# Degree-2 Abel maps and hyperelleptic curves

Alex Abreu, Sally Andria, and Marco Pacini

###### Abstract.

In this paper we resolve the degree-2 Abel map for nodal curves. Our results
are based on a previous work of the authors reducing the problem of the
resolution of the Abel map to a combinatorial problem via tropical geometry.
As an application, we characterize when the (symmetrized) degree-2 Abel map is
not injective, a property that, for a smooth curve, is equivalent to the curve
being hyperelliptic.

MSC (2020): 14H10, 14H40, 14T90.

Keywords: Algebraic curve, hyperelliptic curve, tropical curve, Abel map.

##  1\. Introduction

This paper is dedicated to the construction of an explicit resolution of the
degree-$2$ Abel-Jacobi map for a regular smoothing of a nodal curve. For a
smooth curve $C$, the degree-$d$ Abel map is an important morphism taking a
$d$-tuple of points on $C$, to the associated invertible sheaf on the curve
(tensored with a fixed invertible sheaf on the curve). When the fixed
invertible sheaf is $\mathcal{O}_{C}(dP_{0})$ for a pont $P_{0}$ on $C$, the
map is usually called Abel-Jacobi map. This map encodes many important
properties of the curve. For instance, the degree-$2$ Abel map detects when
the curve is hyperelliptic. More precisely, a smooth curve is hyperelliptic if
and only the degree-$2$ Abel map is not injective, and in this case the curve
is endowed with a $g^{1}_{2}$, which can be identified with the fiber of the
Abel map (up to the action of the symmetric group). In this paper we
investigate how we can extend the above description to singular curves.

We construct an explicit resolution of the degree-$2$ Abel-Jacobi map using
the results in [1], where the general problem of resolving Abel maps is
reduced to checking a certain combinatorial property of the tropical Abel map.
More precisely, this translates into the problem of showing the existence of a
compatibility between the polyhedral structures of the tropical Jacobian of a
curve and the product of the relevant tropical curve. This is the result
contained in [1, Theorem A]. In degree $2$, this yields a very explicit
combinatorial condition describing how to blow up the domain of the geometric
Abel map to get a resolution. This is summarized in Theorem 3.1.

Let $\pi\colon\mathcal{C}\to B$ be a regular smoothing of a curve $C$ with a
section $\sigma$ through its smooth locus, and $\mu$ be a polarization on
$\mathcal{C}$. We denote by $\overline{\mathcal{J}}_{\mu}^{\sigma}$ the
Esteves compactified Jacobian parametrizing $(\sigma,\mu)$-quasistable
torsion-free rank-$1$ sheaves on $\mathcal{C}$ (see [7]). As usual, we write
$\mathcal{C}^{2}:=\mathcal{C}\times_{B}\mathcal{C}$. Let $\mathcal{L}$ be an
invertible sheaf on $\mathcal{C}/B$ of degree-$(k+2)$. We define the
degree-$2$ Abel (rational) map $\alpha^{2}_{\mathcal{L}}$ as

$\displaystyle\alpha^{2}_{\mathcal{L}}\colon\mathcal{C}^{2}$
$\displaystyle\dashrightarrow\overline{\mathcal{J}}_{\mu}^{\sigma}$
$\displaystyle(Q_{1},Q_{2})$
$\displaystyle\longmapsto[\mathcal{L}|_{\pi^{-1}(Q_{1})}(-Q_{1}-Q_{2})].$

Our main result holds when $\mu$ is the trivial degree-$0$ polarization and
$\mathcal{L}$ is the trivial sheaf $\mathcal{O}_{\mathcal{C}}$. An important
ingredient to describe the resolution of the degree-$2$ Abel-Jacobi map is the
notion of tail of a nodal curve. A subcurve of a nodal curve is a
$\delta$-tail if the subcurve and its complementary curves are connected and
intersect each other in $\delta$ nodes.

######  Theorem (Theorem 4.7).

Let $Z_{1},\dots,Z_{N}$ be the 2-tails and the 3-tails of $C$ which do not
contain $\sigma(0)$. Consider the sequence of blowups

$\widetilde{\mathcal{C}}^{2}_{N}\stackrel{{\scriptstyle\phi_{N}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{1}\stackrel{{\scriptstyle\phi_{1}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{0}\stackrel{{\scriptstyle\phi_{0}}}{{\longrightarrow}}\mathcal{C}^{2}$

where $\phi_{0}$ is the blowup of $\mathcal{C}^{2}$ along its diagonal
subscheme and $\phi_{i}$ is the blowup of $\widetilde{\mathcal{C}}^{2}_{i-1}$
along the strict transform of the divisor $Z_{i}\times Z_{i}$ of
$\mathcal{C}^{2}$ via $\phi_{0}\circ\cdots\circ\phi_{i-1}$. Then the rational
map

$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}\circ\phi_{0}\circ\cdots\circ\phi_{N}\colon\widehat{\mathcal{C}}^{2}_{N}\dashrightarrow\overline{\mathcal{J}}^{\sigma}_{\mu}$

is a morphism, i.e., it is defined everywhere.

Next we investigate the relation between the degree-2 Abel map and
hyperelliptic (nodal) curves. More precisely, we study when the (symmetrized)
degree-2 Abel map is not injective. The upshot is that this happens exactly
when the curve has a simple torsion-free rank-1 sheaf of degree 2 with non-
negative degree over every component of the curve and at least two sections.
We call a curve satisfying all these condition a _pseudo-hyperelliptic_ curve.
It is easy to see that if a curve is hyperelliptic, then it is pseudo-
hyperelliptic.

It is worth noticing that a variation of the condition of hyperelliptic curve
was already given by Caporaso in [4]. She introduced and study the notion of
weakly-hyperelliptic curve, which is the condition of the existence of a
balanced degree-2 invertible sheaf on a curve with at least 2 sections. Again,
if a curve is hyperelliptic, then it is weakly-hyperelliptic. We study the
relation between weakly-hyperelliptic and pseudo-hyperelliptic.

######  Theorem (Theorem 4.20).

Let $C$ be a curve with no separating nodes. The following properties hold.

  1. (1)

The curve $C$ is pseudo-hyperelliptic if and only if, for some (every) regular
smoothing $\mathcal{C}\to B$ of $C$, the symmetrized degree-$2$ Abel map of
$\mathcal{C}$ is not injective.

  2. (2)

If $C$ is stable and weakly-hyperelliptic, then $C$ is pseudo-hyperelliptic.

##  2\. Preliminaries

Throughout the paper, we will use the notations introduced in [3, Sections 2
and 3] and [1, Section 3]. In this section we just recall some basic
definitions and constructions.

Given a graph $\Gamma$, we denote by $V(\Gamma)$ and $E(\Gamma)$ the sets of
vertices and edges of $\Gamma$. Given a subset $V\subset V(\Gamma)$, we set
$V^{c}=V(\Gamma)\setminus V$. For an orientation $\overrightarrow{\Gamma}$ on
$\Gamma$, we denote by $s(e)$ and $t(e)$ the source and target of an edge
$e\in E(\Gamma)$. Given subsets $V,W\subset V(\Gamma)$, we let $E(V,W)$ be the
set of edges of $\Gamma$ connecting a vertex of $V$ with a vertex of $W$. A
_refinement_ of a graph $\Gamma$ is a graph obtained by inserting a non-
negative number $n_{e}$ (depending on $e$) of vertices in the interior of each
edge $e$ of $\Gamma$. If a vertex $v$ of the refinement is inserted in the
interior of an edge $e$ of $\Gamma$, we say that $v$ is a vertex _over_ $e$.

A metric graph is a pair $(\Gamma,\ell)$ where $\Gamma$ is a graph and $\ell$
is a function $\ell\colon E(\Gamma)\rightarrow\mathbb{R}_{>0}$, called the
length function. Let $(\Gamma,\ell)$ be a metric graph. If
$\overrightarrow{\Gamma}$ is an orientation on $\Gamma,$ we define the
tropical curve $X$ associated to $(\overrightarrow{\Gamma},\ell)$ as

$X=\frac{\left(\bigcup_{e\in E(\overrightarrow{\Gamma})}I_{e}\cup
V(\overrightarrow{\Gamma})\right)}{\sim}$

where $I_{e}=[0,\ell(e)]\times\\{e\\}$ and $\sim$ is the equivalence relation
generated by $(0,e)\sim s(e)$ and $(\ell(e),e)\sim t(e).$ We usually just
write $e$ to represent $I_{e}$ in $X$, and denote by $e^{\circ}$ the interior
of $e$. We say that $(\Gamma,\ell)$ is a _model_ of the tropical curve $X$. We
will identify tropical curves with isometric models.

Let $\Gamma$ be a graph and define $\ell\colon E(\Gamma)\to\mathbb{R}$ by
$\ell(e)=1$ for every $e\in E(\Gamma)$. We denote by $X_{\Gamma}$ the tropical
curve induced by the metric graph $(\Gamma,\ell)$.

Let $X$ be a tropical curve and $\Gamma$ be a graph. A divisor on $X$
(respectively, on $\Gamma$) is a function $\mathcal{D}\colon
X\rightarrow\mathbb{Z}$ (respectively, $D\colon V(\Gamma)\to\mathbb{Z}$) such
that $\mathcal{D}(p)\neq 0$ only for finitely many points $p\in X.$ Given a
divisor $\mathcal{D}$ on $X$, we define the support of $\mathcal{D}$ as the
set of points $p$ of $X$ such that $\mathcal{D}(p)\neq 0$ and denote it by
$\text{supp}(\mathcal{D})$. A _polarization_ on $X$ (respectively, on
$\Gamma$) is a function $\mu\colon X\rightarrow\mathbb{R}$ (respectively,
$\mu\colon V(\Gamma)\to\mathbb{R}$) such that $\mu(p)\neq 0$ only for finitely
many points $p\in X$ and such that $\sum_{p\in X}\mu(p)$ (respectively,
$\sum_{v\in V(v\in V(\Gamma)}\mu(v)$) is an integer, called the _degree_ of
the polarization $\mu$.

Given a point $p_{0}$ in $X$, (respectively, a vertex $v_{0}\in V(\Gamma)$), a
divisor $\mathcal{D}$ on $X$ (respectively, $D$ on $\Gamma$) is called
$(p_{0},\mu)$-quasistable (respectively, $(v_{0},\mu)$-quasistable) if:

$\sum_{p\in Y}(\mathcal{D}(p)-\mu(p))+\frac{\delta_{Y}}{2}\geq 0$

for every tropical subcurve $Y$ of $X$ (respectively, every subset $Y\subset
V(\Gamma)$), where the inequality is strict if $p_{0}\in Y\neq X$. Here,
$\delta_{Y}$ is the number of tangent direction outgoing from $Y$ in the case
of a tropical curve (see [3, Section 3.1] for the precise definition), while
it is equal to $|E(Y,Y^{c})|$ in the case of a graph.

Let $X$ be a tropical curve and $p_{0}$ be a point of $X$. Let $\mu$ be a
polarization on $X$. Recall that in an equivalence class of a divisor on a
tropical curve there is just one $(p_{0},\mu)$-quasistable divisor (see [3,
Theorem 5.6]). For a degree-$d$ divisor $\mathcal{D}$ on $X$, we denote by
$\operatorname{qs}(\mathcal{D})$ the unique $(p_{0},\mu)$-quasistable divisor
on $X$ which is equivalent to $\mathcal{D}$. Given an oriented model
$(\Gamma,\ell)$ of $X$, for every edge $e\in E(\Gamma)$ and every real number
$t\in[0,\ell(e)]$, we let by $p_{e,t}$ the point on $e$ at distance $t$ from
the source of $e$.

