# Bounding the number of graph refinements for Brill-Noether existence

Karl Christ  Institute of algebraic geometry, Leibniz University Hannover,
Welfengarten 1, 30167 Hannover, Germany<EMAIL_ADDRESS>and
Qixiao Ma  Institute of algebraic geometry, Leibniz University Hannover,
Welfengarten 1, 30167 Hannover, Germany<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a finite graph of genus $g$. Let $d$ and $r$ be non-negative
integers such that the Brill-Noether number is non-negative. It is known that
for some $k$ sufficiently large, the $k$-th homothetic refinement $G^{(k)}$ of
$G$ admits a divisor of degree $d$ and rank at least $r$. We use results from
algebraic geometry to give an upper bound for $k$ in terms of $g,d,$ and $r$.

##  1\. Introduction

Let $g,d,$ and $r$ be non-negative integers such that the Brill-Noether number
is non-negative,

$\rho(g,d,r):=(r+1)(d-r)-g\cdot r\geq 0.$

Let $C$ be a smooth proper curve of genus $g$ defined over $\mathbb{C}$. The
classical Brill-Noether existence theorem asserts that there always exists a
degree $d$ divisor $D$ on $C$, whose rank $r(D)=h^{0}(C,\mathcal{O}_{C}(D))-1$
is at least $r$.

In [BN07], Baker and Norine introduced a divisor theory on finite graphs, that
mirrors large parts of the algebro-geometric behavior. In [Bak08], Baker
provided the first link between the algebro-geometric and combinatorial worlds
via his specialization lemma, and since then this relationship has been
extensively studied. Among the original conjectures in [Bak08], Brill-Noether
existence for finite graphs is the last one that remains open:

######  Conjecture ([Bak08] Conjecture 3.9(1)).

Fix integers $g,d,r\geq 0$, such that $\rho(g,d,r)=(r+1)(d-r)-g\cdot r\geq 0$.
Then every graph of genus $g$ admits a divisor $D$ with $\mathrm{deg}(D)=d$
and $r(D)\geq r$.

The conjecture is not known even for $r=1$, despite considerable recent
interest and many results for specific classes of graphs; see, for example,
[Cap12], [AR18], [Man22], [vDdBSvdW22] and [Duo22]. While the problem itself
is purely combinatorial, it leads to a circle of interesting algebro-geometric
questions. Some of them are discussed in Section 4.

The existence of divisors predicted by Conjecture Conjecture is known when we
treat $G$ as a metric $\mathbb{Q}$-graph [Bak08, Theorem 3.12], or
equivalently, the existence of such divisors is known on the $k$-th
homothetical refinement $G^{(k)}$ for some $k$ sufficiently large [HKN13,
Theorem 1.3]. The refinement $G^{(k)}$ of $G$, however, admits more divisors
than $G$ and recent examples of [vDdBSvdW22] show that they can behave quite
differently. See Section 4.3 for their simplest example and a discussion of
the relation to the algebro-geometric side.

The main aim of this note is to use known results from algebraic geometry to
establish the following upper bound on $k$, which significantly improves a
previous bound on $k$ obtained in [vDdBSvdW22] with combinatorial methods. See
Section 3.7 for a comparison.

######  Theorem 1.

Let $G$ be any graph of genus $g$. Let $d$ and $r$ be non-negative integers so
that $\rho:=(r+1)(d-r)-g\cdot r\geq 0$. Then there always exists a divisor
$D\in\mathrm{Div}^{d}(G^{(k)})$ with $r_{G^{(k)}}(D)\geq r$ for some

$k<g!\cdot\prod_{i=0}^{r}\frac{i!}{(g-d+r+i)!}.$

##  2\. Preliminaries

###  2.1. Basics on graphs

We collect some basic concepts and notations from [Bak08]:

  1. (1)

Graph: By a graph $G$, we always mean a finite weightless connected multigraph
without loop edges. We denote the set of vertices and edges of $G$ by $V(G)$
and $E(G)$, respectively. The genus $g$ of the graph is given by

$g(G)=1-|V(G)|+|E(G)|.$

  2. (2)

Divisors and degree: Let $\mathrm{Div}(G)$ be the free abelian group on
$V(G)$. An element $D\in\mathrm{Div}(G)$ is called a divisor on $G$.
Explicitly, we can write

$D=\sum_{v\in V(G)}a_{v}\cdot(v),\ a_{v}\in\mathbb{Z}.$

The degree of $D$ is defined as $\mathrm{deg}(D)=\sum_{v\in V(G)}a_{v}$. For a
fixed integer $d$, we denote the set of divisors of degree $d$ on $G$ by
$\mathrm{Div}^{d}(G)$. We call a divisor $D\in\mathrm{Div}(D)$ effective, if
$D=\sum_{v\in V(G)}a_{v}\cdot(v)$ with $a_{v}\geq 0$ for all $v\in V(G)$.

