# A metric graph satisfying w41=1 that cannot be lifted to a curve satisfying dim⁡ (W41)=1

Marc Coppens
From the journal Open Mathematics

## Abstract

For all integers g ≥ 6 we prove the existence of a metric graph G with w41=1 such that 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.

MSC 2010: 14H51; 14T05; 05C99

### 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 mR be its maximal ideal and k = R/mR 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 = Γ𝔛 which is a finite metric subgraph of the Berkovich analytification Xan 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 xk are rational then this augmentation function is identically zero and we can consider Γ as a metric graph. This is the situation we consider in this paper. In this situation, from the point of view of the metric graph Γ, we say the curve Xis 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 Xis a lift of Γ, those theories on Xand Γ are related by means of a specialisation map

τ*:Div(X)Div(Γ).

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])

dim(|D|)rk(τ*(D)).

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 g2rr then Γ has a g21 and g2rr=rg21 (see [4, 5]). Hence the theory of linear systems gdr of Clifford index 0 (meaning d – 2r = 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 gdr satisfying d – 2r = 1, r ≥ 1 and dr + 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 gdr with Clifford index 1 and dg – 1 then either d = 3 or d = 5. In [7] we obtain for all r ≥ 1 and d = 2r + 1 ≤ g – 1 the existence of a non-hyperelliptic graph Γ of genus g having a linear system gdr.

For a curve X the complete linear systems gdn with nr are parametrized by a closed subscheme Wdr of the Jacobian J(X) and d2rdim(Wdr) gives a kind of generalisation of the Clifford index for moving linear systems on X. In particular in the case rg – 1 then dim(Wdr)d2r and dim(Wdr)=d2r for some 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 wdr as a replacement for dim(Wdr) in the case of graphs which is more close to the definition of the rank of a divisor on a graph (see Section 2.2). In [10] it is proved that wdrdim(Wdr) in the case Γ is a metric graph and the curve X is a lift of Γ. At the moment is seems not known whether wdrd2r for some 0 < 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 dim(Wdr)=d2r1. In the case C is a non-hyperelliptic curve such that dim(Wdr)=d2r1 for some r ≥ 1 and dg – 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: CE 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 g41 then 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 dim(W41)=1 is different from the theory of graphs satisfying w41=1. In particular for all genus g> 10 we prove the existence of a metric graph Gn of genus g (n = g – 3) satisfying w41=1 that cannot be lifted to a curve satisfying dim(W41)=1.

Related to this result it should be mentioned that non-liftable linear systems on graphs are also known. We say that a linear system gdr on a graph G is liftable if there exists a lift X of G and a divisor D on X with dim(|D|) = r such that τ*(D)gdr. As an example, from [13, Theorem 4.8] it is known that the linear system g21 on a hyperelliptic graph having a vertex 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 g31 that cannot be lifted to a curve. Also in [7] one finds lots of types of linear systems gdr that cannot be lifted to curves, e.g. so-called free linear systems g52 of graphs that cannot be lifted to a curve because a curve with a plane model of degree 5 has genus at most 6.

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 Gn (n an integer at least equal to 2) and we prove it satisfies w41=1 and it has Clifford index equal to 2. In Section 4 first we explain that in the case Gn could be lifted to a curve satisfying dim(W41)=1 then for some tropical modification Γ˜ of Gn there would exist a finite harmonic morphism (see Section 2.3) π:Γ˜Γ of degree 2 with Γ a metric graph of genus 1. Finally in Section 4 we prove that such harmonic morphism does not exist.

### 2 Generalities

#### 2.1 Graphs

A topological graph Γ is a compact topological space such that for each P ∈ Γ there exists nP ∈ ℤ+ and ε0+ such that some neighborhood Up of P in Γ is homeomorphic to {z = re2πik/nP: 0 ≤ r ≤ ∊ and k is an integer satisfying 0 ≤ knP – 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 np ≠ 2. We only consider finite topological graphs. We call nP the valence of P on Γ. Those finitely many points P of Γ with np ≠ 2 are called the essential vertices of Γ. The tangent space Tp(Γ) of Γ at P is the set of nP connected components of UP\{P} for UP as above. In this definition, using another such neighborhood UP then we identify connected components of UP \{P} and UP\{P} in the case their intersection is not empty.