Given a tropical curve $X$, we let $J^{\text{trop}}_{p_{0},\mu}(X)$ be the
tropical Jacobian parametrizing $(p_{0},\mu)$-quasistable divisors on $X$.
Recall that $J^{\text{trop}}_{p_{0},\mu}(X)$ is homeomorphic to the usual
tropical Jacobian (see [3, Theorem 5.10]). We set $X^{2}:=X\times X$. Given a
divisor $\mathcal{D}^{\dagger}$ on $X$, we define the tropical Abel map

$\displaystyle\alpha_{2,\mathcal{D}^{\dagger}}^{\text{trop}}\colon X^{2}$
$\displaystyle\to J^{\text{trop}}_{p_{0},\mu}(X)$ $\displaystyle(p_{1},p_{2})$
$\displaystyle\longmapsto[\mathcal{D}^{\dagger}-p_{1}-p_{2}],$

where $[-]$ denotes the class of a divisor in the tropical Jacobian.
Alternatively, the map $\alpha_{2,\mathcal{D}^{\dagger}}^{\text{trop}}$ takes
$(p_{1},p_{2})$ to the unique $(p_{0},\mu)$-quasistable divisor in the class
of $\mathcal{D}^{\dagger}-p_{1}-p_{2}$.

######  Remark 2.1.

Let $X$ be a tropical curve with a point $p_{0}\in X$. Let $\Gamma$ be a model
of $X$. Let $\mu$ be a polarization on $X$ induced by a polarization on
$\Gamma$ and $\mathcal{D}$ a degree-$d$ divisor on $X$. We let
$\widehat{\Gamma}$ the minimal refinement of $\Gamma$ such that
$\text{supp}(\mathcal{D})\subset V(\widehat{\Gamma})$. We denote by $D$ the
divisor on $\widehat{\Gamma}$ induced by $\mathcal{D}$. We call the pair
$(\widehat{\Gamma},D)$ on $\widehat{\Gamma}$ the _combinatorial type_ of
$\mathcal{D}$. By [3, Proposition 5.3], the degree-$d$ divisor $\mathcal{D}$
on $X$ is $(p_{0},\mu)$-quasistable if and only if $\widehat{\Gamma}$ is
obtained by inserting at most one vertex in the interior of each edge of
$\Gamma$ and $D$ is $(p_{0},\mu)$-quasistable on $\widehat{\Gamma}$.

##  3\. Degree-2 Abel maps

Let $C$ be a nodal curve over an algebraically closed filed $k$. A subcurve
$Z$ of $C$ is a reduced union of components of $C$. Given a subcurve $Z$ of
$C$, we let $Z^{c}:=\overline{C\setminus Z}$. Throughout this section we will
fix a regular smoothing $\pi\colon\mathcal{C}\rightarrow B$ of a nodal curve
$C$ with a section $\sigma\colon B\rightarrow\mathcal{C}$ of $\pi$ through its
smooth locus. We denote by
$\mathcal{C}^{2}:=\mathcal{C}\times_{B}\mathcal{C}$.

Let $\mu$ be a degree-$k$ polarization on $\mathcal{C}$. We denote by
$\overline{\mathcal{J}}_{\mu}^{\sigma}$ the Esteves compactified Jacobian
parametrizing $(\sigma,\mu)$-quasistable torsion-free rank-$1$ sheaves on the
curves of the family $\pi$ (see [7] for more details). Let $\mathcal{L}$ be an
invertible sheaf on $\mathcal{C}/B$ of degree-$(k+2)$. As in [2], we define
the degree-$2$ Abel (rational) map $\alpha^{2}_{\mathcal{L}}$ as

$\displaystyle\alpha^{2}_{\mathcal{L}}\colon\mathcal{C}^{2}$
$\displaystyle\dashrightarrow\overline{\mathcal{J}}_{\mu}^{\sigma}$
$\displaystyle(Q_{1},Q_{2})$
$\displaystyle\longmapsto[\mathcal{L}|_{\pi^{-1}(\pi(Q_{1}))}(-Q_{1}-Q_{2})].$

We let $\Gamma$ be the dual graph of $C$ and $X_{\Gamma}$ be the tropical
curve induced by $\Gamma$ (with unitary lengths). Given an invertible sheaf
$\mathcal{L}$ on $\mathcal{C}$, we denote by $D^{\dagger}_{\mathcal{L}}$ the
divisor on $\Gamma$ given by the multidegree of $\mathcal{L}|_{C}$. We also
let $\mathcal{D}^{\dagger}_{\mathcal{L}}$ be the divisor on $X_{\Gamma}$
induced by $D^{\dagger}_{\mathcal{L}}$.

Given a point $\mathcal{N}=(N_{1},N_{2})$ of $\mathcal{C}^{2}$, where $N_{i}$
is a node of $C$, we will consider the following two ways of blowing up
$\mathcal{C}^{2}$ locally around $\mathcal{N}$. If $N_{1}\in Z_{1}\cap
Z_{1}^{c}$ and $N_{2}\in Z_{2}\cap Z_{2}^{c}$ for subcurves $Z_{1}$ and
$Z_{2}$ of $C$, we can consider the blowups
$\phi\colon\widetilde{\mathcal{\mathcal{C}}}^{2}_{\phi}\rightarrow\mathcal{C}^{2}$
and
$\phi^{\prime}\colon\widetilde{\mathcal{\mathcal{C}}}^{2}_{\phi^{\prime}}\rightarrow\mathcal{C}^{2}$
respectively along $Z_{1}\times Z_{2}$, or along $Z_{1}\times Z_{2}^{c}$. The
first one is also equivalent to the blowup along $Z_{1}^{c}\times Z_{2}^{c}$
and the second one is equivalent to the blowup along $Z_{1}^{c}\times Z_{2}$.
In both cases, the inverse image of $\mathcal{N}$ is isomorphic to
$\mathbb{P}^{1}_{k}$. The situation is illustrated in Figure 1, where
$\operatorname{st}_{\phi}$ and $\operatorname{st}_{\phi^{\prime}}$ applied to
a divisor of $\mathcal{C}^{2}$ denote the strict transform of this divisor.
These blowups induce a dual picture on the product $X_{\Gamma}^{2}$: we
illustrate the relation between these blowups and the dual picture in Figure
2.

$Z_{1}$$Z_{1}^{c}$$Z_{2}$$Z_{2}^{c}$$N_{2}$$N_{1}$$\scriptstyle{\operatorname{st}_{\phi}(Z^{c}_{1}\times
Z^{c}_{2})}$$\scriptstyle{\operatorname{st}_{\phi}(Z_{1}\times
Z_{2})}$$\scriptstyle{\operatorname{st}_{\phi}(Z_{1}\times
Z^{c}_{2})}$$\scriptstyle{\operatorname{st}_{\phi}(Z^{c}_{1}\times
Z_{2})}$$\mathbb{P}^{1}_{k}$$Z_{1}$$Z_{1}^{c}$$Z_{2}$$\scriptstyle{\operatorname{st}_{\phi^{\prime}}(Z^{c}_{1}\times
Z^{c}_{2})}$$\scriptstyle{\operatorname{st}_{\phi^{\prime}}(Z_{1}\times
Z^{c}_{2})}$$Z_{2}^{c}$$\scriptstyle{\operatorname{st}_{\phi^{\prime}}(Z^{c}_{1}\times
Z_{2})}$$\scriptstyle{\operatorname{st}_{\phi^{\prime}}(Z_{1}\times
Z_{2})}$$N_{2}$$N_{1}$$\mathbb{P}^{1}_{k}$ Figure 1. The two types of blowup
of $\mathcal{C}^{2}$ around $(N_{1},N_{2})$.

######  Theorem 3.1.

Let $\pi\colon\mathcal{C}\rightarrow B$ be a regualar smoothing of a nodal
curve $C$ with smooth components. Let $\sigma\colon B\rightarrow\mathcal{C}$
be a section of $\pi$ through its smooth locus. Let $\mu$ be a polarization of
degree $k$ over the family and $\mathcal{L}$ be an invertible sheaf on
$\mathcal{C}$ of degree $k+2$. Let $(N_{1},N_{2})$ be a point of
$\mathcal{C}^{2}$, with $N_{i}$ a node of $C$. Let $Z_{1}$ and $Z_{2}$ be
subcurves of $C$ such that $N_{1}\in Z_{1}\cap Z_{1}^{c}$ and $N_{2}\in
Z_{2}\cap Z_{2}^{c}$. Let $e_{1}$ and $e_{2}$ be the edges in the dual graph
$\Gamma$ of $C$ that correspond to $N_{1}$ and $N_{2}$, where $e_{i}$ is
oriented from $Z_{i}$ to $Z_{i}^{c}$. Consider the divisor
$\mathcal{D}_{x,y}=\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{1},x}-p_{e_{2},y}$
on $X_{\Gamma}$, for some $x,y\in[0,1]$.

  1. (1)

If the combinatorial type of $\operatorname{qs}(\mathcal{D}_{x,y})$ is
constant on the sets

$\\{(x,y);\ 0<x<y<1\\}\text{ and }\\{(x,y);\ 0<y<x<1\\},$

then the blowup of $\mathcal{C}^{2}$ along $Z_{1}\times Z_{2}$ resolves the
Abel map $\alpha^{2}_{\mathcal{L}}$ locally around the point $(N_{1},N_{2})$.

  2. (2)

If the combinatorial type of $\operatorname{qs}(\mathcal{D}_{x,y})$ is
constant on the sets

$\\{(x,y);\ 0<x<1-y<1\\}\text{ and }\\{(x,y);\ 0<1-y<x<1\\},$

then the blowup of $\mathcal{C}^{2}$ along $Z_{1}\times Z_{2}^{c}$ resolves
the Abel map $\alpha^{2}_{\mathcal{L}}$ locally around the point
$(N_{1},N_{2})$.

$Z_{1}$$Z_{1}^{c}$$Z_{2}$$Z_{2}^{c}$$N_{2}$$N_{1}$$e_{2}$$e_{1}$ Figure 2. The
sets $\\{(x,y);\ 0<x<1-y<1\\}$ and $\\{(x,y);\ 0<1-y<x<1\\}$ and the
corresponding blow-up.

  3. (3)

If the combinatorial type of $\operatorname{qs}(\mathcal{D}_{x,y})$ is
constant on the set

$\\{(x,y);\ 0<x,y<1\\},$

then the Abel map $\alpha^{2}_{\mathcal{L}}$ is defined at the point
$(N_{1},N_{2})$.

###### Proof.

Items (1) and (2) follow directly from [1, Theorem 5.4].

Let us prove Item (3). Let $\mathcal{N}=(N_{1},N_{2})\in\mathcal{C}^{2}$. Let
$\phi\colon\mathcal{X}\rightarrow\mathcal{C}^{2}$ and
$\phi^{\prime}\colon\mathcal{Y}\rightarrow\mathcal{C}^{2}$ be the blowups
respectively along $Z_{1}\times Z_{2}$ and $Z_{1}\times Z_{2}^{c}$ (see Figure
1). By items (1) and (2), we know that $\alpha^{2}_{\mathcal{L}}\circ\phi$ and
$\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}$ are defined respectively over the
inverse images $\phi^{-1}(\mathcal{N})\cong\mathbb{P}^{1}_{k}$ and
$\phi^{\prime-1}(\mathcal{N})\cong\mathbb{P}^{1}_{k}$. Let $y_{0}$ be the
distinguished point on $\phi^{-1}(\mathcal{N})$ given by

$y_{0}=\operatorname{st}_{\phi}(Z_{1}\times
Z_{2})\cap\operatorname{st}_{\phi}(Z_{1}\times
Z_{2}^{c})\cap\operatorname{st}_{\phi}(Z_{1}^{c}\times Z_{2}^{c}).$

Let $x_{1},x_{2}$ be any two points on $\phi^{\prime-1}(\mathcal{N})$. We know
that $\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}$ is defined at $x_{1}$ and
$x_{2}$. For $i=1,2$, consider a map
$\rho^{\prime}_{i}\colon\operatorname{Spec}k[[t]]\rightarrow\mathcal{Y}$ such
that $\rho^{\prime}_{i}(0)=x_{i}$ and $\rho_{i}(\eta)$ is contained in
$\operatorname{st}_{\phi^{\prime}}(Z_{1}\times Z_{2}^{c})$, as in Figure 3,
where $0$ and $\eta$ are the special and generic points of
$\operatorname{Spec}k[[t]]$, respectively. In particular, we have
$\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}(x_{i})=\alpha^{2}_{\mathcal{L}}\circ\overline{\rho}_{i}(0)$,
where
$\overline{\rho}_{i}=\phi^{\prime}\circ\rho^{\prime}_{i}\colon\operatorname{Spec}(k[[t]])\rightarrow\mathcal{C}^{2}$.
By construction, we can lift $\overline{\rho}_{i}$ to maps
$\rho_{i}\colon\operatorname{Spec}k[[t]]\to\mathcal{X}$ such that
$\rho_{1}(0)=\rho_{2}(0)=y_{0}$. By the same reasoning, we have
$\alpha^{2}_{\mathcal{L}}\circ\phi(y_{0})=\alpha^{2}_{\mathcal{L}}\circ\rho_{i}(0)$
for $i=1,2$. Then we get:

$\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}(x_{1})=\alpha^{2}_{\mathcal{L}}\circ\phi(y_{0})=\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}(x_{2}).$

Hence $\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}$ contracts all fibers of
$\phi^{\prime}$. Moreover, arguing as in the proof of [9, Corollary III 11.4],
we have
$\phi^{\prime}_{*}\mathcal{O}_{\mathcal{Y}}\cong\mathcal{O}_{\mathcal{C}^{2}}$,
since $\phi^{\prime}$ is birational and $\mathcal{C}^{2}$ is normal. Hence by
the Rigidity Lemma (see [6, Lemma 1.15, pag.12]) the map
$\alpha^{2}_{\mathcal{L}}\circ\phi^{\prime}$ factors through $\phi^{\prime}$,
so $\alpha^{2}_{\mathcal{L}}$ is defined at $\mathcal{N}$.

