambda-2\,\Delta_{2}+\Delta_{3})\,,$

where we use the formulas of Section 10 and Section 12.

We can now compare the class of the hyperelliptic hypertangent divisor $H_{h}$
with that of the pull back of the hypertangent divisor $H$ to the Hurwitz
space. By Proposition 16.3 the pull back of $H$ has class
$[\mathcal{O}(24)]+108\,\lambda-24\,(\Delta_{2}+\Delta_{4})-12\,\Delta_{1}$.
Since the $24$ Weierstrass points collapse with multiplicity $3$ to the $8$
ramification points we compare the class (of the pull back of) $[H]$ with that
of $3\,[H_{h}]$. Substituting the formula for $\lambda$ we get

$[H]-3\,[H_{h}]=9\,\Delta_{1}\,,$

which means that the pull back of $H$ vanishes with multiplicity $9$ at the
hyperelliptic boundary component $\Delta_{1}$.

##  17\. Genus Four

For a smooth curve $C$ of genus $4$ the natural map ${\rm
Sym}^{2}(H^{0}(C,\omega_{C}))\to H^{0}(C,\omega_{C}^{\otimes 2})$ is
surjective and the kernel has dimension $1$. It determines a quadric in
${\mathbb{P}}^{3}={\mathbb{P}}(H^{0}(C,\omega_{C}))$ containing the canonical
curve. Over $\overline{\mathcal{M}}_{4}$ we find a corresponding exact
sequence

$0\to U\to{\rm Sym}^{2}({\mathbb{E}})\to\pi_{*}\omega_{C}^{\otimes 2}\to 0\,.$

The line bundle $U$ has first Chern class
$5\,\lambda-(13\,\lambda-\delta)=-8\,\lambda+\delta$ by Mumford’s calculation
of $c_{1}(\pi_{*}\omega_{c}^{\otimes 2})$ [24, Thm. 5.10]. In the bundle
${\mathbb{P}}({\mathbb{E}})$ the quadric containing the canonical curve
determines a divisor $Q$. Let
$u:{\mathbb{P}}({\mathbb{E}})\to\overline{\mathcal{M}}_{4}$ be the projection.

######  Lemma 17.1.

The divisor class of $Q$ satisfies:
$[Q]=[\mathcal{O}(2)]+u^{*}(8\,\lambda-\delta)$.

###### Proof.

Observe that $u^{*}u_{*}\mathcal{O}(2)=u^{*}({\rm Sym}^{2}({\mathbb{E}}))$.
The natural morphism $u^{*}u_{*}\mathcal{O}(2)\to\mathcal{O}(2)$ induces
$u^{*}U\to\mathcal{O}(2)$. The divisor $Q$ is the vanishing locus of this
morphism, hence has class $[\mathcal{O}(2)]+u^{*}(8\,\lambda-\delta)$. ∎

######  Corollary 17.2.

The effective divisor $Q$ defines a Teichmüller modular cusp form $\chi$ of
genus $4$ and weight $(2,0,0,8)$.

If we view a section of ${\rm Sym}^{2}({\mathbb{E}})$ as a quadratic form on
${\mathbb{E}}^{\vee}$ we can take the discriminant, cf. [7]. Doing this with
the form $\chi$ of weight $(2,0,0,8)$ just constructed we get a scalar-valued
modular form $D(\chi)$ of weight $34$. This modular form vanishes on the
closure of the locus of curves whose canonical model lies on a quadric cone.
This locus has class $34\lambda-4\,\delta_{0}-14\,\delta_{1}-18\,\delta_{2}$
by Teixidor i Bigas [27, Prop. 3.1] and equals the divisor of curves with a
vanishing thetanull. The modular form $D(\chi)$ is the square root of the
restriction to $\mathcal{M}_{4}$ of the product of the even theta
characteristics on $\mathcal{A}_{4}$.