  3. (3)

Linear equivalence: To a $\mathbb{Z}$-valued function $f$ on $V(G)$, we may
associate a divisor on $G$ whose coefficient at $v\in V(G)$ is

$\sum_{e=vw\in E(G)}(f(v)-f(w)).$

Divisors of this form have degree $0$ and are called principal divisors. Two
divisors $D,D^{\prime}\in\mathrm{Div}(G)$ are linearly equivalent, written
$D\sim D^{\prime}$, if $D-D^{\prime}$ is a principal divisor. The complete
linear system associated to $D$ is given by

$|D|=\\{E\in\mathrm{Div}(G)\colon E\sim D\text{ and }E\text{ is effective
}\\}.$

  4. (4)

Rank of divisors: Given a divisor $D$ on $G$, its rank $r_{G}(D)$ is $-1$ if
$|D|=\emptyset$, and otherwise given as

$r_{G}(D)=\mathrm{max}\\{k\in\mathbb{Z}\colon|D-E|\neq\emptyset,\ \text{ for
all effective divisors }E\in\mathrm{Div}^{k}(G)\\}.$

We call a divisor of degree $d$ and rank $r$ a $\mathfrak{g}_{d}^{r}$ on $G$.

###  2.2. Refinements of graphs

Let $G$ be a graph. A refinement of $G$ is a graph $G^{\prime}$ obtained by
inserting a finite set of vertices in the interior of the edges of $G$. The
natural inclusion of vertices

$V(G)\subset V(G^{\prime})$

induces an injective group homomorphism

$\iota_{G,G^{\prime}}\colon\mathrm{Div}(G)\to\mathrm{Div}(G^{\prime}).$

The map $\iota_{G,G^{\prime}}$ does not necessarily preserve the rank of the
divisors, i.e., in general $r_{G}(D)\neq
r_{G^{\prime}}(\iota_{G,G^{\prime}}(D)),$ see [Cap12, Example 5.4]. The rank
however is preserved, if we insert the same number of vertices in each edge:

######  Definition 1.

For an integer $n\geq 1$, let $G^{(n)}$ be the refinement of $G$ by inserting
$n$ vertices in the interior of every edge of $G$. We call the refinement
$G^{(n)}$ of $G$ the $n$-th homothetic refinement of $G$.

######  Proposition 1 ([HKN13] Corollary 3.4).

For every $D\in\mathrm{Div}(G)$ and every $n\geq 0$, we have

$r_{G}(D)=r_{G^{(n)}}(\iota_{G,G^{(n)}}(D)).$

Therefore, if the Brill-Noether existence conjecture holds for a graph $G$, it
holds for any homothetic refinement $G^{(n)}$ of $G$.

##  3\. Proof of Theorem 1

###  3.1. Regular one-parameter smoothing

Let $G$ be a graph of genus $g$ and $X_{0}$ a nodal curve over $\mathbb{C}$
with dual graph $G$. By this we mean in particular, that every irreducible
component of $X_{0}$ is isomorphic to $\mathbb{P}^{1}_{\mathbb{C}}$ since $G$
is weightless.

As usual, we denote by $\mathbb{C}[[t]]$ the ring of formal power series with
coefficients in $\mathbb{C}$ and by $\mathbb{C}((t))$ the field of Laurent
series, its field of fractions.

By [Bak08, Theorem B.2], we may spread out the curve $X_{0}/\mathbb{C}$ to a
family of nodal curves over $\mathrm{Spec}(\mathbb{C}[[t]])$:

$\pi\colon\mathcal{X}\to T=\mathrm{Spec}(\mathbb{C}[[t]]),$

  1. (1)

with special fiber $X_{0}$,

  2. (2)

with generic fiber $\mathcal{X}_{\eta}$ a smooth proper curve of genus $g$,
and

  3. (3)

such that the total space $\mathcal{X}$ is regular.