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)0+{}. Moreover 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 S1 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 rkΓ(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 gdr on Γ if deg(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 Don 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 Wdr parametrizing linear systems gdr on a curve. In the case Γ has no linear system gdr then wdr=1. Otherwise wdr is the maximal integer 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 EF ≥ 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 eE(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 ϕe′ : e′ → e is a dilation with some factor de'(ϕ)0+. In the case ϕ(e′) is a vertex then we write de′ (ϕ) = 0. We call de′ (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′ ∈ Tρ′(Γ′) and e′ an edge of (Γ′, V′) such that v′ is defined by some connected component of e′ \ {P′} we set dν′ (ϕ) = de′(ϕ). Such morphism is called finite in the case de′ (ϕ) > 0 for all e′ ∈ E(Γ′, V′). There is a natural map dϕ(P′): TP′(Γ′) \ {ν′: dν′ (ϕ) = 0} → Tϕ(P′)(Γ) defined as follows. The connected component of e′ \ {P′} defining υ′ ∈ Tp(Γ′) with dυ(ϕ) ≠ 0 is mapped to a connected component of ϕ(e′) \ {ϕ(P′)} and this defines υTϕ(P′)(Γ), then dϕ(P′)(υ′) = υ.

The morphism ϕ : Γ′ → Γ of metric graphs is called harmonic at P′ ∈ Γ′ if for each υTϕ(P′) (Γ) the number

{dv(ϕ):vTP(Γ) and dϕ(P)(v)=v}

is independent of υ. In that case this sum is denoted by dP/(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 ∑(dP′(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 Γ1 and Γ2 are called tropically equivalent if there is a common tropical modification Γ of Γ1 and Γ2. 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 3.1

(Lemma 1 in [7]). Let Γ0be a metric graph and let Γ be a graph obtained from Γ0by attaching loops at some different points of valence 2 on Γ0. Let γ be such a loop attached to Γ0. Let E and Ebe linearly equivalent divisors on Γ0or on γ then E and Eare linearly equivalent divisors on Γ.

Lemma 3.2

(Lemma 2 in [7]). Assume Γ0and Γ are as in Lemma 3.1. Let E and Ebe effective divisors on Γ0such that E and Eare linearly equivalent as divisors on Γ. Then E and Eare linearly equivalent as divisors on Γ0.

The proofs of Lemmas 3.1 and 3.2 do not depend on the particular graph Γ0 used in [7].

Lemma 3.3

(Corollary 1 in [7]). Let Γ0be a metric graph and let Γ be the graph obtained from Γ0by attaching a loop γ at some point υ ∈ Γ0. Let P be a point of γ \ {υ} and let D be an effective divisor on Γ0. If rkΓ(D + P) ≥ r then rkΓ0(D) ≥ r.

Lemma 3.4

(Main Theorem in [4]). Let Γ be a metric graph of genus g ≥ 4 and let r be an integer satisfying 2 ≤ rg − 2 such that Γ has a linear systemg2rrthen Γ has a linear systemg21.

The metric graph G0 we start with has genus 2 and can be seen in figure 1.

### Fig. 1

The graph G0

Here υ1 and υ2 are two points of valence 3 (all other points have valence 2) and they are connected by three edges e0, e1 and e2 of mutually different lengths. For 0 ≤ / ≤ 2 the point mi is the midpoint of ei.

Lemma 3.5.

The graph G0has a uniqueg21given by |υ1 + υ2| = |2m0| = |2m1| = |2m2| and in the case υG0such that 2υ ∈ |υ1 + υ2| then υ = mifor some 0 ≤ i ≤ 2.

