## Abstract

For all integers *g* ≥ 6 we prove the existence of a metric graph *G* with *G* has Clifford index 2 and there is no tropical modification *G*′ of *G* such that there exists a finite harmonic morphism of degree 2 from *G*′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

### 1 Introduction

Let *K* be an algebraically closed field complete with respect to some non-trivial non-Archimedean valuation. Let *R* be the valuation ring of *K*, let *m*_{R} be its maximal ideal and *k* = *R*/*m*_{R} the residue field. Let *X* be a smooth complete curve of genus *g* defined over *K*. Associated to a semistable formal model 𝔛 over *R* of *X* there exists a so-called skeleton Γ = Γ_{𝔛} which is a finite metric subgraph of the Berkovich analytification *X*^{an} of *X* together with an augmentation function *a*: Γ → ℤ^{+} such that *a*(*v*) = 0 except for at most finitely many points (see e.g. [1]). In the case all components of the special fiber *X*is a lift of Γ.

There exists a theory of divisors and linear equivalence on Γ very similar to the theory on curves and, in case *X*is a lift of Γ, those theories on *X*and Γ are related by means of a specialisation map

For a divisor *E* on Γ one defines a rank rk(*E*) and for a divisor *D* on *X* the specialisation theorem says (see e.g. [2], since we restrict to the case of zero augmentation map this is in principle considered in [3])

In the hyperelliptic case many classical results on linear systems on curves also do hold for linear systems on metric graphs. As an example, if the graph Γ has a very special linear system *d* – 2*r* = 0) is the same for graphs as for curves (see Section 2.2 for this terminology). This is not true for the theory of linear systems of Clifford index more than 0. As an example: the theory of linear systems of Clifford index 1 concerns the theory of linear systems *d* – 2*r* = 1, *r* ≥ 1 and *d* ≤ *r* + *g* – 2. In the case of curves, H. Martens Theorem (see [6]) states that if *C* is a non-hyperelliptic curve and *C* has a linear system *d* ≤ *g* – 1 then either *d* = 3 or *d* = 5. In [7] we obtain for all *r* ≥ 1 and *d* = 2*r* + 1 ≤ *g* – 1 the existence of a non-hyperelliptic graph Γ of genus *g* having a linear system

For a curve *X* the complete linear systems *n* ≥ *r* are parametrized by a closed subscheme *J*(*X*) and *X*. In particular in the case *r* ≤ *g* – 1 then *r* < *g* – 1 if and only if *X* is hyperelliptic (see [8]). In [9] it is shown that using the dimension of a similar subspace of the Jacobian *J*(Γ) of a metric graph Γ this statement is not true. Moreover in that paper the authors do introduce a much better invariant *X* is a lift of Γ. At the moment is seems not known whether *r* < *g* – 1 implies Γ is hyperelliptic.

Concerning the next case, in [11, Appendix] one finds Mumford’s classification of all curves *X* such that *C* is a non-hyperelliptic curve such that *r* ≥ 1 and *d* ≤ *g* – 1 then either *C* is a trigonal curve or *C* is a smooth plane curve of degree 5 or *C* is a bi-elliptic curve (meaning that there exists a double covering *π*: *C* → *E* with *E* an elliptic curve). In [11] the author assumes char(*k*) ≠ 2 although it is not so clear whether this is also necessary for the arguments in the appendix of that paper. In the appendix of this paper I give a very short proof that in the case *C* is neither hyperelliptic, not trigonal of genus *g* ≥ 10 and if *C* has two different linear systems *C* is bi-elliptic not using any assumption on char(*k*). Many more generalisations are proved by different authors (see e.g. [12]). In this paper we show that the theory of curves satisfying *g*> 10 we prove the existence of a metric graph *G*_{n} of genus *g* (*n* = *g* – 3) satisfying

Related to this result it should be mentioned that non-liftable linear systems on graphs are also known. We say that a linear system *G* is liftable if there exists a lift *X* of *G* and a divisor *D* on *X* with dim(|*D*|) = *r* such that *v* adjacent to at least 3 different bridges (a bridge of a graph *G* is an edge *e* such that *G* \ *e* is disconnected) is not liftable to a hyperelliptic curve because of the violation of some Hurwitz condition. In [14, Example 5.13] one finds an example due to Luo of a graph with a

In Section 2 we recall some generalities on graphs and the theory of divisors on graphs. For generalities on the specialisation map and the relation between the metric graphs and skeleta inside Berkovich curves we refer to the references. It is not needed to understand the arguments used in this paper, it is important for the motivation. In Section 3 we give the description of the graph denoted by *G*_{n} (*n* an integer at least equal to 2) and we prove it satisfies *G*_{n} could be lifted to a curve satisfying *G*_{n} there would exist a finite harmonic morphism (see Section 2.3)

### 2 Generalities

#### 2.1 Graphs

A *topological graph* Γ is a compact topological space such that for each *P* ∈ Γ there exists *n*_{P} ∈ ℤ^{+} and *U*_{p} of *P* in Γ is homeomorphic to {*z* = *re*^{2πik/nP}: 0 ≤ *r* ≤ ∊ and *k* is an integer satisfying 0 ≤ *k* ≤*n*_{P} – 1} ⊂ ℂ with *P* corresponding to 0. Such a topological graph Γ is called *finite* in the case there are only finitely many points *P* ∈ Γ satisfying *n*_{p} ≠ 2. We only consider finite topological graphs. We call *n*_{P} the *valence* of *P* on Γ. Those finitely many points *P* of Γ with *n*_{p} ≠ 2 are called the *essential vertices* of Γ. The *tangent space T*_{p}(Γ) of Γ at *P* is the set of *n*_{P} connected components of *U*_{P}\{*P*} for *U*_{P} as above. In this definition, using another such neighborhood *U*_{P} \{*P*} and

A metric graph Γ is a finite topological graph Γ together with a finite subset *V*_{∞} (Γ) of the set of 1-valent points of Γ and a complete metric on Γ \ *V*_{∞}(Γ). A *vertex set V* of a metric graph Γ is a finite subset of Γ containing all essential vertices. The pair (Γ, *V*) is called a *metric graph with vertex set V*. The elements of *V* are called the vertices of (Γ, *V*). The connected components of Γ \ *V* are called the *edges* of (Γ, *V*). The elements of *ē* \ *e* are called the *end vertices* of *e*(*ē* is the closure of *e*). We always choose *V* such that each edge has two different end vertices. Using the metric on Γ \ *V*_{∞}(Γ) each edge *e* of Γ has a length *l*(*e*) = ∞ if and only if some end vertex of *e* belongs to *V*_{∞}(Γ). We write *E*(Γ, *V*) to denote the set of edges of (Γ, *V*). The *genus* of (Γ, *V*) is defined by |*E*(Γ)| – |*V*(Γ)| + 1 and it is independent of the choice of *V*. Therefore it is denoted by *g*(Γ) and called the genus of Γ.

