We now return to the proof. We need to compute the rank of the matrix (24)
with respect to the basis (26):

$(\Psi^{d})_{ji}=\frac{1}{2}\left\langle\widetilde{\gamma}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle=\frac{1}{2}\left\langle\widetilde{\gamma}^{+}_{j}-\iota_{*}(\widetilde{\gamma}^{+}_{j}),\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle=\left\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle.$

To compute the intersection numbers
$\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}\rangle$ and
$\langle\widetilde{\gamma}^{+}_{j},\iota(\widetilde{f}_{i})\rangle$, we pass
to the contracted graph $\widetilde{G}^{c}$. First of all, we note that none
of edges $\widetilde{f}_{i}$ or $\iota(\widetilde{f}_{i})$ on $\widetilde{G}$
are contracted. Therefore, for any cycle $\widetilde{\gamma}$ on
$\widetilde{G}$, its intersection with $\widetilde{f}_{i}$ or
$\iota(\widetilde{f}_{i})$ can be computed on the contracted graph
$\widetilde{G}^{c}$:

$\langle\widetilde{\gamma},\widetilde{f}_{i}\rangle=\langle\widetilde{\gamma}^{c},\widetilde{f}_{i}\rangle,\quad\langle\widetilde{\gamma},\iota(\widetilde{f}_{i})\rangle=\langle\widetilde{\gamma}^{c},\iota(\widetilde{f}_{i})\rangle,\quad
i=1,\ldots,d.$

Furthermore, we observe that, since $\widetilde{\gamma}^{+}_{j}$ is the unique
cycle on $\widetilde{T}\cup\\{\widetilde{e}_{j}\\}$ such that
$\langle\widetilde{\gamma}^{+}_{j},\widetilde{e}_{j}\rangle=1$, the contracted
cycle $(\widetilde{\gamma}_{j})^{c}$ is the unique cycle on
$\widetilde{T}^{c}\cup\\{\widetilde{e}_{j}\\}$ such that
$\langle(\widetilde{\gamma}^{+}_{j})^{c},\widetilde{e}_{j}\rangle=1$.

We first look at the cycles $(\widetilde{\gamma}^{+}_{j})^{c}$ for
$j=1,\ldots,r-1$. The edge $e_{j}\in E(G^{c})$ is a loop at $v_{j}$. Its lift
$\widetilde{e}_{j}\in E(\widetilde{G}^{c})$ starts at $t(\widetilde{f}_{j})$,
which is one of the two vertices $\widetilde{v}^{\pm}_{j}$ (say
$\widetilde{v}^{+}_{j}$), and ends at the other vertex $\widetilde{v}^{-}_{j}$
(by our labeling convention
$\widetilde{v}^{-}_{j}=t(\iota(\widetilde{f}_{j}))$). The contracted cycle
$(\widetilde{\gamma}^{+}_{j})^{c}$ is the unique cycle of the graph
$\widetilde{T}^{c}\cup\\{\widetilde{e}_{j}\\}$ containing
$+\widetilde{e}_{j}$: it starts at $\widetilde{v}^{+}_{j}$, proceeds to
$\widetilde{v}^{-}_{j}$ via $\widetilde{e}_{j}$, then to $\widetilde{v}_{0}$
via the unique path on $\widetilde{T}^{c}$ from $\widetilde{v}^{+}_{j}$ (the
first edge of this path being $-\iota(\widetilde{f}_{j})$), and then from
$\widetilde{v}_{0}$ to $\widetilde{v}^{-}_{j}$ via a unique path (the last
edge of this path being $+\widetilde{f}_{j}$). By the ordering convention that
we chose for $T^{c}$ and hence $\widetilde{T}^{c}$, the only edges that can
occur on this (other that the generating edge $\widetilde{e}_{j}$, which does
not lie on $\widetilde{T}^{c}$) are $\widetilde{f}_{i}$ and
$\iota(\widetilde{f}_{i})$ with $i\leq j$. Furthermore, $\widetilde{f}_{j}$
and $\iota(\widetilde{f}_{j})$ occur, as we have seen, with coefficients $+1$
and $-1$, respectively. It follows that for $j=1,\ldots,r-1$ we have

$(\Psi^{d})_{ji}=\left\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle=\langle(\widetilde{\gamma}^{+}_{j})^{c},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle=\left\\{\begin{array}[]{cc}0\mbox{
or }\pm 2,&i<j,\\\ 2,&i=j,\\\ 0,&i>j.\end{array}\right.$

We now calculate the intersection numbers
$\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle$
for $j=r,\ldots,d$. Recall that we have chosen
$\widetilde{e}_{j}=\widetilde{f}_{j}$, so the cycle
$(\widetilde{\gamma}^{+}_{j})^{c}$ is the unique cycle on
$\widetilde{T}^{c}\cup\\{\widetilde{f}_{j}\\}$ containing
$+\widetilde{f}_{j}$. By our ordering convention, the edges of the tree
$\widetilde{T}^{c}$ are $\widetilde{f}_{i}$ and $\iota(\widetilde{f}_{i})$ for
$1\leq i\leq r-1$. Hence $(\widetilde{\gamma}^{+}_{j})^{c}$ intersects
$\widetilde{f}_{j}$ with multiplicity $+1$, and does not intersect any
$\widetilde{f}_{i}$ or $\iota(\widetilde{f}_{i})$ with $i\geq r$. Therefore,
for $j\geq r$ we have

$(\Psi^{d})_{ji}=\left\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle=\langle(\widetilde{\gamma}^{+}_{j})^{c},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle=\left\\{\begin{array}[]{cc}0,\pm
1,\mbox{ or }\pm 2,&i\leq r-1,\\\ 1,&i=j,\\\ 0,&i\geq r,i\neq
j.\end{array}\right.$

Putting everything together, we see that the $d\times d$ minor of
$(\Psi^{d})_{ji}$ corresponding to the partial basis
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{d}$ is a lower-triangular
matrix, whose first $r-1$ diagonal entries are 2, and the remaining equal to
1. Hence, $\Psi^{d}$ has rank $d$.

∎

We now restrict our attention to the Abel–Prym map in degree $d=g-1$, which we
denote $\Psi$:

$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma),\quad\Psi(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D}),$

In this case the source has the same dimension as the target, and we can
compute the determinant of the matrix (24) of $\Psi$ on any top-dimensional
cell $C(\widetilde{F})$. This determinant depends on a choice of basis
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$ for
$\operatorname{Ker}\pi_{*}$, but only up to sign, hence the quantity

$\deg_{\Psi}(\widetilde{F})=|\det\Psi(\widetilde{D})|,\quad\widetilde{D}\in
C(\widetilde{F}),$ (27)

which we call the _degree_ of $\Psi$ on $C(\widetilde{F})$, is well-defined.

We now compute the degree of $\Psi$ on the top-dimensional cells of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$. We recall from Sec. 3.2 that,
given a connected free double cover $p:\widetilde{G}\to G$ of a graph $G$ of
genus $g$, a subset $F\subset E(G)$ of $g-1$ edges of $G$ is called an _odd
genus one decomposition of rank $r$_ if $G\backslash F$ consists of $r$
connected components of genus one, and each of them has connected preimage in
$\widetilde{G}$.

######  Corollary 4.2.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover with model
$p:\widetilde{G}\to G$, let
$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$
be the Abel–Prym map, let
$C(\widetilde{F})=C^{g-1}(\widetilde{F},0)\subset\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$
be a top-dimensional cell corresponding to the multiset
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-1}\\}\subset
E(\widetilde{G})$, and let $F=p(\widetilde{F})$. Then
$\deg_{\Psi}(\widetilde{F})$ is equal to

$\deg_{\Psi}(\widetilde{F})=\left\\{\begin{array}[]{cc}2^{r-1},&\mbox{edges of
}F\mbox{ are distinct and form an odd genus one decomposition of rank }r,\\\
0&\mbox{otherwise.}\end{array}\right.$ (28)

In particular, the volume of the image of $C(\widetilde{F})$ in
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ is equal to

$\operatorname{Vol}(\Psi(C(\widetilde{F})))=\frac{2^{r(\widetilde{F})-1}w(F)}{\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))},\quad
w(F)=w(\widetilde{F})=\ell(\widetilde{f}_{1})\cdots\ell(\widetilde{f}_{g-1})$
(29)

if $F$ is an odd genus one decomposition of rank $r(\widetilde{F})$, and zero
otherwise.

###### Proof.

This follows directly from the proof of Theorem 4.1. If the edges of
$F=p(\widetilde{F})$ are not all distinct, then $\Psi$ contracts the cell
$C(\widetilde{F})$ and hence $\det\Psi=0$ on $C(\widetilde{F})$. If $F$
consists of distinct edges, let $G\backslash F=G_{0}\cup\cdots\cup G_{r-1}$ be
the decomposition into connected components. By (25) we have that
$g_{0}+\cdots+g_{r-1}=r$, hence either $g_{k}=0$ for some $k$, or all
$g_{k}=1$. In the former case $G_{k}$ is a tree, so the restriction of the
cover $p$ to $G_{k}$ is trivial and hence $\det\Psi=0$ on $\widetilde{F}$. In
the latter case, $\Psi$ has rank $d=g-1$ if and only if the restriction of $p$
to each $G_{k}$ is nontrivial, which is true precisely when $F$ is an odd
genus one decomposition. Furthermore, the matrix of $\Psi$ with respect to the
basis (26) is lower triangular, with the first $r-1$ diagonal entries equal to
2, and the remaining equal to 1. Hence $\det\Psi=2^{r-1}$ on
$C(\widetilde{F})$, as required.

To prove (4.2), it is sufficient to note that $C(\widetilde{F})$ is a
parallelotope with volume $w(\widetilde{F})$, and that
$\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))^{-1}$ is
the volume of the unit cube in the coordinate system on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ that we used to compute the
matrix of $\Psi$ (see Equation (9)). ∎

##  5\. Harmonicity of the Abel–Prym map

In this section, we consider the degree $g-1$ Abel–Prym map
$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$
associated to a free double cover $\pi:\widetilde{\Gamma}\to\Gamma$. The
cellular decomposition of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$
induces a decomposition of
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ (which is locally
modelled on $\mathbb{R}^{g-1}$). Pulling this decomposition back to
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$ and refining cells as
needed, the Abel–Prym map $\Psi$ is a map of polyhedral spaces. We show that
$\Psi$ is a harmonic map of polyhedral spaces of global degree $2^{g-1}$, with
respect to the degree function (28).

######  Theorem 5.1.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover of metric graphs.
Then the Abel–Prym map

$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma),\quad\Psi(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D})$

is a harmonic map of polyhedral spaces of global degree $2^{g-1}$, with
respect to the degree function $\deg_{\Psi}$ defined on the codimension zero
cells of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ by (28).

The proof consists of two parts. First, we show that $\Psi$ is harmonic at
each codimension one cell of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$, and hence has a well-
defined global degree $d$ because the polyhedral space
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is connected through
codimension one. We then show that $d=2^{g-1}$. The proof of the first part is
a somewhat involved calculation. We separate this result into a Proposition,
and give its proof after the proof of the main Theorem 5.1.

######  Proposition 5.2.

The degree $g-1$ Abel–Prym map

$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma),\quad\Psi(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D})$

is harmonic at each codimension one cell of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$.

###### Proof of Theorem 5.1.

By Proposition 5.2, the Abel–Prym map has a certain global degree $d$. It may
be possible to directly show that $d=2^{g-1}$, by somehow counting the
preimages in a single fiber of $\Psi$, but we employ a different method.
Namely, we use the harmonicity of the map $\Psi$ to give an alternative
calculation of the volume of the Prym variety, in terms of the unknown global
degree $d$ (similarly to how the volume of $\operatorname{Jac}(\Gamma)$ is
computed in [ABKS14]). However, we have already computed the volume of the
Prym variety in Theorem 3.4, using an entirely different method. Comparing the
two formulas, we find that in fact $d=2^{g-1}$.

Let $M_{i}$ for $i=1,\ldots,N$ be the codimension zero cells of
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$, and let
$\widetilde{M}_{ij}$ for $j=1,\ldots,k_{i}$ be the codimension zero cells of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ mapping surjectively to
$M_{i}$. The cells $\widetilde{M}_{ij}$ are obtained by refining the natural
cellular decomposition of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$, in
other words each $\widetilde{M}_{ij}$ is a subset of a cell
$C(\widetilde{F}_{ij})$, where $\widetilde{F}_{ij}\subset
E(\widetilde{\Gamma})$ is a subset of edges such that $p(\widetilde{F}_{ij})$
is an odd genus one decomposition of $\Gamma$ of some rank
$r_{ij}=r(p(\widetilde{F}_{ij}))$. Equation (29) gives the volume dilation
factor of $\Psi$ on the cell $C(\widetilde{F}_{ij})$, and hence on
$\widetilde{M}_{ij}$. Therefore

$\operatorname{Vol}(M_{i})=\frac{2^{r_{ij}-1}\operatorname{Vol}(\widetilde{M}_{ij})}{\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))}$

for all $i$ and $j$. On the other hand, the harmonicity condition implies that
for each $i$ we have

$d=\sum_{j=1}^{k_{i}}\deg_{\Psi}(\widetilde{M}_{ij})=\sum_{j=1}^{k_{i}}2^{r_{ij}-1}.$

Putting this together, we can write

$\operatorname{Vol}(M_{i})=\frac{1}{d}\sum_{j=1}^{k_{i}}2^{r_{ij}-1}\cdot\operatorname{Vol}(M_{i})=\frac{1}{d\cdot\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))}\sum_{j=1}^{k_{i}}4^{r_{ij}-1}\operatorname{Vol}(\widetilde{M}_{ij}).$

The sum of the volumes of the $M_{i}$ is the volume of the Prym variety:

$\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\operatorname{Vol}(\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma))=\sum_{i=1}^{N}\operatorname{Vol}(M_{i})=\frac{1}{d\cdot\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))}\sum_{i,j}4^{r_{ij}-1}\operatorname{Vol}(\widetilde{M}_{ij}).$

On the other hand, corresponding to each odd genus one decomposition $F$ of
$\Gamma$ there are $2^{g-1}$ subsets $\widetilde{F}\subset
E(\widetilde{\Gamma})$ such that $p(\widetilde{F})=F$, because each
decomposition has exactly $g-1$ edges. The volume
$\operatorname{Vol}(C(\widetilde{F}))$ of each of these cells is equal to
$w(F)$. Each cell $C(\widetilde{F})$ corresponding to an odd genus one
decomposition $F=p(\widetilde{F})$ is a disjoint union of some of the
$M_{ij}$, and each $M_{ij}$ lies in some $C(\widetilde{F})$. Hence in fact the
sum in the right hand side can be written as

$\sum_{i,j}4^{r_{ij}-1}\operatorname{Vol}(M_{ij})=\sum_{\widetilde{F}\subset
E(\widetilde{\Gamma})}4^{r(p(\widetilde{F}))-1}\operatorname{Vol}(\widetilde{F})=2^{g-1}\sum_{F\subset
E(\Gamma)}4^{r(F)-1}w(F),$

where the last sum is over all odd genus one decompositions $F$ of $\Gamma$.
Therefore,

$\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\frac{2^{g-1}}{d\cdot\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))}\sum_{F\subset
E(\Gamma)}4^{r(F)-1}w(F).$

Comparing this formula with (17), we see that $d=2^{g-1}$.