Proof. Clearly 2mi ∈ |υ1 + υ2 | for 0 ≤ i ≤ 2 and in the case υei \ {υ1, υ2, mi} then taking υ′ on ei such that the distance on ei from υ to υ1 is equal to the distance of υ′ to υ2, then υ + υ′ ∈ |υ1 + υ2| (clearly υυ′). This proves rk(υ1 + υ2) = 1 (it cannot have rank 2 because g(G0) ≠ 0). It is well-known that a graph of genus at least 2 has at most one g21 (see [15, Proposition 5.5], the proof given for a finite graph also holds for a metric graph). Indeed, for this graph G0, if υυ2 then υ1 + υ is clearly υ2-reduced, hence |v1+vv2|=0 and therefore rk(υ1 + υ) = 0. This proves the uniqueness of g21 on G0.

Finally for υ + υ′ ∈ |υ1 + υ2| as before (including the possibility υ + υ′ = υ1 + υ2), since υ′ is a u-reduced divisor one has |vv|=0 hence |v1+v22v|=0. This proves 2υ ∈ |υ1 + υ2| implies A = mi for some 0 ≤ i ≤ 2. □

As indicated in figure 1 we fix qi ∈]υi, mi, [⊂ ei, for i = 1,2.

Lemma 3.6.

There is nog31onG0such that|g312m0|0, |g312q1|0and|g312q2|0.

Proof. For i = 1,2 we take qiei such that the distance on ei from υ1 to qi is equal to the distance on ei, from υ2 to qi. Assume g31 on G0 with |g312m0|0, hence there exists υG0 such that |g31=|g21+v|.

First assume υ ∈ (e0e2) \ {υ1}. Then |q1+v+q1|=|g31-2q1|= and clearly q1+v is a q1-reduced divisor. This implies |q'1+v-q1|=|g31-2q1|=. In the case υ ∉ (e0e2) \ {υ1} then certainly υ ∈ (e0e1) \ {υ2} and using similar arguments we find |g312q2|=0. This finishes the proof of the lemma. □

Now for an integer n ≥ 1 we make a graph Gn as follows. In case n ≥ 3 we fix some more different points q3, … , qn on G0 \ {υ1, υ2, m0, m1, m2, q1, q3}. Then, for all n ≥ 1, the graph Gn is obtained from G0 by attaching a loop γ0 at m0 and loops γi, at qi for each 1 ≤ in (we also are going to denote m0 by q0). As an example see a possible picture of G6 in figure 2. Clearly g(Gn) = n + 3. We prove that the Clifford index of Gn is at least 2 in case n ≥ 2.

### Fig. 2

The graph G6

Proposition 3.7.

Let r be an integer with 1 ≤ rn. Then Gn has nog2r+1rin the case n ≥ 2.

Proof. First we show Gn has no linear system g21 in the case n ≥ 1. Assume there is a g21 on Gn. Take 1 ≤ in and let υγi \ {qi} and υ′ ∈ Gn such that v+vg21. Let Gn0 be the closure of Gn \ γi, and assume vGn0. It follows from Lemma 3.3 that rkGn0(v)1, but because g(Gn0)>0 this is impossible. Hence υ′ ∈ γi. On γi there exists υ″ such that υ + υ′ is linearly equivalent to qi + υ″ as divisors on γi, hence qi+vg21 because of Lemma 3.1. But we proved this implies υ″ = qi, hence 2qig21. This implies rkG0 (2qi) = 1. Indeed, take pG0 \ {qi} and let Dp be the p-reduced divisor on Go linearly equivalent to 2qi. We need to show Dpp ≥ 0. The burning algorithm applied to Gn implies Dp is a p-reduced divisor on Gn too. Since rkGn(2qi ) = 1 it follows Dpp ≥ 0. So we obtain rkG0 (2qi) = 1, but from Lemma 3.5 we know this cannot be true, hence Gn has no linear system g21.

From now on assume n ≥ 2. Fix some integer r satisfying 1 ≤ rn − 1 and assume Gn has a linear system g2r+1r. For 0 ≤ ir + 1 fix υiγi with υiqi. For 0 ≤ i ≤ 2 there exists Eig2r+1r satisfying

Ei(vi+v3+vr+1)0.