##  18\. Appendix on base-point freeness

The relative dualizing sheaf $\omega_{\pi}$ of the universal family
$\pi:{\mathcal{C}}_{g}\to{\mathcal{M}}_{g}$ of genus $g$ smooth curves is base
point free and the surjection $\pi^{*}{\mathbb{E}}\to\omega_{\pi}$ gives a map
$\varphi:{\mathcal{C}}_{g}\to{\mathbb{P}}(\mathbb{E})$ over
${\mathcal{M}}_{g}$, which is generically an embedding. Let $\Gamma$ be the
image $\varphi({\mathcal{C}}_{g})$. We wish to describe the closure of the
image over the generic points of the boundary components $\Delta_{i}$ for
$i=0,\ldots,[g/2]$. Over the general point of $\Delta_{0}$ the sheaf
$\omega_{\pi}$ is base point free and the map $\varphi$ extends over this
locus. But over the general point of $\Delta_{i}$, $i\geq 1$, which represents
a nodal curve of the form $C_{1}\cup C_{2}$, with $C_{1}$, $C_{2}$ smooth
curves of genus $i$ and $g-i$ meeting at a nodal point $x$, the sheaf
$\omega_{\pi}$ has a base point at $x$. We consider a family $\pi:Y\to B$ of
stable curves of genus $g$ with $B$ the spectrum of a discrete valuation ring.
We assume that the central fibre $C$ is a nodal curve $C=C_{1}\cup C_{2}$ of
genera $i$ and $g-i$ and smooth generic fibre. After a degree $2$ base change
$B^{\prime}\to B$ we get an $A_{1}$-singularity which we resolve resulting in
a semistable family $\pi^{\prime}:X\to B^{\prime}$ with a special fibre which
is a chain of three curves $C=C_{1}^{\prime}\cup R\cup C_{2}^{\prime}$ with
$C_{1}^{\prime}\cong C_{1}$ and $C_{2}^{\prime}\cong C_{2}$ smooth curves of
genus $i$ and $g-i$ and $R$ a rational $(-2)$-curve. We have the commutative
diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\scriptstyle{\pi^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{B}$

The morphism $v$ is $(2:1)$ ramified at $C_{1},C_{2}$. We have
$v^{*}\omega_{\pi}=\omega_{\pi^{\prime}}$, and
$\sigma^{*}\mathbb{E}_{B}=\mathbb{E}_{B^{\prime}}$ and
$v^{*}C_{j}=2\,C^{\prime}_{j}+R$ for $j=1,2$. There is then a natural $(2:1)$
map $\mathbb{P}(\mathbb{E}_{B^{\prime}})\to{\mathbb{P}}(\mathbb{E}_{B})$.

Now we will show that the system $\omega_{\pi^{\prime}}(-R)$ defines a map
$X\to\mathbb{P}(\mathbb{E}_{B^{\prime}})$ which combined with the above
$(2:1)$ map gives a $(2:1)$ map
$\varphi^{\prime}:X\to\mathbb{P}(\mathbb{E}_{B})$ mapping the curves
$C^{\prime}_{1}$ and $C_{2}^{\prime}$ to their canonical image and $R$ to a
double line. The reduced image of the map $\varphi^{\prime}$ describes the
closure of $D$ over $b_{0}$, the special point of $B$.

To avoid unnecessary notation we now write $\pi:X\to B$ for the semi-stable
family denoted by $\pi^{\prime}:X\to B^{\prime}$ above.

######  Proposition 18.1.

Let $\omega$ be the relative dualizing sheaf of ${\pi}:X\to B$. Then we have
${\pi}_{*}(\omega(-R))\cong\pi_{*}(\omega)$ and the central fibre of
$\pi_{*}(\omega(-R))$ is of codimension $1$ in $H^{0}(C,\omega(-R))$ and
defines a base point free linear system on $C$.

###### Proof.

We let $q=C_{1}\cap R$ and $p=C_{2}\cap R$. The exact sequence
$0\to\omega(-R)\to\omega\to\omega_{|R}\to 0$ induces a sequence

$0\to\pi_{*}(\omega(-R))\to\pi_{*}(\omega)\xrightarrow{r}\pi_{*}(\omega)_{|R}$

and the map $r$ is zero because
$\omega_{|C}=(\omega_{C_{1}}(q),\mathcal{O}_{R},\omega_{C_{2}}(p))$, therefore
the restrictions to $C_{1}$ (resp. $C_{2})$ must vanish at $q$ (resp. $p$),
hence extend by $0$ on $R$. We thus see by the exactness that
$\pi_{*}(\omega(-R))\cong\pi_{*}(\omega)$.