We will call $\mathcal{X}$ a regular one-parameter smoothing of $X_{0}$. We
denote by $\eta\in T$ the generic point and by $\mathcal{X}_{\eta}$ the
generic fiber.

###  3.2. The $k$-th root fibration

Let $k\geq 1$ be an integer. Let $\pi\colon\mathcal{X}\to T$ be a regular one-
parameter smoothing of $X_{0}$. Let $r_{k}\colon
T=\mathrm{Spec}(\mathbb{C}[[t]])\to T$ be the morphism induced by $t\mapsto
t^{k}$. Let
$\rho_{k}\colon\mathcal{X}_{1}=\mathcal{X}\times_{T,r_{k}}T\to\mathcal{X}$ be
the base change of $\pi$ along $r_{k}$. Let
$\nu\colon\mathcal{X}_{2}\to\mathcal{X}_{1}$ be the normalization, and let
$d\colon\mathcal{X}^{\prime}\to\mathcal{X}_{2}$ be the minimal
desingularization. We call

$\pi^{(k)}:=\pi\circ\rho_{k}\circ\nu\circ d\colon\mathcal{X}^{\prime}\to T$

the $k$-th root fibration of $\pi\colon\mathcal{X}\to T$, see [BPVdV84,
III.9]. The total space $\mathcal{X}^{\prime}$ is regular by our construction.
The generic fiber of $\pi^{(k)}$, denoted by $\mathcal{X}^{\prime}_{\eta}$, is
a smooth proper curve of genus $g$. Let $X_{0}^{\prime}$ be the special fiber
of $\pi^{(k)}$.

######  Proposition 2.

The curve $X^{\prime}_{0}$ is a nodal curve with smooth rational components,
and has dual graph $G^{(k-1)}$.

###### Proof.

We keep track of the special fiber through each step in the construction of
the $k$-th root fibration. Clearly the special fiber remains unchanged by base
change. Since $X_{0}$ is reduced, it remains unchanged by normalization. The
effect of the minimal desingularization follows from a local calculation: Each
node of $X_{0}$ is locally defined by $xy-t=0$. After base change and
normalization, the local equation becomes $xy-t^{k}=0$, which is an $A_{k-1}$
singularity. The minimal resolution of an $A_{k-1}$ singularity is given by
iterated blow-ups, which inserts a chain of $k-1$ smooth rational curves to
each node of the special fiber. ∎

###  3.3. The Picard scheme

We denote by $\pi^{\prime}\colon\mathrm{Pic}^{d}_{\mathcal{X}/T}\to T$ the
relative Picard scheme representing families of line bundles of total degree
$d$ on $\pi\colon\mathcal{X}\to T$. Since we work over a strictly henselian
base, the set $\mathrm{Pic}_{\mathcal{X}/T}(T)$ of $T$-sections of the Picard
scheme represent line bundles on $\mathcal{X}$.

Let us denote the the generic fiber of $\pi^{\prime}$ by
$\mathrm{Pic}^{d}(\mathcal{X}_{\eta})$. The special fiber is a countable union
of tori, indexed by the multi-degrees of line bundles on $X_{0}$; the multi-
degree $\underline{\mathrm{deg}}(L_{0})$ of a line bundle $L_{0}$ on $X_{0}$
is a tuple of integers, one for each irreducible component of $X_{0}$, given
by the degree of the restriction of $L_{0}$ to that component. We may view
such a multi-degree as an element of $\mathrm{Div}(G)$, where $G$ is the dual
graph of $X_{0}$.

###  3.4. The specialization lemma

Our proof relies on Baker’s specialization lemma. As before, we let
$\pi\colon\mathcal{X}\to T=\mathrm{Spec}(\mathbb{C}[[t]])$ be a regular one-
parameter smoothing of a nodal curve $X_{0}$ that has dual graph $G$. Consider
the specialization map

$\tau\colon\mathrm{Pic}_{\mathcal{X}/T}^{d}(T)\to\mathrm{Div}^{d}(G),\ \
\mathcal{L}\mapsto\underline{\mathrm{deg}}(\mathcal{L}|_{X_{0}})$

which sends a relative line bundle to the multidegree of its restriction to
the central fiber. Let $\eta\in T$ denote the generic point and
$\mathcal{X}_{\eta}$ the generic fiber:

######  Proposition 3 (Specialization lemma, [Bak08] Lemma 2.8).

In the above setting, we have

$r(\mathcal{X}_{\eta},\mathcal{L}|_{\mathcal{X}_{\eta}})\leq
r_{G}(\tau(\mathcal{L}))$

for any $\mathcal{L}\in\mathrm{Pic}_{\mathcal{X}/T}^{d}(T)$.