A *subgraph* of a metric graph (Γ, *V*) with vertex set is a closed subset Γ′ ⊂ Γ such that (Γ′, Γ′ ∩ *V*) is a metric graph with vertex set. In the case Γ′ is homeomorphic to the unit circle *S*^{1} in ℂ then it is called a *loop* in (Γ, *V*). A metric graph Γ is called a *tree* if, *g*(Γ) = 0.

#### 2.2 Linear systems on graphs

We refer to Section 2 of [7] for the definitions of a divisor, an effective divisor, a rational function, linear equivalence of divisors, the canonical divisor and the rank of a divisor on a metric graph. The rank of a divisor *D* on a metric graph Γ is denoted by rk(*D*) and if it is necessary to add the graph then we write rk_{Γ}(*D*). For a divisor *D* on a graph Γ we write *D* ≥ 0 to indicate it is an effective divisor on Γ. A very important tool in the study of divisors on a metric graph Γ is the concept of a reduced divisor at some point *P* of Γ (see [7]*Section 2.1) and the burning algorithm to decide whether a given divisor on Γ is reduced at *P* (see [7, Section 2.2]).

For a divisor *D* on a metric graph Γ we write |*D*| to denote the set of effective divisors linearly equivalent to *D*. As in the case of curves we call it the *complete linear system* defined by *D*. The rank rk(*D*) replaces the concept of the dimension of a complete linear system on a curve. As in the case of curves we say the complete linear system |*D*| is a linear system *D*) = *d* and rk(*D*) = *r*. From of the Riemann-Roch Theorem for divisors on graphs it follows as in the case of curves that divisors *D* on Γ such that rk(*D*) cannot be computed in a trivial way are exactly those divisors satisfying rk(*D*) > max{0, deg(*D*) – *g*(Γ) + 1}. Those divisors are called *very special*. The *Clifford index* of a very special divisor *D* on Γ is defined by *c*(*D*) = deg(*Z*) – 2rk(*D*). Clifford’s Theorem for metric graphs implies *c*(*D*) ≥ 0 for all very special divisors *D* on Γ. The Clifford index *c*(Γ) of Γ is the minimal value *c*(*D*) for a very special divisor *D*on Γ.

Motivated by the definition of the rank of a divisor on a metric graph one introduces the following replacement for the dimension of the space *w* > 0 such that for each effective divisor *F* of degree *r* + *w* there exists an effective divisor *E* of degree *d* with rk(*E*) ≥ *r* such that *E* – *F* ≥ 0.

#### 2.3 Harmonic morphism

Let Γ and Γ′ be two metric graphs and let *ϕ*: Γ′ → Γ be a continuous map. In the case *V*(resp. *V*′) is a vertex set of Γ (resp. Γ′) then *ϕ* is called a *morphism* from (Γ′, *V*′) to (Γ, *V*) if *ϕ*(*V*′) ⊂ *V* and for each *e* ∈ *E*(Γ, *V*) the set *ϕ*^{−1} (ē) is a union of closures of edges of (Γ′, *V*′). Moreover if *e*′ ∈ *E*(Γ′, *V*′) with *e*′ ⊂ *ϕ*^{−1} (ē) then either *ϕ*(*e*′) is a vertex in *V*′ or the restriction *ϕ*_{e}′ : *e*′ → *e* is a dilation with some factor *ϕ*(*e*′) is a vertex then we write *d*_{e}′ (*ϕ*) = 0. We call *d*_{e}′ (*ϕ*) the *degree* of *ϕ* along *e*′.

We say *ϕ* is a morphism of metric graphs if there exist vertex sets *V* (resp. *V*′) of Γ (resp. Γ′) such that *ϕ* is a morphism from (Γ′, *V*′) to (Γ, *V*). In that case, for *P* ∈ Γ′, *ν*′ ∈ *T*_{ρ}′(Γ′) and *e*′ an edge of (Γ′, *V*′) such that *v*′ is defined by some connected component of *e*′ \ {*P*′} we set *d*_{ν′} (*ϕ*) = *d*_{e}′(*ϕ*). Such morphism is called *finite* in the case *d*_{e}′ (*ϕ*) > 0 for all *e*′ ∈ *E*(Γ′, *V*′). There is a natural map *d*_{ϕ}(*P*′): *T*_{P}′(Γ′) \ {*ν*′: *d*_{ν}′ (*ϕ*) = 0} → *T*_{ϕ(P′)}(Γ) defined as follows. The connected component of *e*′ \ {*P*′} defining *υ*′ ∈ *T*_{p′}(Γ′) with *d*_{υ′}(*ϕ*) ≠ 0 is mapped to a connected component of *ϕ*(*e*′) \ {*ϕ*(*P*′)} and this defines *υ* ∈ *T*_{ϕ(P′)}(Γ), then *d _{ϕ}*(

*P*′)(

*υ*′) =

*υ*.

The morphism *ϕ* : Γ′ → Γ of metric graphs is called *harmonic* at *P*′ ∈ Γ′ if for each *υ* ∈ *T*_{ϕ(P′)} (Γ) the number

is independent of *υ*. In that case this sum is denoted by *d*_{P′}/(*ϕ*) and it is called the *degree* of *ϕ* at *P*′. We say the morphism *ϕ* is harmonic if *ϕ* is surjective and *ϕ* is harmonic at each point *P*′ ∈ Γ′. In this case for *P* ∈ Γ one has ∑(*d*_{P}′(*ϕ*) : *ϕ*(*P*′) = *P*) is independent of *P* and it is called the *degree* of *ϕ* denoted by deg(*ϕ*).