Next we observe that $\dim H^{0}(C,\omega(-R))=g_{1}+g_{2}+1$ with $g_{i}$ the
genus of $C_{i}$. This follows directly from
$\omega(-R)_{|C}=(\omega_{C_{1}},\mathcal{O}_{R}(2),\omega_{C_{2}})$.

We have the exact sequence

$None$ $0\to\omega(-R-C_{1})\to\omega(-R)\to\omega(-R)_{|C_{1}}\to 0\,,$

where $\omega(-R)_{|C_{1}}\cong\omega_{C_{1}}$ and
$\omega(-R-C_{1})_{|C}=(\omega_{C_{1}}(q),\mathcal{O}_{R}(1),\omega_{C_{2}})$.
For a section $(s_{1},s,s_{2})\in H^{0}(C,\omega(-R-C_{1}))$ the section $s$
is the unique section of $\mathcal{O}_{R}(1)$ that vanishes at $q$ and with
$s(p)=s_{2}(p)$. We thus see $\dim H^{0}(C,\omega(-R-C_{1}))=g_{1}+g_{2}$.
Therefore $h^{0}(\omega(-R-C_{1}))$ has constant rank $g_{1}+g_{2}$ on the
fibres of $\pi$, hence $R^{1}\pi_{*}(\omega(-R-C_{1}))$ is a line bundle. We
conclude that the special fibre of $\pi_{*}(\omega(-R-C_{1}))$ equals
$H^{0}(C,\omega(-R-C_{1}))$. But $\pi_{*}(\omega(-R)_{|C_{1}})$ is a torsion
sheaf, hence the connecting homomorphism $\pi_{*}(\omega(-R)_{|C_{1}})\to
R^{1}\pi_{*}(\omega(-R-C_{1}))$ of (15) must be zero and we get an induced
exact sequence

$0\to\pi_{*}(\omega(-R-C_{1}))\xrightarrow{\iota}\pi_{*}(\omega(-R))\xrightarrow{j}\pi_{*}(\omega(-R))_{|C_{1}})\to
0\,.$

Consider now a section $\sigma$ of $\pi_{*}(\omega(-R-C_{1}))$ with
restriction $(s_{1},s,s_{2})$ to $C$. Suppose that $s\neq 0$. If we multiply
$\sigma$ with a local section $\tau$ of $\mathcal{O}(C_{1})$ on $X$ with
divisor $C_{1}$ then $\iota(\sigma)=\sigma\cdot\tau_{|C}$ has as restriction
to $R$ a section of $\mathcal{O}_{R}(2)$ vanishing with multiplicity $2$ at
$q$ and therefore it does not vanish anywhere else. Hence the subspace of the
special fibre $V$ of $\pi_{*}(\omega(-R))$ of sections vanishing on $C_{1}$
has $q$ as only base point on $R$. Furthermore, the map $j$ is surjective, and
choosing a section $s_{1}\in H^{0}(C_{1},\omega_{C_{1}})$ with $s_{1}(q)\neq
0$ we see that $q$ is not a base point. Therefore there are no base points on
$R$. By the surjectivity of $j$ the restriction of $V$ to $C_{1}$ is
$H^{0}(C_{1},\omega_{C_{1}})$ and therefore there are no base points on
$C_{1}$. By symmetry the same holds for $C_{2}$.

Similarly to (15) we have an exact sequence

$0\to\omega(-R-C_{1}-C_{2})\to\omega(-R)\to\omega(-R)_{|C_{1}+C_{2}}\to 0$

and by a similar reasoning we see that we get an exact sequence

$0\to\pi_{*}(\omega(-R-C_{1}-C_{2}))\xrightarrow{\iota}\pi_{*}(\omega(-R))\xrightarrow{j}\pi_{*}(\omega(-R))_{|C_{1}+C_{2}})\to
0\,.$