###  3.5. The Brill-Noether locus

Let $r$ and $d$ be non-negative integers that satisfy $\rho=(r+1)(d-r)-g\cdot
r\geq 0$. Over an algebraically closed field $k$, by the classical Brill-
Noether theory [ACGH85, V.(1.1)], we know that for a smooth curve $C$ of genus
$g$, the Brill-Noether locus

$W_{d}^{r}(C)=\\{L\in\mathrm{Pic}^{d}(C)\colon h^{0}(C,L)\geq
r+1\\}\subset\mathrm{Pic}^{d}(C)$

is nonempty, and can be naturally equipped with a scheme structure. The closed
points of $W_{d}^{r}(C)\backslash W_{d}^{r+1}(C)$ are called
$g_{d}^{r}$’s.111We use $\mathfrak{g}_{d}^{r}$ for divisors on graphs (as
defined in Section 2.1) and $g_{d}^{r}$ for divisors on algebraic curves.

For any nodal curve $X_{0}$ with dual graph $G$, we consider the Brill-Noether
locus
$W_{d}^{r}(\mathcal{X}_{\eta})\subset\mathrm{Pic}^{d}(\mathcal{X}_{\eta})$ on
the generic fiber of a regular one-parameter smoothing $\mathcal{X}$ of
$X_{0}$.

######  Proposition 4.

If there exists a closed point $Q$ in $W_{d}^{r}(\mathcal{X}_{\eta})$ of
degree $k$, then taking the $k$-th root fibration
$\pi^{(k)}\colon\mathcal{X}^{\prime}\to T$, the generic fiber
$W_{d}^{r}(\mathcal{X}^{\prime}_{\eta})$ admits a $\mathbb{C}((t))$-rational
point.

###### Proof.

Let $Q\subset W_{d}^{r}(\mathcal{X}_{\eta})$ be a closed point of degree $k$.
It corresponds to a degree-$k$ extension of $\mathbb{C}((t))$. By [Kol07,
Theorem 1.94], we know that $Q\cong\mathrm{Spec}(\mathbb{C}((t^{1/k})))$ and
therefore splits after we take the $k$-th root base change $r_{k}\colon T\to
T$. Since the formation of Brill-Noether schemes commutes with base change,
this yields a $\mathbb{C}((t))$-rational point of
$W_{d}^{r}(\mathcal{X}_{\eta}^{\prime})$. ∎

###  3.6. Proof of Theorem 1

By assumption we have $\rho:=(r+1)(d-r)-g\cdot r\geq 0.$ Thus by the classical
Brill-Noether theory, we know that for a general curve $C$, the Brill-Noether
locus $W_{r}^{d}(C)$ has dimension exactly $\rho$.

Let $X_{0}$ be a nodal curve with smooth rational components and dual graph
$G$. Let $\pi\colon\mathcal{X}\to T=\mathrm{Spec}(\mathbb{C}[[t]])$ be a
regular one-parameter smoothing of $X_{0}$. Let $\mathcal{X}_{\eta}$ be the
generic fiber of $\pi$. By the proof of [Bak08, Theorem B.2], we may further
ask $\mathcal{X}_{\eta}$ to be Brill-Noether general.

By Hensel’s lemma, $\mathbb{C}$-points in the smooth locus of $X_{0}$ lift to
$\mathbb{C}[[t]]$-points of $\mathcal{X}$, and therefore the set
$\mathcal{X}_{\eta}(\mathbb{C}((t)))$ of $\mathbb{C}((t))$-points is Zariski
dense in $\mathcal{X}_{\eta}$. Note that
$\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$ is naturally dominated by the $g$-th
self product of $\mathcal{X}_{\eta}$, so the set of $\mathbb{C}((t))$-points
is Zariski dense in $\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$.