An *elementary tropical modification* of a metric graph Γ is a metric graph Γ′ obtained by attaching an infinite closed edge to Γ at some point *P* ∈ Γ \ *V*_{∞}(Γ). A metric graph obtained from Γ as a composition of finitely many elementary tropical modifications is called a *tropical modification* of Γ. Two metric graphs Γ_{1} and Γ_{2} are called *tropically equivalent* if there is a common tropical modification Γ of Γ_{1} and Γ_{2}. This terminology can be found in e.g. [14] together with some examples.

### 3 The example

In the proof of this section we are going to use some lemmas concerning linear systems on graphs.

(Lemma 1 in [7]). *Let* Γ_{0}*be a metric graph and let* Γ *be a graph obtained from* Γ_{0}*by attaching loops at some different points of valence 2 on* Γ_{0}. *Let γ be such a loop attached to* Γ_{0}. *Let E and E*′ *be linearly equivalent divisors on* Γ_{0}*or on γ then E and E*′ *are linearly equivalent divisors on* Γ.

(Lemma 2 in [7]). *Assume* Γ_{0}*and* Γ *are as in Lemma 3.1. Let E and E*′ *be effective divisors on* Γ_{0}*such that E and E*′ *are linearly equivalent as divisors on* Γ. *Then E and E*′ *are linearly equivalent as divisors on* Γ_{0}.

The proofs of Lemmas 3.1 and 3.2 do not depend on the particular graph Γ_{0} used in [7].

(Corollary 1 in [7]). *Let* Γ_{0}*be a metric graph and let* Γ *be the graph obtained from* Γ_{0}*by attaching a loop γ at some point υ* ∈ Γ_{0}. *Let P be a point of γ* \ {*υ*} *and let D be an effective divisor on* Γ_{0}. *If* rk_{Γ}(*D* + *P*) ≥ *r then* rk_{Γ0}(*D*) ≥ *r*.

(Main Theorem in [4]). *Let* Γ *be a metric graph of genus g* ≥ 4 *and let r be an integer satisfying* 2 ≤ *r* ≤ *g* − 2 *such that* Γ *has a linear system**then* Γ *has a linear system*

The metric graph *G*_{0} we start with has genus 2 and can be seen in figure 1.

### Fig. 1

Here *υ*_{1} and *υ*_{2} are two points of valence 3 (all other points have valence 2) and they are connected by three edges *e*_{0}, *e*_{1} and *e*_{2} of mutually different lengths. For 0 ≤ / ≤ 2 the point *m _{i}* is the midpoint of

*e*.

_{i}*The graph G*_{0}*has a unique**given by* |*υ*_{1} + *υ*_{2}| = |2*m*_{0}| = |2*m*_{1}| = |*2*_{m}_{2}| *and in the case υ* ∈ *G*_{0}*such that* 2*υ* ∈ |*υ*_{1} + *υ*_{2}| *then* υ = *m _{i}*

*for some*0 ≤

*i*≤ 2.

*Proof.* Clearly 2*m _{i}* ∈ |

*υ*

_{1}+

*υ*

_{2}| for 0 ≤

*i*≤ 2 and in the case

*υ*∈

*e*\ {

_{i}*υ*

_{1},

*υ*

_{2},

*m*} then taking

_{i}*υ*′ on

*e*such that the distance on

_{i}*e*from

_{i}*υ*to

*υ*

_{1}is equal to the distance of

*υ*′ to

*υ*

_{2}, then

*υ*+

*υ*′ ∈ |

*υ*

_{1}+

*υ*

_{2}| (clearly

*υ*≠

*υ*′). This proves rk(

*υ*

_{1}+

*υ*

_{2}) = 1 (it cannot have rank 2 because

*g*(

*G*

_{0}) ≠ 0). It is well-known that a graph of genus at least 2 has at most one

*G*

_{0}, if

*υ*≠

*υ*

_{2}then

*υ*

_{1}+

*υ*is clearly

*υ*

_{2}-reduced, hence

*υ*

_{1}+

*υ*) = 0. This proves the uniqueness of

*G*

_{0}.

Finally for *υ* + *υ*′ ∈ |*υ*_{1} + *υ*_{2}| as before (including the possibility *υ* + *υ*′ = *υ*_{1} + *υ*_{2}), since *υ*′ is a u-reduced divisor one has *υ* ∈ |*υ*_{1} + *υ*_{2}| implies *A* = *m _{i}* for some 0 ≤

*i*≤ 2. □

As indicated in figure 1 we fix *q _{i}* ∈]

*υ*,

_{i}*m*, [⊂

_{i}*e*, for

_{i}*i*= 1,2.

*There is no**on**G*_{0}*such that**and*

*Proof.* For *i* = 1,2 we take *e _{i}* from

*υ*

_{1}to

*q*is equal to the distance on

_{i}*e*, from

_{i}*υ*

_{2}to

*G*

_{0}with

*υ*∈

*G*

_{0}such that

First assume *υ* ∈ (*e*_{0} ∪ *e*_{2}) \ {*υ*_{1}}. Then *q*_{1}-reduced divisor. This implies *υ* ∉ (*e*_{0} ∪ *e*_{2}) \ {*υ*_{1}} then certainly *υ* ∈ (*e*_{0} ∪ *e*_{1}) \ {*υ*_{2}} and using similar arguments we find

Now for an integer *n* ≥ 1 we make a graph *G _{n}* as follows. In case

*n*≥ 3 we fix some more different points

*q*

_{3}, … ,

*q*on

_{n}*G*

_{0}\ {

*υ*

_{1},

*υ*

_{2},

*m*

_{0},

*m*

_{1},

*m*

_{2},

*q*

_{1},

*q*

_{3}}. Then, for all

*n*≥ 1, the graph

*G*is obtained from

_{n}*G*

_{0}by attaching a loop

*γ*

_{0}at

*m*

_{0}and loops

*γ*, at

_{i}*q*for each 1 ≤

_{i}*i*≤

*n*(we also are going to denote

*m*

_{0}by

*q*

_{0}). As an example see a possible picture of

*G*

_{6}in figure 2. Clearly

*g*(

*G*) =

_{n}*n*+ 3. We prove that the Clifford index of

*G*is at least 2 in case

_{n}*n*≥ 2.

### Fig. 2

*Let r be an integer with* 1 ≤ *r* ≥ *n*. *Then G _{n} has no*

*in the case n*≥ 2.