$\phi$$\phi^{\prime}$$y_{0}$$x_{2}$$x_{1}$ Figure 3. The blowups $\phi$ and
$\phi^{\prime}$.

∎

##  4\. The resolution of the degree-2 Abel map

###  4.1. Local resolutions

Throughout this section we will fix a regular smoothing
$\pi\colon\mathcal{C}\rightarrow B$ of a nodal curve $C$ with a section
$\sigma\colon B\rightarrow\mathcal{C}$ of $\pi$ through its smooth locus. We
will perform blowups of $\mathcal{C}^{2}$ along divisors of type $Z\times Z$,
where $Z$ is a subcurve of the special fiber $C$. Actually, we will restrict
our attention to a special class of subcurves, called _tails_.

######  Definition 4.1.

A _$\delta$ -tail_ of a nodal curve $C$ is a connected subcurve $Z$ such that
$Z^{c}$ is connected and $|Z\cap Z^{c}|=\delta$.

######  Proposition 4.2.

Let $\mu$ be a polarization of degree $k$ and $\mathcal{L}$ be an invertible
sheaf of degree $k+2$ over $\mathcal{C}/B$. Assume that the components of $C$
are smooth. Consider a point $\mathcal{N}=(N_{1},N_{2})\in\mathcal{C}^{2}$,
where $N_{1},N_{2}$ are nodes of $C$, with $N_{1}=Z\cap Z^{c}$ for a $1$-tail
$Z$ of $C$. Then the degree-$2$ Abel map
$\alpha^{2}_{\mathcal{L}}\colon\mathcal{C}^{2}\dashrightarrow\overline{\mathcal{J}}_{\mu}^{\sigma}$
is defined at $\mathcal{N}$.

###### Proof.

Let $\Gamma$ be the dual graph of $C$ and $X=X_{\Gamma}$ the associated
tropical curve with edges of unitary lengths. Let $v_{0}$ be the vertex of
$\Gamma$ corresponding to $P_{0}=\sigma(0)$, and $p_{0}\in X$ be the point
corresponding to $v_{0}$. We let $e_{1}$ and $e_{2}$ be the edges of $\Gamma$
corresponding to $N_{1}$ and $N_{2}$. The tropical Abel map
$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}\colon
X^{2}\rightarrow J^{\text{trop}}_{p_{0},\mu}$ takes a pair
$(p_{e_{1},t_{1}},p_{e_{2},t_{2}})$, for real numbers $t_{1},t_{2}\in(0,1)$,
to the class of the divisor on $X$ given by:

(1)
$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}(p_{e_{1},t_{1}},p_{e_{2},t_{2}})=[\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}].$

We define the divisor $\mathcal{P}=p_{e_{1},t_{1}}-p_{e_{1},0}$ on $X$. Since
$N_{1}=Z\cap Z^{c}$ for a $1$-tail $Z$ of $X$, we have that the graph obtained
from $\Gamma$ by removing the edge $e_{1}$ is not connected. Hence the divisor
$\mathcal{P}$ on $X$ is principal. So we can write:

$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}(p_{e_{1},t_{1}},p_{e_{2},t_{2}})=[\mathcal{\widehat{D}}^{\dagger}-p_{e_{2},t_{2}}],$

where
$\mathcal{\widehat{D}}^{\dagger}=\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{1},0}$
(which is a divisor on $X$ induced by a divisor on $\Gamma$). So we reduce
ourselves to the case of the degree-$1$ Abel map. As explained in [1, Lemma
5.10] and in the proof of [1, Theorem 5.8], the combinatorial type of the
quasistable divisor on $X$ equivalent to
$\mathcal{\widehat{D}}^{\dagger}-p_{e_{1},t_{2}}$ is independent of $t_{2}$.
Hence the combinatorial type of the quasistable divisor on $X$ equivalent to
$\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}$ is
independent of the pairs $t_{1},t_{2}\in(0,1)$. By Theorem 3.1 (3), we deduce
that the Abel map $\alpha^{2}_{\mathcal{L}}$ is already defined at
$(N_{1},N_{2})$. ∎

######  Proposition 4.3.

Let $\mu$ be a polarization of degree $k$ and $\mathcal{L}$ an invertible
sheaf of degree $k+2$ over $\mathcal{C}/B$. Assume that the components of $C$
are smooth. Let $Z$ be a $2$-tail of $C$ and write $\\{N_{1},N_{2}\\}=Z\cap
Z^{c}$. Consider the point

$\mathcal{N}=(N_{1},N_{2})\in(Z\cap Z^{c})\times(Z\cap
Z^{c})\subset\mathcal{C}^{2}.$

Let $\phi\colon\widetilde{\mathcal{C}}^{2}\rightarrow\mathcal{C}^{2}$ be the
blowup of $\mathcal{C}^{2}$ with center $Z\times Z$. Then the rational map

$\widetilde{\alpha}^{2}_{\mathcal{L}}\colon\widetilde{\mathcal{C}}^{2}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\mathcal{C}^{2}\stackrel{{\scriptstyle\alpha^{2}_{\mathcal{L}}}}{{\dashrightarrow}}\overline{\mathcal{J}}^{\sigma}_{\mu}$

is defined along the rational curve
$\phi^{-1}(\mathcal{N})\cong\mathbb{P}^{1}_{k}$.

###### Proof.

We can keep the set-up of Proposition 4.2. The tropical Abel map
$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}\colon
X^{2}\rightarrow J^{\text{trop}}_{p_{0},\mu}$ is as in Equation (1). Assume
that $t_{1}>t_{2}$. We define the divisor on $X$:

$\mathcal{P}=p_{e_{1},t_{1}}-p_{e_{1},t_{1}-t_{2}}+p_{e_{2},t_{2}}-p_{e_{2},0}.$

Since $\\{N_{1},N_{2}\\}=Z\cap Z^{c}$ for a $2$-tail $Z$ of $C$, we have that
the graph obtained from $\Gamma$ by removing the edges $e_{1},e_{2}$ is not
connected. Hence $\mathcal{P}$ is a principal divisor. Then we have

$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}(p_{e_{1},t_{1}},p_{e_{2},t_{2}})=[\mathcal{\widehat{D}}^{\dagger}-p_{e_{1},t}],$

where $t=t_{1}-t_{2}$ and
$\mathcal{\widehat{D}}^{\dagger}=\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{2},0}$
(which is a divisor induced by a divisor on $\Gamma$). So we reduce ourselves
to the case of the degree-$1$ Abel map: as explained in [1, Lemma 5.10] and in
the proof of [1, Theorem 5.8], the combinatorial type of the quasistable
divisor on $X$ equivalent to $\mathcal{\widehat{D}}^{\dagger}-p_{e_{1},t}$ is
independent of $t$. Hence the combinatorial type of the quasistable divisor on
$X$ equivalent to
$\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}$ is
independent of $(t_{1},t_{2})$ whenever $t_{1}>t_{2}$. A similar reasoning can
be done for the case $t_{1}<t_{2}$. Hence, using Theorem 3.1 (1), we conclude
that the blowup along $Z\times Z$ gives rise to a resolution of
$\alpha^{2}_{\mathcal{L}}$ locally around $\mathcal{N}=(N_{1},N_{2})$. ∎

######  Proposition 4.4.

Let $\mu$ be a polarization of degree $k$ and $\mathcal{L}$ an invertible
sheaf of degree $k+2$ over $\mathcal{C}/B$. Assume that the components of $C$
are smooth. Consider the point $\mathcal{N}=(N,N)\in\mathcal{C}^{2}$, for a
node $N$ of $C$. Let
$\phi\colon\widetilde{\mathcal{C}}^{2}\rightarrow\mathcal{C}^{2}$ be the
blowup of $\mathcal{C}^{2}$ with center the diagonal subscheme of
$\mathcal{C}^{2}$. Then the rational map

$\widetilde{\alpha}^{2}_{\mathcal{L}}\colon\widetilde{\mathcal{C}}^{2}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\mathcal{C}^{2}\stackrel{{\scriptstyle\alpha^{2}_{\mathcal{L}}}}{{\dashrightarrow}}\overline{\mathcal{J}}^{\sigma}_{\mu}$

is defined along the rational curve
$\phi^{-1}(\mathcal{N})\cong\mathbb{P}^{1}_{k}$.

###### Proof.

We can keep the set-up of Proposition 4.2. The tropical Abel map
$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}\colon
X^{2}\rightarrow J^{\text{trop}}_{p_{0},\mu}$ is as in Equation (1), where
$e:=e_{1}=e_{2}$. Assume that $t_{1}+t_{2}<1$. We define the principal divisor
on $X$:

$\mathcal{P}=p_{e,0}-p_{e,t_{1}}-p_{e,t_{2}}+p_{e,t_{1}+t_{2}}.$

Then we have

$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}(p_{e,t_{1}},p_{e,t_{2}})=[\mathcal{\widehat{D}}^{\dagger}-p_{e,t}],$

where
$\mathcal{\widehat{D}}^{\dagger}=\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e,0}$
and $t=t_{1}+t_{2}$. So we reduce ourselves to the case of the degree-$1$ Abel
map: as explained in [1, Lemma 5.10] and in the proof of [1, Theorem 5.8], the
combinatorial type of the quasistable divisor on $X$ equivalent to
$\mathcal{\widehat{D}}^{\dagger}-p_{e,t}$ is independent of $t$. Hence the
combinatorial type of the quasistable divisor on $X$ equivalent to
$\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e,t_{1}}-p_{e,t_{2}}$ is independent
of $(t_{1},t_{2})$ whenever $t_{1}+t_{2}<1$.

The reasoning is similar for $t_{1}+t_{2}>1$: we just consider

$\mathcal{P}=p_{e,1}-p_{e,t_{1}}-p_{e,t_{2}}+p_{e,t_{1}+t_{2}-1},$

$\mathcal{\widehat{D}}^{\dagger}=\mathcal{D}^{\dagger}_{\mathcal{L}}-p_{e,1}$
and $t=t_{1}+t_{2}-1$, so that

$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{L}}}(p_{e,t_{1}},p_{e,t_{2}})=[\mathcal{\widehat{D}}^{\dagger}-p_{e,t}].$

By Theorem 3.1 (2), we deduce that the blowup along the diagonal subscheme of
$\mathcal{C}^{2}$ gives rise to a resolution of
$\alpha^{2}_{\mathcal{L}}\circ\phi$ locally around $\mathcal{N}=(N,N)$. ∎

###  4.2. The resolution of the Abel-Jacobi map

Our main goal is to give a complete resolution of the degree-$2$ Abel-Jacobi
map of any nodal curve, namely the map taking a pair $(Q_{1},Q_{2})$ of points
on a curve $C$ to $\mathcal{O}_{C}(2P_{0}-Q_{1}-Q_{2})$ for a given smooth
point $P_{0}$ of $C$. This is done in Theorem 4.7. Before, we need two
results.