∎

######  Remark 5.3.

Given a Prym divisor class represented by
$\Psi(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D})$, the degree of $\Psi$
at $\widetilde{D}\in\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ in general
depends on the choice of representative (in other words, the degree of $\Psi$
is not constant in fibers). We give an example of a free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ and effective divisors $\widetilde{D}_{1}$
and $\widetilde{D}_{2}$ on the source, such that
$\widetilde{D}_{1}-\iota(\widetilde{D}_{1})\simeq\widetilde{D}_{2}-\iota(\widetilde{D}_{2})$,
but the degrees of $\Psi$ at $\widetilde{D}_{1}$ and $\widetilde{D}_{2}$ are
different.

Figure 3. A double cover with a Prym divisor with representatives of distinct
degrees.

Consider the free double cover $\pi:\widetilde{\Gamma}\to\Gamma$ shown on
Figure 3. The curves $\widetilde{\Gamma}$ and $\Gamma$ have genera $5$ and
$3$, respectively. The edge $\widetilde{e}_{3}$ has length at least $3$, while
all other edges have length $1$. Fix real numbers $0<y<x<1$. Let
$\widetilde{P}_{1}$, $\widetilde{Q}_{1}$, $\widetilde{P}_{2}$, and
$\widetilde{Q}_{2}$ be the points on the edges $\widetilde{e}_{1}^{+}$,
$\widetilde{e}_{2}^{+}$, $\widetilde{e}^{-}_{1}$, and $\widetilde{e}_{3}^{+}$,
respectively, located at the following distances from the corresponding end
vertices:

$d(\widetilde{v}_{1}^{+},\widetilde{P}_{1})=x,\quad
d(\widetilde{v}_{3}^{+},\widetilde{Q}_{1})=y,\quad
d(\widetilde{v}_{2}^{-},\widetilde{P}_{2})=x-y,\quad
d(\widetilde{v}_{2}^{+},\widetilde{Q}_{2})=1+2y.$

Let $\widetilde{D}_{1}=\widetilde{P}_{1}+\widetilde{Q}_{1}$ and
$\widetilde{D}_{2}=\widetilde{P}_{2}+\widetilde{Q}_{2}$. The divisor
$(\widetilde{D}_{1}-\iota\widetilde{D}_{1})-(\widetilde{D}_{2}-\iota\widetilde{D}_{2})$
is seen to be equivalent to 0 by repeatedly applying Dhar’s burning algorithm
[BS13, Section 5.1] at each point where the divisor has a negative number of
chips. As a consequence, $\widetilde{D}_{1}-\iota\widetilde{D}_{1}$ and
$\widetilde{D}_{2}-\iota\widetilde{D}_{2}$ are linearly equivalent. However,
$\pi(\widetilde{D}_{1})$ is supported on the odd genus one decomposition
$\\{\widetilde{e}_{1},\widetilde{e}_{2}\\}$ of rank $2$, while
$\pi(\widetilde{D}_{2})$ is supported on the odd genus one decomposition
$\\{\widetilde{e}_{1},\widetilde{e}_{3}\\}$ of rank $1$, so the degrees of
$\Psi$ at $\widetilde{D}_{1}$ and $\widetilde{D}_{2}$ are distinct.

By varying $x$ and $y$, we obtain two polyhedral cells in
$\operatorname{Sym}^{2}(\widetilde{\Gamma})$ having the same image in
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$: the subset
$C_{1}=\\{(x-y,1+2y):0<y<x<1\\}$ of
$\widetilde{e}^{-}_{1}\times\widetilde{e}^{+}_{3}$ and the subset
$C_{2}=\\{(x,y):0<y<x<1\\}$ of
$\widetilde{e}^{+}_{1}\times\widetilde{e}^{+}_{2}$. The volumes of $C_{1}$ and
$C_{2}$ are equal to $1$ and $1/2$, respectively, which agrees with the fact
that the degree of $\Psi$, equal to $1$ on $C_{1}$ and $2$ on $C_{2}$, is the
volume dilation factor. We also observe that the global degree of $\Psi$ is
equal to $2^{g-1}=4$. Therefore, Theorem B implies that there is a third
divisor $\widetilde{D}_{3}$ (effective of degree two) such that
$\widetilde{D}_{1}-\iota\widetilde{D}_{1}\simeq\widetilde{D}_{3}-\iota\widetilde{D}_{3}$,
and such that $\Gamma\setminus\pi(\widetilde{D}_{3})$ consists of a single
connected component.

Before giving the proof of Proposition 5.2, we describe the structure of the
Abel–Prym map for the covers of a genus two dumbbell graph.

######  Example 5.4.

Consider the two covers $\pi_{1}:\widetilde{\Gamma}_{1}\to\Gamma$ and
$\pi_{2}:\widetilde{\Gamma}_{2}\to\Gamma$ of the dumbbell graph $\Gamma$
described in Example 3.7. In this case $g-1=1$, and the Abel–Prym maps
$\Psi_{1}:\widetilde{\Gamma}_{1}\to\operatorname{Prym}(\widetilde{\Gamma}_{1}/\Gamma)$
and
$\Psi_{2}:\widetilde{\Gamma}_{2}\to\operatorname{Prym}(\widetilde{\Gamma}_{2}/\Gamma)$
are harmonic morphisms of metric graphs of degree two, which we now describe.

With respect to the cover $\pi_{1}$, each edge of $\Gamma$ is an odd genus one
decomposition, hence $\Psi_{1}$ does not contract any edges. The edges
$\widetilde{e}^{\pm}_{1}$ and $\widetilde{e}^{\pm}_{2}$ are mapped onto edges
$f_{1}$ and $f_{2}$, respectively. The degree of $\Psi_{1}$ on these edges is
equal to one, hence the lengths of $f_{1}$ and $f_{2}$ are $x_{1}$ and
$x_{2}$, respectively. Each of the two edges $\widetilde{e}^{\pm}_{3}$ is
mapped onto an edge $f^{\pm}_{3}$, of length $2x_{3}$ because the degree of
$\Psi_{1}$ is equal to two. Hence
$\operatorname{Prym}(\widetilde{\Gamma}_{1}/\Gamma)$ is a circle of
circumference $x_{1}+x_{2}+4x_{3}$, as we have already seen in Example 3.7.

The map $\Psi_{2}$, on the other hand, contracts the edges
$\widetilde{e}^{\pm}_{1}$ and $\widetilde{e}^{\pm}_{3}$ because $\\{e_{1}\\}$
and $\\{e_{3}\\}$ are not genus one decompositions, and maps
$\widetilde{e}^{\pm}_{2}$ to a unique loop edge $f_{2}$ of
$\operatorname{Prym}(\widetilde{\Gamma}_{2}/\Gamma)$ of length $x_{2}$. The
morphisms $\Psi_{1}$ and $\Psi_{2}$ are given in Figure 4.

$\widetilde{\Gamma}_{1}$$\widetilde{e}^{+}_{1}$$\widetilde{e}^{-}_{1}$$\widetilde{e}^{-}_{3}$$\widetilde{e}^{+}_{3}$$\widetilde{e}^{+}_{2}$$\widetilde{e}^{-}_{2}$$f_{1}$$f^{+}_{3}$$f^{-}_{3}$$f_{2}$$\operatorname{Prym}(\widetilde{\Gamma}_{1}/\Gamma)$$\widetilde{\Gamma}_{2}$$\widetilde{e}^{+}_{1}$$\widetilde{e}^{-}_{1}$$\widetilde{e}^{-}_{3}$$\widetilde{e}^{+}_{3}$$\widetilde{e}^{-}_{2}$$\widetilde{e}^{+}_{2}$$f_{2}$$\operatorname{Prym}(\widetilde{\Gamma}_{2}/\Gamma)$
Figure 4. Abel–Prym maps corresponding to the covers in Example 3.7

###### Proof of Proposition 5.2.

Let $\widetilde{C}$ be a codimension one cell of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ such that its image
$C=\Psi(\widetilde{C})$ is a codimension one cell in
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. Since
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is a torus, it locally looks
like $\mathbb{R}^{g-1}$, and we can think of the cell $C$ as lying in a
hyperplane $H_{0}\subset\mathbb{R}^{g-1}$, with respect to an appropriate
local coordinate system. There are exactly two codimension zero cells
$M^{\pm}$ attached to $C$, each contained in a corresponding half-space of
$\mathbb{R}^{g-1}$, which we also denote $M^{\pm}$ by abuse of notation. To
show that $\Psi$ is harmonic at $\widetilde{C}$, we need to show that the sum
of $|\det\Psi|$ over the codimension zero cells of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$ mapping to $M^{+}$ is
the same as the sum over those mapping to $M^{-}$, in which case this sum is
the degree of $\Psi$ on $\widetilde{C}$.

If $\Psi$ contracts every codimension zero cell of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ attached to $\widetilde{C}$,
then the harmonicity condition is trivially verified, and we set
$\deg_{\Psi}(\widetilde{C})=0$. Hence we assume that $\Psi$ does not contract
some codimension zero cell attached to $\widetilde{C}$. By Corollary 4.2, this
cell is a subset of $C(\widetilde{F})$, where $F=p(\widetilde{F})$ is an odd
genus one decomposition of $G$. If $\widetilde{C}$ lies in the interior of
$C(\widetilde{F})$, then $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ also
locally looks like $\mathbb{R}^{n}$ in a neighborhood of $\widetilde{C}$, the
map $\Psi$ is simply an affine linear map near $\widetilde{C}$, and therefore
harmonic (such cells $\widetilde{C}$ do not occur in the standard polyhedral
decomposition (22) of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$,
but may occur in the refined decomposition induced by the map $\Psi$).

We therefore assume that $\widetilde{C}$ lies on the boundary of a cell
$C(\widetilde{F})$, where
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-1}\\}$ is a set of
edges of $\widetilde{G}$ such that $F=\\{f_{1},\ldots,f_{g-1}\\}$,
$f_{i}=p(\widetilde{f}_{i})$ is an odd genus one decomposition of $G$ of rank
$r$. To simplify notation, we assume that in fact $\widetilde{C}$ is a
codimension one cell of $C(\widetilde{F})$ with respect to the standard
polyhedral decomposition (22) of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$. In other words, we
assume that
$\widetilde{C}=C^{g-2}(\widetilde{F}\backslash\\{\widetilde{f}_{a}\\},\widetilde{v})$
for some $a\in\\{1,\ldots,g-1\\}$, and where
$\widetilde{v}=s(\widetilde{f}_{a})$ is the starting vertex of
$\widetilde{f}_{a}$ with respect to an appropriate orientation (we shall later
specify which edge $\widetilde{f}_{a}$ we pick, in order to make our notation
consistent with Construction C).

The top dimensional cells of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$
that are adjacent to $\widetilde{C}$ have the form
$C(\widetilde{F}^{\prime})$, where
$\widetilde{F}^{\prime}=(\widetilde{F}\backslash\\{\widetilde{f}_{a}\\})\cup\\{\widetilde{f}^{\prime}\\}$
and where $\widetilde{f}^{\prime}$ is any edge rooted at $\widetilde{v}$. We
assume that all edges $\widetilde{f}^{\prime}$ are oriented in such a way that
$s(\widetilde{f}^{\prime})=\widetilde{v}$. By Corollary 4.2, $\Psi$ does not
contract $C(\widetilde{F}^{\prime})$ if and only if
$p(\widetilde{F}^{\prime})$ is an odd genus one decomposition of $G$. To prove
harmonicity, we need to show that the sum of $|\det\Psi|$ on those cells
$C(\widetilde{F}^{\prime})$ mapping to $M^{+}$ is equal to the sum of those
that map to $M^{-}$. By Corollary 4.2, the value of $|\det\Psi|$ on a non-
contracted cell $C(\widetilde{F}^{\prime})$ is a power of two. In fact, as we
shall see, adjacent to any cell $\widetilde{C}$ there are either two, three,
or four non-contracted cells $C(\widetilde{F})$, with the degrees distributed
as shown on Figure 5 (plus an arbitrary number of contracted cells).

$2^{k}$$2^{k}$ | $2^{k+1}$$2^{k}$$2^{k}$ | $2^{k}$$2^{k}$$2^{k}$$2^{k}$
---|---|---
Two non-contracted cells | Three non-contracted cells | Four non-contracted cells
Figure 5. The Abel–Prym map near a non-contracted codimension one cell.

We calculate the matrix of $\Psi$ (or, rather, some of its entries) on each
cell $C(\widetilde{F}^{\prime})$ with respect to an appropriate coordinate
system, in the same way that we proved part (2) of Theorem 4.1. First, we
choose local coordinates on
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$. As before, we identify
$C(\widetilde{F})$ with the parallelotope
$[0,\ell(\widetilde{f}_{1})]\times\cdots\times[0,\ell(\widetilde{f}_{g-1})]$
lying in the half-space $H^{+}=\\{x:x_{a}\geq 0\\}\subset\mathbb{R}^{g-1}$.
Under this identification, the cell $\widetilde{C}$ lies in the hyperplane
$H^{0}=\\{x:x_{a}=0\\}$, and the corresponding cells of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ are
$C=\Psi(\widetilde{C})\subset\Psi(H^{0})$ and $M^{\pm}=\Psi(H^{\pm})$, where
$H^{-}=\\{x:x_{a}\leq 0\\}\subset\mathbb{R}^{g-1}$. Similarly, we think of
each of the other $C(\widetilde{F}^{\prime})$ as lying in its own $H^{+}$.

To construct coordinates on $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$,
we apply Construction C to the set $\widetilde{F}$. The output is a basis
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$ of
$\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to
H_{1}(\Gamma,\mathbb{Z})$ given by Equation (26). As explained in Subsection
2.6, the basis $\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$
defines a coordinate system on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, with respect to which the
map $\Psi$ on the cell $C(\widetilde{F})$ is affine linear, and the
$(g-1)\times(g-1)$ matrix of the linear part is given by Equation (24):

$\Psi(\widetilde{F})_{ji}=\frac{1}{2}\langle\widetilde{\gamma}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle=\frac{1}{2}\langle\widetilde{\gamma}^{+}_{j}-\iota_{*}(\widetilde{\gamma}^{+}_{j}),\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle=\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\rangle$

We recall that we showed in Theorem 4.1 and Corollary 4.2 that
$\Psi(\widetilde{F})_{ij}$ is a lower triangular matrix with determinant
$2^{r-1}$, where $r$ is the rank of $\widetilde{F}$.