For j ∈ {3, … , r + 1} ∪ {i} let Di,j = Ei ∩ (γj \ {qj}), hence Di, jυj > 0. In the case that for some j the qj-reduced divisor on γj linearly equivalent to Di, j contains a point vj different from qj (then the point vj is unique) then there is an effective divisor E′ on Gn\γj¯ of degree 2r such that E+vjg2r+1r. From Lemma 3.3 it follows rkGn\γj¯(E)=r, hence Gn\γj¯ has a linear system g2rr. Since (Gn\γj¯)=n+2 and 2 ≤ 2r ≤ 2n − 2 this would imply Gn\γj¯ is a hyperelliptic graph (Lemma 3.4). But we proved Gn\γj¯ is not hyperelliptic (it is a graph Gn − 1; the proof of that argument also works on Gn\γ0¯ in the case j = 0). Since υj is not linearly equivalent to qj as a divisor on γj it follows Di, j is linearly equivalent to mi, jqj for some mi, j ≥ 2 on γi.

So we obtain Eig2r+1r on Gn such that

Ei'(2qi+2q3++2qr+1)0

for 0 ≤ i ≤ 2 and Ei is contained in G0. Because of Lemma 3.2 those divisors Ei are linearly equivalent as divisors on G0. It follows Ei=Ei(2q3++2qr+1) with 0 ≤ i ≤ 2 are effective linearly equivalent divisors on G0 of degree 3 with Ei2qi0. Since g(G0) = 2 each divisor of degree 3 on G0 defines a g31 and we obtain a contradiction to Lemma 3.6.

Finally, if Gn has an g2n+1n then because of the Theorem of Riemann-Roch |KGng2n+1n|=g31, we already excluded this case. □

Proposition 3.8.

On Gn we havew41=1.

In order to prove this proposition we need the existence of many linear systems g41 on Gn. Those can be obtained from divisors of degree 4 on Go using the following lemma.

Lemma 3.9.

Let D be an effective divisor of degree 4 on G0. Then rkGn(D) ≥ 1.

Proof. Since D is an effective divisor of degree 4 on G0 from the Riemann-Roch Theorem it follows rkG0(D) = 2. This implies for each 0 ≤ in there is an effective divisor D′ ≥ 2qi linearly equivalent to D on G0. Because of Lemma 3.1 the divisor D′ is linearly equivalent to D on Gn. Moreover, for υγi, there is an effective divisor on γi linearly equivalent to 2qi containing υ and using the same lemma we obtain the existence of an effective divisor on Gn linearly equivalent to D containing υ. Similarly, for υG0 we obtain an effective divisor on G0 linearly equivalent to D and containing υ and again this divisor is also linearly equivalent to D as a divisor on Gn. This proves rkGn(D) ≥ 1. □

Proof ofProposition 3.8. Fix υ1, υ2Gn. We need to prove that there exists a g41 on Gn such that |g41v1v2|0.

In the case υ1, υ2G0 we can use any effective divisor D on G0 containing υ1 + υ2. Then we have rkGn (D) ≥ 1 because of Lemma 3.9 and |Dv1v2|0. Next assume υ1G0 and υ2γi \ {qi} for some 0 ≤ in. On G0 take any effective divisor D of degree 4 containing υ1 + 2qi. Since 2qi is linearly equivalent to v2+v2 for some v2γi as a divisor on λi, again using Lemma 3.1 we find that D2qi+v2+v2 is an effective divisor linearly equivalent to D on Gn. Hence rkGn(D) ≥ 1 because of Lemma 3.9 and |Dv1v2|0. Assume υ1λi1\{qi1} and υ2λi2\{qi2} for some i1i2. Then one makes a similar argument using the divisor 2qi1 + 2qi1 on Go. Finally, if υ1, υ2λi, \ {qi} for some i then one uses any effective divisor D of degree 4 on G0 containing 3qi. Since 3qi is linearly equivalent on λi to an effective divisor containing υ1 + υ2 again we find rkGn(D) ≥ 1 and |Dv1v2|0.