######  Lemma 4.5.

Let $Z$ be a $\delta$-tail of a curve $C$.

  * (1)

If $Z\cap Z^{c}\subset Z^{\prime}$ for some tail $Z^{\prime}$ of $C$, then
either $Z\subset Z^{\prime}$ or $Z^{c}\subset Z^{\prime}$.

  * (2)

If $|(Z\cap Z^{c}\cap Z^{\prime}\cap(Z^{\prime})^{c}|=\delta-1$ for some tail
$Z^{\prime}$ of $C$, then one of the following conditions holds:

$Z\subset Z^{\prime},\;\;Z^{\prime}\subset Z,\;\;Z^{c}\subset
Z^{\prime},\;\;Z^{\prime}\subset Z^{c}.$

###### Proof.

See [10, Lemma 2.4]. ∎

######  Lemma 4.6.

Let $P_{0}$ be a smooth point of $C$. Let $\mathcal{T}=(Z_{1},\dots,Z_{h})$ be
a sequence of tails of $C$, where $Z_{i}$ is a $k_{i}$-tail with
$k_{i}\in\\{2,3\\}$ and $P_{0}\not\in Z_{i}$. Consider the sequence of blowups

$\phi_{\mathcal{T}}\colon\widetilde{\mathcal{C}}^{2}_{h}\stackrel{{\scriptstyle\phi_{h}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\phi_{3}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{2}\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{1}\stackrel{{\scriptstyle\phi_{1}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{0}\stackrel{{\scriptstyle\phi_{0}}}{{\longrightarrow}}\mathcal{C}^{2}$

where $\phi_{0}$ is the blowup of $\mathcal{C}^{2}$ along its diagonal
subscheme and $\phi_{i}$ is the blowup of $\widetilde{\mathcal{C}}^{2}_{i-1}$
along the strict transform of the divisor $Z_{i}\times Z_{i}$ of
$\mathcal{C}^{2}$ via $\phi_{1}\circ\cdots\circ\phi_{i-1}$. Then
$\phi_{\mathcal{T}}$ is independent of the ordering of the sequence
$\mathcal{T}$.

###### Proof.

Assume that a permutation $\mathcal{T}^{\prime}$ of $\mathcal{T}$ gives rise
to a blowup $\phi_{\mathcal{T}^{\prime}}$ different from $\phi_{\mathcal{T}}$.
This implies that, locally at a point $\mathcal{N}=(N_{1},N_{2})$ with
$N_{1},N_{2}$ distinct nodes of $C$, the blowups $\phi_{\mathcal{T}}$ and
$\phi_{\mathcal{T}^{\prime}}$ are different. We can assume that locally at
$\mathcal{N}$, the blowup $\phi_{\mathcal{T}}$ has center $Z_{i}\times Z_{i}$
and the blowup $\phi_{\mathcal{T}^{\prime}}$ has center $Z_{j}\times Z_{j}$,
for $i,j\in\\{1,\dots,h\\}$, so that we have $Z_{i}\neq Z_{j}$ and

$\\{N_{1},N_{2}\\}\subset Z_{i}\cap Z_{i}^{c}\cap Z_{j}\cap Z_{j}^{c}.$

By Lemma 4.5, one of the following conditions holds:

(2) $Z_{i}\subset Z_{j},\;\;Z_{j}\subset Z_{i},\;\;Z^{c}_{i}\subset
Z_{j},\;\;Z_{j}\subset Z_{i}^{c}.$

Let $C_{1}$, $C^{\prime}_{1}$, $C_{2}$, $C^{\prime}_{2}$ be the components of
$C$ such that $N_{1}\in C_{1}\cap C^{\prime}_{1}$ and $N_{2}\in C_{2}\cap
C^{\prime}_{2}$. Since $\phi_{\mathcal{T}}$ and $\phi_{\mathcal{T}^{\prime}}$
are different locally at $\mathcal{N}$, we can assume, without loss of
generality, that $C_{1}\cup C_{2}\subset Z_{i}$ and $C_{1}\cup
C_{2}^{\prime}\subset Z_{j}$. Then we have

$C_{1}\subset Z_{i}\cap Z_{j},\;\;C_{2}\subset Z_{i}\cap
Z_{j}^{c},\;\;C^{\prime}_{2}\subset Z_{i}^{c}\cap Z_{j}.$

On the other hand:

  1. (1)

since $C_{1}\subset Z_{i}\cap Z_{j}$, it follows that $Z_{j}\not\subset
Z_{i}^{c}$.

  2. (2)

since $C_{2}\subset Z_{i}\cap Z_{j}^{c}$, it follows that $Z_{i}\not\subset
Z_{j}$.

  3. (3)

since $C_{2}^{\prime}\subset Z_{i}^{c}\cap Z_{j}$, it follows that
$Z_{j}\not\subset Z_{i}$.

  4. (4)

since $P_{0}\in Z_{i}^{c}\setminus Z_{j}$, it follows that
$Z_{i}^{c}\not\subset Z_{j}$.

This contradicts Equation (2). ∎

######  Theorem 4.7 (Degree-$2$ Abel-Jacobi map).

Let $\pi\colon\mathcal{C}\rightarrow B$ be a regular smoothing of a nodal
curve $C$. Let $\sigma\colon B\rightarrow\mathcal{C}$ be a section of $\pi$
through its smooth locus and $\mu$ be the trivial degree-$0$ polarization. Let
$Z_{1},\dots,Z_{N}$ be the 2-tails and the 3-tails of $C$ which do not contain
$\sigma(0)$. Consider the sequence of blowups

$\widetilde{\mathcal{C}}^{2}_{N}\stackrel{{\scriptstyle\phi_{N}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{1}\stackrel{{\scriptstyle\phi_{1}}}{{\longrightarrow}}\widetilde{\mathcal{C}}^{2}_{0}\stackrel{{\scriptstyle\phi_{0}}}{{\longrightarrow}}\mathcal{C}^{2}$

where $\phi_{0}$ is the blowup of $\mathcal{C}^{2}$ along its diagonal
subscheme and $\phi_{i}$ is the blowup of $\widetilde{\mathcal{C}}^{2}_{i-1}$
along the strict transform of the divisor $Z_{i}\times Z_{i}$ of
$\mathcal{C}^{2}$ via $\phi_{0}\circ\cdots\circ\phi_{i-1}$. Then the rational
map

$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}\circ\phi_{0}\circ\cdots\circ\phi_{N}\colon\widehat{\mathcal{C}}^{2}_{N}\dashrightarrow\overline{\mathcal{J}}^{\sigma}_{\mu}$

is a morphism, i.e., it is defined everywhere.

Before proving the theorem, we need to recall a result in [10] describing how
to convert the sheaf $\mathcal{O}_{C}(2P_{0}-Q_{1}-Q_{2})$ into a
$(\sigma,\mu)$-quasistable sheaf, where $P_{0},Q_{1},Q_{2}$ are smooth points
of the nodal curve $C$. We will give the graph-theoretical equivalent of this
result, which suits better with our purposes. More precisely, given a graph
$\Gamma$ and vertices $v_{0},v_{1},v_{2}\in V(\Gamma)$, we will describe the
$(v_{0},\mu)$-quasistable divisor on $\Gamma$ equivalent to
$2v_{0}-v_{1}-v_{2}$ (see Theorem 4.16).

Let $\Gamma$ be a graph. Given a subset $V\subset V(\Gamma)$, we denote by
$\Gamma(V)$ the subgraph of $\Gamma$ whose set of vertices is $V$ and whose
edges are the edges of $\Gamma$ connecting two (possibly coinciding) vertices
of $V$.

######  Definition 4.8.

A _hemisphere_ of $\Gamma$ is a subset $H\subset V(\Gamma)$ such that
$\Gamma(H)$ and $\Gamma(H^{c})$ are connected subgraphs of $\Gamma$. A
$\delta$-hemisphere of $\Gamma$ is a hemisphere $H$ such that
$|E(H,H^{c})|=\delta$.

We denote by $\mathcal{H}_{\Gamma,\delta}$ the set of $\delta$-hemispheres of
$\Gamma$. Given subsets $V,W\subset V(\Gamma)$, we define:

$\mathcal{H}_{\Gamma,\delta}(V,W):=\\{H\in\mathcal{H}_{\Gamma,\delta}|V\subset
H^{c}\text{ and }W\subset H\\}.$

Let $S$ be a finite set. We say that a set $\mathcal{H}$ of subsets of $S$ is
union-closed (respectively, intersection-closed) if $H_{1}\cup
H_{2}\in\mathcal{H}$ (respectively, $H_{1}\cap H_{2}\in\mathcal{H}$) for every
$H_{1},H_{2}\in\mathcal{H}$. We note that every non-empty intersection-closed
set has a unique minimal element and every non-empty union-closed set has a
unique maximal element.

######  Proposition 4.9.

Let $\Gamma$ be a graph and $v_{0},v_{1},v_{2}$ vertices of $\Gamma$. Then the
sets $\mathcal{H}_{\Gamma,1}(v_{0},v_{1})$ and
$\mathcal{H}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})$ are union-closed and
intersection-closed.

###### Proof.

See [5, Lemma 4.3] and [10, Section 3 and Proposition 3.1]. ∎

######  Definition 4.10.

Given subsets $V,W\subset V(\Gamma)$, we say that $W$ is _$V$ -free_ if
$E(V,V^{c})\cap E(W,W^{c})=\emptyset$.

######  Remark 4.11.

A $1$-hemisphere $H$ is $H^{\prime}$-free for every $\delta$-hemisphere
$H^{\prime}\neq H,H^{c}$. If $H_{1},H_{2}\subset V(\Gamma)$ are $V$-free
hemispheres, for some $V\subset V(\Gamma)$, then $H_{1}\cap H_{2}$ and
$H_{1}\cup H_{2}$ are also $V$-free.

######  Definition 4.12.

Let $\Gamma^{\prime}$ be a subdivision of $\Gamma$. Let $V$ be a subset of
$V(\Gamma^{\prime})$. We say that an edge $e\in E(\Gamma)$ is _fully
contained_ in $V$ if $V$ contains the vertices incident to $e$ and all the
vertices over $e$. Note that when $\Gamma^{\prime}=\Gamma$, this simply means
that $e\in E(V,V)$.

######  Lemma 4.13.

Let $H_{2}$ and $H_{3}$ be a $2$-hemisphere and a $3$-hemisphere. Write
$E(H_{2},H_{2}^{c})=\\{f_{1},f_{2}\\}$ and
$E(H_{3},H_{3}^{c})=\\{e_{1},e_{2},e_{3}\\}$. Assume that the intersection
$H=H_{2}\cap H_{3}$ is non-empty and properly contained in $H_{2}$ and
$H_{3}$. Assume that $H_{2}\cup H_{3}\neq V(\Gamma)$. Then, up to reordering
the indices, one of the following properties hold

  1. (1)

$H$ is a $2$-hemisphere such that $E(H,H^{c})=\\{f_{1},e_{1}\\}$, with $f_{1}$
fully contained in $H_{3}$ and $e_{1}$ fully contained in $H_{2}$, while
$f_{2}$ is fully contained in $H_{3}^{c}$ and $e_{2},e_{3}$ are fully
contained in $H_{2}^{c}$.

  2. (2)

$H$ is a $3$-hemisphere such that $E(H,H^{c})=\\{f_{1},e_{1},e_{2}\\}$, with
$f_{1}$ fully contained in $H_{3}$ and $e_{1},e_{2}$ fully contained in
$H_{2}$, while $f_{2}$ is fully contained in $H_{3}^{c}$ and $e_{3}$ is fully
contained in $H_{2}^{c}$.