This implies that given $s_{1}\in H^{0}(C_{1},\omega_{C_{1}})$ and $s_{2}\in
H^{0}(C_{2},\omega_{C_{2}})$ there is a unique element $(s_{1},s,s_{2})$ in
the special fibre $V$ of $\pi_{*}(\omega(-R))$ mapping to $(s_{1},s_{2})$
under $j$. The morphism $X\to{\mathbb{P}}(\pi_{*}(\omega(-R)))$ is given by
the surjection $\pi^{*}\pi_{*}(\omega(-R))\to\omega(-R)$. The image of the
curve $C$ in the special fibre of ${\mathbb{P}}({\mathbb{E}})$ consists of the
canonical images of $C_{1}$ and $C_{2}$, provided with with image of $p$ and
$q$ and the image of $R$, that is, the line connecting the images of $p$ and
$q$. If the genus $g(C_{i})=1$ then the image of $C_{i}$ is a point. ∎

##  19\. Appendix: divisor classes of Gheorghita-Tarasca and Korotkin-
Sauvaget-Zograf

Here we apply the method employed in Section 16 to determine in a relatively
straightforward way the divisor classes of two divisors in
${\mathbb{P}}({\mathbb{E}}^{\vee}_{k})$ with
${\mathbb{E}}_{k}=\pi_{*}(\omega_{\pi}^{k})$, thus reproving a theorem of
Gheorghita-Tarasca ([14, Thm 1]) and a theorem of Korotkin-Sauvaget-Zograf
([22, Thm. 1.12]). The first divisor is a generalization of a divisor in
${\mathbb{P}}({\mathbb{E}}^{\vee})$ considered by Gheorghita in [13]. We
consider in ${\mathbb{P}}({\mathbb{E}}^{\vee}_{k})$ over $\mathcal{M}_{g}$ the
divisor

$G_{k}=\\{(C,\omega)\in{\mathbb{P}}({\mathbb{E}}_{k}^{\vee}):\text{${\rm
div}(\omega)$ contains a Weierstrass point}\\}$

and let $\overline{G}_{k}$ be the closure of $G_{k}$ in
${\mathbb{P}}({\mathbb{E}}^{\vee}_{k})$ over $\overline{\mathcal{M}}_{g}$. We
let
$\check{u}:{\mathbb{P}}({\mathbb{E}}_{k}^{\vee})\to\overline{\mathcal{M}}_{g}$
be the natural morphism and $\check{h}$ the hyperplane class on
${\mathbb{P}}({\mathbb{E}}_{k}^{\vee})$.

######  Theorem 19.1.

(Gheorghita-Tarasca) The class of $\overline{G}_{k}$ is given by

$\frac{1}{k}[\overline{G}_{k}]=g(g^{2}-1)\check{h}+2(3g^{2}+2g+1)\check{u}^{*}\lambda-\binom{g+1}{2}\check{u}^{*}\delta_{0}-\sum_{i=1}^{[g/2]}(g-i)i(g+3)\check{u}^{*}\delta_{i}\,.$

The second divisor is the divisor $Z_{k}$ in
${\mathbb{P}}({\mathbb{E}}_{k}^{\vee})$ over $\overline{\mathcal{M}}_{g}$ of
regular $k$-differentials for $k\geq 2$ possessing a double zero.

######  Theorem 19.2.

(Korotkin-Sauvaget-Zograf) The class of the divisor $Z_{k}$ for $k\geq 2$ for
$k\geq 2$ and $(g,k)\neq(2,2)$ is given by

$[Z_{k}]=(4k+2)(g-1)\,\check{h}+k(k+1)\,\check{u}^{*}(12\lambda-\sum_{i=0}^{[g/2]}\delta_{i})\,.$

For the proof of both theorems we use, as in Section 16, the incidence variety
$I_{k}$ between $\mathbb{P}(\mathbb{E}_{k})$ and
$\mathbb{P}(\mathbb{E}^{\vee}_{k})$ which fits in the following commutative
diagram:

$\textstyle{I_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\check{\rho}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathbb{P}}(\mathbb{E}_{k}^{\vee})}$$\scriptstyle{\check{u}}$$\textstyle{\mathbb{P}(\mathbb{E}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\overline{\mathcal{M}}_{g}}$

We denote by $h$ (resp. $\check{h}$) the first Chern class of the hyperplane
line bundle on ${\mathbb{P}}(\mathbb{E}_{k})$ (resp.
${\mathbb{P}}(\mathbb{E}_{k}^{\vee})$), thus suppressing the dependence on
$k$. As explained in Section 16, we have
$I_{k}={\mathbb{P}}(\check{F}_{k}^{\vee})$ as a bundle over
${\mathbb{P}}(\mathbb{E}_{k}^{\vee})$, with $\check{F_{k}}$ defined by the
exact sequence on ${\mathbb{P}}({\mathbb{E}}_{k}^{\vee})$