### 4 The lifting problem

We now consider the following lifting problem associated to Gn. Let K be an algebraically closed complete non-archimedean valued field and let X be a smooth algebraic curve of genus g. Let Xan be the analytification of 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 = Gn and dim(W41(X))=1? In that case, taking into account the result from [10] mentioned in the introduction, this would give a geometric explanation for w41(Gn)=1. This lifting problem will be the motivation for considering the existence of a certain harmonic morphism associated to Gn. 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 w41=1 is different from the classification of smooth curves satisfying dim(W41)=1.

Assume the lifting problem has a solution. The curve X of that solution cannot be hyperelliptic since Gn 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 dim(W41(X))=1 (for arbitrary characteristic and g ≥ 10 see the Appendix): X is trigonal (has a g31), X is a smooth plane curve of degree 5 (hence has genus 6 and has g52) or X is bi-elliptic (there exists a double covering π: XE with g(E) = 1). From Proposition 3.7 we know Gn has no g31 and no g52, hence the curve X has to be bi-elliptic.

So assume there exists a morphism π: XE with g(E) = 1 of degree 2. This induces a map πan: XanEan between the Berkovich analytifications. In the case 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 Gn, 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 Γ˜ (resp. Γ) of X (resp. E) such that π induces a finite harmonic morphism Γ˜Γ of degree 2 (Section 4 of [16] uses no assumption on the characteristic of k). Since Γ˜ is a metric graph (augmentation map identically zero) this is also the case for Γ hence Γ is a metric graph. So in the case Gn is liftable to smooth curve X satisfying dim(W41(X))=1 then there exist a tropical modification Γ˜ of Gn and a metric graph Γ of genus 1 such that there exists a finite harmonic morphism π˜:Γ˜Γ of degree 2. We are going to prove that such finite harmonic morphism does not exist. In the proof the following lemma will be useful.

Lemma 4.1.

Let ϕ: (Γ1, V1) → (Γ1, V2) be a finite harmonic morphism between metric graphs with vertex sets. Let (T′, V′) ⊂ (Γ1, V1) be a subgraph such that Tis a tree, Γ1\T¯Γ1is connected andΓ1\T¯Tconsists of a unique point t (in particular tV1). There is no subtree (T, V) of (T′, V′) different from a point such that ϕ(T) is contained in a loop Γ ⊂ Γ2.

Proof. Assume T is a subtree of T′ not being one point and assume ϕ(T) is contained in a loop Γ of Γ2. 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 qV be a point of valence 1 on T such that qt and let f be the edge of T having q as a vertex point. This edge f defines vTq1), let w = dϕ(q)(v), hence ϕ(q) ∈ Γ and wTϕ(q>)(Γ). Since Γ is a loop there is a unique w′ ∈ Tϕ(q)(Γ) with w′ ≠ w and since ϕ is harmonic there exists v′ ∈ Tq1) with dϕ(q)(v′) = w′. Let f′ be the edge in Γ1 having q as a vertex point and defining v′. From dϕ(q)(v′) ∈ Tϕ(q)(Γ) it follows ϕ(f′) ⊂ Γ, hence l(f′) is finite. Since f′ ≠ f and qt one has f′ is an edge of T\T¯. Since T′ is a tree, also Tf′ is a tree and one has ϕ(Tf′) ⊂ Γ. Moreover l(Tf′) = l(T) + l(f′) and l(f′) has as a fixed lower bound the minimal length of an edge contained in Γ1. □

Theorem 4.2.

There does not exist a tropical modificationΓ˜of Gn such that there exists a graph Γ with g(T) = 1 and a finite harmonic morphismϕ:Γ˜Γof degree 2.

Proof. Assume Γ˜ is a tropical modification of Gn and ϕ:Γ˜Γ is a finite harmonic morphism of degree 2 of metric graphs with g(Γ) = 1. Since g(Γ) = 1 we know g(ϕ(G0)) ≤ 1. In case g(ϕ(G0)) = 0 then in step 1 we are going to prove that the restriction of ϕ to G0 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(G0)) = 0. Then ϕ(G0) looks as in Figure 3 with ϕ|ei: ei → [ϕ(v1), ϕ(mi)] having degree 2, ♯((ϕ|ei)−1(q)) = 2 for all q ∈ [ϕ{v1), ϕ(mi)[and (ϕ|ei) (ϕ(mi)) = {mi} for 0 ≤ i < 2.