Now let $C(\widetilde{F}^{\prime})$ be another codimension zero cell of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma}/\Gamma)$ adjacent to
$\widetilde{C}$, so
$\widetilde{F}^{\prime}=(\widetilde{F}\backslash\\{\widetilde{f}_{a}\\})\cup\\{\widetilde{f}^{\prime}\\}$,
where $\widetilde{f}^{\prime}$ is an edge rooted at $\widetilde{v}$ other than
$f_{a}$. We calculate the matrix of $\Psi$ on $C(\widetilde{F}^{\prime})$
using the same basis $\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$
(in other words, we do not recalculate the basis by replacing $\widetilde{F}$
with $\widetilde{F}^{\prime}$ in Construction C). The resulting matrix differs
from $\Psi(\widetilde{F})_{ji}$ by a single column only:

$\Psi(\widetilde{F}^{\prime})_{ji}=\left\\{\begin{array}[]{cc}\Psi(\widetilde{F})_{ji},&i\neq
a,\\\
\langle\widetilde{\gamma}^{+}_{j},\widetilde{f}^{\prime}-\iota(\widetilde{f}^{\prime})\rangle,&i=a.\end{array}\right.$
(30)

To check the harmonicity of $\Psi$ around $\widetilde{C}$, it suffices to
compute the determinants $\det\Psi(\widetilde{F}^{\prime})$ for all
$\widetilde{F}^{\prime}$. Indeed, the Abel–Prym map $\Psi$ contracts the cell
$C(\widetilde{F}^{\prime})$ if and only if
$\det\Psi(\widetilde{F}^{\prime})=0$. Furthermore, $C(\widetilde{F}^{\prime})$
maps to $M^{+}$ if $\det\Psi(\widetilde{F}^{\prime})>0$ and to $M^{-}$ if
$\det\Psi(\widetilde{F}^{\prime})<0$, and to prove harmonicity we need to
check that the positive determinants exactly cancel the negative determinants.

The set $F$ is an odd genus one decomposition of $G$ of some rank $r$, and we
denote

$G\backslash F=G_{0}\cup\cdots\cup G_{r-1}$

the decomposition into connected components. Each $G_{k}$ has genus one, and
each $p^{-1}(G_{k})$ is connected. We denote

$\widetilde{u}=t(\widetilde{f}_{a}),\quad v=p(\widetilde{v})=s(f_{a}),\quad
u=p(\widetilde{u})=t(f_{a}).$

There are two separate cases that we need to consider: either both endpoints
$u$ and $v$ of the edge $f_{a}=p(\widetilde{f}_{a})$ that we are removing lie
on one connected component, or the edge $f_{a}$ connects two different
components.

Both endpoints of the edge $f_{a}$ lie on a single connected component of
$G\backslash F$. Without loss of generality, we assume that $f_{a}$ is rooted
on the component $G_{0}$. The edge $f_{a}$ is a loop on the contracted graph
$G^{c}$ rooted at the vertex $v_{0}$. Since a loop cannot be part of a
spanning tree, we can further assume without loss of generality that
$\widetilde{f}_{a}=\widetilde{f}_{g-1}$. We observe that, on the contracted
graph $\widetilde{G}^{c}$, the edge $\widetilde{f}_{g-1}$ is a loop rooted at
$\widetilde{v}_{0}$. The contraction of the cycle
$\widetilde{\gamma}^{+}_{g-1}$ is the unique cycle containing
$+\widetilde{f}_{g-1}$, but since this is already a loop we see that
$(\widetilde{\gamma}^{+}_{g-1})^{c}=\widetilde{f}_{g-1}$.

It follows that the intersection of $\widetilde{\gamma}^{+}_{g-1}$ with all
other edges $\widetilde{f}_{i}$ and $\iota(\widetilde{f}_{i})$ for
$i=1,\ldots,g-2$ is zero. Hence the matrix $\Psi_{ji}$ is block upper
triangular, having a $(g-2)\times(g-2)$ lower triangular block with
determinant $2^{r-1}$ in the upper left corner, and a 1 in the lower right
corner. Therefore, the images of the subspaces $H^{\pm}$ and the hyperplane
$H^{0}$ are

$M^{+}=\Psi(H^{+})=\\{y:y_{g-1}\geq 0\\},\quad
M^{-}=\Psi(H^{-})=\\{y:y_{g-1}\leq 0\\},\quad\Psi(H^{0})=\\{y:y_{g-1}=0\\}.$

Now let $\widetilde{f}^{\prime}$ be an edge at $\widetilde{v}$, so that
$\widetilde{F}^{\prime}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{f}^{\prime}\\}$
defines a cell $C(\widetilde{F}^{\prime})$ adjacent to $C(\widetilde{F})$ via
$C^{\prime}$. The matrix $\Psi(\widetilde{F}^{\prime})_{ji}$ is given by (30),
and is obtained from the matrix $\Psi(\widetilde{F})$ by replacing the last
column. Hence it is also block upper triangular, and to compute
$\det\Psi(\widetilde{F}^{\prime})$ it suffices to find the new entry

$\Psi(\widetilde{F}^{\prime})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{f}^{\prime}-\iota(\widetilde{f}^{\prime})\rangle$
(31)

in the lower right corner. Furthermore, the sign of this entry determines the
sign of $\det\Psi(\widetilde{F}^{\prime})$ and hence the image cell
$\Psi(C(\widetilde{F}^{\prime}))$ of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$: if the entry is positive,
then $\Psi$ maps $C(\widetilde{F}^{\prime})$ to the same half-space $M^{+}$ as
$C(\widetilde{F})$, while if it is negative then
$\Psi(C(\widetilde{F}^{\prime}))\subset M^{-}$, and if it is zero then
$C(\widetilde{F}^{\prime})$ is contracted.

There are several possibilities to consider, depending on the relative
positions of $v=s(f_{g-1})$ and $u=t(f_{g-1})$ on the component $G_{0}$. Let
$\gamma(G_{0})$ denote the unique cycle on $G_{0}$ (oriented in any
direction), then any vertex of $G_{0}$ has a unique (possibly trivial)
shortest path to $\gamma(G_{0})$. For two distinct vertices $v_{1},v_{2}\in
V(G_{0})$, we write $v_{1}<v_{2}$ if the unique path from $v_{2}$ to
$\gamma(G_{0})$ passes through $v_{1}$; this defines a partial order on
$V(G_{0})$.

  1. (1)

The vertex $v$ does not lie on $\gamma(G_{0})$, and $v\nless u$. In other
words, $v$ lies on a tree attached to $\gamma(G_{0})$, and $u$ does not lie
higher up on the same tree.

Let $g_{1}$ be the unique edge rooted at $v$ that points in the direction of
the cycle $\gamma(G_{0})$. Since the unique path from $u$ to $\gamma(G_{0})$
avoids $v$, the graph
$G_{0}^{\prime}=G_{0}\cup\\{f_{g-1}\\}\backslash\\{g_{1}\\}$ is connected, has
genus one, and has connected preimage, since the unique cycle of
$G_{0}^{\prime}$ is $\gamma(G_{0})$. Therefore,
$F_{1}=\\{f_{1},\ldots,f_{g-2},g_{1}\\}$ is an odd genus one decomposition of
$G$, of the same length $r$ as $F$. For any other edge $e^{\prime}$ rooted at
$v$, removing it disconnects the corresponding branch of the tree from
$G_{0}$, and attaching $f_{g-1}$ does not reconnect this branch. Hence
$G\backslash\\{f_{1},\ldots,f_{g-2},e^{\prime}\\}$ has a connected component
of genus zero, and $\\{f_{1},\ldots,f_{g-2},e^{\prime}\\}$ is not a genus one
decomposition. The graph $G_{0}$ and its preimage $\pi^{-1}(G_{0})$ are shown
on Figure 6.

Figure 6. The cycle $\widetilde{\gamma}^{+}_{g-1}$ in Case (1)

We see that the only cell $C(\widetilde{F}^{\prime})$ adjacent to
$C(\widetilde{F})$ through $C^{\prime}$ on which $|\det\Psi|$ is nonzero
corresponds to
$\widetilde{F}^{\prime}=\widetilde{F}_{1}=\widetilde{F}\cup\\{\widetilde{g}_{1}\\}\backslash\\{\widetilde{f}_{g-1}\\}$,
where $\widetilde{g}_{1}$ is the unique edge rooted at $\widetilde{v}$ that
maps to $g_{1}$. Furthermore, $F$ and $F_{1}=p(\widetilde{F}_{1})$ have the
same rank $r$, hence the value of $|\det\Psi|$ on the two cells
$C(\widetilde{F})$ and $C(\widetilde{F}_{1})$ is equal to $2^{r-1}$, so to
prove harmonicity we only need to show that $\Psi$ maps $C(\widetilde{F}_{1})$
to the half-space $M^{-}$. As explained above, it suffices to compute the last
diagonal entry (31) of $\Psi(\widetilde{F}_{1})$, where
$\widetilde{f}^{\prime}=\widetilde{g}_{1}$.

The cycle $\widetilde{\gamma}^{+}_{g-1}$ is the unique cycle of the graph
$\widetilde{T}\cup\\{\widetilde{f}_{g-1}\\}$ containing
$+\widetilde{f}_{g-1}$. It starts at the vertex
$\widetilde{v}=s(\widetilde{f}_{g-1})$, proceeds to
$\widetilde{u}=t(\widetilde{f}_{g-1})$ via $+\widetilde{f}_{g-1}$, and then
from $\widetilde{u}$ back to $\widetilde{v}$ via the unique path in the tree
$\widetilde{T}$. This path actually lies in the spanning tree
$\widetilde{T}_{0}^{+}\cup\widetilde{T}_{0}^{-}\cup\\{\widetilde{e}^{+}_{0}\\}$
of $p^{-1}(G_{0})$. The last edge of the path is $\widetilde{g}_{1}$, oriented
in the opposite direction since we’ve assumed that
$s(\widetilde{g}_{1})=\widetilde{v}$, hence
$\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}\rangle=-1$. In
addition, the path does not contain $\iota(\widetilde{g}_{1})$. It follows
that

$\Psi(\widetilde{F}_{1})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=-1.$

Therefore $\Psi$ maps the cell $C(\widetilde{F}_{1})$ to the half-space
$M^{-}$, hence $\Psi$ is harmonic.

  2. (2)

The vertex $v$ does not lie on $\gamma(G_{0})$, and $v<u$. As before, let
$g_{1}$ denote the unique edge at $v$ pointing towards $\gamma(G_{0})$, and
let $g_{2}$ be the unique edge rooted at $v$ which lies on the path from $v$
to $u$ (this path, when reversed, is part of the unique path from $u$ to
$\gamma(G_{0})$). Attaching $f_{g-1}$ to $G_{0}$ produces a graph of genus
two. Any edge $e^{\prime}$ rooted at $v$ other than $f_{g-1}$, $g_{1}$, or
$g_{2}$ is the starting edge of a separate branch of $G_{0}\cup\\{f_{g-1}\\}$,
so removing $e^{\prime}$ creates a genus zero connected component. Hence the
only genus one decompositions of the form
$(F\backslash\\{f^{\prime}\\})\cup\\{f_{g-1}\\}$ are
$F_{1}=\\{f_{1},\ldots,f_{g-2},g_{1}\\}$,
$F_{2}=\\{f_{1},\ldots,f_{g-2},g_{2}\\}$, and $F$ itself. The decompositions
$F$ and $F_{2}$ have length $r$, while $F_{1}$ has length $r+1$, because the
edge $g_{1}$ is a bridge edge of $G_{0}\cup\\{f_{g-1}\\}$, and removing it
produces two genus one connected components.

We now consider the edges $\widetilde{g}_{1}$, $\widetilde{g}_{2}$, and
$\widetilde{f}_{g-1}$ on $\widetilde{G}$, lying above $g_{1}$, $g_{2}$, and
$f_{g-1}$ and rooted at $\widetilde{v}=s(\widetilde{f}_{g-1})$. Denote
$\widetilde{F}_{1}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{g}_{1}\\}$
and
$\widetilde{F}_{2}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{g}_{2}\\}$.
The edges $g_{1}$ and $g_{2}$ lie on the same tree attached to the cycle
$\gamma(G_{0})$ as the vertex $v$, and the lift of a tree is a tree. Hence the
endpoints of the edges $\widetilde{g}_{1}$ and $\widetilde{g}_{2}$ both lie on
the same subtree $\widetilde{T}_{0}^{\pm}$ of $p^{-1}(G_{0})$ as
$\widetilde{v}$, and we assume without loss of generality that this component
is $\widetilde{T}_{0}^{+}$. For $t(\widetilde{f}_{g-1})$, however, there are
two sub-possibilities, as shown on Figure 7.

Figure 7. The cycle $\widetilde{\gamma}^{+}_{g-1}$ in the two sub-cases of
Case (2)

    1. (a)

The target vertex $\widetilde{u}=t(f_{g-1})$ lies on $\widetilde{T}_{0}^{+}$.
In this case, the unique cycle $\widetilde{\gamma}^{+}_{g-1}$ of the graph
$\widetilde{T}\cup\\{\widetilde{f}_{g-1}\\}$ actually lies on
$\widetilde{T}_{0}^{+}\cup\\{\widetilde{f}_{g-1}\\}$: it starts at
$\widetilde{v}$, proceeds to $\widetilde{u}$ via $+\widetilde{f}_{g-1}$, and
then returns to $\widetilde{v}$ via the unique path that ends with the edge
$-\widetilde{g}_{2}$. It follows that
$\langle\widetilde{\gamma}^{+}_{g-1},g_{2}\rangle=-1$ and
$\langle\widetilde{\gamma}^{+}_{g-1},\iota(g_{2})\rangle=0$. Furthermore, the
cycle $\widetilde{\gamma}^{+}_{g-1}$ does not intersect the edges
$\widetilde{g}_{1}$ and $\iota(\widetilde{g}_{1})$. Hence we can compute the
last diagonal entries of the upper-triangular matrices
$\Psi(\widetilde{F}_{1})$ and $\Psi(\widetilde{F}_{2})$:

$\Psi(\widetilde{F}_{1})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=0,\quad\Psi(\widetilde{F}_{2})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{2}-\iota(\widetilde{g}_{2})\rangle=-1.$

It follows that $|\det\Psi(\widetilde{F}_{1})|=0$, hence the cell
$C(\widetilde{F}_{1})$ is contracted. Also, $\Psi$ maps the cell
$C(\widetilde{F}_{2})$ to the opposite half-space $M^{-}$ as
$C(\widetilde{F})$, but with the same determinant, since $\widetilde{F}$ and
$\widetilde{F}_{2}$ have the same rank $r$. Hence $\Psi$ is harmonic.