$0\to\check{F_{k}}\to\check{u}^{*}\mathbb{E}_{k}^{\vee}\to\mathcal{O}_{\mathbb{P}(\mathbb{E}_{k}^{\vee})}(1)\to
0.$

Then $\rho^{*}h={\mathcal{O}}_{\mathbb{P}(\check{F_{k}}^{\vee})}(1)$.
Similarly, $I_{k}=\mathbb{P}(F^{\vee}_{k})$ as a bundle over
$\mathbb{P}(\mathbb{E}_{k})$, with $F_{k}$ the tautological rank $r-1$ bundle
on $\mathbb{P}(\mathbb{E}_{k})$. Then
$\check{\rho}^{*}(\check{h})={\mathcal{O}}_{\mathbb{P}({F_{k}}^{\vee})}(1)$.

We let

$\gamma=\check{\rho}_{*}\rho^{*}:{\rm
CH}_{\mathbb{Q}}^{*}({\mathbb{P}}({\mathbb{E}}_{k}))\to{\rm
CH}_{\mathbb{Q}}^{*}({\mathbb{P}}({\mathbb{E}}_{k}^{\vee}))$

be the induced map.

######  Lemma 19.3.

We have $\gamma(h^{i})=0$ for $i\leq r-3$, $\gamma(h^{r-2})=1$ and
$\gamma(h^{r-1})=\check{h}+\check{u}^{*}{\rm c}_{1}(\mathbb{E}_{k})$.

###### Proof.

For dimension reasons $\gamma(h^{i})=0$ for $i\leq r-3$. Moreover
$\gamma(h^{r-2})=1$ by construction. Applying $\check{\rho}_{*}$ to the Chern
class relation
$\sum_{i=0}^{r-1}(-1)^{r-1-i}(\rho^{*}h)^{i}\check{\rho}^{*}c_{r-1-i}(\check{F}_{k}^{\vee})=0$
we get $\gamma(h^{r-1})=c_{1}(\check{F}_{k}^{\vee})$ and this equals
$\check{h}+\check{u}^{*}{\rm c}_{1}(\mathbb{E}_{k})$ by the exact sequence. ∎

Proof of Theorem 19.1. Let $W$ be the Weierstrass divisor on
$\overline{\mathcal{C}}_{g}$. This is an irreducible divisor. We denote by
$\varphi_{k}:\overline{\mathcal{C}}_{g}\to{\mathbb{P}}({\mathbb{E}}_{k})$ the
morphism defined in Section 3. For $k\geq 2$ we have
$G_{k}=\rho_{*}\rho^{*}(\varphi_{k}(W))$ over $\mathcal{M}_{g}$ and
$\rho_{*}\rho^{*}$ sends an irreducible divisor over
$\overline{\mathcal{M}}_{g}$ to an irreducible divisor. Therefore we have
$[\overline{G}_{k}]=\gamma({\varphi_{k}}_{*}[W])$. The group ${\rm
Pic}_{\mathbb{Q}}(\overline{\mathcal{C}}_{g})$ is generated by
$\omega_{\pi},\pi^{*}\lambda$, $\pi^{*}\delta_{0}$ and $\gamma_{i}$ (for
$i=1,\ldots,g-1$) with $\gamma_{i}$ the divisor class defined by the component
of genus $i$ lying over $\Delta_{\min(i,g-i)}$. By Cuckierman [9] the divisor
class $[W]$ can be written as $w_{1}-w_{2}$ with

$w_{1}=\binom{g+1}{2}\omega_{\pi}-\pi^{*}\lambda,\quad
w_{2}=\binom{g-i+1}{2}\sum_{i=1}^{g-1}\gamma_{i}\,.$