*Proof.* First we show *G _{n}* has no linear system

*n*≥ 1. Assume there is a

*G*. Take 1 ≤

_{n}*i*≤

*n*and let

*υ*∈

*γ*\ {

*i**q*} and

_{i}*υ*′ ∈

*G*such that

_{n}*G*\

_{n}*γ*, and assume

_{i}*υ*′ ∈

*γ*. On

_{i}*γ*there exists

_{i}*υ*″ such that

*υ*+

*υ*′ is linearly equivalent to

*q*+

_{i}*υ*″ as divisors on

*γ*, hence

_{i}*υ*″ =

*q*, hence

_{i}_{G0}(2

*q*) = 1. Indeed, take

_{i}*p*″

*G*

_{0}\ {

*q*} and let

_{i}*D*be the

_{p}*p*-reduced divisor on Go linearly equivalent to 2

*q*. We need to show

_{i}*D*−

_{p}*p*≥ 0. The burning algorithm applied to

*G*implies

_{n}*D*is a

_{p}*p*-reduced divisor on

*G*too. Since rk

_{n}*(2*

_{Gn}*q*) = 1 it follows

_{i}*D*−

_{p}*p*≥ 0. So we obtain rk

*G*

_{0}(2

*q*) = 1, but from Lemma 3.5 we know this cannot be true, hence

_{i}*G*has no linear system

_{n}From now on assume *n* ≥ 2. Fix some integer *r* satisfying 1 ≤ *r* ≥ *n* − 1 and assume *G _{n}* has a linear system

*i*≤

*r*+ 1 fix

*υ*∈

*i**γ*with

_{i}*υ*≠

_{i}*q*. For 0 ≤

_{i}*i*≤ 2 there exists

For *j* ∈ {3, … , *r* + 1} ∪ {*i*} let *D*_{i,j} = *E _{i}* ∩ (

*γ*\ {

_{j}*q*}), hence

_{j}*D*

_{i, j}−

*υ*> 0. In the case that for some

_{j}*j*the

*q*-reduced divisor on

_{j}*γ*linearly equivalent to

_{j}*D*

_{i, j}contains a point

*q*(then the point

_{j}*E*′ on

*r*such that

*r*≤ 2

*n*− 2 this would imply

*G*

_{n − 1}; the proof of that argument also works on

*j*= 0). Since

*υ*is not linearly equivalent to

_{j}*q*as a divisor on

_{j}*γ*it follows

_{j}*D*

_{i, j}is linearly equivalent to

*m*,

_{i}*for some*

_{j}q_{j}*m*

_{i, j}≥ 2 on

*γ*.

_{i}So we obtain *G _{n}* such that

for 0 ≤ *i* ≤ 2 and *G*_{0}. Because of Lemma 3.2 those divisors *G*_{0}. It follows *i* ≤ 2 are effective linearly equivalent divisors on *G*_{0} of degree 3 with *g*(*G*_{0}) = 2 each divisor of degree 3 on *G*_{0} defines a

Finally, if *G _{n}* has an

*On G _{n} we have*

In order to prove this proposition we need the existence of many linear systems *G _{n}*. Those can be obtained from divisors of degree 4 on Go using the following lemma.

*Let D be an effective divisor of degree 4 on G*_{0}. Then rk_{Gn}(*D*) ≥ 1.

*Proof*. Since *D* is an effective divisor of degree 4 on *G*_{0} from the Riemann-Roch Theorem it follows rk_{G}_{0}(*D*) = 2. This implies for each 0 ≤ *i* ≤ *n* there is an effective divisor *D*′ ≥ 2*q _{i}* linearly equivalent to

*D*on

*G*

_{0}. Because of Lemma 3.1 the divisor

*D*′ is linearly equivalent to

*D*on

*G*. Moreover, for

_{n}*υ*∈

*γ*, there is an effective divisor on

_{i}*γ*linearly equivalent to 2

_{i}*q*containing

_{i}*υ*and using the same lemma we obtain the existence of an effective divisor on

*G*linearly equivalent to

_{n}*D*containing

*υ*. Similarly, for

*υ*∈

*G*

_{0}we obtain an effective divisor on

*G*

_{0}linearly equivalent to

*D*and containing

*υ*and again this divisor is also linearly equivalent to

*D*as a divisor on

*G*. This proves rk

_{n}*G*(

_{n}*D*) ≥ 1. □

*Proof of**Proposition 3.8*. Fix *υ*_{1}, *υ*_{2} ∈ *G _{n}*. We need to prove that there exists a

*G*such that

_{n}In the case *υ*_{1}, *υ*_{2} ∈ *G*_{0} we can use any effective divisor *D* on *G*_{0} containing *υ*_{1} + *υ*_{2}. Then we have rk*G _{n}* (

*D*) ≥ 1 because of Lemma 3.9 and

*υ*

_{1}∈

*G*

_{0}and

*υ*

_{2}∈

*γ*

_{i}\ {

*q*} for some 0 ≤

_{i}*i*≤

*n*. On

*G*

_{0}take any effective divisor

*D*of degree 4 containing

*υ*

_{1}+ 2

*q*. Since 2

_{i}*q*is linearly equivalent to

_{i}*λ*, again using Lemma 3.1 we find that

_{i}*D*on

*G*. Hence rk

_{n}*G*(

_{n}*D*) ≥ 1 because of Lemma 3.9 and

*υ*

_{1}∈

*λ*

_{i1}\{

*q*

_{i1}} and

*υ*

_{2}∈

*λ*

_{i2}\{

*q*

_{i2}} for some

*i*

_{1}≠

*i*

_{2}. Then one makes a similar argument using the divisor 2

*q*

_{i1}+ 2

*q*

_{i1}on Go. Finally, if

*υ*

_{1},

*υ*

_{2}∈

*λ*, \ {

_{i}*q*} for some

_{i}*i*then one uses any effective divisor

*D*of degree 4 on

*G*

_{0}containing 3

*q*. Since 3

_{i}*q*is linearly equivalent on

_{i}*λ*to an effective divisor containing

_{i}*υ*

_{1}+

*υ*

_{2}again we find rk

*G*(

_{n}*D*) ≥ 1 and

### 4 The lifting problem

We now consider the following lifting problem associated to *G _{n}*. Let

*K*be an algebraically closed complete non-archimedean valued field and let

*X*be a smooth algebraic curve of genus

*g*. Let

*X*be the analytification of

^{an}*X*(as a Berkovich curve). Let

*R*be the valuation ring of

*K*, assume 𝔛 is a strongly semistable model of

*X*over

*R*(meaning the special fiber is nodal with smooth irreducible components) such that the special fiber has only rational components and let Γ be the associated skeleton. Is it possible to obtain this situation such that Γ =