    2. (b)

The target vertex $\widetilde{u}=t(f_{g-1})$ lies on $\widetilde{T}_{0}^{-}$.
In this case, the cycle $\widetilde{\gamma}^{+}_{g-1}$ starts at
$\widetilde{v}$, proceeds to $\widetilde{u}$ via $\widetilde{f}_{g-1}$, and
proceeds to $\iota(\widetilde{v})$ via a unique path that ends with the edge
$-\iota(\widetilde{g}_{2})$. From there the path returns from
$\iota(\widetilde{v})$ to $\widetilde{v}$ via the unique path that passes
through the edge $\widetilde{e}^{+}_{0}$ that links the two trees
$\widetilde{T}^{\pm}_{0}$; this path starts with the edge
$\iota(\widetilde{g}_{1})$ and ends with $-\widetilde{g}_{1}$. Summarizing, we
see that

$\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}\rangle=-1,\quad\langle\widetilde{\gamma}^{+}_{g-1},\iota(\widetilde{g}_{1})\rangle=1,\quad\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{2}\rangle=0,\quad\langle\widetilde{\gamma}^{+}_{g-1},\iota(\widetilde{g}_{1})\rangle=-1.$

Hence we calculate the final diagonal entries of $\Psi(\widetilde{F}_{1})$ and
$\Psi(\widetilde{F}_{2})$:

$\Psi(\widetilde{F}_{1})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=-2,\quad\Psi(\widetilde{F}_{2})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{2}-\iota(\widetilde{g}_{2})\rangle=1.$

It follows that $\Psi$ maps $C(\widetilde{F}_{2})$ to the same half-space
$M^{+}$ with the same determinant $|\det d\Psi(\widetilde{F}_{2})|=|\det
d\Psi(\widetilde{F})|=2^{r-1}$, while $C(\widetilde{F}_{1})$ is mapped to the
opposite space $M^{-}$ with determinant $|\det
d\Psi(\widetilde{F}_{1})|=2^{r}$. Since $2^{r}=2^{r-1}+2^{r-1}$, the map
$\Psi$ is harmonic.

  3. (3)

The vertex $v$ lies on $\gamma(G_{0})$, and $v\nless u$. Let $g_{1}$ and
$g_{2}$ be the two edges of $G_{0}$ rooted at $v$ that lie on the cycle
$\gamma(G_{0})$, then $F_{1}=\\{f_{1},\ldots,f_{g-2},g_{1}\\}$ and
$F_{2}=\\{f_{1},\ldots,f_{g-2},g_{2}\\}$ are genus one decompositions of
$G_{0}$ of the same rank $r$ as $F$, since removing $g_{1}$ or $g_{2}$ from
$G_{0}\cup\\{f_{g-1}\\}$ gives a connected graph of genus one. Any edge
$f^{\prime}\in T_{v}G_{0}$ other than $g_{1}$ and $g_{2}$ is the starting edge
of a separate tree which does not contain $u=t(f_{g-1})$, so
$G_{0}\cup\\{f_{g-1}\\}\backslash\\{f^{\prime}\\}$ has a genus zero connected
component, and $\\{f_{1},\ldots,f_{g-2},f^{\prime}\\}$ is not a genus one
decomposition.

Let $\widetilde{g}_{1}$ and $\widetilde{g}_{2}$ be the edges of
$\widetilde{G}$ at $\widetilde{v}$ lying above $g_{1}$ and $g_{2}$,
respectively, and denote
$\widetilde{F}_{1}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{g}_{1}\\}$
and
$\widetilde{F}_{2}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{g}_{2}\\}$.
The preimage of the cycle $\gamma(G_{0})$ is the unique cycle
$\gamma(p^{-1}(G_{0}))$ of the genus one graph $p^{-1}(G_{0})$. We orient this
cycle so that it starts with the edge $\widetilde{g}_{1}$, passes through
$\iota(\widetilde{v})$, and ends with $-\widetilde{g}_{2}$. Let
$\widetilde{u}^{\prime}$ be the end vertex of the unique shortest path from
$\widetilde{u}$ to $\gamma(p^{-1}(G_{0}))$; this vertex may be $\widetilde{u}$
itself but cannot be $\widetilde{v}$ or $\iota(\widetilde{v})$, since we have
assumed that the shortest path from $u$ to $\gamma(G_{0})$ does not pass
through $v$. We now assume without loss of generality that
$\widetilde{u}^{\prime}$ lies on the same path from $\widetilde{v}$ to
$\iota(\widetilde{v})$ as $\widetilde{g}_{1}$, otherwise exchange
$\widetilde{g}_{1}$ and $\widetilde{g}_{2}$ (see Figure 8).

Figure 8. The cycle $\widetilde{\gamma}^{+}_{g-1}$ in Case (3)

All cells adjacent to $\widetilde{C}$ other than $C(\widetilde{F})$,
$C(\widetilde{F}_{1})$, and $C(\widetilde{F}_{2})$ are contracted. For the
last two, we need to compute the matrix entry (31). We now calculate the
relevant intersection numbers. The path $\widetilde{\gamma}^{+}_{g-1}$ starts
at $\widetilde{v}$, proceeds via $+\widetilde{f}_{g-1}$ to $\widetilde{u}$,
then to $\widetilde{u}^{\prime}$, and then back to $\widetilde{v}$ along a
path lying in $\gamma(p^{-1}(G_{0}))$ that ends with $-\widetilde{g}_{1}$, and
does not contain $\iota(\widetilde{g}_{1})$, $\widetilde{g}_{2}$, or
$\iota(\widetilde{g}_{2})$. It follows that

$\Psi(\widetilde{F}_{1})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=-1,\quad\Psi(\widetilde{F}_{2})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{2}-\iota(\widetilde{g}_{2})\rangle=0.$

Therefore $\Psi$ maps $C(\widetilde{F}_{1})$ to the opposite side $M^{-}$ as
$C(\widetilde{F})$, but with the same determinant $|\det\Psi|=2^{r-1}$. On the
other hand, $C(\widetilde{F}_{2})$ is contracted (this can also be seen by
noting that the preimage of the graph
$G_{0}\cup\\{f_{g-1}\\}\backslash\\{g_{2}\\}$ is disconnected). Hence $\Psi$
is harmonic.

  4. (4)

Finally, we consider the possibility that $v$ lies on $\gamma(G_{0})$ and that
$v<u$, in other words $u$ lies on a tree attached to $v$. In this case, there
are three edges at $v$ that give genus one decompositions: the two edges
$g_{1}$ and $g_{2}$ lying on the cycle $\gamma(G_{0})$, and the edge $g_{3}$
that starts the unique path from $v$ to $u$. All other edges $e^{\prime}$ at
$v$ support trees, and their removal from $G_{0}\cup\\{f_{g-1}\\}$ produces a
connected component of genus zero.

For $i=1,2,3$ denote $\widetilde{g}_{i}$ the lift of $g_{i}$ rooted at
$\widetilde{v}$, and denote
$\widetilde{F}_{i}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{g-2},\widetilde{g}_{i}\\}$.
As in Case 2 above, there are two subcases, depending on whether the target
vertex $\widetilde{u}$ lies on the same tree $T^{\pm}_{0}$ as $\widetilde{v}$
(say $T^{+}_{0}$), or on the other tree. The two possibilities are shown on
Figure 9.

Figure 9. The cycle $\widetilde{\gamma}^{+}_{g-1}$ in the two sub-cases of
Case (4)

    1. (a)

The vertex $\widetilde{u}$ lies on $T^{+}_{0}$. In this case, any path on the
graph $p^{-1}(G_{0}\cup\\{f_{g-1}\\})$ starting at $\widetilde{v}$ and ending
at $\iota(\widetilde{v})$ passes through the preimage $p^{-1}(\gamma(G_{0}))$
of the unique cycle of $G_{0}$. Removing either
$\\{\widetilde{g}_{1},\iota(\widetilde{g}_{1})\\}$ or
$\\{\widetilde{g}_{2},\iota(\widetilde{g}_{2})\\}$ from
$p^{-1}(\gamma(G_{0}))$ disconnects the cycle, and therefore the entire
preimage graph $p^{-1}(G_{0}\cup\\{f_{g-1}\\})$. It follows that $F_{1}$ and
$F_{2}$ are not odd genus one decompositions. To prove harmonicity, we need to
compute $\Psi(\widetilde{F}_{3})_{g-1,g-1}$. The cycle
$\widetilde{\gamma}^{+}_{g-1}$ starts at $\widetilde{v}$, proceeds to
$\widetilde{u}$ via $\widetilde{f}_{g-1}$, and then back to $\widetilde{v}$
via a path in $T_{0}^{+}$ that ends in $-\widetilde{g}_{3}$ and does not
contain $\iota(\widetilde{g}_{3})$. It follows that

$\Psi(\widetilde{F}_{3})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{3}-\iota(\widetilde{g}_{3})\rangle=-1.$

Therefore $\Psi$ maps $C(\widetilde{F}_{3})$ to the opposite side $M^{-}$ as
$C(\widetilde{F})$, but with the same determinant $2^{r-1}$. Hence $\Psi$ is
harmonic.

    2. (b)

The vertex $\widetilde{u}$ lies on $T^{-}_{0}$. In this case, all three genus
one decompositions $F_{1}$, $F_{2}$, and $F_{3}$ are odd. There are two paths
from $\widetilde{v}$ to $\iota(\widetilde{v})$ along the cycle
$p^{-1}(\gamma(G_{0}))$, starting with edges $\widetilde{g}_{1}$ and
$\widetilde{g}_{2}$. We assume without loss of generality that the path that
contains the edge $\widetilde{e}^{+}_{0}$ (and hence lies in the spanning tree
$\widetilde{T}$) begins with $\widetilde{g}_{1}$. In this case, the path
$\widetilde{\gamma}^{+}_{g-1}$ begins at $\widetilde{v}$, moves to
$\widetilde{u}$ via $\widetilde{f}_{g-1}$, then to $\iota(\widetilde{v})$ via
a path ending in $-\iota(\widetilde{g}_{3})$, and finally from
$\iota(\widetilde{v})$ via the path (passing through $\widetilde{e}^{+}_{0}$)
that starts with $\iota(\widetilde{g}_{2})$ and ends with
$-\widetilde{g}_{1}$. Hence $\widetilde{\gamma}^{+}_{g-1}$ contains
$-\widetilde{g}_{1}+\iota(\widetilde{g}_{2})-\iota(\widetilde{g}_{3})$ and
does not contain the edges $\iota(\widetilde{g}_{1})$, $\widetilde{g}_{2}$, or
$\widetilde{g}_{3}$, and therefore the diagonal entries are

$\Psi(\widetilde{F}_{1})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=-1,\quad\Psi(\widetilde{F}_{2})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{2}-\iota(\widetilde{g}_{2})\rangle=-1,$
$\Psi(\widetilde{F}_{3})_{g-1,g-1}=\langle\widetilde{\gamma}^{+}_{g-1},\widetilde{g}_{3}-\iota(\widetilde{g}_{3})\rangle=1.$

Therefore, $\Psi$ maps the two cells $C(\widetilde{F})$ and
$C(\widetilde{F}_{3})$ to the half-space $M^{+}$ and the two cells
$C(\widetilde{F}_{1})$ and $C(\widetilde{F}_{2})$ to the half-space $M^{-}$,
all with the same determinant $2^{r-1}$. Hence $\Psi$ is harmonic.

The endpoints of $f_{a}$ lie on different connected components of $G\backslash
F$. We assume without loss of generality that $v=s(f_{a})$ lies on $G_{0}$ and
that $u=t(f_{a})$ lies on $G_{1}$. Furthermore, we assume that $f_{a}$ lies in
the spanning tree $T^{c}$, and the ordering convention then implies that
$f_{a}=f_{1}$ and $\widetilde{f}_{a}=\widetilde{f}_{1}$. Since the matrix
$\Psi(\widetilde{F})$ is lower triangular, we see that
$M^{+}=\Psi(H^{+})=\\{y:y_{1}\geq 0\\}$ and $M^{-}=\Psi(H^{-})=\\{y:y_{1}\leq
0\\}$.

Let $\widetilde{f}^{\prime}$ be an edge at $\widetilde{v}$, and let
$\widetilde{F}^{\prime}=\\{\widetilde{f}^{\prime},\widetilde{f}_{2},\ldots,\widetilde{f}_{g-1}\\}$
define a cell $C(\widetilde{F}^{\prime})$ adjacent to $C(\widetilde{F})$ via
$C^{\prime}$. The matrix $\Psi(\widetilde{F}^{\prime})$ is obtained from the
matrix $\Psi(\widetilde{F})$ by replacing the first column, so we are only
interested in the new entry
$\Psi(\widetilde{F}^{\prime})_{11}=\langle\widetilde{\gamma}^{+}_{1},\widetilde{f}^{\prime}-\iota(\widetilde{f}^{\prime})\rangle$
in the top left: if it is zero then $p(\widetilde{F}^{\prime})$ is not an odd
genus one decomposition, and if it is nonzero then its sign determines whether
$\Psi$ maps $C(\widetilde{F}^{\prime})$ to $M^{+}$ or $M^{-}$.

The edge $f_{1}$ is a bridge edge of the graph $G_{0}\cup
G_{1}\cup\\{f_{1}\\}$. We need to consider two possibilities:

  1. (1)

The vertex $v=s(f_{1})$ does not lie on the unique cycle $\gamma(G_{0})$ of
the graph $G_{0}$. There is a unique edge $g_{1}$ at $v$ pointing in the
direction of $\gamma(G_{0})$, and the graph $G_{0}\cup
G_{1}\cup\\{f_{1}\\}\backslash\\{g_{1}\\}$ has two connected components of
genus one, namely $G_{0}\backslash\\{g_{1}\\}$ and $G_{1}$. Therefore,
$F_{1}=\\{g_{1},f_{2},\ldots,f_{g-1}\\}$ is an odd genus one decomposition of
the same length $r$ as $F$. Any other edge $f^{\prime}$ at $v$ supports a tree
rooted at $v$, hence removing $f^{\prime}$ from $G_{0}\cup G_{1}\cup\\{f\\}$
separates a genus zero connected component, and the corresponding
decomposition is not genus one (see Figure 10).

Figure 10. The cycle $\widetilde{\gamma}^{+}_{1}$ in Case (1)

Let $\widetilde{g}_{1}$ denote the lift of $g_{1}$ at $\widetilde{v}$, and
denote
$\widetilde{F}_{1}=\\{\widetilde{g}_{1},\widetilde{f}_{2},\ldots,\widetilde{f}_{g-1}\\}$.
To show that $\Psi$ is harmonic, it remains to show that $\Psi$ maps the cell
$C(\widetilde{F}_{1})$ to the opposite side $M^{-}$, in other words we need to
show that the diagonal entry
$\Psi(\widetilde{F}_{1})_{11}=\langle\widetilde{\gamma}^{+}_{1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle$
is negative.