Note that $\varphi_{k}^{*}h=k\,\omega_{\pi}$. We thus get

$\displaystyle{\varphi_{k}}_{*}w_{1}$
$\displaystyle={\varphi_{k}}_{*}(\frac{g(g+1)}{2}\omega_{\pi}-\pi^{*}\lambda)={\varphi_{k}}_{*}\varphi_{k}^{*}\left(\frac{g(g+1)}{2k}h-u^{*}\lambda\right)$
$\displaystyle=\left(\frac{g(g+1)}{2k}h-u^{*}\lambda\right)\,{\varphi_{k}}_{*}[1]=\left(\frac{g(g+1)}{2k}h-u^{*}\lambda\right)(\sum_{i=0}^{r-2}h^{i}\beta_{r-2-i})$
$\displaystyle=\sum_{i=0}^{r-1}h^{i}u^{*}(\frac{g(g+1)}{2k}\beta_{r-1-i}-\lambda\beta_{r-2-i})\,,$

where we used (1). Lemma 19.3 implies that under applying $\gamma$ only the
terms with $h^{j}$ where $j=r-1$ or $r-2$ contribute and we get

$\gamma({\varphi_{k}}_{*}w_{1})=\gamma(h^{r-1})\frac{g(g+1)}{2k}\beta_{0}+\gamma(h^{r-2})\check{u}^{*}(\frac{g(g+1)}{2k}\beta_{1}-\lambda\beta_{0})\,.$

Substituting the expressions for $\beta_{0}$ and $\beta_{1}$ from Proposition
3.1 leads to

$\gamma({\varphi_{k}}_{*}w_{1})=g(g^{2}-1)\,\check{h}+2\,k\,(3g^{2}+2g+1)\check{u}^{*}\lambda-{k\over
2}\,g(g+1)\sum_{i=0}^{[g/2]}\check{u}^{*}\delta_{i}\,.$

For the term $\gamma({\varphi_{k}}_{*}w_{2})$ we first observe
$\gamma({\varphi_{k}}_{*}\gamma_{i})=(2i-1)k\check{u}^{*}\delta_{i}$ because
the component of genus $i$ over $\Delta_{\min(i,g-i)}$ has degree $(2i-1)k$ in
${\mathbb{P}}^{r-1}$ and thus maps under $\gamma$ to $(2i-1)k$ times the class
$[1]$ of $({\mathbb{P}}^{r-1})^{\vee}$ over $\Delta_{\min(i,g-i)}$. This gives

$\displaystyle\gamma(\varphi_{k*}w_{2})$
$\displaystyle=\frac{k}{2}\sum_{i=1}^{g-1}(g-i)(g-i+1)(2i-1)\,\check{u}^{*}\delta_{i}$
$\displaystyle=\frac{k}{2}\sum_{i=1}^{[g/2]}((g-i)(g-i+1)(2i-1)+i(i+1)(2g-2i-1))\check{u}^{*}\delta_{i}$
$\displaystyle=\frac{k}{2}\sum_{i=1}^{[g/2]}(2i(g-i)(g+3)-g(g+1))\check{u}^{*}\delta_{i}.$

Together this gives the correct expression for class of $[\overline{G}_{k}]$
as in Theorem 19.1.

When $k=1$, over ${\mathcal{M}}_{g}\cup\Delta_{0}$ we work as above and the
coefficients of $\lambda$ and $\delta_{0}$ in the formula are as in the case
$k\geq 2$. To find the contribution of $\delta_{i}$ in the formula of
$\beta_{1}$, we work over the family over a base $B$, as in Appendix 18, where
we have the $(2:1)$ morphism $\varphi^{\prime}:X\to\mathbb{P}(\mathbb{E})$
defined by the $\omega_{\pi^{\prime}}(-R)$. We follow the notation of Appendix
18 and in the formulas we only need to consider terms that contribute to the
boundary class $\delta_{i}$. By [9] the Weierstrass divisor does not pass
through the node of a general element over $\Delta_{i}$ and thus $v^{*}W$ does
not contain the ‘exceptional’ divisor $R$. We have by Cuckierman’s formula
$[v^{*}W]=w_{1}-w_{2}$ with
$w_{1}=\binom{g+1}{2}\omega_{\pi^{\prime}}-{\pi^{\prime}}^{*}\lambda$, and as
contribution to $w_{2}$ over $\Delta_{i}$ (for $i\leq[g/2]$) we have the
expression

$None$
$i(i+1)(2\gamma_{1}+\mathfrak{r})+(g-i)(g-i+1)(2\gamma_{2}+\mathfrak{r})\,,$

where $\gamma_{1}$ (resp. $\gamma_{2}$) is the class of the component
$C_{1}^{\prime}$ of genus $i$ (resp. $C_{2}^{\prime}$ of genus $g-i$) over
$\Delta_{i}\cap B$ and $\mathfrak{r}$ the class of $R$.