Let
$\Theta=W_{g-1}^{0}(\mathcal{X}_{\eta})\subset\mathrm{Pic}^{g-1}(\mathcal{X}_{\eta})$
be the theta divisor. For any $L\in\mathrm{Pic}^{1}(\mathcal{X}_{\eta})$, let
$t_{L}\Theta\subset\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$ be the translate of
$\Theta$ by $L$. Note that for any irreducible subscheme
$Z\subset\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$, the open subscheme

$U_{Z,\Theta}:=\\{L\in\mathrm{Pic}^{1}(\mathcal{X}_{\eta})|Z\nsubseteq
t_{L}\Theta\\}\subset\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$

is non-empty: this can be checked over an algebraic closed field, where the
inverse translates of $Z$ by $\mathrm{Pic}^{1}(\mathcal{X}_{\eta})$ cover
$\mathrm{Pic}^{g-1}(\mathcal{X}_{\eta})$, so some of them are not contained in
$\Theta$, and intersect the ample divisor $\Theta$ properly.

For an equi-dimensional subscheme
$Z=\cup_{i=1}^{s}Z_{i}\subset\mathrm{Pic}^{g}(\mathcal{X}_{\eta})$, let
$U_{Z,\Theta}:=\cap_{i=1}^{s}U_{Z_{i},\Theta}$. Then the intersection $Z\cap
t_{L}\Theta$ is an equal-dimensional subscheme of dimension $\dim Z-1$ for any
$L\in U_{Z,\Theta}$.

We fix a set of points
$L_{1},\cdots,L_{\rho}\in\mathrm{Pic}^{1}(\mathcal{X}_{\eta})$, that give a
set of translates $t_{L_{1}}\Theta,\cdots,t_{L_{\rho}}\Theta$ of $\Theta$. By
[ACGH85, Theorem V.1.3], we have for the intersection number of effective
cycles on $\mathrm{Pic}^{d}(\mathcal{X}_{\eta})$:

$W_{d}^{r}(\mathcal{X}_{\eta})\cdot(t_{L_{1}}\Theta)\cdots(t_{L_{\rho}}\Theta)=g!\cdot\prod_{i=0}^{r}\frac{i!}{(g-d+r+i)!}.$

Let us view the left hand side as iterated intersections of
$W_{d}^{r}(\mathcal{X}_{\eta})$ with translates of $\Theta$. By a suitable
choice of the $L_{i}$’s, we can ask at each step, that the intersection is
dimensionally transverse. Therefore, the intersection product is represented
by an effective zero cycle, and $W_{d}^{r}(\mathcal{X}_{\eta})$ admits a
closed point of degree

$k\leq g!\cdot\prod_{i=0}^{r}\frac{i!}{(g-d+r+i)!}.$

Let us take the $k$-th root fibration $\pi^{(k)}\colon\mathcal{X}^{\prime}\to
T$. By Proposition 2, the dual graph of the central fiber of
$\mathcal{X}^{\prime}$ is $G^{(k-1)}$. By Proposition 4, the generic fiber
$W_{d}^{r}(\mathcal{X}_{\eta}^{\prime})$ admits a $\mathbb{C}((t))$-rational
point. Since $\mathcal{X}^{\prime}_{\eta}(\mathbb{C}((t)))\neq\emptyset$, the
rational point is represented by a degree-$d$ line bundle $\mathcal{L}_{\eta}$
on $\mathcal{X}^{\prime}_{\eta}$ of rank
$r(\mathcal{X}^{\prime}_{\eta},\mathcal{L}_{\eta})$ at least $r$. Since
$\mathcal{X}^{\prime}$ is regular, $\mathcal{L}_{\eta}$ extends to some line
bundle $\mathcal{L}$ on $\mathcal{X}^{\prime}$. By Proposition 3, the divisor
$\tau(\mathcal{L})\in\mathrm{Div}^{d}(G^{(k-1)})$ has rank
$r_{G^{(k-1)}}(\tau(\mathcal{L}))\geq
r(\mathcal{X}^{\prime}_{\eta},\mathcal{L}_{\eta})\geq r$. This finishes the
proof of Theorem 1.

###  3.7. Comparison to previous bound

In [vDdBSvdW22, Remark 3.7] a different upper bound for $k$ is given by

(1) $k\leq(m+n^{r}d)!\cdot d^{m+n^{r}d},$

where $n=|V(G)|$, $m=|E(G)|$. The bound in Theorem 1 improves this bound
significantly.