###### Proof.

By the hypothesis, the sets $H$, $H_{2}\setminus H$, $H_{3}\setminus H$ and
$H_{2}^{c}\cap H_{3}^{c}$ are nonempty and form a partition of $V(\Gamma)$.
Since $H_{3}$ is connected, we have that $E(H,H_{3}\setminus H)$ is nonempty.
However $E(H,H_{3}\setminus H)\subseteq E(H_{2},H_{2}^{c})=\\{f_{1},f_{2}\\}$.
We assume that $f_{1}\in E(H,H_{3}\setminus H)$. Arguing in a similar manner,
using that $H_{3}^{c}$ is connected, we have that $f_{2}\in E(H_{2}\setminus
H,H_{2}^{c}\cap H_{3}^{c})$. Even more, we can conclude that $e_{1}\in
E(H,H_{2}\setminus H)$ and $e_{3}\in E(H_{3}\setminus H,H_{2}^{c}\cap
H_{3}^{c})$ (see Figure 4).

$H$$H_{2}\setminus H$$H_{3}\setminus H$$H_{2}^{c}\cap
H_{3}^{c}$$e_{1}$$e_{3}$$f_{1}$$f_{2}$ Figure 4. The edges
$e_{1},e_{3},f_{1},f_{2}$.

Since $H_{2}$ is a $2$-hemisphere, we have that

$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad E(H,H_{3}\setminus H)$
$\displaystyle=\\{f_{1}\\},$ $\displaystyle E(H_{2}\setminus H,H_{3}\setminus
H)$ $\displaystyle=\emptyset,\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad E(H,H_{2}^{c}\cap
H_{3}^{c})$ $\displaystyle=\emptyset,$ $\displaystyle\quad E(H_{2}\setminus
H,H_{2}^{c}\cap H_{3}^{c})$
$\displaystyle=\\{f_{2}\\}.\quad\quad\quad\quad\quad\quad\quad\quad$

Hence $E(H_{3},H_{3}^{c})=E(H,H_{2}\setminus H)\cup E(H_{3}\setminus
H,H_{2}^{c}\cap H_{3}^{c})$. Therefore, the edge $e_{2}$ only has two
possibilities: it belongs to either $E(H_{3}\setminus H,H_{2}^{c}\cap
H_{3}^{c})$ or $E(H,H_{2}\setminus H)$. In the former case $H$ satisfies the
conditions in item (1), while in the latter case it satisfies the conditions
in item (2).

Note that the sets $H$, $H_{3}\setminus H$, $H_{2}\setminus H$ and
$H_{2}^{c}\cap H_{3}^{c}$ are connected because $H_{3}=H\cup(H_{3}\setminus
H)$ is connected and $H$ and $H_{3}\setminus H$ are connected by a single
edge, hence each $H$ and $H_{3}\setminus H$ must be connected. The same
reasoning holds for $H_{2}\setminus H$ and $H_{2}^{c}\cap H_{3}^{c}$, using
the fact that $H_{3}^{c}$ is connected. Hence $H^{c}=(H_{3}\setminus
H)\cup(H_{2}\setminus H)\cup(H_{2}^{c}\cap H_{3}^{c})$ is connected, which
means that $H$ is a hemisphere. ∎

We let $H_{2,1}$ be the minimal element of
$\mathcal{H}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})$ (which exists and is unique
by Proposition 4.9). Define

$\mathcal{H}^{\operatorname{free}}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})=\\{H_{2,1},\dots,H_{2,m_{2}}\\},$

where $H_{2,i}$ is the minimal element of the set of hemispheres of
$\mathcal{H}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})$ that are $H_{2,j}$-free for
every $j<i\leq m_{2}$ and containing $H_{2,i-1}$. The hemisphere $H_{2,i}$
exists and is unique since the set

$\\{H\in\mathcal{H}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})|H_{2,i-1}\subset
H,H\text{ is }H_{2,j}\text{-free for $j=1,\ldots,i-1$}\\}$

is intersection-closed by Proposition 4.9 and Remark 4.11.

Notice that we have a sequence of nested $2$-hemispheres

$H_{2,1}\subset H_{2,2}\subset\ldots\subset H_{2,m_{2}}.$

We let $\mathcal{H}^{\prime}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})$ be the
subset of $\mathcal{H}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})$ of the hemispheres
that are $H$-free for every
$H\in\mathcal{H}_{2}^{\operatorname{free}}(v_{0},\\{v_{1},v_{2}\\})$.

######  Proposition 4.14.

The subset $\mathcal{H}^{\prime}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})$ is
intersection-closed.

###### Proof.

See [10, Proposition 3.5]. ∎

We let $H_{3,1}$ be the minimal element of
$\mathcal{H}^{\prime}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})$ and define

$\mathcal{H}^{\operatorname{free}}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})=\\{H_{3,1},\dots,H_{3,m_{3}}\\},$

where $H_{3,i}$ is the minimal element of the set of hemispheres
$\mathcal{H}^{\prime}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\})$ that are
$H_{3,j}$-free for every $j<i\leq m_{3}$ and containing $H_{3,i-1}$. As
before, we have that $\mathcal{H}_{\Gamma,3}^{\operatorname{free}}$ is well-
defined. Notice that we have a sequence of nested $3$-hemispheres

$H_{3,1}\subset H_{3,2}\subset\dots\subset H_{3,m_{3}}.$

######  Remark 4.15.

Let $k=2,3$. Notice that we have a natural orientation on every edge $e\in
E(H_{k,i},H_{k,i}^{c})$ such that $s(e)\in H_{k,i}$ and $t(e)\in H_{k,i}^{c}$.
Moreover, if $e\in E(H_{k,i},H_{k,i}^{c})$, then $t(e)\in H_{k,i+1}$.

Finally we set:

$\displaystyle\mathcal{F}_{\Gamma}(v_{0},v_{1},v_{2})=$ $\displaystyle\
\mathcal{H}_{\Gamma,1}(v_{0},v_{1})\ \sqcup\
\mathcal{H}_{\Gamma,1}(v_{0},v_{2})$ $\displaystyle\sqcup\
\mathcal{H}^{\operatorname{free}}_{\Gamma,2}(v_{0},\\{v_{1},v_{2}\\})\ \sqcup\
\mathcal{H}^{\operatorname{free}}_{\Gamma,3}(v_{0},\\{v_{1},v_{2}\\}).$

Notice that the same $1$-hemisphere could belong in both
$\mathcal{H}_{\Gamma,1}(v_{0},v_{1})$ and
$\mathcal{H}_{\Gamma,1}(v_{0},v_{2})$.

For every subset $V\subset V(\Gamma)$, we let $\textnormal{div}(V)$ be the
principal divisor on $\Gamma$ given by:

$\textnormal{div}(V)=\sum_{e\in E(V,V^{c})}(s(e)-t(e)),$

where the orientation is chosen such that $s(e)\in V$ for every $e\in
E(V,V^{c})$. Let $v_{0},v_{1},v_{2}$ be vertices of $\Gamma$ and $\mu$ the
trivial degree-$0$ polarization on $\Gamma$. The following result tells us how
to find the $(v_{0},\mu)$-quasistable divisor equivalent to the divisor
$2v_{0}-v_{1}-v_{2}$.

######  Theorem 4.16.

Let $\Gamma$ be a graph and $v_{0},v_{1},v_{2}$ vertices of $\Gamma$. Then

$2v_{0}-v_{1}-v_{2}-\sum_{V\in\mathcal{F}_{\Gamma}(v_{0},v_{1},v_{2})}\textnormal{div}(V)$

is the $(v_{0},\mu)$-quasistable divisor equivalent to $2v_{0}-v_{1}-v_{2}$.

###### Proof.

See [10, Theorem 5.3]. ∎

Before going on with the proof of Theorem 4.7, we need to introduce a divisor
on a tropical curve $X$ attached to a hemisphere of its underlying graph.

######  Definition 4.17.

Let $X$ be a tropical curve and $(\Gamma,\ell)$ be a model of $X$. Let $v_{0}$
be a vertex of $\Gamma$. For a hemisphere $H$ of $\Gamma$, consider the
orientation on an edge $e\in E(H,H^{c})$ from the vertex of $e$ contained in
$H$ to the vertex of $e$ contained in $H^{c}$ (see Remark 4.15). We define the
divisor

$\mathcal{P}_{H}=\sum_{e\in E(H,H^{c})}p_{e,0}-\sum_{e\in
E(H,H^{c})}p_{e,\ell(e)}.$

######  Remark 4.18.

Notice that if $\ell(e)=\ell(e^{\prime})$ for every $e,e^{\prime}\in
E(H,H^{c})$, then $\mathcal{P}_{H}$ is a principal divisor on $X$.

###### Proof of Theorem 4.7.

We can assume that all the components of $C$ are smooth. Indeed, the general
case follows from the case in which the components of $C$ are smooth arguing
as in the last part of the proof of [1, Theorem 5.8], and using [10, Theorem
1.3].

Since quasistability is an open property by [7, Proposition 34], it is enough
to check that the global blowup of $\mathcal{C}^{2}$ described in the
statement is a blowup resolving the Abel map
$\alpha^{2}_{\mathcal{O}_{C}}\colon\mathcal{C}^{2}\rightarrow\overline{\mathcal{J}}^{\sigma}_{\mu}$
locally around any point $\mathcal{N}=(N_{1},N_{2})$ of $\mathcal{C}^{2}$ for
$N_{i}$ a node of $C$.

If $N_{1}=Z\cap Z^{c}$ for a $1$-tail $Z$ of $C$, the result follows from
Proposition 4.2. If $N_{1}=N_{2}$, the result follows from Proposition 4.4. So
we will assume, through the rest of the proof, that $N_{1}\neq N_{2}$ and
neither $N_{1}$ nor $N_{2}$ disconnects $C$.

We will use Theorem 3.1. Let $\Gamma_{0}$ be the dual graph of $C$, and let
$X=X_{\Gamma_{0}}$, namely, the tropical curve whose underlying graph is
$\Gamma_{0}$ with all unitary lengths. Let $v_{0}$ be the vertex of
$\Gamma_{0}$ corresponding to $P_{0}=\sigma(0)$, and $p_{0}\in X$ be the point
corresponding to $v_{0}$. We have
$\mathcal{D}^{\dagger}_{\mathcal{O}_{\mathcal{C}}}=2p_{0}$. The tropical Abel
map $\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{O}_{C}}}\colon
X^{2}\rightarrow J^{\text{trop}}_{p_{0},\mu}$ takes a pair
$(p_{e_{1},t_{1}},p_{e_{2},t_{2}})$, for edges $e_{1},e_{2}\in E(\Gamma_{0})$
and real numbers $t_{1},t_{2}\in[0,1]$, to:

$\alpha^{\text{trop}}_{2,\mathcal{D}^{\dagger}_{\mathcal{O}_{C}}}(p_{e_{1},t_{1}},p_{e_{2},t_{2}})=[2p_{0}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}].$

Let $(\Gamma,\ell)$ be the model of $X$ such that $\Gamma$ is the refinement
of $\Gamma_{0}$ obtained by inserting vertices $v_{e_{1}},v_{e_{2}}$ in the
interior of $e_{1},e_{2}$, respectively, with
$\ell([s(e_{i}),v_{e_{i}}])=t_{i}$. We let $K_{2,1}$ be the minimal element of
$\mathcal{H}^{\operatorname{free}}_{\Gamma,2}(v_{0},\\{v_{e_{1}},v_{e_{2}}\\})$
and $K_{3,1}$ be the minimal element of
$\mathcal{H}^{\operatorname{free}}_{\Gamma,3}(v_{0},\\{v_{e_{1}},v_{e_{2}}\\})$.

We have three cases to consider.

  1. (1)

We have that $E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap
E(K_{2,1},K_{2,1}^{c})\neq\emptyset$, for every $i=1,2$.