If we denote by $T$ the closure of the reduced image of $v^{*}W$ under the
$(2:1)$ map $\varphi^{\prime}:X^{\prime}\to\mathbb{P}(\mathbb{E}_{B})$, then
$[G_{1}]=\gamma([T])$. Recall that
$\varphi^{\prime*}h=\omega_{\pi^{\prime}}-R$. Thus the
$\delta_{i}$-contribution in $2[T]$ coming from from $w_{1}$ is

$\frac{g(g+1)}{2}\varphi^{\prime}_{*}(\omega_{\pi^{\prime}})=\frac{g(g+1)}{2}\varphi^{\prime}_{*}({\varphi^{\prime}}^{*}h+\mathfrak{r})=\frac{g(g+1)}{2}\left(h\varphi^{\prime}_{*}[1]+\varphi^{\prime}_{*}\mathfrak{r})\right)\,.$

If we apply (1) the contribution to $\delta_{i}$ in

$\gamma(h{\varphi^{\prime}}_{*}[1])=\gamma(\sum_{i=0}^{g-2}h^{i+1}u^{*}\beta_{g-2-i})=2(g-1)(\check{h}+\check{u}^{*}\lambda)+\check{u}^{*}\beta_{1}$

comes from $\check{u}^{*}\beta_{1}$ alone and equals
$-4\check{u}^{*}\delta_{i}$, as $\delta_{i}$ appears in the formula of
$\beta_{1}$ with coefficient $-2$. Since
$\gamma(\varphi^{\prime}_{*}\mathfrak{r})=2\,\check{u}^{*}\delta_{i}$ we get
from $w_{1}$ together the contribution $-g(g+1)\check{u}^{*}\delta_{i}$. From
$w_{2}$ we get by applying $\gamma$ to (16), using
$\gamma(\varphi^{\prime}_{*}\gamma_{1})=(2i-2)\check{u}^{*}\delta_{i}$,
$\gamma(\varphi^{\prime}_{*}\gamma_{2})=(2g-2i-2)\check{u}^{*}\delta_{i}$ and
$\gamma(\varphi^{\prime}\mathfrak{r})=2\check{u}^{*}\delta_{i}$, the
contribution $2(g+3)i(i-g)+g(g+1)$. Together $w_{1}-w_{2}$ thus contribute
$-2(g+3)i(i-g)$ to the coefficient of $\delta_{i}$, as required.

$\square$

Proof of Theorem 19.2. Here $k\geq 2$, hence we have the morphism
$\varphi_{k}:\overline{\mathcal{C}}_{g}\to\mathbb{P}(\mathbb{E}_{k})$. Let
$\pi_{1}:\overline{\mathcal{C}}_{g,1}\to\overline{\mathcal{C}}_{g}$ be the
universal curve over $\overline{\mathcal{C}}_{g}$ and
$s:\overline{\mathcal{C}}_{g}\to\overline{\mathcal{C}}_{g,1}$ the tautological
section the image of which we denote by $S$.

We claim: $\varphi_{k}^{*}F_{k}=\pi_{1*}(\omega^{\otimes k}_{\pi_{1}}(-S))$.
Indeed, by our assumptions on $g$ and $k$, we have
$R^{1}\pi_{1*}(\omega^{\otimes k}_{\pi_{1}}(-S))=0$, so
$\pi_{1*}(\omega^{\otimes k}_{\pi_{1}}(-S))$ is a vector bundle on
$\overline{\mathcal{C}}_{g}$. For a point $x\in\mathbb{P}(\mathbb{E}_{k})$ the
fibre of $F_{k}$ is the hyperplane in the corresponding fibre of
$\mathbb{E}_{k}$ representing the point $x$. When $x=\varphi_{k}(p)$ with
$p\in\overline{\mathcal{C}}_{g}$, then
$(F_{k})_{x}=H^{0}(C_{p},\omega^{\otimes k}_{\pi_{1}}(-p))$, with $C_{p}$ the
corresponding firbre of $\pi_{1}$ over $p$. Hence the claim. We now have on
$\overline{\mathcal{C}}_{g}$ the sequence