### Fig. 3

In the case g(ϕ(G0) = 0

Assume g(ϕ(G0)) = 0. Consider the loop c1 = e1e0 (remember Figure 1). Since ϕ(G0) has genus 0 it follows ϕ(ci) is a subtree T1 of ϕ(G0). Since T1 is the image of a loop and deg(ϕ) = 2, it follows that ♯((ϕ|c1)−1(q′)) = 1 for some q′ ∈ ϕ(c1) if and only if q′ is a point of valence 1 of ϕ(c1). Since deg(ϕ) = 2 it follows dq(ϕ) = 2 for qc1 such that ϕ(c1) has valence 1 on ϕ(c1) and dq(ϕ) = 1 for qc1 if ϕ(q) does not have valence 1 on ϕ(c1). In the case ϕ(c1) would have a point q′ of valence 3 then there exist at least 3 different points q on e1 with ϕ(q) = q′, contradicting deg(ϕ) = 2. Hence ϕ(c1) can be considered as a finite edge with two vertices. Also for each qc1 and vTq(c1) one has db(ϕ) = 1.

Assume ϕ(v1) ≠ ϕ(v2). Then ϕ(e2) is a path from ϕ(v1) to ϕ(v2) outside of ϕ(c1). This would imply g(ϕ(G0)) ≥ 1, contradicting g(ϕ(G0)) = 0, hence ϕ(v1) = ϕ(v2). This also implies that ϕ(m0) and ϕ(m1) are the two points of valence 1 on ϕ(c1). Repeating the previous arguments for the loop c2 = e2e0 one obtains the given description for ϕ|G0: G0ϕ(G0).

Step 2:g(ϕ(G0)) = 1.

In case g(ϕ(G0)) = 0 then we have the description for ϕ|G0:G0ϕ(G0) obtained in Step 1. We are going to prove that this description cannot hold. Since g(ϕ(G0)) ≤ 1, this implies g(ϕ(G0)) = 1.

Consider ϕ(q1) ∈ ϕ(e1) and q1q1 on e1 with ϕ(q1)=ϕ(q1) (see Figure 2). Assume ϕ(γ1) is a tree. If ϕ(γ1) would have valence 1 at ϕ(q1) then from the arguments used in Step 1 it follows dq1(ϕ) = 2. But from Step 1 we know dq1(ϕ) = 1, therefore ϕ(q1) cannot be a point of valence 1 on ϕ(γ1). Hence there exists q1γ1\{q1} with ϕ(q1)=ϕ(q1) and we obtain ♯(ϕ−1(ϕ(q1))) ≥ 3, contradicting deg(ϕ) = 2. It follows g(ϕ(γ1)) = 1, hence ϕ(γ1) contains a loop e1 in Γ. In case ϕ(q1)e1 then again we obtain ♯(ϕ−1(ϕ(q1))) ≥ 3, contradicting deg(ϕ) = 2. Therefore ϕ(q1) ∈ ϕ(q1)e1 and e1ϕ(G0)={ϕ(q1)}.

Repeating the arguments using q2 and γ2 we obtain a loop e2 in Γ such that e2ϕ(G0)={ϕ(q2)}, hence e1e2=0. Since g(Γ) = 1 this is impossible. As a conclusion we obtain g(ϕ(G0)) = 1 finishing the proof of step 2.

In the case ϕ(c1) would have genus 0 (c1 as in the proof of Step 1), then from the arguments used in Step 1 it follows that for qe2 \ {v1, v2} one has ϕ(q) ∉ ϕ(c1). In the case ϕ(v1) ≡ ϕ{v2) it implies ϕ(c2) has genus 1 (again c2 as in the proof of Step 1). In the case ϕ(v1) = ϕ(v2) and g(ϕ(c2)) = O too, it would imply g(ϕ(G0)) = 0

so this cannot occur. Therefore without loss of generality, we can assume ϕ(c1) has genus 1 (but then ϕ(c2) could have genus 0).