  2. (2)

We have that

$E(v_{e_{1}},\\{v_{e_{1}}\\}^{c})\cap E(K_{2,1},K_{2,1}^{c})\neq\emptyset.$
$E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap E(K_{2,1},K_{2,1}^{c})=\emptyset$

We distinguish 3 subcases:

    1. (2.a)

There exists a $3$-hemisphere $K_{3}$ containing $v_{e_{1}}$ and $v_{e_{2}}$
and not containing $v_{0}$ such that $E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap
E(K_{3},K_{3}^{c})\neq\emptyset$, for every $i=1,2$.

    2. (2.b)

Every $3$-hemisphere $K_{3}$ containing $v_{e_{1}}$ and $v_{e_{2}}$ and not
containing $v_{0}$ satisfies the condition
$E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap E(K_{3},K_{3}^{c})=\emptyset$ and there
exists a $3$-hemisphere $K_{3}^{\prime}$ such that
$E(v_{e_{1}},\\{v_{e_{1}}\\}^{c})\cap E(K_{3}^{\prime},K_{3}^{\prime
c})\neq\emptyset$.

    3. (2.c)

Every $3$-hemisphere $K_{3}$ containing $v_{e_{1}}$ and $v_{e_{2}}$ and not
containing $v_{0}$ satisfies the condition
$E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap E(K_{3},K_{3}^{c})=\emptyset$, for every
$i=1,2$.

Notice that there are no other subcases to consider, because if the two
conditions:

$E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap E(K_{3},K_{3}^{c})\neq\emptyset.$
$E(v_{e_{1}},\\{v_{e_{1}}\\}^{c})\cap E(K_{3},K_{3}^{c})=\emptyset$

hold for some $3$-hemisphere $K_{3}$, then, by Lemma 4.13, we have that
$K_{2,1}\cap K_{3}$ is a $3$-hemisphere (because $K_{2,1}$ is minimal) and it
would satisfy the condition in case (2.a).

  3. (3)

We have

$E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap
E(K_{2,1},K_{2,1}^{c})=\emptyset,\;\text{for every }i=1,2.$

We distinguish 3 subcases:

    1. (3.a)

We have that $E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap
E(K_{3,1},K_{3,1}^{c})\neq\emptyset$, for every $i=1,2$.

    2. (3.b)

We have that

$E(v_{e_{1}},\\{v_{e_{1}}\\}^{c})\cap E(K_{3,1},K_{3,1}^{c})\neq\emptyset.$
$E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap E(K_{3,1},K_{3,1}^{c})=\emptyset$

    3. (3.c)

We have that $E(v_{e_{i}},\\{v_{e_{i}}\\}^{c})\cap
E(K_{3,1},K_{3,1}^{c})=\emptyset$, for every $i=1,2$.

We discuss the above cases. Case (1) follows from Proposition 4.3.

Case (2.a). We assume that the orientation of $e_{1}$ and $e_{2}$ satisfies
the condition $s(e_{1}),s(e_{2})\in K_{3}$. Recall that
$t_{i}=\ell([s(e_{i}),v_{e_{i}}])$.

We consider the refinement $\Gamma^{\prime}$ of $\Gamma_{0}$ by adding two
vertices over each edge. Of course, $\Gamma^{\prime}$ is a refinement of
$\Gamma$. We denote by $\psi$ the natural function

$\psi\colon E(\Gamma^{\prime})\rightarrow E(\Gamma_{0})$

taking an edge $e$ of $\Gamma^{\prime}$ to the edge $f$ of $\Gamma_{0}$ if $e$
is obtained by subdividing $f$.

For $i=1,2$, we already have the vertex $v_{e_{i}}$ over the edge $e_{i}$, so
we will only add another vertex $v^{\prime}_{e_{i}}$. As illustrated in Figure
5, if $t_{1}<t_{2}$, the vertices over $e_{1}$ will be ordered as follows:
$s(e_{1}),v_{e_{1}},v_{e_{1}}^{\prime},t(e_{1})$, while the vertices over
$e_{2}$ are ordered as follows:
$s(e_{2}),v^{\prime}_{e_{2}},v_{e_{2}},t(e_{2})$. On the other hand, if
$t_{1}>t_{2}$, then the orderings become
$s(e_{1}),v_{e_{1}}^{\prime},v_{e_{1}},t(e_{1})$ and
$s(e_{2}),v_{e_{2}},v^{\prime}_{e_{2}},t(e_{2})$.

$s(e_{1})$$v_{e_{1}}$$v_{e_{1}}^{\prime}$$t(e_{1})$$s(e_{2})$$v_{e_{2}}^{\prime}$$v_{e_{2}}$$t(e_{2})$$s(e_{1})$$v_{e_{1}}^{\prime}$$v_{e_{1}}$$t(e_{1})$$s(e_{2})$$v_{e_{2}}$$v_{e_{2}}^{\prime}$$t(e_{2})$
Figure 5.

Throughout the proof of case (2a), we will assume that $t_{1}<t_{2}$, leaving
to the reader the case $t_{1}>t_{2}$. Our goal will be to find a length
function $\ell^{\prime}$ on $\Gamma^{\prime}$ so that
$(\Gamma^{\prime},\ell^{\prime})$ is a model of $X$, and the divisors
$\mathcal{P}_{H}$ for every
$H\in\mathcal{F}_{\Gamma^{\prime}}(v_{0},v_{e_{1}},v_{e_{2}})$ are principal
on $X$. This allows us to conclude the proof. Indeed, using Remark 2.1 and
Theorem 4.16 we get that the divisor

(3)
$\mathcal{D}_{t_{1},t_{2}}:=2p_{0}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}+\sum_{H\in\mathcal{F}_{\Gamma^{\prime}}(v_{0},v_{e_{1}},v_{e_{2}})}\mathcal{P}_{H}$

is $(p_{0},\mu)$-quasistable. Hence Theorem 3.1 (1) tells us that the blowup
illustrated on the left hand side of Figure 1 with $Z_{1}=Z_{2}=\widehat{Z}$,
where $\widehat{Z}$ is the $3$-tail of $C$ induced by $K_{3}\cap V(\Gamma)$,
gives rise to a resolution of the Abel map
$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}$ locally at $(N_{1},N_{2})$. This is
the blowup locally around $(N_{1},N_{2})$ prescribed by the global blowup in
the statement of Theorem 4.7.

We proceed with the construction of the length function $\ell^{\prime}$. We
write

(4)
$\displaystyle\mathcal{H}_{\Gamma^{\prime},2}^{\operatorname{free}}(v_{0},\\{v_{e_{1}},v_{e_{2}}\\})$
$\displaystyle=\\{H_{2,1},\ldots,H_{2,{m_{2}}}\\}$
$\displaystyle\mathcal{H}_{\Gamma^{\prime},3}^{\operatorname{free}}(v_{0},\\{v_{e_{1}},v_{e_{2}}\\})$
$\displaystyle=\\{H_{3,1},\ldots,H_{3,m_{3}}\\}.$

We define a sequence $f_{1},f_{2},\ldots,f_{k}$ of edges of $\Gamma_{0}$ as
illustrated in Figure 8, with $f_{1}=e_{1}$ and $f_{2}\in
E(H_{2,1},H_{2,1}^{c})$, and where the other edges are chosen as follows.
Assume that $t_{1}<t_{2}$. The edges of the sequence satisfy

$f_{2i+1},f_{2i+2}\in\psi(E(H_{2,{3i+1}},H^{c}_{2,{3i+1}})),\;\;f_{2i+1},f_{2i+2}\in\psi(E(H_{2,{3i+2}},H_{2,{3i+2}}^{c}))$
$f_{2i},f_{2i+1}\in\psi(E(H_{2,{3i}},H^{c}_{2,{3i}})).$

Notice that if $k$ is odd with $k=2k^{\prime}+1$, then

$\psi(E(H_{2,3i+1},H_{2,3i+1}^{c}))=\psi(E(H_{2,3i+2},H_{2,3i+2}^{c}))=\psi(E(H_{2,3i+3},H_{2,3i+3}^{c}))$

for every $i\geq k^{\prime}$. If $k$ is even with $k=2k^{\prime}$, then for
every $i\geq k^{\prime}$ we have

$\psi(E(H_{2,3i},H_{2,3i}^{c}))=\psi(E(H_{2,3i+1},H_{2,3i+1}^{c}))=\psi(E(H_{2,3i+2},H_{2,3i+2}^{c})).$

Now we consider the $3$-hemispheres. If $\psi(E(H_{3,1},H_{3,1}^{c}))$ and
$\psi(E(H_{2,i},H_{2,i}^{c}))$ are disjoint for every $i$, let us define a
length function $\ell^{\prime}$ on the set of edges of $\Gamma^{\prime}$ so
that $(\Gamma^{\prime},\ell^{\prime})$ is a model for $X$. We will assume that
$k=2k^{\prime}$ is even (see Figure 6, also see Figure 7 for the case
$t_{1}>t_{2}$), leaving to the reader to work out the other case (see Figure
8). For every $e\in E(\Gamma^{\prime})$, we define:

$\ell^{\prime}(e)=\begin{cases}\begin{array}[]{ll}\frac{1-t_{1}}{2}&\text{ if
}e\in E(H_{2,{3i+1}},H_{2,3i+1}^{c})\text{ or }e\in
E(H_{2,3i+2},H_{2,3i+2}^{c})\text{ for some }i<k^{\prime}\\\ t_{1}&\text{ if
}\psi(e)=f_{k}\text{ and }e\notin
E(H_{2,3k^{\prime}-2},H_{2,3k^{\prime}-2}^{c})\cup
E(H_{2,3k^{\prime}-1},H_{2,3k^{\prime}-1}^{c})\\\ t_{1}&\text{ if either }e\in
E(H_{2,3i},H^{c}_{2,3i})\text{ for some $i<k^{\prime}$, or }t(e)=v_{1}\\\
1/3&\text{ otherwise}.\end{array}\end{cases}$

For every edge $f\in E(\Gamma)$ we have that
$\sum_{e\in\psi^{-1}(f)}\ell^{\prime}(e)=1$. Indeed if $f=f_{i}$, then the sum
of the lengths will be $t_{1}+\frac{1-t_{1}}{2}+\frac{1-t_{1}}{2}=1$, while if
$f\notin\\{f_{1},\ldots,f_{k}\\}$, then the 3 edges in $\psi^{-1}(f)$ will
have length $1/3$. So $(\Gamma^{\prime},\ell^{\prime})$ is a model of $X$.
Notice that the divisors $\mathcal{P}_{H_{2,i}}$ and $\mathcal{P}_{H_{3,i}}$
are principal divisors by Remark 4.18. This conclude the proof in this case.

$v_{e_{1}}$$f_{1}$$H_{2,1}$$H_{2,2}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{2}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{3}$$H_{2,3}$$H_{2,4}$$H_{2,5}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{4}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$
Figure 6. Attributing lengths to the edges of $\Gamma^{\prime}$ for
$t_{1}<t_{2}$ and $k$ even. In this case, $v_{e_{2}}$ is contained in
$H_{2,i}$ for every $i=1,\ldots,m_{2}$.
$v_{e_{1}}$$f_{1}$$H_{2,1}$$\frac{t_{1}}{2}$$\frac{t_{1}}{2}$${1-t_{1}}$$f_{2}$$\frac{t_{1}}{2}$${1-t_{1}}$$\frac{t_{1}}{2}$$f_{3}$$H_{2,2}$$H_{2,3}$$H_{2,4}$$\frac{t_{1}}{2}$$\frac{t_{1}}{2}$$1-t_{1}$$f_{4}$$\frac{t_{1}}{2}$${1-t_{1}}$$\frac{t_{1}}{2}$
Figure 7. Attributing lengths to the edges of $\Gamma^{\prime}$ for
$t_{1}>t_{2}$ and $k$ even. In this case, $v_{e_{2}}$ is contained in
$H_{2,i}$ for every $i=1,\ldots,m_{2}$.

We are left to consider the case in which $\psi(E(H_{3,1},H_{3,1}^{c}))$ and
$\psi(E(H_{2,i},H_{2,i}^{c}))$ have a common edge for some $i$. In this case,
this edge must be $f_{k}$.