$None$ $0\to\pi_{1*}(\omega^{\otimes
k}_{\pi_{1}}(-2S))\to\pi_{1*}(\omega^{\otimes k}_{\pi_{1}}(-S))\to
s^{*}(\omega^{\otimes k}_{\pi_{1}}(-S))\to 0\,,$

with $s^{*}(\omega_{\pi_{1}}^{\otimes k}(-S))\cong\omega_{\pi}^{k+1}$, and
this sequence is exact up to codimension $2$ because
$R^{1}\pi_{1*}(\omega^{\otimes k}_{\pi_{1}}(-2S))$ vanishes in codimension $2$
in view of the conditions on $(g,k)$.

Let $F_{k}(\nu)=\pi_{1*}(\omega^{\otimes k}(-\nu S))$ for $\nu=1,2$ with
$F_{k}(1)\cong\varphi^{*}F_{k}$. Let
$j:\mathbb{P}(F_{k}(1)^{\vee})\to\mathbb{P}(F_{k}^{\vee})$ be the natural map.
Then $\tilde{h}=j^{*}\check{\rho}^{*}(\check{h})$ is the class of the
hyperplane line bundle on $\mathbb{P}(F_{k}(1)^{\vee})$. The inclusion
$F_{k}(2)\hookrightarrow F_{k}(1)$ induces a map
$\sigma:\mathbb{P}(F_{k}(2)^{\vee})\to\mathbb{P}(F_{k}(1)^{\vee})$. We have
the commutative diagram:

$\textstyle{\mathbb{P}(F_{k}(2)^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{P}(F_{k}(1)^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}}$$\scriptstyle{j}$$\textstyle{I=\mathbb{P}(F_{k}^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\check{\rho}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{P}(\mathbb{E}_{k}^{\vee})}$$\scriptstyle{\check{u}}$$\textstyle{\overline{\mathcal{C}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{k}}$$\textstyle{\mathbb{P}(\mathbb{E}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\overline{\mathcal{M}}_{g}}$

Let $\alpha=j\sigma:\mathbb{P}(F_{k}(2)^{\vee})\to I$ and $A={\rm Im}(\alpha)$
the image of this map. Then $[Z_{k}]=\check{\rho}_{*}[A]$. By the exact
sequence (17) we have
$\sigma_{*}[\mathbb{P}(F_{k}(2)^{\vee})]=\tilde{h}+(k+1)\rho_{1}^{*}\omega_{\pi}$.
We observe that $j_{*}[1]=\rho^{*}[\overline{\Gamma}_{k}]$ and
$k\,\omega_{\pi}=\varphi_{k}^{*}h$ and we find

$\displaystyle k{[A]}$
$\displaystyle=k\,j_{*}(j^{*}\check{\rho}^{*}\check{h})+(k+1)j_{*}(\rho_{1}^{*}\phi_{k}^{*}h)=k\,\check{\rho}^{*}\check{h}\,j_{*}[1]+(k+1)\rho^{*}(h)j_{*}[1]$
$\displaystyle=k\,\check{\rho}^{*}\check{h}\,\rho^{*}[\overline{\Gamma}_{k}]+(k+1)\rho^{*}(h[\overline{\Gamma}_{k}])\,.$

By Proposition 3.1 and Lemma 19.3 we have
$\check{h}\gamma([\overline{\Gamma}_{k}])=2k(g-1)\check{h}$ and

$\gamma(h[\overline{\Gamma}_{k}])=\gamma(\sum_{i=0}^{r-2}h^{i+1}u^{*}\beta_{r-2-i})=2k(g-1)(\check{h}+\check{u}^{*}{\rm
c}_{1}(\mathbb{E}_{k}))+\check{u}^{*}\beta_{1}=2k(g-1)\check{h}+k^{2}\check{u}^{*}\kappa_{1}$

and thus $[Z_{k}]=2(2k+1)(g-1)\check{h}+k(k+1)\check{u}^{*}\kappa_{1}$ in
agreement with the formula of Theorem 19.2. $\square$

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