*G*and

_{n}*G*. Making this motivation we are going to refer to some suited papers for terminology and some definitions. The definitions necessary to understand the question on the existence of the harmonic morphism are given in Section 2.3. Finally we are going to prove that the harmonic morphism does not exist, proving that the lifting problem has no solution. In particular we obtain that the classification of metric graphs satisfying

_{n}Assume the lifting problem has a solution. The curve *X* of that solution cannot be hyperelliptic since *G _{n}* is not hyperelliptic. This follows from the specialisation Theorem from [2] or [3] already mentioned in the introduction. From [11] one obtains the following classification in case char(

*k*) ≠ 2 of non-hyperelliptic curves

*X*of genus at least 6 satisfying

*g*≥ 10 see the Appendix):

*X*is trigonal (has a

*X*is a smooth plane curve of degree 5 (hence has genus 6 and has

*X*is bi-elliptic (there exists a double covering

*π*:

*X*→

*E*with

*g*(

*E*) = 1). From Proposition 3.7 we know

*G*has no

_{n}*X*has to be bi-elliptic.

So assume there exists a morphism *π*: *X* → *E* with *g*(*E*) = 1 of degree 2. This induces a map *π ^{an}*:

*X*→

^{an}*E*between the Berkovich analytifications. In the case

^{an}*E*is not a Tate curve then each strong semistable reduction of

*E*contains a component of genus 1 in its special fiber, in particular the augmentation map of the associated skeleton has a unique point with value 1. Otherwise such skeleton can be considered as a metric graph of genus 1. Each skeleton associated to a semistable reduction of

*X*is tropically equivalent to the graph

*G*, in particular it can be considered as a metric graph. From the results in [16, Section 4, especially Corollaries 4.26 and 4.28] it follows that there exist skeletons

_{n}*X*(resp.

*E*) such that

*π*induces a finite harmonic morphism

*k*). Since

*G*is liftable to smooth curve

_{n}*X*satisfying

*G*and a metric graph Γ of genus 1 such that there exists a finite harmonic morphism

_{n}*Let ϕ*: (Γ_{1}, *V*_{1}) → (Γ_{1}, *V*_{2}) *be a finite harmonic morphism between metric graphs with vertex sets. Let* (*T*′, *V*′) ⊂ (Γ_{1}, *V*_{1}) *be a subgraph such that T*′ *is a tree*, *is connected and**consists of a unique point t* (*in particular t* ∈ *V*_{1}). *There is no subtree* (*T, V*) *of* (*T*′, *V*′) *different from a point such that ϕ*(*T*) *is contained in a loop* Γ ⊂ Γ_{2}.

*Proof*. Assume *T* is a subtree of *T*′ not being one point and assume *ϕ*(*T*) is contained in a loop Γ of Γ_{2}. Let *l*(*T*) (resp. *l*(Γ)) be the sum of the lengths of all the edges of *T* (resp. Γ). By definition one has *l*(*T*) ≤ deg(*ϕ*)*l*(Γ), in particular *l*(*T*) is finite. We are going to prove that we have to be able to enlarge *T* such that *l*(*T*) grows with a fixed lower bound. Repeating this a few times gives a contradiction to the upper bound deg(*ϕ*)*l*(*Γ*).

Let *q* ∈ *V* be a point of valence 1 on *T* such that *q* ≠ *t* and let *f* be the edge of *T* having *q* as a vertex point. This edge *f* defines *v* ∈ *T _{q}*(Γ

_{1}), let

*w*=

*d*(

_{ϕ}*q*)(

*v*), hence

*ϕ*(

*q*) ∈ Γ and

*w*∈

*T*

_{ϕ(q>)}(Γ). Since Γ is a loop there is a unique

*w*′ ∈

*T*

_{ϕ(q)}(Γ) with

*w*′ ≠

*w*and since

*ϕ*is harmonic there exists

*v*′ ∈

*T*(Γ

_{q}_{1}) with

*d*(

_{ϕ}*q*)(

*v*′) =

*w*′. Let

*f*′ be the edge in Γ

_{1}having

*q*as a vertex point and defining

*v*′. From

*d*(

_{ϕ}*q*)(

*v*′) ∈

*T*

_{ϕ(q)}(Γ) it follows

*ϕ*(

*f*′) ⊂ Γ, hence

*l*(

*f*′) is finite. Since

*f*′ ≠

*f*and

*q*≠

*t*one has

*f*′ is an edge of

*T*′ is a tree, also

*T*∪

*f*′ is a tree and one has

*ϕ*(

*T*∪

*f*′) ⊂ Γ. Moreover

*l*(

*T*∪

*f*′) =

*l*(

*T*) +

*l*(

*f*′) and

*l*(

*f*′) has as a fixed lower bound the minimal length of an edge contained in Γ

_{1}. □

*There does not exist a tropical modification**of G _{n} such that there exists a graph* Γ

*with g*(T) = 1

*and a finite harmonic morphism*

*of degree 2.*

*Proof*. Assume *G _{n}* and

*g*(Γ) = 1. Since

*g*(Γ) = 1 we know

*g*(

*ϕ*(

*G*

_{0})) ≤ 1. In case

*g*(

*ϕ*(

*G*

_{0})) = 0 then in step 1 we are going to prove that the restriction of

*ϕ*to

*G*

_{0}has a very particular description and next in step 2 we are going to prove that this does not occur.

Step 1: Assume *g*(*ϕ*(*G*_{0})) = 0. Then *ϕ*(*G*_{0}) looks as in Figure 3 with *ϕ*|_{ei}: *e _{i}* → [

*ϕ*(

*v*

_{1}),

*ϕ*(

*m*)] having degree 2, ♯((

_{i}*ϕ*|

_{ei})

^{−1}(

*q*)) = 2 for all

*q*∈ [

*ϕ*{

*v*

_{1}),

*ϕ*(

*m*)[and (

_{i}*ϕ*|

_{ei})

^{−}(

*ϕ*(

*m*)) = {

_{i}*m*} for 0 ≤

_{i}*i*< 2.