Step 3:ϕ|c1: c1ϕ(c1) is an isomorphism (meaning it is finite harmonic of degree 1)

Since g(ϕ(c1)) = 1 it follows there is a loop e in ϕ(c1), finitely many points r1, · · ·, rt on e and finitely many trees Ti inside ϕ(c1) such that

Tie = {r1}

TiTj=0 in the case ij

ϕ(c1) = eT1 ∪ · · · ∪ Tt

(of course t = 0, hence ϕ(c1) is a loop, is also possible; we are going to prove that t = 0).

### Fig. 4

g(ϕ(c1)) = 1

In the case valϕ(c1) (ri) > 3 for some 1 ≤ it then ϕ−1(vi) contain at least 3 different points on c1, contradicting deg(ϕ) = 2. So we obtain a situation like in Figure 4. Let r ∈ {r1, · · · , rt} and let T be the associated subtree of ϕ(c1). Then ϕ−1(r) = {r′, r″} ⊂ c1 with r′ ≡ r″. The tangent space Tr(ϕ(c1)) consists of 3 elements (see Figure 5). Hence there exists wTr(G0) \ Tr(c1) such that dϕ(r′)(w) ∈ Tr(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 G'n defining w hence ϕ(f) ⊂ e. Because of Lemma 4.1 this implies r′ is one of the points qi on G0 and fγi. Since deg(ϕ) = 2 and g(Γ) = 1 it follows eϕ(γi). Repeating the same argument using r′ instead of r′ one obtains a contradiction to deg(ϕ) = 2. This proves t = 0, hence ϕ(c1) = e is a loop.

### Fig. 5

In the case t ≡ 0

Because deg(ϕ) = 2 we cannot go back and forth on e moving along c1 and taking the image under ϕ. In principle it could be the case that there exist different points q′, q″ on c1 such that the image of the closure of both components of c1 \ {q′, q″} is equal to e with ϕ(q′) = ϕ(q″) and dq(ϕ) = dq(ϕ) = 2. This would correspond to something like shown in Figure 6. This figure has to be understood as follows. Moving along c1 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 c1 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

A case that cannot occur

In that case there should exist vTq'(G'n) with dϕ(q′)(υ) ∈ Tϕ(q′)(e) \ dϕ(q′)(Tq (c1)). Hence the edge f of Gn' defining 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 c1 such that ϕ(q′) = ϕ(q″) then ϕ|c1 : c1e is harmonic of degree 2 and dq (ϕ|c1) = 1 for all qc1. In this case ϕ(e2) ∩ e = {ϕ{υ1), ϕ{υ2)} since deg(ϕ) = 2. In the case ϕ(υ1) ≠ ϕ(υ2) then this contradicts g(Γ) = 1. In the case ϕ(υ1) = ϕ(υ2) then because of the description of ϕ|c1 one has l(e1) = l(e0). We assume this is not the case, so we can assume ϕ:c1e is bijective.

In the case for each edge f on Γ˜ with fc1 one has df(ϕ) = 2 then again, since ϕ(υ1) ≠ ϕ(υ2) we have ϕ(e2) ∩ e = {ϕ(υ1), ϕ(υ2)}, contradicting g(Γ) = 1. Assume there exists qc1 being a vertex of Γ˜ and two edges e′, e″ of Γ˜ contained in c1 with vertex end point q such that de (ϕ) = 1 and de = 2. In particular it follows that dq (ϕ) = 2. Let v'Tq(Γ˜) correspond to e′ then there exists vTq(Γ˜) with υTq (c1) such that dϕ(q)(υ) = dϕ(q)(υ′). Let f be the edge of Γ˜ defining υ, then ϕ(f) ⊂ e. From Lemma 4.1 it follows that q is one of the points qi and fγi. Since g(Γ) = 1 and deg(ϕ) = 2 it follows that eϕ(γi), but this is impossible because de″ (ϕ) = 2 and deg(ϕ) = 2. This proves ϕ|c1 : c1e is an isomorphism of metric graphs.

Step 4: Finishing the proof of the theorem.