We claim that $k$ is odd. First we prove that $H_{3,1}^{c}$ contains the
vertices of $\Gamma^{\prime}$ incident to $f_{2i}$ and the vertices over
$f_{2i}$ for every $i$. Let us denote by $e_{2},f,f_{k}$ the edges of
$\psi(E(H_{3,1},H_{3,1}^{c}))$. The intersection $H_{3,1}\cap H_{2,1}$ cannot
be a $2$-hemisphere, otherwise we would contradict the minimality of
$H_{2,1}$. Indeed, the fact that $E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap
E(H_{3,1},H_{3,1}^{c})\neq\emptyset$ implies that

$E(v_{e_{2}},\\{v_{e_{2}}\\}^{c})\cap E(H_{3,1}\cap H_{2,1},(H_{3,1}\cap
H_{2,1})^{c})\neq\emptyset,$

so $H_{3,1}\cap H_{2,1}\subsetneqq H_{2,1}$. By Lemma 4.13, we see that
$\psi(E(H_{3,1}\cap H_{2,1},(H_{3,1}\cap H_{2,1})^{c}))=\\{e_{2},f,f_{1}\\}$,
with $e_{2}$ and $f$ fully contained in $H_{2,1}$ and $f_{2}$ fully contained
in $H_{3,1}^{c}$.

We now iterate the reasoning. Intersecting $H_{3,1}\cap H_{2,3}$ (see Figure
8), we must have that $e_{2},f\in\psi(E(H_{3,1}\cap H_{2,3},(H_{3,1}\cap
H_{2,3})^{c}))$, hence, by Lemma 4.13, $H_{3,1}\cap H_{2,3}$ is a
$3$-hemisphere, and $f_{3}$ is fully contained in $H_{3,1}$ (as neither
$f_{2}$ nor $f_{k}$ is fully contained in $H_{3,1}$). Considering $H_{3,1}\cap
H_{2,4}$, we have that $f_{4}$ must be fully contained in $H_{3,1}^{c}$, and
iterating this process we see that $f_{2i}$ is fully contained in $H_{3}^{c}$
for every $i=1,\ldots,\lfloor\frac{k}{2}\rfloor$. So $k$ must be odd and we
write $k=2k^{\prime}+1$.

As illustrated in Figure 8, for every $e\in E(\Gamma^{\prime})$, we define:

$\ell^{\prime}(e)=\begin{cases}\begin{array}[]{ll}\frac{1-t_{1}}{2}&\text{ if
}e\in E(H_{2,3i+1},H_{2,3i+1}^{c})\text{ or
}E(H_{2,3i+2},H_{2,3i+2}^{c})\text{ for some }i<k^{\prime},\\\ t_{1}&\text{ if
}e\in E(H_{2,3i},H_{2,3i}^{c})\text{ or }t(e)=v_{1},\\\ 1-t_{2}&\text{ if
}e\in E(H_{3,3i+1},H_{2,3i+1}^{c})\text{ for any }i,\\\ t_{2}-t_{1}&\text{ if
}e\in E(H_{3,3i+2},H_{2,3i+2}^{c})\text{ for any }i,\\\ t_{1}&\text{ if }e\in
E(H_{3,{3i}},H^{c}_{2,{3i}})\text{ for any }i,\\\ 1/3&\text{ if
}\psi(e)\notin\psi(E(H_{j,i},H^{c}_{j,i}))\text{ for any $j=2,3$ and
$i$}.\end{array}\end{cases}$

The remaining edges can be assigned lengths in a such way that
$\sum_{e\in\psi^{-1}(f)}\ell^{\prime}(e)=1$ for every $f\in E(\Gamma_{0})$, so
$(\Gamma^{\prime},\ell^{\prime})$ is a model of $X$. Again, by Remark 4.18,
the divisors $\mathcal{P}_{H_{2,i}}$ and $\mathcal{P}_{H_{3,i}}$ are principal
divisors, finishing the proof.

$v_{e_{1}}$$f_{1}$$H_{2,1}$$H_{2,2}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{2}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{3}$$H_{2,3}$$H_{2,4}$$H_{2,5}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{4}$$t_{1}$$\frac{1-t_{1}}{2}$$\frac{1-t_{1}}{2}$$f_{5}$$H_{2,6}$$H_{3,1}$$H_{3,2}$$t_{1}$$1-t_{2}$$t_{2}-t_{1}$$t_{1}$$1-t_{2}$$t_{2}-t_{1}$$v_{e_{2}}$$t_{1}$$1-t_{2}$$t_{2}-t_{1}$$t_{1}$$1-t_{2}$$t_{2}-t_{1}$
Figure 8. Attributing lengths to the edges of $\Gamma^{\prime}$, for
$t_{1}<t_{2}$ and $k$ odd. In this case, $v_{e_{2}}$ is contained in $H_{2,i}$
for every $i=1,\ldots,m_{2}$.
$v_{e_{1}}$$f_{1}$$H_{2,1}$$\frac{t_{1}}{2}$$\frac{t_{1}}{2}$${1-t_{1}}$$f_{2}$$\frac{t_{1}}{2}$${1-t_{1}}$$\frac{t_{1}}{2}$$f_{3}$$H_{2,2}$$H_{2,3}$$H_{2,4}$$\frac{t_{1}}{2}$$\frac{t_{1}}{2}$$1-t_{1}$$f_{4}$$\frac{t_{1}}{2}$${1-t_{1}}$$\frac{t_{1}}{2}$$f_{5}$$H_{2,5}$$H_{2,6}$$H_{3,1}$$\frac{t_{1}}{2}$$\frac{t_{1}}{2}$$1-t_{1}$$H_{3,2}$$t_{2}$$1-t_{1}$$t_{1}-t_{2}$$v_{e_{2}}$$1-t_{1}$$t_{1}-t_{2}$$t_{2}$$t_{2}$$1-t_{1}$$t_{1}-t_{2}$
Figure 9. Attributing lengths to the edges of $\Gamma^{\prime}$ for
$t_{1}>t_{2}$ and $k$ odd. In this case, $v_{e_{2}}$ is contained in $H_{2,i}$
for every $i=1,\ldots,m_{2}$.

Case (2.b). Consider the refinement $\Gamma^{\prime}$ of $\Gamma_{0}$ by
adding one vertex over each edge. Notice that $\Gamma^{\prime}$ is a
refinement of $\Gamma$. Let $H_{2,i}$ and $H_{3,j}$ be defined as in Equation
(4). Let $k$ be the integer such that
$|\psi(E(H_{2,i},H_{2,i}^{c}))\cap\psi(E(H_{2,i+1},H_{2,i+1}^{c}))|=1$ for
every $i\leq k-1$, and

$\psi(E(H_{2,k+2i+1},H_{2,k+2i+1}^{c}))=\psi(E(H_{2,k+2i+2},H_{2,k+2i+2}^{c})),\text{
for }i\geq 0.$

If $k$ is even, we define the length $\ell^{\prime}$ on $\Gamma^{\prime}$ as
follows:

$\ell^{\prime}(e)=\begin{cases}\begin{array}[]{ll}1-t_{1}&\text{ if }e\in
E(H_{2,2i+1},H_{2,2i+1})\text{ with
}i=0,\dots,\lfloor\frac{m_{2}-1}{2}\rfloor\\\ t_{1}&\text{ if }e\in
E(H_{2,2i},H_{2,2i}^{c})\text{ with
}i=1,\dots,\lfloor\frac{m_{2}}{2}\rfloor\text{ or }t(e)=v_{1}\\\
1-t_{1}&\text{ if }e\in E(H_{3,2i},H_{3,2i}^{c})\text{ with
}i=1,\dots,\lfloor\frac{m_{3}}{2}\rfloor\\\ t_{1}&\text{ if }e\in
E(H_{3,2i+1},H_{3,2i+1}^{c})\text{ with
}i=0,\dots,\lfloor\frac{m_{3}-1}{2}\rfloor\\\ 1/2&\text{ if
}\psi(e)\notin\psi(E(H_{r,i},H_{r,i}^{c})\text{ for any $r=2,3$ and
$i$}.\end{array}\end{cases}$

The remaining edges can be assigned lengths in a such way that
$\sum_{e\in\psi^{-1}(f)}\ell^{\prime}(e)=1$ for every $f\in E(\Gamma_{0})$, so
$(\Gamma^{\prime},\ell^{\prime})$ is a model of $X$.

When $k$ is odd the situations is similar (see Figure 10): the unique
difference is that we define $\ell^{\prime}(e)=1-t_{1}$ for $e\in
E(H_{3,2i+1},H_{3,2i+1}^{c})$ and $\ell^{\prime}(e)=t_{1}$ for $e\in
E(H_{3,2i},H_{3,2i}^{c})$.

$v_{e_{1}}$$H_{2,1}$$t_{1}$$1-t_{1}$$t_{1}$$1-t_{1}$$H_{2,2}$$H_{2,3}$$t_{1}$$1-t_{1}$$t_{1}$$1-t_{1}$$H_{2,4}$$H_{3,1}$$t_{1}$$1-t_{1}$$t_{1}$$1-t_{1}$$t_{1}$$1-t_{1}$$H_{3,2}$$t_{1}$$1-t_{1}$$t_{1}$$1-t_{1}$
Figure 10. Attributing lengths to the edges of $\Gamma^{\prime}$ for $k$ odd.

As in Case (2.a), we have that the combinatorial type of the divisor
$\mathcal{D}_{t_{1},t_{2}}$ defined in Equation (3) does not depend on
$0<t_{1},t_{2}<1$. Hence Theorem 3.1 (3) ensures that the Abel map
$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}$ is already defined at
$(N_{1},N_{2})$, as given by the global blowup in the statement of Theorem
4.7.

Case (2.c). This case is the same as Case (2.b) except that
$\psi(E(H_{2,i},H^{c}_{2,i})\cap\psi(E(H_{3,j},H^{c}_{3,j})=\emptyset$ for
every $i=1,\ldots,m_{2}$ and $j=1,\ldots,m_{3}$. So we can freely assign
lengths to the edges in $E(H_{3,j},H^{c}_{3,j})$. The conclusion is the same
as in Case (2.b).

Case (3.a). This case follows the same steps in Case (2.a): the difference is
that $k=0$ and the sequence of edges $f_{1},\ldots,f_{k}$ is empty. The
conclusion is the same as in Case (2.a).

Case (3.b). This case follows the same steps in Case (2.b): the difference is
that $k=0$. The conclusion is the same as in Case (2.b).

Case (3.c). In this case, we do not have to further refine $\Gamma_{0}$ as
$E(H,H^{c})$ does not contain any edge incident to $v_{e_{1}}$ or $v_{e_{2}}$,
for every $H$ in $\mathcal{F}_{\Gamma}(v_{0},v_{e_{1}},v_{e_{2}})$. So $P_{H}$
is principal on $X$ for every
$H\in\mathcal{F}_{\Gamma}(v_{0},v_{e_{1}},v_{e_{2}})$. As in the previous
cases, using Remark 2.1 and Theorem 4.16, the divisor

$\mathcal{D}_{t_{1},t_{2}}:=2p_{0}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}+\sum_{H\in\mathcal{F}_{\Gamma}(v_{0},v_{e_{1}},v_{e_{2}})}\mathcal{P}_{H}$

on the tropical curve $X$ is $(p_{0},\mu)$-quasistable and equivalent to
$2p_{0}-p_{e_{1},t_{1}}-p_{e_{2},t_{2}}$. Since the combinatorial type of
$\mathcal{D}_{t_{1},t_{2}}$ is independent of $t_{1}$ and $t_{2}$, it follows
from Theorem 3.1 (3) that the Abel map
$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}$ is already defined at
$(N_{1},N_{2})$, as prescribed by the global blowup in the statement of
Theorem 4.7. ∎

Given a regular smoothing $f\colon\mathcal{C}\rightarrow B$ of a curve,
consider the blowup $\widetilde{\mathcal{C}}\rightarrow\mathcal{C}$ giving
rise to a resolution of the degree-2 Abel map
$\alpha^{2}_{\mathcal{O}_{\mathcal{C}}}$, as in Theorem 4.7. Since the locus
we are blowing up is invariant under the natural action of $S_{2}$ on
$\mathcal{C}^{2}$, we can take the quotient

$\operatorname{Sym}^{2}(\widetilde{\mathcal{C}})=\widetilde{\mathcal{C}}/S_{2}.$

We thus obtain a map:

(5)
$\beta_{2}\colon\operatorname{Sym}^{2}(\widetilde{\mathcal{C}})\rightarrow\overline{\mathcal{J}}_{\mu}(\mathcal{C})$

resolving the rational “symmetrized” Abel map
$\operatorname{Sym}(\mathcal{C}^{2})\dashrightarrow\overline{\mathcal{J}}_{\mu}(\mathcal{C})$.