Indeed, the bound in Theorem 1 depends only on $g,r$ and $d$. If we fix the
value of $g$, the minimal value of $n$ is $2$ since $G$ is weightless and
without loops; the minimal value of $m$ is $g+1$, since each cycle contains at
least two edges, and any two cycles need to differ by at least one edge (both
values are realized for the graph with $2$ vertices with $g+1$ edges between
them). Hence the minimal value for fixed $g$ of (1) is

$(g+1+2^{r}d)!\cdot d^{g+1+2^{r}d}.$

We claim that the bound in Theorem 1 is better already in this case. Indeed,
we can estimate the bound in Theorem 1, as follows:

$\displaystyle
g!\cdot\prod_{i=0}^{r}\frac{i!}{(g-d+r+i)!}<g!(r!)^{r}<g!(d!)^{r}<g!d^{dr}.$

Now $g!<(g+1+2^{r}d)!$ and $d^{dr}<d^{g+1+2^{r}d}$, which gives the claim.

##  4\. Further discussion

###  4.1. Sufficient algebro-geometric claims

In [Cap12, Theorem 6.3] a proof of Conjecture Conjecture is given, that
however contains a gap as pointed out in [BJ16, Footnote 5, p. 379]. The
approach of [Cap12, Theorem 6.3] is to reduce Conjecture Conjecture to known
algebro-geometric statements via Baker’s specialization lemma, along similar
lines we used in the proof of Theorem 1.

Here are three related algebro-geometric statements, that would suffice to
complete the argument (though none of them is necessary for Brill-Noether
existence on finite graphs). Namely, Conjecture Conjecture would follow, if
one could choose for every graph $G$ a regular one-parameter smoothing
$\mathcal{X}\to T=\mathrm{Spec}(\mathbb{C}[[t]])$ of a curve $X_{0}$ with dual
graph $G$ such that

  1. (1)

the special fiber of the closure of the Brill-Noether scheme
$W_{d}^{r}(\mathcal{X}_{\eta})$ in $\mathrm{Pic}^{d}_{\mathcal{X}/T}$ is not
empty;

  2. (2)

the Brill-Noether scheme $W_{d}^{r}(\mathcal{X}_{\eta})$ admits
$\mathbb{C}((t))$-rational points; or

  3. (3)

the special fiber of the closure of the Brill-Noether locus
$W_{d}^{r}(\mathcal{X}_{\eta})$ in some compactification of
$\mathrm{Pic}^{d}_{\mathcal{X}/T}$ has a reduced component.

These assertions are in increasing strength. That is, (3) implies (2) since if
the special fiber of the closure of $W_{d}^{r}(\mathcal{X}_{\eta})$ contains a
reduced component, it contains smooth points that can be lifted to sections of
$W_{d}^{r}(\mathcal{X}_{\eta})$ by Hensel’s lemma. And (2) implies (1), since
the Néron model of $\mathrm{Pic}^{d}(\mathcal{X}_{\eta})$ may be realized as a
sublocus of the special fiber of $\mathrm{Pic}^{d}_{\mathcal{X}/T}$. Assertion
(1) in turn implies Brill-Noether existence for graphs, via the argument
proposed in [Cap12, Theorem 6.3].

We conclude our discussion with two examples, that illustrate these
conditions.

###  4.2. Dependence on choice of $X_{0}$ and on its smoothing

Our first example illustrates the aforementioned conditions in the simplest
nontrivial case: $g_{3}^{1}$’s on genus $4$ curves. Recall that by [Har77, IV.
Example 5.2.2], a non-hyperelliptic smooth genus $4$ curve $C$ is the complete
intersection of a unique quadric surface $Q$ and a cubic surface $T$ in
$\mathbb{P}^{3}$ via its canonical embedding.

Let $\pi\colon\mathcal{U}\to
P=|\mathcal{O}_{\mathbb{P}^{3}}(2)|\times|\mathcal{O}_{\mathbb{P}^{3}}(3)|$ be
the universal family of $(2,3)$-complete intersections in $\mathbb{P}^{3}$.
Let $S\subset P$ be the open subscheme over which $\pi$ is a family of stable
curves, and the quadric $Q$ has at worst a node. Let $R\subset S$ be the locus
where $Q$ is singular. Let $f\colon\mathcal{F}\to S$ be the universal Fano
scheme of lines on $Q$. The morphsim $f$ factors through its scheme of
components $g\colon\mathcal{W}\to S$, which is a double covering of $S$
ramified over $R$.