It follows that ϕ(υ1) and ϕ(υ2) do split e into two parts e′ and e″ of lengths l(e1) and l(e0). Since g(Γ) = 1 it follows ϕ(e2) contains e′ or e″, we assume it contains e′. In the case ϕ(e2) would contain q˜e"\{ϕ(v1),ϕ(v2)} then because of g(Γ) = 1 it follows ϕ(e2) contains one of the connected components of e"\{q˜}. On that connected component we get a contradiction to deg(ϕ) = 2. Indeed, for a point on that connected component the inverse image under ϕ contains one point of c1 and two different points of e2.

So we obtain different points r1, … , rt on e′ and trees T1, … , Tt with Tie′ = ri for 1 ≤ it and TiTj=0 for ij such that ϕ(e2) = e′ ∪ T1 ∪ … ∪ Tt. It is possible (and we are going to prove) that t = 0, hence e′ = ϕ(e2). In the case ri ∉ {ϕ(υ1), ϕ(υ2)} then there exist two different points r′, r″ on e2 such that ϕ(r′) = ϕ(r″) = ri. Since ri is also the image of a point on c1 we get a contradiction to deg(ϕ) = 2. Hence t ≤ 2 and ri ∈ {ϕ(υ1), ϕ(υ2)}.

Assume ri = ϕ(υ1). We obtain qe2 with q ∉ {υ1, υ2} and ϕ(q) = ϕ(υ1). There exists υTq(Γ) such that dϕ(q)(υ) is the element of Tϕ(q) (Γ) defined by e″. Let f be the edge of Γ˜ defining υ. From Lemma 4.1 it follows q is one of the points qi and fγi. Since g(Γ) = 1 and deg(ϕ) = 2 we obtain ϕ(γi) contains e. This implies that for Pe′ there are at least 3 points contained in ϕ−1(P), a contradiction. This proves t = 0, hence ϕ(e2) = e′. Since deg(ϕ) = 2 it also implies df(ϕ) = 1 for each edge f of Γ˜ contained in e2, hence l(e′) = l(e2). Since l(e2) ∉ {l(e0), l(e1)} we obtain a contradiction, finishing the proof of the theorem. □

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

Corollary 4.3.

For each genus g ≥ 5 there is metric graph G of genus g satisfyingw411that has no divisor of Clifford index at most 1 and is not tropically equivalent to a metric graphΓ˜such that there exists a finite harmonic morphism π:Γ˜Γof degree 2 with g(Γ) = 1. In particular in the case g ≥ 10 the graph G cannot be lifted to a curve X of genus g satisfying dim(W41)=1.

### Appendix

We give a very easy proof of a statement implying Mumford’s Theorem in [11, Appendix] in the case of (W41)=1 and g ≥ 10 not using any assumption on the characteristic. This case corresponds to the situation considered in the paper.

Proposition 4.4.

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 g41 then C is bi-elliptic.

Proof. Let g1 and g2 be two linear systems g41 on C. Since C has no g31 both linear systems are base point free and gi defines a morphism ϕi : C —> → ℙ1. Those morphisms give rise to a morphism ϕ = (ϕ1, ϕ2): C → ℙ1 × ℙ1. The projections to the factors induce ϕ1 and ϕ2.

The Picard group of ℙ1 × ℙ1 is equal to ℤ × ℤ with (a, b) being represented by the divisor a (P × ℙ) + b(ℙ × P) (with P ∈ ℙ1). Using (1,1) one gets an embedding of ℙ1 × ℙ1 as a smooth quadric Q in ℙ3. By composition we have a morphism ϕ: CQ ⊂ ℙ3 defined by a linear subsystem of degree 8 of |g + g2|.

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 ϕi is constant, a contradiction. Otherwise this hyperplane section is a smooth conic γ on Q and ϕ: Cγ ≅ ℙ1 has degree 4. This case implies both ϕ1 and ϕ2 are projectively equivalent to ϕ, therefore g1 = g2 and again we obtain a contradiction.

It follows that ϕ : CQ ⊂ ℙ3 is non-degenerated (defined by some g83). In particular 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 : Cϕ(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 : Cϕ(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 : CE. □

## 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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