We have chosen an edge $e_{1}$ lying on the unique cycle $\gamma(G_{1})$ of
$G_{1}$, and a lift $\widetilde{e}^{+}_{1}$ lying on the unique cycle of
$p^{-1}(G_{1})$, with the property that the path from
$\widetilde{u}=t(\widetilde{f})$ to $\iota(\widetilde{u})$ that passes through
$\widetilde{e}^{+}_{1}$ has the same orientation as $\widetilde{e}^{+}_{1}$.
Hence the path $\widetilde{\gamma}^{+}_{1}$ is constructed as follows: it
starts at $\widetilde{v}$, proceeds via $\widetilde{f}_{1}$ to
$\widetilde{u}$, then via the aforementioned path to $\iota(\widetilde{u})$,
then to $\iota(\widetilde{v})$ via $-\iota(\widetilde{f}_{1})$, and then from
$\iota(\widetilde{v})$ to $\widetilde{v}$ via the unique path in
$p^{-1}(G_{0})$ containing the edge $\widetilde{e}^{+}_{0}$. This path begins
with $\iota(\widetilde{g}_{1})$ and ends with $-\widetilde{g}_{1}$, hence

$\Psi(\widetilde{F}_{1})_{11}=\langle\widetilde{\gamma}^{+}_{1},\widetilde{g}_{1}-\iota(\widetilde{g}_{1})\rangle=-2.$

Therefore, $\Psi$ maps $C(\widetilde{F})$ and $C(\widetilde{F}_{1})$ to
different sides of $\widetilde{C}$ with the same determinant, so $\Psi$ is
harmonic.

Figure 11. The configuration in Case (2)

  2. (2)

The vertex $v=s(f_{1})$ lies on the unique cycle $\gamma(G_{0})$ of $G_{0}$.
It is easy to see that this case is in fact a relabeling of Case 2b described
above. Indeed, let $g^{\prime}_{1}$ and $g^{\prime}_{2}$ be the two edges at
$v$ lying on $\gamma(G_{0})$. Replacing $f_{1}$, $g^{\prime}_{1}$, and
$g^{\prime}_{2}$ with respectively $g_{1}$, $f_{g-1}$, and $g_{2}$, we obtain
the same picture as in Case 2b (see Figure 11).

This completes the proof of Proposition 5.2. ∎

$\Gamma$$h_{1}$$h_{3}$$h_{4}$$h_{5}$$h_{7}$$h_{6}$$\widetilde{\Gamma}$$\widetilde{h}^{+}_{1}$$\widetilde{h}^{-}_{1}$$\widetilde{h}^{-}_{3}$$\widetilde{h}^{+}_{3}$$\widetilde{h}^{-}_{5}$$\widetilde{h}^{+}_{5}$$\widetilde{h}^{-}_{4}$$\widetilde{h}^{+}_{4}$$\widetilde{h}^{-}_{6}$$\widetilde{h}^{+}_{6}$$\widetilde{h}^{+}_{7}$$\widetilde{h}^{-}_{7}$
Figure 12. Free double cover with $g=3$.

######  Example 5.5.

We now illustrate Thm. 5.1, the structure of the Abel–Prym map, as well as
Constructions A, B, and C, using the free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ shown in Fig. 12. We note that $g=3$, hence
the Abel–Prym map

$\Psi:\operatorname{Sym}^{2}(\widetilde{\Gamma})\to\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$

maps to the identity connected component of the Prym variety. The graph
$\widetilde{\Gamma}$ has no loops, so it is sufficient to use the minimal
model $p:\widetilde{G}\to G$, which is obtained by not subdividing any edges.

We begin by using Construction C to define a basis
$\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ for
$\operatorname{Ker}p_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to
H_{1}(G,\mathbb{Z})$. This basis defines a coordinate system on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. We will write down the
matrix of $\Psi$ with respect to this coordinate system on each non-contracted
cell of $\operatorname{Sym}^{2}(\widetilde{\Gamma})$. First, we need to choose
two edges $\widetilde{f}_{1},\widetilde{f}_{2}\in E(\widetilde{G})$ mapping to
distinct edges of $G$. It is convenient to set
$\widetilde{f}_{1}=\widetilde{h}^{+}_{3}$ and
$\widetilde{f}_{2}=\widetilde{h}^{-}_{6}$. Removing $f_{1}=h_{3}$ and
$f_{2}=h_{6}$ decomposes $G$ into three connected components, which we denote
as follows:

$G_{0}=\\{h_{4},h_{5}\\},\quad G_{1}=\\{h_{1}\\},\quad G_{2}=\\{h_{7}\\}.$

For $G_{0}$ we choose the spanning tree $T_{0}=\\{h_{5}\\}$, while $G_{1}$ and
$G_{2}$ have trivial spanning trees. The corresponding spanning tree $T$ of
$G$ is

$T=\\{h_{3},h_{5},h_{6}\\}.$

To construct a spanning tree $\widetilde{T}$ for $\widetilde{G}$, we join the
two lifts $\widetilde{T}^{\pm}$ of $T$ with one of the lifts of $h_{4}$. We
choose $\widetilde{e}^{+}_{0}=\widetilde{h}^{+}_{4}$, so that

$\widetilde{T}=\widetilde{T}^{+}\cup\widetilde{T}^{-}\cup\\{\widetilde{e}^{+}_{0}\\}=\\{\widetilde{h}^{\pm}_{3},\widetilde{h}^{+}_{4},\widetilde{h}^{\pm}_{5},\widetilde{h}^{\pm}_{6}\\}.$

In order to agree with the notation of Constructions A-C, we denote
$e_{0}=h_{4}$, $e_{1}=h_{1}$, $e_{2}=h_{7}$,
$\widetilde{e}^{+}_{1}=\widetilde{h}^{+}_{1}$, and
$\widetilde{e}^{+}_{2}=\widetilde{h}^{+}_{7}$ (recall that we require
$t(\widetilde{f}_{k})=s(\widetilde{e}_{k})$).

According to Construction B, the cycles $\widetilde{\gamma}^{+}_{1}$ and
$\widetilde{\gamma}^{+}_{2}$ are the unique cycles of $\widetilde{G}$
containing the spanning tree $\widetilde{T}$ and the edges
$\widetilde{e}^{+}_{1}=\widetilde{h}^{+}_{1}$ and
$\widetilde{e}^{+}_{2}=\widetilde{h}^{+}_{7}$, respectively. In other words,

$\widetilde{\gamma}^{+}_{1}=\widetilde{h}_{1}^{+}-\widetilde{h}^{-}_{3}-\widetilde{h}^{+}_{4}-\widetilde{h}^{+}_{5}+\widetilde{h}^{+}_{3},\quad\widetilde{\gamma}^{+}_{2}=\widetilde{h}^{+}_{7}-\widetilde{h}^{+}_{6}+\widetilde{h}^{+}_{4}+\widetilde{h}^{-}_{5}+\widetilde{h}^{-}_{6}.$

The cycles $\widetilde{\gamma}^{+}_{1}-\iota_{*}(\widetilde{\gamma}^{+}_{1})$
and $\widetilde{\gamma}^{+}_{2}-\iota_{*}(\widetilde{\gamma}^{+}_{2})$ form a
basis for $\operatorname{Ker}p_{*}$.

We now determine the matrix of the linear map $\Psi$ on each non-contracted
cell of $\operatorname{Sym}^{2}(\widetilde{\Gamma})$. There are 15 two-element
subsets of $E(G)$, and all of them are ogods except for $\\{h_{1},h_{3}\\}$
and $\\{h_{6},h_{7}\\}$. Therefore, there are 52 cells of
$\operatorname{Sym}^{2}(\widetilde{\Gamma})$ on which $\Psi$ has maximal rank,
corresponding to the 13 ogods. The matrix of $\Psi$ on a cell
$\widetilde{h}^{+}_{i}\times\widetilde{h}^{+}_{j}$ is the $2\times 2$ matrix
whose columns are obtained by intersecting respectively
$\widetilde{h}^{+}_{i}-\widetilde{h}^{-}_{i}$ and
$\widetilde{h}^{+}_{j}-\widetilde{h}^{-}_{j}$ with the cycles
$\widetilde{\gamma}^{+}_{1}$ and $\widetilde{\gamma}^{+}_{2}$, and the matrix
of $\Psi$ on the other three cells
$\widetilde{h}^{\pm}_{i}\times\widetilde{h}^{\pm}_{j}$ is obtained by
appropriately changing the signs of the columns.

The following table lists the 13 ogods of the graph $G$. For each ogod
$F=\\{h_{i},h_{j}\\}$, we provide the rank $r(F)$ (the number of connected
components of $G\backslash F)$, the local degree $\deg\Psi=2^{r(F)-1}$ on each
cell of $\operatorname{Sym}^{2}(\widetilde{\Gamma})$ corresponding to the
ogod, and the matrix of $\Psi$ on the cell
$\widetilde{h}^{+}_{i}\times\widetilde{h}^{-}_{i}$ (the absolute value of its
determinant is $\deg\Psi$). We also assign a color to each ogod.

$F$ | $r(F)$ | $\deg\Psi|_{\widetilde{h}^{\pm}_{i}\times\widetilde{h}^{\pm}_{j}}$ | $\Psi|_{\widetilde{h}^{+}_{i}\times\widetilde{h}^{+}_{j}}$ | color
---|---|---|---|---
$\\{h_{1},h_{4}\\}$ | 1 | 1 | $\left[\begin{array}[]{cc}1&-1\\\ 0&1\end{array}\right]$ |
$\\{h_{1},h_{5}\\}$ | 1 | 1 | $\left[\begin{array}[]{cc}1&-1\\\ 0&-1\end{array}\right]$ |
$\\{h_{1},h_{6}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}1&0\\\ 0&-2\end{array}\right]$ |
$\\{h_{1},h_{7}\\}$ | 1 | 1 | $\left[\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right]$ |
$\\{h_{3},h_{4}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}2&-1\\\ 0&1\end{array}\right]$ |
$\\{h_{3},h_{5}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}2&-1\\\ 0&-1\end{array}\right]$ |
$\\{h_{3},h_{6}\\}$ | 3 | 4 | $\left[\begin{array}[]{cc}2&0\\\ 0&-2\end{array}\right]$ |
$\\{h_{3},h_{7}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}2&0\\\ 0&1\end{array}\right]$ |
$\\{h_{4},h_{5}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}-1&-1\\\ 1&-1\end{array}\right]$ |
$\\{h_{4},h_{6}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}-1&0\\\ 1&-2\end{array}\right]$ |
$\\{h_{4},h_{7}\\}$ | 1 | 1 | $\left[\begin{array}[]{cc}-1&0\\\ 1&1\end{array}\right]$ |
$\\{h_{5},h_{6}\\}$ | 2 | 2 | $\left[\begin{array}[]{cc}-1&0\\\ -1&-2\end{array}\right]$ |
$\\{h_{5},h_{7}\\}$ | 1 | 1 | $\left[\begin{array}[]{cc}-1&0\\\ -1&1\end{array}\right]$ |

We now explain how to construct the Abel–Prym map
$\Psi:\operatorname{Sym}^{2}(\widetilde{\Gamma})\to\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$.
We first choose a starting cell; it is convenient to begin with the cell
$\widetilde{f}_{1}\times\widetilde{f}_{2}=\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6}$.
According to our calculations, the matrix of $\Psi$ on this cell is

$\Psi|_{\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6}}=\left[\begin{array}[]{cc}2&0\\\
0&2\end{array}\right];$

note that this matrix is lower-triangular with $r(\\{h_{3},h_{6})\\})-1=2$
diagonal entries equal to $2$, in accordance with Cor. 4.2. Viewing
$\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6}$ as the rectangle
$[0,\ell(h_{3})]\times[0,\ell(h_{6})]$ in $\mathbb{R}^{2}$, we see that
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$ is a rectangle with
sides $2\ell(h_{3})$ and $2\ell(h_{6})$ in
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ (with respect to the
coordinate system defined by $\widetilde{\gamma}_{1}$ and
$\widetilde{\gamma}_{2}$). We note that since $\deg\Psi=4$ on the cell
$\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6}$, the Abel–Prym map is one-
to-one over the interior of the cell; in other words, there are no other cells
of $\operatorname{Sym}^{2}(\widetilde{\Gamma})$ lying over
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$.

We now look at the graph to locate the cells in
$\operatorname{Sym}^{2}(\widetilde{\Gamma})$ that are adjacent to
$\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6}$, by changing one of the two
edges to an adjacent edge. We then use the table to determine their images in
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. For example, the terminal
vertex of $\widetilde{h}^{+}_{3}$ is the starting vertex of
$\widetilde{h}^{+}_{1}$ and the terminal vertex of $\widetilde{h}^{-}_{1}$.
Therefore, the images of the two cells
$\widetilde{h}^{+}_{1}\times\widetilde{h}^{-}_{6}$ and
$\widetilde{h}^{-}_{1}\times\widetilde{h}^{-}_{6}$ are glued to the right side
of the rectangle $\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$.
According to the table, each of the images
$\Psi(\widetilde{h}^{+}_{1}\times\widetilde{h}^{-}_{6})$ and
$\Psi(\widetilde{h}^{+}_{1}\times\widetilde{h}^{-}_{6})$ is a rectangle with
sides $\ell(h_{1})$ and $2\ell(h_{6})$; this implies that these images
coincide. Similarly, along the top edge of the rectangle
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$ we glue two identical
rectangles $\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{+}_{7})$ and
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{7})$. On the other hand,
the two cells that are glued along the left side of the rectangle
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$ are the non-congruent
parallelograms $\Psi(\widetilde{h}^{+}_{5}\times\widetilde{h}^{-}_{6})$ and
$\Psi(\widetilde{h}^{-}_{4}\times\widetilde{h}^{-}_{6})$, and similarly there
are two non-congruent parallelograpms along the bottom side of
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{6})$.

We then proceed in the same way for all non-contracted cells. For example, the
right sides of the two identical rectangles
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{+}_{7})$ and
$\Psi(\widetilde{h}^{+}_{3}\times\widetilde{h}^{-}_{7})$ are attached to the
left side of the rectangle
$\Psi(\widetilde{h}^{-}_{3}\times\widetilde{h}^{-}_{6})$, and so on. The
result is a collection of overlapping parallelograms
$\Psi(\widetilde{h}^{\pm}_{i}\times\widetilde{h}^{\pm}_{j})$ tiling the torus
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. Each parallelogram tile
comes equipped with the degree of $\Psi$ (equal to 1, 2, or 4), and the total
degree of all tiles above each point is equal to 4, so any point not lying on
the boundary of a tile lies in anywhere between one and four tiles. The
structure of the overlaps strongly depends on the exact values of the edge
lengths $\ell(h_{i})$.