Let $p\colon\mathcal{X}\to S$ be the restriction of $\pi$ to $S$. Let
$\mathrm{Pic}_{\mathcal{X}/S}$ be the relative Picard scheme. Restriction of
Cartier divisors induces a morphism
$\mathcal{W}\to\mathrm{Pic}^{3}_{\mathcal{X}/S}$. Note that the classifying
morphism $S\to\overline{\mathcal{M}}_{4}$ is dominant, that
$\rho(4,3,1)=2\cdot 2-1\cdot 4=0$ and that a general $W_{3}^{1}$ consists of
$4!\frac{0!}{2!}\frac{1!}{3!}=2$ points. Therefore, $\mathcal{W}$ coincides
with the universal Brill-Noether locus $\mathcal{W}_{3}^{1}$ over the generic
point of $S$.

Since $g$ is finite and $S$ is irreducible, we know that $\mathcal{W}$ is an
irreducible closed subscheme in $\mathrm{Pic}^{3}_{\mathcal{X}/S}$. Therefore
for any separated relative compactification
$i\colon\mathrm{Pic}^{3}_{\mathcal{X}/S}\to\overline{\mathrm{Pic}}^{3}_{\mathcal{X}/S}$,
the closure of the Brill-Noether loci $\mathcal{W}_{3}^{1}$ coincides with the
image of $\mathcal{W}$, and does not meet the boundary
$\overline{\mathrm{Pic}}^{3}_{\mathcal{X}/S}\backslash
i(\mathrm{Pic}^{3}_{\mathcal{X}/S})$. It follows, that condition (1) is always
satisfied in this setting.

The following example shows how the claims of the list in Section 4.1 depend
on the choice of a curve $X_{0}$ with dual graph $G$ and its smoothing:

######  Example 2.

Let $G$ be the graph with $3$ vertices and two edges between each two of the
vertices. Let $\mathcal{M}_{G}\subset\overline{\mathcal{M}}_{4}$ be the
substack of curves with dual graph $G$. Cross ratios on the componets give a
finite chart of $\mathcal{M}_{G}$, hence $\mathcal{M}_{G}$ is irreducible of
dimension $3$. Counting parameters of moduli, we know that a general curve
with dual graph $G$ is the intersection of a smooth quadric
$Q_{0}\subset\mathbb{P}^{3}$ with a union of three general planes $T=H_{1}\cup
H_{2}\cup H_{3}$. The two rulings on $Q_{0}$ give two distinct $g_{3}^{1}$’s
on $X_{0}=Q_{0}\cap T$ and for an arbitrary smoothing, all three conditions in
Section 4.1 are satisfied.

Suppose now, that $Q_{0}$ is a nodal quadric. We already observed that
condition (1) is still satisfied; which of the remaining conditions are
satisfied now depends on the choice of the smoothing. Let $B$ be the local
ring of $S$ at $[X_{0}]$, and let $b\in B$ be the defining equation of
$R\subset S$. The morphism $g\colon\mathcal{W}\to S$ is of the form $B\to
B[T]/(T^{2}-b)$ locally over $[X_{0}]\in S$.

Let $\mathcal{X}\to D=\mathrm{Spec}(\mathbb{C}[[t]])$ be a smoothing of
$X_{0}$ induced by $f\colon D\to\mathrm{Spec}(B)$ with $f^{*}\colon
B\to\mathbb{C}[[t]]$. Let $\mathcal{X}_{\eta}$ be the generic fiber. Then
$W_{3}^{1}(\mathcal{X}_{\eta})=\mathcal{W}_{\eta}$ has coordinate ring
$\mathbb{C}((t))[T]/(T^{2}-f^{*}b)$.

  * •

If $f$ is transversal to $V(b)$, then $f^{-1}(V(b))$ is reduced, $f^{*}b$ is
not a square and $T^{2}-f^{*}b$ is irreducible. Hence $\mathcal{W}_{\eta}$
admits no $\mathbb{C}((t))$-points, and condition (2) (hence also (3)) fails.

  * •

If $f$ is simply tangent to $V(b)$, then $f^{*}b$ is a square in
$\mathbb{C}((t))$ and $\mathcal{W}_{\eta}$ splits. But the special fiber
$\mathcal{W}|_{[X_{0}]}\cong\mathrm{Spec}(\mathbb{C}[[T]]/(T^{2}))$ is non-
reduced. So condition (3) fails for $X_{0}$, while condition (2) still holds.