######  Definition 4.19.

Let $C$ be a curve. We say that $C$ is _pseudo-hyperelliptic_ if it has a
simple torsion-free rank-1 sheaf $I$ of degree 2 with non-negative degree over
every component such that $h^{0}(C,I)\geq 2$.

Recall that a curve is weakly-hyperelliptic if it has a degree-$2$ balanced
invertible sheaf (see [4] for more details). If a stable curve is
hyperelliptic, then it is weakly-hyperelliptic.

######  Theorem 4.20.

Let $C$ be a curve with no separating nodes. The following properties hold.

  1. (1)

$C$ is pseudo-hyperelliptic if and only if, for some (every) regular smoothing
$\mathcal{C}\to B$ of $C$, the map
$\beta_{2}\colon\operatorname{Sym}^{2}(\mathcal{C})\to\overline{\mathcal{J}}_{\mu}(\mathcal{C})$
is not injective.

  2. (2)

if $C$ is stable and weakly-hyperelliptic, then $C$ is pseudo-hyperelliptic.

  3. (3)

if $C$ is stable and has a simple torsion-free rank-1 sheaf $I$ of degree 2
with non-negative degree over every component such that $h^{0}(C,I)\geq 2$,
then $I$ is invertible.

###### Proof.

If $C$ has a rational component $E$ such that $|E\cap E^{c}|\leq 2$ then it is
easy to see that $C$ is pseudo-hyperelliptic and weakly-hyperellpitic, and
that $\beta_{2}$ is not injective. So, we will assume that $C$ is stable.

Assume that $C$ is stable and pseudo-hyperelliptic. Let $I$ be a torsion free
rank-1 sheaf satisfying the condition in Definition 4.19. Let
$\mathbb{P}(I):=\operatorname{Proj}(\operatorname{Sym}(I))\to C$ be the
semistable modification of $C$ where we add a rational curve over the nodes of
$C$ where $I$ is not locally free. We consider the invertible sheaf
$L:=\mathcal{O}_{\mathbb{P}(I)}(1)$, so that we have $I=f_{*}(L)$ and $L$ has
degree $1$ on the exceptional components (see [8, Section 5]). Then, $L$ has
non-negative degree on every component of $\mathbb{P}(I)$ and
$h^{0}(\mathbb{P}(I),L)\geq 2$.

We will apply [4, Theorem 5.9] to $\mathbb{P}(I)$ and $L$. We have two cases.
In the first case, there is a component $C_{0}$ of $\mathbb{P}(I)$ satisfying
the following property. Let $Z_{1},\dots Z_{n}$ be the connected components of
$C_{0}^{c}$. Then

(6) $h^{0}(L|_{C_{0}})\geq
2,\;\;\;L|_{C_{0}^{c}}=\mathcal{O}_{C_{0}^{c}},\;\;\;L|_{C_{0}}=\mathcal{O}_{C_{0}}(C_{0}\cap
Z_{i}),\;\;\;|C_{0}\cap Z_{i}|=2.$

This means that the component $C_{0}$ is not exceptional, because $L$ has
degree $2$ on $C_{0}$. Moreover, $L$ has degree $0$ on every other component,
which implies that $I$ is an invertible sheaf, hence $L=I$ and
$\mathbb{P}(I)=C$. We can consider smooth points
$q_{1},q_{2},q_{1}^{\prime},q_{2}^{\prime}$ of $C$ lying over $C_{0}$ such
that

$L|_{C_{0}}\cong\mathcal{O}_{C_{0}}(q_{1}+q_{2})\cong\mathcal{O}_{C_{0}}(q_{1}^{\prime}+q_{2}^{\prime}).$

By Condition (6), we have that
$L\cong\mathcal{O}_{C}(q_{1}+q_{2})\cong\mathcal{O}_{C}(q^{\prime}_{1}+q^{\prime}_{2})$,
hence
$\mathcal{O}_{C}(2p_{0}-q_{1}-q_{2})\cong\mathcal{O}_{C}(2p_{0}-q_{1}^{\prime}-q_{2}^{\prime})$
which means that
$\beta_{2}(q_{1}+q_{2})=\beta_{2}(q_{1}^{\prime}+q_{2}^{\prime})$, where
$\beta_{2}$ is the map in Equation 5 for some (every) regular smoothing of
$C$.

In the second case, there are two components $C_{1}$ and $C_{2}$ of
$\mathbb{P}(I)$ such that $(C_{1},C_{2})$ is a special $\mathcal{B}$-pair (in
the sense of [4, Definition 5.8]). By [4, Theorem 5.9], we have

$\deg L_{C_{1}}=\deg L|_{C_{2}}=1,\;\;\;L|_{(C_{1}\cup
C_{2})^{c}}\cong\mathcal{O}_{(C_{1}\cup C_{2})^{c}}.$

Notice that, if one between $C_{1}$ and $C_{2}$ is exceptional, then the other
must be exceptional as well, and in particular this implies that $I$ is not
simple, which is a contradiction. We deduce that $I$ is an invertible sheaf,
hence $L=I$ and $\mathbb{P}(I)=C$. We can repeat the argument used in the
first case, now taking $q_{1},q_{1}^{\prime}\in C_{1}$ and
$q_{2},q_{2}^{\prime}\in C_{2}$. We leave the details to the reader. Notice
that we proved (3) and the “only if” part of (1).

Now assume that there is a regular smoothing $\mathcal{C}\rightarrow B$ of $C$
such that $\beta_{2}$ is not injective. We have different cases to consider.

In the first case, there are smooth points
$q_{1},q_{2},q^{\prime}_{1},q^{\prime}_{2}$ of $C$ such that
$\beta_{2}(q_{1}+q_{2})=\beta_{2}(q_{1}^{\prime}+q_{2}^{\prime})$. This means
that there exists an invertible sheaf $T$ on $C$ of type
$T=\mathcal{O}_{\mathcal{C}}(\sum a_{i}C_{i})|_{C}$, where
$a_{i}\in\mathbb{Z}$ and $C_{i}$ are the components of $C$, such that

$\mathcal{O}_{C}(2p_{0}-q_{1}-q_{2})\cong\mathcal{O}_{C}(2p_{0}-q_{1}^{\prime}-q_{2}^{\prime})\otimes
T.$

We deduce that

$\mathcal{O}_{C}(q_{1}^{\prime}+q_{2}^{\prime}-q_{1}-q_{2})\cong T.$

Let $\Gamma$ be the dual graph of $C$. Notice that $\Gamma$ has no separating
edge. If $v_{1},v_{2},v_{1}^{\prime},v_{2}^{\prime}$ are the vertices of
$\Gamma$ corresponding to the components containing the points
$q_{1},q_{2},q_{1}^{\prime},q_{2}^{\prime}$, we have that
$v_{1}^{\prime}+v_{2}^{\prime}-v_{1}-v_{2}$ is a principal divisor on
$\Gamma$. Let $f\colon V(\Gamma)\to\mathbb{Z}$ be the rational function on
$\Gamma$ such that
$\textnormal{div}(f)=v_{1}^{\prime}+v_{2}^{\prime}-v_{1}-v_{2}$ (notice that
$v^{\prime}_{i}\neq v_{j}$ for every $i,j=1,2$ because $\Gamma$ has no
separating edge). We denote by $Z$ the subcurve of $C$ corresponding to the
vertices of $\Gamma$ where $f$ attains its minimum. In particular,
$q_{1},q_{2}\in Z$, $q_{1}^{\prime},q_{2}^{\prime}\in Z^{c}$ and $|Z\cap
Z^{c}|=2$. Moreover we have $T|_{Z}=\mathcal{O}_{Z}(-Z\cap Z^{c})$, which
implies that

$\mathcal{O}_{Z}(Z\cap
Z^{c})\otimes\mathcal{O}_{C}(-q_{1}-q_{2})|_{Z}\cong(\mathcal{O}_{C}(q_{1}^{\prime}+q_{2}^{\prime}-q_{1}-q_{2})\otimes
T^{-1})|_{Z}\cong\mathcal{O}_{C}|_{Z}=\mathcal{O}_{Z}.$

Define $L:=\mathcal{O}_{C}(q_{1}+q_{2})$. We see that $L$ satisfies
$h^{0}(L,C)\geq 2$ (indeed $L$ has the trivial section that vanishes only over
$q_{1},q_{2}$ and a section that vanishes on the whole $Z^{c}$). Thus $C$ is
pseudo-hyperellitic in the sense of Definition 4.19.

In the second case, we have nodes $n,n^{\prime}$ and smooth points
$q,q^{\prime}$ of $C$ such that
$\beta_{2}(n+q)=\beta_{2}(n^{\prime}+q^{\prime})$. Let $\widetilde{C}$ be the
semistable modification of $C$ obtained by adding an exceptional component
over each node of $C$. Then, there exists a twister $T$ on $\widetilde{C}$
such that

$\mathcal{O}_{\widetilde{C}}(2p_{0}-\widetilde{n}-q)\cong\mathcal{O}_{\widetilde{C}}(2p_{0}-\widetilde{n}^{\prime}-q^{\prime})\otimes
T$

where $\widetilde{n}$ and $\widetilde{n}^{\prime}$ are any smooth points of
$\widetilde{C}$ lying over the exceptional component over $n$ and
$n^{\prime}$. Arguing as before, we see that
$L:=\mathcal{O}_{\widetilde{C}}(\widetilde{n}+q)$ satisfies
$h^{0}(L,\widetilde{C})\geq 2$, and hence $h^{0}(f_{*}(L),C)\geq 2$. Thus $C$
is pseudo-hyperelliptic in the sense of Definition 4.19.

In the third case, we have a node $n$ and smooth points
$q,q^{\prime}_{1},q^{\prime}_{2}$ such that
$\beta_{2}(n+q)=\beta_{2}(q^{\prime}_{1}+q^{\prime}_{2})$. This case is not
possible, since the sheaf represented by
$\beta_{2}(q^{\prime}_{1}+q^{\prime}_{2})$ is invertible, while the one
represented by $\beta_{2}(n+q)$ is not.

The remaining cases are the following ones:

  * •

$\beta(n_{1}+n_{2})=\beta(q^{\prime}_{1}+q^{\prime}_{2})$;

  * •

$\beta(n_{1}+n_{2})=\beta(n^{\prime}+q^{\prime})$;

  * •

$\beta(n_{1}+n_{2})=\beta(n_{1}^{\prime}+n_{2}^{\prime})$.

where $n_{1}$, $n_{2}$, $n^{\prime}$, $n^{\prime}_{1}$, $n^{\prime}_{2}$ are
nodes of $C$, and $q$, $q^{\prime}_{1}$, $q^{\prime}_{2}$ and smooth points.
All these cases are done in an similar manner as the second case: first, we
change $C$ by a suitable semistable modification $f\colon\widetilde{C}\to C$
and find a line bundle $L$ such that $h^{0}(L,\widetilde{C})\geq 2$. This
implies that $h^{0}(f_{*}(L),C)\geq 2$ which proves that $C$ is pseudo-
hyperelliptic in the sense of Definition 4.19.

Finally, item (2) of the statement readily follows by [4, Theorem 5.9]. ∎

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