### Fig. 3

Assume *g*(*ϕ*(*G*_{0})) = 0. Consider the loop *c*_{1} = *e*_{1} ∪ *e*_{0} (remember Figure 1). Since *ϕ*(*G*_{0}) has genus 0 it follows *ϕ*(*c*_{i}) is a subtree *T*_{1} of *ϕ*(*G*_{0}). Since *T*_{1} is the image of a loop and deg(*ϕ*) = 2, it follows that ♯((*ϕ*|_{c1})^{−1}(*q*′)) = 1 for some *q*′ ∈ *ϕ*(*c*_{1}) if and only if *q*′ is a point of valence 1 of *ϕ*(*c*_{1}). Since deg(*ϕ*) = 2 it follows *d _{q}*(

*ϕ*) = 2 for

*q*∈

*c*

_{1}such that

*ϕ*(

*c*

_{1}) has valence 1 on

*ϕ*(

*c*

_{1}) and

*d*(

_{q}*ϕ*) = 1 for

*q*∈

*c*

_{1}if

*ϕ*(

*q*) does not have valence 1 on

*ϕ*(

*c*

_{1}). In the case

*ϕ*(

*c*

_{1}) would have a point

*q*′ of valence 3 then there exist at least 3 different points

*q*on

*e*

_{1}with

*ϕ*(

*q*) =

*q*′, contradicting deg(

*ϕ*) = 2. Hence

*ϕ*(

*c*

_{1}) can be considered as a finite edge with two vertices. Also for each

*q*∈

*c*

_{1}and

*v*∈

*T*(

_{q}*c*

_{1}) one has

*d*(

_{b}*ϕ*) = 1.

Assume *ϕ*(*v*_{1}) ≠ *ϕ*(*v*_{2}). Then *ϕ*(*e*_{2}) is a path from *ϕ*(*v*_{1}) to *ϕ*(*v*_{2}) outside of *ϕ*(*c*_{1}). This would imply *g*(*ϕ*(*G*_{0})) ≥ 1, contradicting *g*(*ϕ*(*G*_{0})) = 0, hence *ϕ*(*v*_{1}) = *ϕ*(*v*_{2}). This also implies that *ϕ*(*m*_{0}) and *ϕ*(*m*_{1}) are the two points of valence 1 on *ϕ*(*c*_{1}). Repeating the previous arguments for the loop *c*_{2} = *e*_{2} ∪ *e*_{0} one obtains the given description for *ϕ*|*G*_{0}: *G*_{0} → *ϕ*(*G*_{0}).

Step 2:*g*(*ϕ*(*G*_{0})) = 1.

In case *g*(*ϕ*(*G*_{0})) = 0 then we have the description for *ϕ*|_{G0}:*G*_{0} → *ϕ*(*G*_{0}) obtained in Step 1. We are going to prove that this description cannot hold. Since *g*(*ϕ*(*G*_{0})) ≤ 1, this implies *g*(*ϕ*(*G*_{0})) = 1.

Consider *ϕ*(*q*_{1}) ∈ *ϕ*(*e*_{1}) and *e*_{1} with *ϕ*(*γ*_{1}) is a tree. If *ϕ*(*γ*_{1}) would have valence 1 at *ϕ*(*q*_{1}) then from the arguments used in Step 1 it follows *d*_{q1}(*ϕ*) = 2. But from Step 1 we know *d*_{q1}(*ϕ)* = 1, therefore *ϕ*(*q*_{1}) cannot be a point of valence 1 on *ϕ*(*γ*_{1}). Hence there exists *ϕ*^{−1}(*ϕ*(*q*_{1}))) ≥ 3, contradicting deg(*ϕ*) = 2. It follows *g*(*ϕ*(*γ*_{1})) = 1, hence *ϕ*(*γ*_{1}) contains a loop *ϕ*^{−1}(*ϕ*(*q*_{1}))) ≥ 3, contradicting deg(*ϕ*) = 2. Therefore *ϕ*(*q*_{1}) ∈

Repeating the arguments using *q*_{2} and *γ*_{2} we obtain a loop *g*(Γ) = 1 this is impossible. As a conclusion we obtain *g*(*ϕ*(*G*_{0})) = 1 finishing the proof of step 2.

In the case *ϕ*(*c*_{1}) would have genus 0 (*c*_{1} as in the proof of Step 1), then from the arguments used in Step 1 it follows that for *q* ∈ *e*_{2} \ {*v*_{1}, *v*_{2}} one has *ϕ*(*q*) ∉ *ϕ*(*c*_{1}). In the case *ϕ*(*v*_{1}) ≡ *ϕ*{*v*_{2}) it implies *ϕ*(*c*_{2}) has genus 1 (again *c*_{2} as in the proof of Step 1). In the case *ϕ*(*v*_{1}) = *ϕ*(*v*_{2}) and *g*(*ϕ*(*c*_{2})) = O too, it would imply *g*(*ϕ*(*G*_{0})) = 0

so this cannot occur. Therefore without loss of generality, we can assume *ϕ*(*c*_{1}) has genus 1 (but then *ϕ*(*c*_{2}) could have genus 0).

Step 3:*ϕ*|_{c1}: *c*_{1} → *ϕ*(*c*_{1}) is an isomorphism (meaning it is finite harmonic of degree 1)

Since *g*(*ϕ*(*c*_{1})) = 1 it follows there is a loop *e* in *ϕ*(*c*_{1}), finitely many points *r*_{1}, · · ·, *r _{t}* on

*e*and finitely many trees

*T*inside

_{i}*ϕ*(

*c*

_{1}) such that

*T _{i}* ∩

*e*= {

*r*

_{1}}

*i* ≡ *j*

*ϕ*(*c*_{1}) = *e* ∪ *T*_{1} ∪ · · · ∪ *T _{t}*

(of course *t* = 0, hence *ϕ*(*c*_{1}) is a loop, is also possible; we are going to prove that *t* = 0).