###  4.3. The gonality of a graph may change after refinement

Recent examples of [vDdBSvdW22] show, that the divisor theory on $G$ and
$G^{(k)}$ can behave quite differently. The simplest of their examples is
depicted in Figure 1; the graph $G$ does not admit a divisor of rank $1$ and
degree $5$, a $\mathfrak{g}_{5}^{1}$, whereas $G^{(1)}$ does [vDdBSvdW22,
Proposition 4.2 and Theorem 4.8]. This is not a counterexample to Conjecture
Conjecture, as the Brill-Noether number in this case is $\rho=2(5-1)-9\cdot
1=-1.$

$1$$1$$1$$2$$G$$G^{(1)}$ Figure 1. On the left, a graph $G$ that does not
admit a $\mathfrak{g}_{5}^{1}$. On the right, the first homothetic refinement
$G^{(1)}$, together with a $\mathfrak{g}_{5}^{1}$.

The example is not an artifact of the combinatorial side of the theory, in the
sense that the $\mathfrak{g}_{5}^{1}$ on $G^{(1)}$ is the specialization of a
$g_{5}^{1}$ on a smooth curve. Recall that the limits of rank $1$ line bundles
can be explicitly described via the theory of admissible covers of Harris and
Mumford [HM82].

$2$$2$$2$$2$$2$$\Gamma$$\Sigma$$v$ Figure 2. On the left, $\Gamma$ and
$\Sigma$. On the right, the map $\Gamma\to\Sigma$ zoomed in at a subgraph of
$\Gamma$; the map is defined analogously on the other parts of $\Gamma$.
Numbers indicate the degree of the map on a vertex, if it is different from
$1$; half-edges indicate branch- or ramification points. The map has
stretching factor $1$ along all edges and edges in the same fiber have the
same colour. The blue and brown edges have half the length of the other edges.

The dual graph of such an admissible cover $f\colon X_{0}^{\prime}\to Y_{0}$
is depicted in Figure 2 (it has no ramification over the nodes and otherwise
only simple ramification points). Also depicted is a metric structure on the
graphs that gives a tropical admissible cover $\Gamma\to\Sigma$, which encodes
some information about possible smoothings (see [ABBR15] and [CMR16]).

Let $p\in Y_{0}$ be a smooth point lying on the irreducible component of
$Y_{0}$ corresponding to the vertex $v$. The underlying graph of $\Gamma$
depicted in Figure 2 can be obtained from $G^{(1)}$ by successively adding
either a vertex in an edge or a vertex together with one adjacent edge. Let
$\pi\colon X_{0}^{\prime}\to X_{0}$ denote the map, which successively
contracts the components of $X_{0}^{\prime}$ corresponding to these added
vertices. The curve $X_{0}$ hence has dual graph $G^{(1)}$.

The admissible cover $f\colon X_{0}^{\prime}\to Y_{0}$ is the limit of a
$g_{5}^{1}$ on some smoothing $\mathcal{X}_{\eta}^{\prime}$ of
$X_{0}^{\prime}$. Since $f^{*}\mathcal{O}_{Y_{0}}(p)$ has degree $0$ on all
components contracted by $\pi$, $\pi_{*}f^{*}\mathcal{O}_{Y_{0}}(p)$ is a line
bundle on $X_{0}$. By construction, its multi-degree is the
$\mathfrak{g}_{5}^{1}$ on $G^{(1)}$ depicted in Figure 1. This gives the
desired limit of a $g_{5}^{1}$ in $\mathrm{Pic}^{5}(X_{0})$. The fact that $G$
does not admit a $\mathfrak{g}_{5}^{1}$ shows, that no such limit of a
$g_{5}^{1}$ (as a line bundle) can exist on a curve with dual graph $G$.

We thus get the following global picture: The closure of the locus of smooth
curves admitting a $g_{5}^{1}$ in $\overline{\mathcal{M}}_{9}$ intersects the
boundary stratum $\mathcal{M}_{G}$ parametrizing curves with dual graph $G$.
The closure of $\mathcal{W}_{5}^{1}$ in the universal compactified Jacobian,
however, does not contain any points parametrizing a line bundle over
$\mathcal{M}_{G}$.

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