The result, for a specific choice of edge lengths, is shown on Fig. 13. The
top part is an exploded view of the non-contracted cells of
$\operatorname{Sym}^{2}(\widetilde{\Gamma})$. Each cell
$\widetilde{h}^{\pm}_{i}\times\widetilde{h}^{\pm}_{j}$ is represented by a
parallelogram and is colored according to the corresponding ogod
$\\{h_{i},h_{j}\\}$ (see table). The cells are arranged in several horizontal
layers and are oriented and aligned according to their images in
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. For example, the four large
dark grey cells in the middle layer represent the four cells
$\widetilde{h}^{\pm}_{3}\times\widetilde{h}^{\pm}_{6}$, which are the only
cells of degree four. No attempt has been made to show the incidence relations
between the cells in $\operatorname{Sym}^{2}(\widetilde{\Gamma})$, since this
complex is not embeddable in $\mathbb{R}^{3}$, and the numerous cells that are
contracted by $\Psi$ are not shown.

The multicolored parallelogram on the bottom layer is
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ with the induced cellular
decomposition, obtained by intersecting all the tiles. A cell corresponding to
a single parallelogram has the same color, while if parallelograms of
different colors intersect, then we blend the corresponding colors. The
central point of the parallelogram is the identity element of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$.

Figure 13. The structure of the Abel–Prym map
$\Psi:\operatorname{Sym}^{2}(\widetilde{\Gamma})\to\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$
of the cover shown in Fig. 12. The tesselated parallelogram on the bottom is
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ with the cell decomposition
induced by $\Psi$. The top and middle parts are an exploded view of the non-
contracted cells of $\operatorname{Sym}^{2}(\widetilde{\Gamma})$. Cells of
$\operatorname{Sym}^{2}(\widetilde{\Gamma})$ are colored according to ogod
type, these colors are mixed in cells of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. Edge lengths are
$\ell(h_{1})=2.4$, $\ell(h_{3})=0.8$, $\ell(h_{4})=1$, $\ell(h_{5})=1.4$,
$\ell(h_{6})=1.1$, $\ell(h_{7})=1.4$.

##  Appendix A The algebraic Abel–Prym map (by Sebastian Casalaina-Martin)

Let $\pi:\widetilde{C}\to C$ be a connected étale double cover of a smooth
projective curve $C$ of genus $g\geq 2$ over an algebraically closed field $k$
of characteristic not equal to $2$, let $\iota:\widetilde{C}\to\widetilde{C}$
be the associated involution, and denote by
$\operatorname{Nm}:J(\widetilde{C})\to J(C)$ the norm map for $\pi$, where for
a smooth projective curve $X$ over $k$ we denote by
$J(X)=\operatorname{Pic}^{0}_{X/k}$ the Jacobian of $X$. For any natural
number $d$ the _Abel–Prym map_ in degree $d$ is defined to be the map

$\delta_{d}:\widetilde{C}^{(d)}\longrightarrow\ker\operatorname{Nm}\subseteq
J(\widetilde{C}),$
$\widetilde{D}\mapsto\mathcal{O}_{\widetilde{C}}(\widetilde{D}-\iota\widetilde{D}),$

where $\widetilde{C}^{(d)}$ is the $d$-fold symmetric product of the curve.
The kernel of the norm map has two connected components, namely the Prym
variety $P=P(\widetilde{C}/C):=(\ker\operatorname{Nm})^{\circ}$, the connected
component of the identity, and the remaining component, which we will denote
by $P^{\prime}$; $P$ admits a principal polarization $\Xi$ with the property
that if $\Theta_{\widetilde{C}}$ is the canonical principal polarization on
$J(\widetilde{C})$, then $\Theta_{\widetilde{C}}|_{P}=2\cdot\Xi$ (e.g.,
[Mum74, §6]). The image of $\delta_{d}$ is contained in $P$ if $d$ is even and
contained in $P^{\prime}$ if $d$ is odd (e.g., [Bea77, Lem. 3.3, and p.159]).

The Abel–Prym map in degree-$1$ has been studied quite extensively, and we
recall this case in §A.1. In particular, the map $\delta_{1}$ is a closed
embedding if and only if $\widetilde{C}$ is not hyperelliptic, and has degree
$2$ otherwise. The purpose of this appendix is to provide a proof of some
basic facts regarding the Abel–Prym map for $d>1$. We expect these are well
known, but are not aware of a reference in the literature.

######  Proposition A.1 (Corollary A.9, Corollary A.13, and Proposition A.14).

The Abel–Prym map $\delta_{d}$ is generically finite if and only if $d\leq
g-1$, and surjects onto $P$ (resp. $P^{\prime}$) if and only if $d\geq g-1$
and $d$ is even (resp. $d$ is odd). Moreover, $\deg\delta_{g-1}=2^{g-1}$, and
if $\operatorname{char}(k)=0$, then for $d\leq g-2$ we have
$\deg\delta_{d}=2^{n}\leq 2^{d}$ for some integer $n\leq d$, with equality
holding if $\widetilde{C}$ is hyperelliptic.

######  Remark A.2 (Degree bound in positive characteristic).

If $\operatorname{char}(k)=p>0$, then for $d\leq g-2$ we show that
$\deg\delta_{d}=p^{m}2^{n}$ for some integers $m$ and $n$ with $n\leq d$. The
reason for the uncontrolled power of $p$ in the formula is that we compute the
degree via a cohomology class computation in $\ell$-adic cohomology, with
$\ell\neq p$. A similar computation in crystalline cohomology allows one to
remove the powers of $p$.

For general covers one has:

######  Proposition A.3 (Corollary A.15).

Let $\widetilde{C}/C$ be a general cover. Then $\deg\delta_{d}=1$ for $d<g/2$.

While $\delta_{1}$ is finite, in contrast, for all $d\geq 2$ there exist
positive dimensional fibers of $\delta_{d}$. Indeed, it suffices to show this
for $d$ even since for all $d$, fibers of $\delta_{d}$ can be included in
fibers of $\delta_{d+1}$ using the observation that if
$\delta_{d}(\widetilde{D})=\delta_{d}(\widetilde{D}^{\prime})$, then
$\delta_{d+1}(\widetilde{D}+\tilde{p})=\delta_{d+1}(\widetilde{D}^{\prime}+\tilde{p})$
for any $\tilde{p}\in\widetilde{C}$. For $d$ even, we note that the
composition
$C^{(d/2)}\stackrel{{\scriptstyle\pi^{*}}}{{\to}}\widetilde{C}^{(d)}\stackrel{{\scriptstyle\delta_{d}}}{{\to}}P\subseteq
J(\widetilde{C})$ has image $\mathcal{O}_{\widetilde{C}}$.

Sometimes in the presentation it will be convenient to fix a divisor
$\widetilde{D}_{0}\in\widetilde{C}^{(d)}$, and then consider the associated
_pointed Abel–Prym map_

$\delta_{d,\widetilde{D}_{0}}:\widetilde{C}^{(d)}\longrightarrow P\subseteq
J(\widetilde{C})$
$D\mapsto\mathcal{O}_{\widetilde{C}}(\widetilde{D}-\iota\widetilde{D})\otimes\mathcal{O}_{\widetilde{C}}(\iota\widetilde{D}_{0}-\widetilde{D}_{0})$

which simply differs from the canonical Abel–Prym map $\delta_{d}$ by
translation by
$\mathcal{O}_{\widetilde{C}}(\iota\widetilde{D}_{0}-\widetilde{D}_{0})$, and
has image contained in the Prym variety. We emphasize for clarity that the
Abel–Prym map and pointed Abel–Prym map defined here are different from the
map obtained by restricting the Abel map in degree $d$ to a chosen translate
of the Prym variety (e.g., [Bea82, SV01]).

### Acknowledgements

The author would like to thank Angela Ortega and Roy Smith for useful comments
that pointed to an error in an earlier draft regarding fibers of the Abel–Prym
map. He would also like to thank Gavril Farkas for pointing out an error in an
earlier draft regarding Brill–Noether theory for connected étale double
covers, and directing the author to [AF12, Sch17] where the correct statements
are proven, providing the needed result to complete the proof of Proposition
A.3.

###  A.1. The Abel–Prym map in degree-$1$

We recall the following well-known result:

######  Proposition A.4.

For any prime number $\ell\neq\operatorname{char}(k)$ and any point
$\tilde{p}_{0}\in\widetilde{C}$, the class of the push forward of
$\widetilde{C}$ by the pointed Abel–Prym map is

$(\delta_{1,\tilde{p}_{0}})_{*}[\widetilde{C}]=2\cdot\frac{[\Xi]^{\rho-1}}{(\rho-1)!}\in
H^{2\rho-2}(P,\mathbb{Z}_{\ell}(\rho-1)),$ (32)

where $\rho=\dim P=g-1$. In addition, if $\widetilde{C}$ is not hyperelliptic,
then the Abel–Prym map $\delta_{1}$ is an embedding, so that
$[\delta_{1,\tilde{p}_{0}}(\widetilde{C})]=2\cdot\frac{[\Xi]^{\rho-1}}{(\rho-1)!}$.
If $\widetilde{C}$ is hyperelliptic, then $\delta_{1}$ has degree $2$, and the
image $\Sigma:=\delta_{1}(\widetilde{C})\subseteq P^{\prime}$ is a smooth
hyperelliptic curve of genus $g-1$ so that setting
$\Sigma_{\tilde{p}_{0}}:=\delta_{1,\tilde{p}_{0}}(\widetilde{C})\subseteq P$,
we have $[\Sigma_{\tilde{p}_{0}}]=\frac{[\Xi]^{\rho-1}}{(\rho-1)!}$, and
$(P,\Xi)$ is isomorphic to the principally polarized Jacobian
$(J(\Sigma),\Theta_{\Sigma})$.

###### Proof.

Computing the degree of $\delta_{1}$ is a basic computation from the
definition; the details can be found in [BL04, Prop. 12.5.2] where the
arguments are made over $\mathbb{C}$, but which hold over any algebraically
closed field of characteristic not equal to $2$. The statements regarding the
differential of $\delta_{1}$ can be found in [BL04, Cor. 12.5.5–7] over
$\mathbb{C}$; those arguments also hold over any algebraically closed field of
characteristic not equal to $2$, and we also review this in §A.2.1. As the map
$\delta_{1}$ is finite, computing the class of
$(\delta_{1,\tilde{p}_{0}})_{*}[\widetilde{C}]$ is a standard argument using
the fact that
$H^{2\rho-2}(P,\mathbb{Z}_{\ell})=\bigwedge^{2\rho-2}H^{1}(P,\mathbb{Z}_{\ell})$,
and facts about first Chern classes of symmetric polarizations on abelian
varieties. This is essentially the same argument that is used to prove
Poincaré’s formula in [ACGH85, p.25], and the arguments there are easily
adapted to the Abel–Prym map, and the positive characteristic case (see
[Mas76, Lem. 3.2] over $\mathbb{C}$).

The fact that $\Sigma$ is smooth and that $(P,\Xi)$ is isomorphic to the
principally polarized Jacobian $(J(\Sigma),\Theta_{\Sigma})$ follows from the
fact that $[\Sigma_{\tilde{p}_{0}}]=\frac{[\Xi]^{\rho-1}}{(\rho-1)!}$, and the
criterion of Matsusaka–Ran [Col84]. One can conclude that $\Sigma$ is
hyperelliptic by considering the image of the $g^{1}_{2}$ on $\widetilde{C}$
under the norm map for $\widetilde{C}\to\Sigma$. ∎

######  Remark A.5.

If $\widetilde{C}$ is hyperellptic, then $C$ is hyperelliptic, as well
(consider the image of the $g^{1}_{2}$ on $\widetilde{C}$ under the norm map
for $\pi$). Thus the conclusion in Proposition A.4 that if $\widetilde{C}$ is
hyperelliptic then $(P,\Xi)$ is a hyperelliptic Jacobian is a special case of
a result of Mumford, which states that for any $\pi:\widetilde{C}\to C$ with
$C$ hyperelliptic, the Prym variety is a product of hyperelliptic Jacobians
[Mum74, p.346].

######  Remark A.6.

In the case where $\widetilde{C}$ is hyperelliptic, we can say more. From the
definition, we have that
$\delta_{1}(\tilde{p})=\delta_{1}(\tilde{p}^{\prime})$ for distinct points
$\tilde{p},\tilde{p}^{\prime}\in\widetilde{C}$ if and only if
$\tilde{p}+\iota(\tilde{p}^{\prime})$ is in the (unique) $g^{1}_{2}$ on
$\widetilde{C}$. By Riemann–Hurwitz, $\delta_{1}$ is ramified at four points.
The ramification points $\tilde{r}\in\widetilde{C}$ of $\delta_{1}$ are
distinguished by the fact that the $2$-torsion line bundle $\eta$ determining
the cover $\pi$ satisfies $\eta=\mathcal{O}_{C}(\pi(\tilde{r})-p)$ for some
$p\in C$. Note that this forces $\pi(\tilde{r})$ and $p$ to be branch points
for the hyperelliptic involution on $C$. In summary, if $\widetilde{C}$ is
hyperelliptic, then $\eta\cong\mathcal{O}_{C}(p^{\prime}-p)$ for some distinct
ramification points $p,p^{\prime}\in C$ for the $g^{1}_{2}$ on $C$, and the
four ramification points of $\delta_{1}$ are the points
$\pi^{-1}(\\{p,p^{\prime}\\})$. The details can be found in [BL04, §12.5],
where again the arguments hold in positive characterstic, as well; see also
§A.2.1.

###  A.2. The differential of the Abel–Prym map

We next show that the Abel–Prym map is generically finite for $d\leq g-1$ by
showing that the differential is generically injective in that range.

######  Proposition A.7.

For $d\leq g-1$, the differential of $\delta_{d}$ is generically injective.
For all $d$ the differential generically has rank equal to $\min(d,g-1)$. The
differential of $\delta_{d}$ is injective if and only if $d=1$ and
$\widetilde{C}$ is not hyperelliptic.

More precisely, the differential $\delta_{d}$ is injective at
$\widetilde{D}\in\widetilde{C}^{(d)}$ if and only if the support of
$\widetilde{D}$ satisfies
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$
and, setting $D=\operatorname{Nm}\widetilde{D}$, there does not exist an
effective divisor $E\in C^{(d)}$ such that $\eta=\mathcal{O}_{C}(D-E)$, where
$\eta$ is the $2$-torsion line bundle determining the cover $\widetilde{C}/C$.

###### Proof.

The case $d=1$ is handled in Proposition A.4. Since there exist positive
dimensional fibers of $\delta_{d}$ for all $d\geq 2$, the differential cannot
be injective for $d\geq 2$. For the statements on the generic rank, once we
have shown that the differential of $\delta_{g-1}$ is generically injective,
by a dimension count it is generically surjective, and one can easily check
that this implies that $\delta_{d}$ surjects onto $P$ or $P^{\prime}$ for all
$d\geq g-1$. Thus, for the generic rank of the differential of $\delta_{d}$,
it only remains to show that for $d\leq g-1$, the differential of $\delta_{d}$
is generically injective.