### Fig. 4

In the case val_{ϕ(c1)} (*r _{i}*) > 3 for some 1 ≤

*i*≤

*t*then

*ϕ*

^{−1}(

*v*) contain at least 3 different points on

_{i}*c*

_{1}, contradicting deg(

*ϕ*) = 2. So we obtain a situation like in Figure 4. Let

*r*∈ {

*r*

_{1}, · · · ,

*r*} and let

_{t}*T*be the associated subtree of

*ϕ*(

*c*

_{1}). Then

*ϕ*

^{−1}(

*r*) = {

*r*′,

*r*″} ⊂

*c*

_{1}with

*r*′ ≡

*r*″. The tangent space

*T*(

_{r}*ϕ*(

*c*

_{1})) consists of 3 elements (see Figure 5). Hence there exists

*w*∈

*T*

_{r′}(

*G*

_{0}) \

*T*

_{r′}(

*c*

_{1}) such that

*d*(

_{ϕ}*r*′)(

*w*) ∈

*T*(

_{r}*e*). (In Figure 5 it is the tangent vector corresponding to the direction on

*e*indicated by the number 2.) Let

*f*be the edge of

*w*hence

*ϕ*(

*f*) ⊂

*e*. Because of Lemma 4.1 this implies

*r*′ is one of the points

*q*on

_{i}*G*

_{0}and

*f*⊂

*γ*. Since deg(

_{i}*ϕ*) = 2 and g(Γ) = 1 it follows

*e*⊂

*ϕ*(

*γ*). Repeating the same argument using

_{i}*r*′ instead of

*r*′ one obtains a contradiction to deg(

*ϕ*) = 2. This proves

*t*= 0, hence

*ϕ*(

*c*

_{1}) =

*e*is a loop.

### Fig. 5

Because deg(*ϕ*) = 2 we cannot go back and forth on *e* moving along *c*_{1} and taking the image under *ϕ*. In principle it could be the case that there exist different points *q*′, *q*″ on *c*_{1} such that the image of the closure of both components of *c*_{1} \ {*q*′, *q*″} is equal to *e* with *ϕ*(*q*′) = *ϕ*(*q*″) and *d*_{q′}(*ϕ*) = *d*_{q″}(*ϕ*) = 2. This would correspond to something like shown in Figure 6. This figure has to be understood as follows. Moving along *c*_{1} from *q*′ to *q*″ in the direction indicated by 1 (left hand side of the figure) the image under *ϕ* is equal to *e* while moving in the direction indicated by 1 (right hand side of the figure). Moving on *c*_{1} from *q*″ to *q*′ in the direction indicated by 2 (left hand side of the figure) the image under *ϕ* is equal to *e* while moving in the direction indicated by 2 (right hand side of the figure).

### Fig. 6

In that case there should exist *d _{ϕ}*(

*q*′)(

*υ*) ∈

*T*

_{ϕ(q′)}(

*e*) \

*d*(

_{ϕ}*q*′)(

*T*

_{q′}(

*c*

_{1})). Hence the edge

*f*of

*υ*satisfies

*ϕ*(

*f*) ⊂

*e*, contradicting deg(

*ϕ*) = 2. Hence the situation from Figure 6 cannot occur.

It follows that in the case there exist *q*′ ≠ *q*″ on *c*_{1} such that *ϕ*(*q*′) = *ϕ*(*q*″) then *ϕ*|_{c1} : *c*_{1} → *e* is harmonic of degree 2 and *d _{q}* (

*ϕ*|

_{c1}) = 1 for all

*q*∈

*c*

_{1}. In this case

*ϕ*(

*e*

_{2}) ∩

*e*= {

*ϕ*{

*υ*

_{1}),

*ϕ*{

*υ*

_{2})} since deg(

*ϕ*) = 2. In the case

*ϕ*(

*υ*

_{1}) ≠

*ϕ*(

*υ*

_{2}) then this contradicts

*g*(Γ) = 1. In the case

*ϕ*(

*υ*

_{1}) =

*ϕ*(

*υ*

_{2}) then because of the description of

*ϕ*|

_{c1}one has

*l*(

*e*

_{1}) =

*l*(

*e*

_{0}). We assume this is not the case, so we can assume

*ϕ*:

_{c1}→

*e*is bijective.

In the case for each edge *f* on *f* ⊂ *c*_{1} one has *d _{f}*(

*ϕ*) = 2 then again, since

*ϕ*(

*υ*

_{1}) ≠

*ϕ*(

*υ*

_{2}) we have

*ϕ*(

*e*

_{2}) ∩

*e*= {

*ϕ*(

*υ*

_{1}),

*ϕ*(

*υ*

_{2})}, contradicting

*g*(Γ) = 1. Assume there exists

*q*∈

*c*

_{1}being a vertex of

*e*′,

*e*″ of

*c*

_{1}with vertex end point

*q*such that

*d*

_{e′}(

*ϕ*) = 1 and

*d*

_{e″}= 2. In particular it follows that

*d*(

_{q}*ϕ*) = 2. Let

*e*′ then there exists

*υ*∉

*T*(

_{q}*c*

_{1}) such that

*d*(

_{ϕ}*q*)(

*υ*) =

*d*(

_{ϕ}*q*)(

*υ*′). Let

*f*be the edge of

*υ*, then

*ϕ*(

*f*) ⊂

*e*. From Lemma 4.1 it follows that

*q*is one of the points

*q*and

_{i}*f*⊂

*γ*. Since

_{i}*g*(Γ) = 1 and deg(

*ϕ*) = 2 it follows that

*e*⊂

*ϕ*(

*γ*), but this is impossible because

_{i}*d*

_{e″}(

*ϕ*) = 2 and deg(

*ϕ*) = 2. This proves

*ϕ*|

_{c1}:

*c*

_{1}→

*e*is an isomorphism of metric graphs.

Step 4: Finishing the proof of the theorem.

It follows that *ϕ*(*υ*_{1}) and *ϕ*(*υ*_{2}) do split *e* into two parts *e*′ and *e*″ of lengths *l*(*e*_{1}) and *l*(*e*_{0}). Since *g*(Γ) = 1 it follows *ϕ*(*e*_{2}) contains *e*′ or *e*″, we assume it contains *e*′. In the case *ϕ*(*e*_{2}) would contain *g*(Γ) = 1 it follows *ϕ*(*e*_{2}) contains one of the connected components of *ϕ*) = 2. Indeed, for a point on that connected component the inverse image under *ϕ* contains one point of *c*_{1} and two different points of *e*_{2}.