For this, we factor $\delta_{d}$ as follows:

$\textstyle{\widetilde{C}^{(d)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times\iota}$$\textstyle{\widetilde{C}^{(d)}\times\widetilde{C}^{(d)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\alpha_{d}}\times{\alpha_{d}}}$$\textstyle{\operatorname{Pic}^{d}_{\widetilde{C}/k}\times\operatorname{Pic}^{d}_{\widetilde{C}/k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-}$$\textstyle{\operatorname{Pic}^{0}_{\widetilde{C}/k}}$

where
${\alpha_{d}}:\widetilde{C}^{(d)}\to\operatorname{Pic}^{d}_{\widetilde{C}/k}$
is the Abel map
$\widetilde{D}\mapsto\mathcal{O}_{\widetilde{C}}(\widetilde{D})$. Then at the
level of the differential, at a divisor $\widetilde{D}\in\widetilde{C}^{(d)}$,
these are given by the maps

$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times\iota}$$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\times
H^{0}(\mathcal{O}_{\iota\widetilde{D}}(\iota\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T{\alpha_{d}}\times
T{\alpha_{d}}}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})\times
H^{1}(\mathcal{O}_{\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}}).}$

The composition of the maps above agrees with the composition of the maps

$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times(-\iota)}$$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\times
H^{0}(\mathcal{O}_{\iota\widetilde{D}}(\iota\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T{\alpha_{d}}\times
T{\alpha_{d}}}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})\times
H^{1}(\mathcal{O}_{\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})}$
(33)

which in turn factors as

$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times(-\iota)}$$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\times
H^{0}(\mathcal{O}_{\iota\widetilde{D}}(\iota\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T{\alpha_{d}}\times
T{\alpha_{d}}}$$\scriptstyle{+}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})\times
H^{1}(\mathcal{O}_{\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{d}}$$\textstyle{H^{0}(\mathcal{O}_{\widetilde{D}+\iota\widetilde{D}}(\widetilde{D}+\iota\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T{\alpha_{2d}}}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})}$
(34)

where we denote by $+:H^{0}(\mathcal{O}_{\widetilde{D}}(\widetilde{D}))\times
H^{0}(\mathcal{O}_{\iota\widetilde{D}}(\iota\widetilde{D}))\to
H^{0}(\mathcal{O}_{\widetilde{D}+\iota\widetilde{D}}(\widetilde{D}+\iota\widetilde{D}))$
the differential of the map
$+:\widetilde{C}^{(d)}\times\widetilde{C}^{(d)}\to\widetilde{C}^{(2d)}$ at the
point $(\widetilde{D},\iota\widetilde{D})$, we set
$\beta_{d}:=+\circ(1\times(-\iota))$, and $T\alpha_{2d}$ is the differential
of the Abel map
$\alpha_{2d}:\widetilde{C}^{(2d)}\to\operatorname{Pic}^{2d}_{\widetilde{C}/k}$
at the point $\widetilde{D}+\iota\widetilde{D}$. Recall that $T\alpha_{2d}$ is
identified with the coboundary map

$\textstyle{\partial_{\widetilde{D}+\iota\widetilde{D}}:H^{0}(\mathcal{O}_{\widetilde{D}+\iota\widetilde{D}}(\widetilde{D}+\iota\widetilde{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathcal{O}_{\widetilde{C}})}$
(35)

from the long exact sequence associated to the short exact sequence

$0\to\mathcal{O}_{\widetilde{C}}\to\mathcal{O}_{\widetilde{C}}(\widetilde{D}+\iota\widetilde{D})\to\mathcal{O}_{\widetilde{D}+\iota\widetilde{D}}(\widetilde{D}+\iota\widetilde{D})\to
0.$ (36)

Returning now to showing that the differential of $\delta_{d}$ is generically
injective, note that $\beta_{d}$ in (34) is injective if and only if
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$.
Thus under the assumption
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$,
it suffices to show that the coboundary map
$\partial_{\widetilde{D}+\iota\widetilde{D}}$ in (35) is injective for general
$\widetilde{D}$ of degree $d\leq g-1$. From the long exact sequence in
cohomology, it is enough to show that
$h^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}+\iota\widetilde{D}))=1$.
For this, set $D=\operatorname{Nm}(\widetilde{D})$ so that we have
$\widetilde{D}+\iota\widetilde{D}=\pi^{*}D$. Then using that
$\pi_{*}\mathcal{O}_{\widetilde{C}}=\mathcal{O}_{C}\oplus\eta$, where
$\eta^{\otimes 2}\cong\mathcal{O}_{C}$ is the $2$-torsion line bundle defining
the cover, and the projection formula, we have
$h^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}+\iota\widetilde{D}))=h^{0}(C,\mathcal{O}_{C}(D))+h^{0}(C,\eta(D))$.
Thus it suffices to show that $h^{0}(C,\mathcal{O}_{C}(D))=1$ and
$h^{0}(C,\eta(D))=0$, if $D$ is general of degree $d\leq g-1$ (since if
$\widetilde{D}$ is generic, then $D$ will be, too). By Riemann–Roch, this is
equivalent to showing that $h^{0}(C,K_{C}(-D))=g-d$, and
$h^{0}(C,K_{C}(-D)\otimes\eta)=(g-1)-d$; if $D$ is general of degree $d\leq
g-1$, then these conditions are satisfied. This completes the proof regarding
generic injectivity of the differential.

For the final statement of the proposition, still under the assumption
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$,
we observe that the coboundary map (35) is identified with the direct sum of
the coboundary maps

$\textstyle{H^{0}(\mathcal{O}_{C}(D)|_{D})\oplus
H^{0}(\eta(D))|_{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{D}\oplus\partial_{\eta(D)}}$$\textstyle{H^{1}(\mathcal{O}_{C})\oplus
H^{1}(\eta)}$

associated to the short exact sequence
$0\to\mathcal{O}_{C}\to\mathcal{O}_{C}(D)\to\mathcal{O}_{C}(D)|_{D}\to 0$, and
the short exact sequence obtained by tensoring with $\eta$. For this, apply
$\pi_{*}$ to (36), and use $R^{1}\pi_{*}\mathcal{O}_{\widetilde{C}}=0$ and the
projection formula to obtain the direct sum of the short exact sequences
together with the identification
$\pi_{*}\mathcal{O}_{\widetilde{D}+\iota\widetilde{D}}(\widetilde{D}+\iota\widetilde{D})=\mathcal{O}_{C}(D)|_{D}\oplus\eta(D)|_{D}$.
Since the image of $\beta_{d}$ in (34) is equal to the anti-invariant part of
the target, which is in turn identified with $H^{0}(\eta(D))|_{D})$, one is
reduced to checking the injectivity of $\partial_{\eta(D)}$. From the long
exact sequence in cohomology, this coboundary map is injective if and only if
$0=h^{0}(C,\eta)=h^{0}(C,\eta(D))$. This fails if and only if
$\eta\cong\mathcal{O}_{C}(D-E)$ for some effective divisor $E\in C^{(d)}$,
completing the proof. ∎

######  Remark A.8.

It is elementary to see from the definition of the Abel–Prym map, or from
(33), that if
$h^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}))\geq 2$, then
the differential of $\delta_{d}$ fails to be injective at $\widetilde{D}$.
Therefore Proposition A.7 implies that if
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$
and $h^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}))\geq 2$,
then $h^{0}(C,\eta(D))>0$. Here we give an elementary proof of this fact.
Indeed, let $\tilde{\sigma}\in
H^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}))$ be such that
its divisor of zeros is $(\tilde{\sigma})_{0}=\widetilde{D}$. Then for general
$\tilde{\sigma}^{\prime}\in
H^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}))$, with divisor
of zeros $(\tilde{\sigma}^{\prime})_{0}=\widetilde{D}^{\prime}$, we have that
the section $\tilde{\sigma}\cdot\iota\tilde{\sigma}^{\prime}\in
H^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}+\iota\widetilde{D}))$,
with divisor of zeros
$(\tilde{\sigma}\cdot\iota\tilde{\sigma}^{\prime})_{0}=\widetilde{D}+\iota\widetilde{D}^{\prime}$,
has support that is not equal to the support of
$\iota\widetilde{D}+\widetilde{D}^{\prime}$; here we are using that
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$.
Therefore,
$\tilde{\sigma}\cdot\iota\tilde{\sigma}^{\prime}\neq\tilde{\sigma}^{\prime}\cdot\iota\tilde{\sigma}$,
since their associated divisors have different supports. Consequently,
$\tilde{\sigma}\cdot\iota\tilde{\sigma}^{\prime}-\tilde{\sigma}^{\prime}\cdot\iota\tilde{\sigma}\neq
0$ gives a non-trivial element of
$H^{0}(\widetilde{C},\mathcal{O}_{\widetilde{C}}(\widetilde{D}+\iota\widetilde{D}))^{-}=H^{0}(C,\eta(D))$.

######  Corollary A.9.

The Abel–Prym map $\delta_{d}$ is generically finite if and only if $d\leq
g-1$, and surjects onto $P$ (resp. $P^{\prime}$) if and only if $d\geq g-1$
and $d$ is even (resp. $d$ is odd). ∎

######  Corollary A.10.

Suppose that $\eta$ is not in the image of the difference map $C^{(d)}\times
C^{(d)}\to J(C)$, which requires $d<g/2$, and holds if in addition
$\widetilde{C}/C$ is general. Then the exceptional locus of $\delta_{d}$ is
exactly the locus of $\widetilde{D}\in\widetilde{C}^{(d)}$ such that
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})\neq\emptyset$.

###### Proof.

First, we explain that if $\widetilde{C}/C$ is general and $d<g/2$, then
$\eta$ is not in the image of the difference map $C^{(d)}\times C^{(d)}\to
J(C)$. Indeed, if $\eta=\mathcal{O}_{C}(D-E)$ for some effective divisors $D$
and $E$ of degree $d$ on $C$, then since
$\pi^{*}\eta\cong\mathcal{O}_{\widetilde{C}}$ (e.g., [Mum74, Lem. p.332]), we
would have $\pi^{*}D\sim\pi^{*}E$, so that in this case $\widetilde{C}$ would
have a $g^{1}_{2d}$, which is not possible due to [AF12, Thm. 1.4], which
states that for a general cover $\widetilde{C}/C$, there is no $g^{1}_{e}$ on
$\widetilde{C}$ for $e<g$ (in positive characteristic combine the proof of
[AF12, Thm. 1.4] given in [AF12, Prop. 3.1] with [Oss14, Thm. 1.1]).

Now assuming that $\eta$ is not in the image of the difference map
$C^{(d)}\times C^{(d)}\to J(C)$, which implies that $d<g/2\leq g-1$ (e.g.,
[ACGH85, Ex. D-1, p.223]; the argument holds in positive characteristic, as
well), then Proposition A.7 implies that the exceptional locus is contained in
the locus of $\widetilde{D}\in\widetilde{C}^{(d)}$ such that
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})\neq\emptyset$.
On the other hand, it is easy to see that the exceptional locus of
$\delta_{d}$ contains the locus of $\widetilde{D}\in\widetilde{C}^{(d)}$ such
that
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})\neq\emptyset$.
Indeed, suppose that
$\widetilde{D}=\tilde{p}+\iota\tilde{p}+\widetilde{E}=\pi^{*}p+\widetilde{E}$
for some $\tilde{p}\in\widetilde{C}$, $p\in C$, and
$\widetilde{E}\in\widetilde{C}^{(d-2)}$. Then for all $p^{\prime}\in C$ set
$\widetilde{D}_{p^{\prime}}=\pi^{*}p^{\prime}+\widetilde{E}$, and we have
$\delta_{d}(\widetilde{D}_{p^{\prime}})=\delta_{d}(\widetilde{D})$. ∎

####  A.2.1. Geometric interpretation of the differential of the Abel–Prym map

The projectivized differential of $\delta_{1}$ factors as

$\textstyle{\widetilde{C}=\mathbb{P}T\widetilde{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\mathbb{P}T\delta_{1}}$$\textstyle{\mathbb{P}TP=P\times\mathbb{P}T_{0}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{K_{C}\otimes\eta}}$$\textstyle{\mathbb{P}H^{0}(C,K_{C}\otimes\eta)^{\vee}=\mathbb{P}T_{0}P}$

where $\eta$ is the $2$-torsion line bundle on $C$ determining the cover
$\pi$, the bottom row is the Prym canonical map, given by the linear system
$|K_{C}\otimes\eta|$, and the right vertical map is the projection onto the
second factor. Moreover $\mathbb{P}T\delta_{1}$ is defined at
$\tilde{p}\in\widetilde{C}$ if and only if $\phi_{K_{C}\otimes\eta}$ is
defined at $p=\pi(\tilde{p})$; i.e., $p$ is not a base point of
$|K_{C}\otimes\eta|$. One can find all of this in [BL04, Prop. 12.5.2] over
$\mathbb{C}$; the arguments hold in positive characteristic, as well. Indeed,
factoring $\delta_{1}$ as
$\widetilde{C}\stackrel{{\scriptstyle\alpha_{1}}}{{\to}}\operatorname{Pic}^{1}_{\widetilde{C}/k}\stackrel{{\scriptstyle
1-\iota}}{{\to}}P^{\prime}\subseteq J(\widetilde{C})$ where $\alpha_{1}$ is
the Abel map, and taking the differential at a point
$\tilde{p}\in\widetilde{C}$, one obtains the maps
$H^{0}(\mathcal{O}_{\tilde{p}}(\tilde{p}))\stackrel{{\scriptstyle
T\alpha_{1}}}{{\to}}H^{1}(\mathcal{O}_{\widetilde{C}})\stackrel{{\scriptstyle
1-\iota}}{{\to}}H^{1}(\mathcal{O}_{\widetilde{C}})^{-}\subseteq
H^{1}(\mathcal{O}_{\widetilde{C}})$. Identifying
$H^{1}(\widetilde{C},\mathcal{O}_{\widetilde{C}})=H^{0}(\widetilde{C},K_{\widetilde{C}})^{\vee}$
and
$H^{1}(\widetilde{C},\mathcal{O}_{\widetilde{C}})^{-}=H^{0}(C,K_{C}\otimes\eta)^{\vee}$,
one uses the fact that the projectivized differential of the Abel map is the
canonical map $\phi_{K_{\widetilde{C}}}$ on $\widetilde{C}$.

A Riemann–Roch computation gives that $|K_{C}\otimes\eta|$ has a base point if
and only if $\eta=\mathcal{O}_{C}(p^{\prime}-p)$ for some $p,p^{\prime}\in C$,
in which case $C$ is hyperelliptic and $p$ and $p^{\prime}$ are the base
points of $|K_{C}\otimes\eta|$ (e.g., [CDCK19, Lem. 2.1(i)]). Note that
$\pi^{*}p^{\prime}\sim\pi^{*}p$ gives a $g^{1}_{2}$ on $\widetilde{C}$. In
particular, this recovers the statements about the differential of
$\delta_{1}$ in Proposition A.4 and Remark A.6.

As a consequence, if $K_{C}\otimes\eta$ is very ample, then given
$\widetilde{D}\in\widetilde{C}^{(d)}$, and setting
$D=\operatorname{Nm}\widetilde{D}$, the projectivization of
$(T_{\widetilde{D}}\delta_{d})(T_{\widetilde{D}}\widetilde{C}^{(d)})$ is
identified with the span of the scheme $\phi_{K_{C}\otimes\eta}(D)$ in
$\mathbb{P}H^{0}(C,K_{C}\otimes\eta)^{\vee}$. The same result holds more
generally so long as the Prym canonical map is defined at the support of $D$
and the Prym canonical model of $C$ is smooth at the image of the support of
$D$. Note that another Riemann–Roch computation gives that if
$|K_{C}\otimes\eta|$ is base point free, then $K_{C}\otimes\eta$ fails to be
very ample if and only if
$\eta\cong\mathcal{O}_{C}(p+q-p^{\prime}-q^{\prime})$ for some
$p,q,p^{\prime},q^{\prime}\in C$, in which case $C$ is tetragonal, and $p$ and
$q$ are not separated by $|K_{C}\otimes\eta|$; as usual, when $p=q$ we mean
the differential drops rank at $p=q$ (e.g., [CDCK19, Lem. 2.1(ii)]).

###  A.3. Push-forward of the fundamental class under the Abel–Prym map

The main result of this subsection is the following proposition:

######  Proposition A.11.

Let $\ell$ be a prime number not equal to $\operatorname{char}(k)$. For $d\leq
g-1$, and taking $\widetilde{D}_{0}\in\widetilde{C}^{(d)}$, the class of the
push forward of the symmetric product under the pointed Abel–Prym map
$\delta_{d,\widetilde{D}_{0}}$ is given by

$(\delta_{d,\widetilde{D}_{0}})_{*}[\widetilde{C}^{(d)}]=2^{d}\frac{[\Xi]^{\rho-d}}{(\rho-d)!}\in
H^{2\rho-2d}(P,\mathbb{Z}_{\ell}(\rho-d)),$

where $\rho=g-1=\dim P$.

While Proposition A.11 can be proven exactly as in the $d=1$ case (i.e., as in
the proof of (32) in Proposition A.4), that computation is somewhat laborious,
and we prefer to give an alternate proof using (32) as the starting point.
Similar computations can be found in [Bea82, §1] and [Smi89].

For this, we take a brief detour. If $X,Y\subseteq A$ are subvarieties of an
abelian variety, and the map $a:X\times Y\to X+Y\subseteq A$ given by addition
is generically finite, then it essentially follows from the definition of the
Pontryagin product that in the Chow ring or in the cohomology ring:

$[X]\star[Y]=a_{*}[X\times Y]=\deg(a)[X+Y].$

We will want a slight generalization. If we suppose that $f_{X}:X^{\prime}\to
X\subseteq A$ and $f_{Y}:Y^{\prime}\to Y\subseteq A$ are generically finite
surjective morphisms, and we set $a^{\prime}=a\circ(f_{X}\times f_{Y})$ to be
the composition:

$\textstyle{a^{\prime}:X^{\prime}\times
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{X}\times
f_{Y}}$$\textstyle{X\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{+}$$\textstyle{X+Y\subseteq
A,}$

then, still under the assumption that $a$ is generically finite, we have

$f_{X,*}[X^{\prime}]\star f_{Y,*}[Y^{\prime}]=a^{\prime}_{*}[X^{\prime}\times
Y^{\prime}]=\deg(a^{\prime})[X+Y].$ (37)

Indeed, we have the following string of equalities:

$\displaystyle a^{\prime}_{*}[X^{\prime}\times Y^{\prime}]$
$\displaystyle=(\deg a^{\prime})[X+Y]=(\deg(f_{X}\times f_{Y}))(\deg
a)[X+Y]=(\deg f_{X})(\deg f_{Y})(\deg a)[X+Y]$ $\displaystyle=(\deg
f_{X})(\deg f_{Y})[X]\star[Y]=((\deg f_{X})[X^{\prime}])\star((\deg
f_{Y})[Y^{\prime}])=f_{X,*}[X^{\prime}]\star f_{Y,*}[Y^{\prime}].$

Finally, we will want to use the standard result that for a principally
polarized abelian variety $(A,\Theta)$ of dimension $g$, in the Chow ring we
have:

$\frac{[\Theta]^{g-a}}{(g-a)!}\star\frac{[\Theta]^{g-b}}{(g-b)!}=\binom{a+b}{a}\frac{[\Theta]^{g-(a+b)}}{(g-(a+b))!},$
(38)

which we will use in the form

$\left(\frac{[\Theta]^{g-1}}{(g-1)!}\right)^{\star
d}=d!\frac{[\Theta]^{g-d}}{(g-d)!}.$ (39)

A reference for (38) over $\mathbb{C}$ is [BL04, Cor. 16.5.8, p.538], which
uses as its starting point [Bea86, Thm., p.647], also proven over
$\mathbb{C}$. However, [DM91, Thm. 2.19] shows that Beauville’s result holds
over any algebraically closed field, and consequently, the arguments for
[BL04, Cor. 16.5.8, p.538] go through in positive characteristic, as well. Of
course, (38) is elementary to prove in $\ell$-adic cohomology, and this is, in
fact, all we need.

###### Proof of Proposition A.11.

Let $\tilde{p}_{0}\in\widetilde{C}$, and set
$\widetilde{D}_{0}=d\tilde{p}_{0}$. From Corollary A.9 we know that
$\delta_{d}$ is generically finite; therefore, if we factor the composition
$\delta_{d,\widetilde{D}_{0}}^{\times}:\widetilde{C}^{d}\stackrel{{\scriptstyle\operatorname{Sym}}}{{\to}}\widetilde{C}^{(d)}\stackrel{{\scriptstyle\delta_{d,\widetilde{D}_{0}}}}{{\to}}P$
as

$\textstyle{\delta^{\times}_{d,\widetilde{D}_{0}}:\widetilde{C}^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1,\tilde{p}_{0}}^{d}}$$\textstyle{P^{\times
d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+}$$\textstyle{P}$

then from the left hand side of (37), (32), and (39), we have

$(\delta^{\times}_{d,\widetilde{D}_{0}})_{*}[\widetilde{C}^{d}]=\left((\delta_{1,\tilde{p}_{0}})_{*}[\widetilde{C}]\right)^{\star^{d}}=\left(2\frac{[\Xi]^{\rho-1}}{(\rho-1)!}\right)^{\star^{d}}=2^{d}d!\frac{[\Xi]^{\rho-d}}{(\rho-d)!}.$

On the other hand, we have

$(\delta^{\times}_{d,\widetilde{D}_{0}})_{*}[\widetilde{C}^{d}]=(\delta_{d,\widetilde{D}_{0}})_{*}\operatorname{Sym}_{*}[\widetilde{C}^{d}]=d!(\delta_{d,\widetilde{D}_{0}})_{*}[\widetilde{C}^{(d)}],$

completing the proof. ∎

###  A.4. The degree of the Abel–Prym map

We start with the following consequence of Proposition A.11:

######  Corollary A.12.

Let $\ell$ be a prime number not equal to $\operatorname{char}(k)$. For $d\leq
g-1$, and taking $\widetilde{D}_{0}\in\widetilde{C}^{(d)}$, the class
$[\operatorname{Im}\delta_{d,\widetilde{D}_{0}}]$ of the image of the pointed
Abel–Prym map $\delta_{d,\widetilde{D}_{0}}$ (as a set, or rather as an
irreducible scheme, with the reduced induced scheme structure) is

$[\operatorname{Im}\delta_{d,\widetilde{D}_{0}}]=\frac{2^{d}}{\deg\delta_{d,\widetilde{D}_{0}}}\frac{[\Xi]^{\rho-d}}{(\rho-d)!}\in
H^{2\rho-2d}(P,\mathbb{Z}_{\ell}(\rho-d)),$

where $\rho=g-1=\dim P$.

###### Proof.

This follows from Proposition A.11 using the fact that $\delta_{d}$ is
generically finite (Corollary A.9) so that
$(\delta_{d,\widetilde{D}_{0}})_{*}[\widetilde{C}^{(d)}]=\deg(\delta_{d,\widetilde{D}_{0}})[\operatorname{Im}\delta_{d,\widetilde{D}_{0}}]$.
∎

This gives the following corollary:

######  Corollary A.13.

We have $\deg\delta_{g-1}=2^{g-1}$, and if $\operatorname{char}(k)=0$, then
for $d\leq g-2$ we have $\deg\delta_{d}=2^{n}\leq 2^{d}$ for some integer
$n\leq d$. If $\operatorname{char}(k)=p>0$, then for $d\leq g-2$ we have
$\deg\delta_{d}=p^{m}2^{n}$ for some integers $m$ and $n$ with $n\leq d$.

###### Proof.

In the case where $d=g-1$, we know, in addition, the class of the image
$\operatorname{Im}(\delta_{g-1,\widetilde{D}_{0}})$; indeed,
$\delta_{g-1,\widetilde{D}_{0}}$ surjects on to $P$ (Corollary A.9), so that
$\operatorname{Im}(\delta_{g-1,\widetilde{D}_{0}})=P$. The fact that
$\deg\delta_{g-1}=2^{g-1}$ then follows immediately from Corollary A.12,
completing the proof.

The case where $d\leq g-2$ follows from the fact that
$\frac{[\Xi]^{\rho-d}}{(\rho-d)!}\in
H^{2\rho-2d}(P,\mathbb{Z}_{\ell}(\rho-d))$ is a minimal cohomology class
(i.e., it is not divisible by $\ell$), so that Corollary A.12 (considered for
all primes $\ell\neq\operatorname{char}(k)$) implies that $\deg\delta_{d}$
must be a power of the characteristic exponent of $k$ times a power of $2$
that is at most $2^{d}$. ∎

###  A.5. The Abel–Prym map for hyperelliptic covers

We now discuss the degree of the Abel–Prym map in the case where
$\widetilde{C}$ is hyperelliptic.

######  Proposition A.14.

Assume that $\widetilde{C}$ is hyperelliptic. Then for $d\leq g-1$, we have
$\deg\delta_{d}=2^{d}$.

###### Proof.

From Corollary A.13 we only need to show that $\deg\delta_{d}\geq 2^{d}$. So
fix $\widetilde{D}=\tilde{p}_{1}+\cdots+\tilde{p}_{d}\in\widetilde{C}^{(d)}$
to be general; in particular, such that the fiber of $\delta_{d}$ over
$\delta_{d}(\widetilde{D})$ is finite. For
$\widetilde{D}^{\prime}=\tilde{p}_{1}^{\prime}+\cdots+\tilde{p}_{d}^{\prime}\in\widetilde{C}^{(d)}$
we have $\delta_{d}(\widetilde{D}^{\prime})=\delta_{d}(\widetilde{D})$ if and
only if
$\tilde{p}_{1}-\iota\tilde{p}_{1}+\cdots+\tilde{p}_{d}-\iota\tilde{p}_{d}\sim\tilde{p}_{1}^{\prime}-\iota\tilde{p}_{1}^{\prime}+\cdots+\tilde{p}_{d}^{\prime}-\iota\tilde{p}_{d}^{\prime}$
or equivalently,

$\tilde{p}_{1}+\iota\tilde{p}_{1}^{\prime}+\cdots+\tilde{p}_{d}+\iota\tilde{p}_{d}^{\prime}\sim\tilde{p}_{1}^{\prime}+\iota\tilde{p}_{1}+\cdots+\tilde{p}_{d}^{\prime}+\iota\tilde{p}_{d}.$

Denoting by $h$ the hyperelliptic involution on $\widetilde{C}$, the
assumption that $\widetilde{D}$ is general means that we can assume that the
$\tilde{p}_{i}$ and $\iota h(\tilde{p}_{j})$ are all distinct. Therefore there
are $2^{d}$ distinct choices of $\widetilde{D}^{\prime}$ such that
$\delta_{d}(\widetilde{D}^{\prime})=\delta_{d}(\widetilde{D})$, determined by
the $2^{d}$ choices of taking $\tilde{p}_{i}^{\prime}$ either equal to
$\tilde{p}_{i}$ or to $\iota h(\tilde{p}_{i})$; note that since the
$g^{1}_{2}$ on $\widetilde{C}$ is unique, $\tilde{p}+h(\tilde{p})$ is in the
$g^{1}_{2}$ if and only if $\iota\tilde{p}+\iota h(\tilde{p})$ is in the
$g^{1}_{2}$. ∎

###  A.6. The Abel–Prym map for general covers

We now prove that for a general cover $\widetilde{C}/C$, the degree of the
Abel–Prym map is $1$ for $d<g/2$.

######  Corollary A.15 (The Abel–Prym map for general covers).

Let $\widetilde{C}/C$ be a general cover. Then $\deg\delta_{d}=1$ for $d<g/2$.

###### Proof.

Let $\widetilde{D}\in\widetilde{C}^{(d)}$ be a general point; in particular
assume that $\widetilde{D}$ is reduced and
$\operatorname{Supp}(\widetilde{D})\cap\operatorname{Supp}(\iota\widetilde{D})=\emptyset$.
Now suppose that $\widetilde{D}^{\prime}\in\widetilde{C}^{(d)}$ and
$\delta_{d}(\widetilde{D})=\delta_{d}(\widetilde{D}^{\prime})$. Then we have
$\widetilde{D}+\iota\widetilde{D}^{\prime}\sim\widetilde{D}^{\prime}+\iota\widetilde{D}$.
If
$\widetilde{D}+\iota\widetilde{D}^{\prime}=\widetilde{D}^{\prime}+\iota\widetilde{D}$,
then considering the supports of the divisors, we must have that
$\widetilde{D}^{\prime}=\widetilde{D}$. Otherwise, $\widetilde{C}$ admits a
$g^{1}_{2d}$. However, by [AF12, Thm. 1.4], for a general cover
$\widetilde{C}/C$, there is no $g^{1}_{e}$ on $\widetilde{C}$ for $e<g$. Thus
if $d<g/2$, then $\widetilde{C}$ does not admit a $g^{1}_{2d}$, and so
$\delta_{d}$ is generically injective. ∎

######  Remark A.16.

The Donagi–Smith [DS81, §3] approach to computing the degree of a generically
finite morphism by computing the local degree along a fiber, together with the
General Position Theorem (e.g., [ACGH85, p.109] or [EH92, Thm. 3.1]) allows
one to compute $\deg\delta_{d}=1$ in characteristic $0$ for $\widetilde{C}/C$
general and $d\leq g-2$.

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