So we obtain different points *r*_{1}, … , *r _{t}* on

*e*′ and trees

*T*

_{1}, … ,

*T*with

_{t}*T*∩

_{i}*e*′ =

*r*for 1 ≤

_{i}*i*≤

*t*and

*i*≠

*j*such that

*ϕ*(

*e*

_{2}) =

*e*′ ∪

*T*

_{1}∪ … ∪

*T*. It is possible (and we are going to prove) that

_{t}*t*= 0, hence

*e*′ =

*ϕ*(

*e*

_{2}). In the case

*r*∉ {

_{i}*ϕ*(

*υ*

_{1}),

*ϕ*(

*υ*

_{2})} then there exist two different points

*r*′,

*r*″ on

*e*

_{2}such that

*ϕ*(

*r*′) =

*ϕ*(

*r*″) =

*r*. Since

_{i}*r*is also the image of a point on

_{i}*c*

_{1}we get a contradiction to deg(

*ϕ*) = 2. Hence

*t*≤ 2 and

*r*∈ {

_{i}*ϕ*(

*υ*

_{1}),

*ϕ*(

*υ*

_{2})}.

Assume *r _{i}* =

*ϕ*(

*υ*

_{1}). We obtain

*q*∈

*e*

_{2}with

*q*∉ {

*υ*

_{1},

*υ*

_{2}} and

*ϕ*(

*q*) =

*ϕ*(

*υ*

_{1}). There exists

*υ*∈

*T*(Γ) such that

_{q}*d*

_{ϕ}(

*q*)(

*υ*) is the element of

*T*

_{ϕ(q)}(Γ) defined by

*e*″. Let

*f*be the edge of

*υ*. From Lemma 4.1 it follows

*q*is one of the points

*q*and

_{i}*f*⊂

*γ*. Since

_{i}*g*(Γ) = 1 and deg(

*ϕ*) = 2 we obtain

*ϕ*(

*γ*) contains

_{i}*e*. This implies that for

*P*∈

*e*′ there are at least 3 points contained in

*ϕ*

^{−1}(

*P*), a contradiction. This proves

*t*= 0, hence

*ϕ*(

*e*

_{2}) =

*e*′. Since deg(

*ϕ*) = 2 it also implies

*d*(

_{f}*ϕ*) = 1 for each edge

*f*of

*e*

_{2}, hence

*l*(

*e*′) =

*l*(

*e*

_{2}). Since

*l*(

*e*

_{2}) ∉ {

*l*(

*e*

_{0}),

*l*(

*e*

_{1})} we obtain a contradiction, finishing the proof of the theorem. □

As a corollary of the theorem we obtain the goal of this paper.

*For each genus g* ≥ 5 *there is metric graph G of genus g satisfying**that has no divisor of Clifford index at most 1 and is not tropically equivalent to a metric graph**such that there exists a finite harmonic morphism *

### Appendix

We give a very easy proof of a statement implying Mumford’s Theorem in [11, Appendix] in the case of *g* ≥ 10 not using any assumption on the characteristic. This case corresponds to the situation considered in the paper.

*Assume C is a smooth non-hyperelliptic, non-trigonal irreducible complete curve defined over an algebraically closed field k of any characteristic. In the case g*(*C*) ≥ 10 *and C has at least two different linear systems *.

*Proof*. Let *g*_{1} and *g*_{2} be two linear systems *C*. Since *C* has no *g _{i}* defines a morphism

*ϕ*:

_{i}*C*—> → ℙ

^{1}. Those morphisms give rise to a morphism

*ϕ*= (

*ϕ*

_{1},

*ϕ*

_{2}):

*C*→ ℙ

^{1}× ℙ

^{1}. The projections to the factors induce

*ϕ*

_{1}and

*ϕ*

_{2}.

The Picard group of ℙ^{1} × ℙ^{1} is equal to ℤ × ℤ with (*a*, *b*) being represented by the divisor *a* (*P* × ℙ) + *b*(ℙ × *P*) (with *P* ∈ ℙ^{1}). Using (1,1) one gets an embedding of ℙ^{1} × ℙ^{1} as a smooth quadric *Q* in ℙ^{3}. By composition we have a morphism *ϕ*: *C* → *Q* ⊂ ℙ^{3} defined by a linear subsystem of degree 8 of |*g* + *g*_{2}|.

In the case the linear system does not have dimension 3 then *ϕ*(*C*) is contained in a hyperplane section of *Q*. In the case this hyperplane section is a union of two lines on *Q* then *ϕ*(*C*) is one of those lines implying some *ϕ _{i}* is constant, a contradiction. Otherwise this hyperplane section is a smooth conic

*γ*on

*Q*and

*ϕ*:

*C*→

*γ*≅ ℙ

^{1}has degree 4. This case implies both

*ϕ*

_{1}and

*ϕ*

_{2}are projectively equivalent to

*ϕ*, therefore

*g*

_{1}=

*g*

_{2}and again we obtain a contradiction.

It follows that *ϕ* : *C* → *Q* ⊂ ℙ^{3} is non-degenerated (defined by some *ϕ*(*C*) is not contained in a plane and therefore deg(*ϕ*(*C*)) ≥ 3. Also deg(*ϕ*(*C*)) divides 8, therefore deg(*ϕ*(*C*)) = 4 or deg(*ϕ*(*C*)) = 8. Then either *ϕ* : *C* → *ϕ*(*C*) has degree 2 or degree 1. In the case the degree is 1 then *C* is birationally equivalent to a curve on *Q* belonging to the linear system (4, 4). Because of the adjunction formula this implies *g*(*C*) ≤ 9, a contradiction to our assumptions. Therefore *ϕ* : *C* → *ϕ*(*C*) has degree 2 and *ϕ*(*C*) is an irreducible curve on *Q* belonging to the linear system (2,2). The adjuction formula implies *g*(*ϕ*(*C*)) ≤ 1. Since *C* is not hyperelliptic we obtain *ϕ*(*C*) is a smooth elliptic curve *E* on *Q* and we obtain a double covering *ϕ* : *C* → *E*. □

## Acknowledgement

Reasearch partially supported by the FWO-grant 1.5.012.13N. The author likes to thank the referee for the suggestions to improve the paper.

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**Received:**2015-4-11

**Accepted:**2015-11-30

**Published Online:**2016-2-9

**Published in Print:**2016-1-1

© 2015 Coppens, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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