A metric graph satisfying $w^1_4=1$ that cannot be lifted to a curve satisfying $\dim (W^1_4)=1$

0.41 MB·English

by  Marc Coppens

A metric graph satisfying w1 = 1 that cannot be
4
lifted to a curve satisfying dim(W1) = 1.
4
5
1
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Marc Coppens
2 ∗

n
a
J
Abstract
5
1 For all integers g ≥ 6 we prove the existence of a metric graph G
with w1 = 1 such that G has Clifford index 2 and there is no tropical
4
] modification G(cid:48) of G such that there exists a finite harmonic morphism
G
of degree 2 from G(cid:48) to a metric graph of genus 1. Those examples show
A
that dimension theorems on the space classifying special linear systems
. forcurvesdo notall ofthemhave immediatetranslationto thetheory of
h
divisors on metric graphs.
t
a
m 2000 Mathematics Subject Classification. 14H51; 14T05; 05C99
[
Keywords. metric graphs, curves, lifting problems, special divisors, Clifford

1 index, dimension theorems
v
0
4 1 Introduction
7
3
LetK beanalgebraicallyclosedfieldcompletewithrespecttosomenon-trivial
0
. non-Archimedean valuation. Let R be the valuation ring of K, let mR be its
1
maximal ideal and k = R/m the residu field. Let X be a smooth complete
0 R
curve of genus g defined over K. Associated to a semistable formal model X
5
1 over R of X there exists a so-called skeleton Γ = ΓX which is a finite metrix
: subgraph of the Berkovich analytification Xan of X together with an augmen-
v
tation function a:Γ→Z+ such that a(v)=0 except for at most finitely many
i
X points (see e.g. [6]). In case all components of the special fiber X are rational
k
r then this augmentation function is identically zero and we can consider Γ as a
a
metric graph. This is the situation we consider in this paper.
There exists a theory of divisors and linear equivalence on Γ very similar to
the theory on curves and those theories on X and Γ are related by means of a
specialisation map
τ :Div(X)→Div(Γ) .
∗
∗KU Leuven, Technologiecampus Geel, Departement Elektrotechniek (ESAT), Kleinhoef-
straat 4, B-2440 Geel, Belgium; email: [email protected]. Partially supported by
theFWO-grant1.5.012.13N
1
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. [1], since we restrict to the case of zero
augmentation map this is in principle considered in [4])
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 gr then Γ has a g1 and gr =rg1 (see [7]; [9]).
2r 2 2r 2
Hence the theory of linear systems of Clifford index 0 is the same for graphs as
for curves. This is not true for the theory of linear systems of Clifford index
more than 0 (see [8]).
For a curve X the complete linear systems gn with n ≥ r are parametrized
d
by a closed subscheme Wr of the Jacobian J(X) and d−2r−dim(Wr) gives a
d d
kind of generalisation of the Clifford index for moving linear systems on X. In
particular in case r ≤g−1 then dim(Wr)≤d−2r and dim(Wr)=d−2r for
d d
some r < g−1 if and only if X is hyperelliptic (see [12]). In [11] it is shown
that using the dimension of a similar subspace Wr of the Jacobian J(Γ) of a
d
metric graph Γ this statement is not true. Moreover in that paper the authors
do introduce a much better invariant wr which is more close to the definition
d
of the rank of a graph. In [10] it is proved that wr ≥ dim(Wr) in case Γ is
d d
a skeleton of a curve X over K. At the moment is seems not known whether
wr ≥d−2r for some 0<r <g−1 implies Γ is hyperelliptic.
d
In [13] one finds a classification of all curves X such that dim(Wr) = d−
d
2r−1. Manymoregeneralisationsareprovedbydifferentauthors. Inthispaper
we show that the theory of curves satisfying dim(W1)=1 is different from the
4
theory of graphs satisfying w1 = 1. In particular for all genus g ≥ 6 we prove
4
theexistenceofametricgraphG ofgenusg (n=g−3)satisfyingw1 =1that
n 4
cannot be a skeleton of a curve satisfying dim(W1)=1.
4
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
themetricgraphsandskeletainsideBerkovichcurveswerefertothereferences.
Itisnotneededtounderstandtheargumentsusedinthispaper,itisimportant
for the motivation. In Section 3 we give the description of the graph denoted
by G (n an integer at least equal to 2) and we prove it satisfies w1 =1 and it
n 4
hasCliffordindexequalto2. InSection4firstweexplainthatincaseG could
n
be lifted to a curve satisfying dim(W1)=1 then for some tropical modification
4
G of G there would exist a finite harmonic morphism π : G → Γ of degree
(cid:48)n n (cid:48)n
2 with Γ a metric graph of genus 1. Finally in Section 4 we prove that such
harmonic morphism does not exist.
2
2 Generalities
2.1 Graphs
A topological graph Γ is a compact topological space such that for each P ∈ Γ
there exists n ∈Z+ and (cid:15)∈R+ such that some neighborhood U of P in Γ is
P 0 P
homeomorphic to {z = re2πik/nP : 0 ≤ r ≤ (cid:15) and k is an integer satisfying 0 ≤
k ≤n −1}⊂CwithP correspondingto0. SuchatopologicalgraphΓiscalled
P
finite in case there are only finitely many points P ∈ Γ satisfying n (cid:54)= 2. We
P
onlyconsiderfinitetopologicalgraphs. Wecalln thevalence ofP onΓ. Those
P
finitelymanypointsofΓarecalledtheessentialvertices ofΓ. Thetangentspace
T (Γ) of Γ at P is the set of n connected components of U \{P} for U as
P P P P
above. In this definition, using another such neighborhood U then we identify
P(cid:48)
connected components of U \{P} and U \{P} in case their intersection is
P P(cid:48)
not empty.
AmetricgraphΓisafinitetopologicalgraphΓtogetherwithafinitesubset
V (Γ) of the set of 1-valent points of Γ and a complete metric on Γ\V (Γ).
A∞vertex set of a metric graph Γ is a finite subset of Γ containing all ess∞ential
vertices. Thepair(Γ,V)iscalledametricgraphwithvertexsetV. Theelements
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 e\e are called the end vertices
of e (e 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
lenght l(e)∈R+∪{∞}. Moreover l(e)=∞ if and∞only if some end vertex of e
0
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. Hence it is denoted by g(Γ) and called the genus of Γ.
Asubgraph ofametricgraph(Γ,V)withvertexsetisaclosedsubsetΓ ⊂Γ
(cid:48)
suchthat(Γ,Γ ∩V)isametricgraphwithvertexset. IncaseΓ ishomeomor-
(cid:48) (cid:48) (cid:48)
phic to S1 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 [8] for the definitions on a metric graph of a divisor,
an effective divisor, a rational function, linearly equivalence of divisors, the
canonicaldivisorandtherankofadivisor. TherankofadivisorD onametric
graph Γ is denoted by rk(D) and if it is necessary to add the graph then we
write rk (D). For a divisor D on a graph Γ we write D ≥ 0 to indicate it is
Γ
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
Section 2.1 in [8]) and the burning algorithm to decide whether a given divisor
on Γ is reduced at P (see Section 2.2 in [8]).
For a divisor D on a metric graph Γ we write |D| to denote the set of
effective divisors linearly equivalent to D. As is the case of curves we call it
the complete linear system defined by D. The rank rk(D) replaces the concept
3
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 gr on Γ if
d
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}. Thosedivisorsarecalledvery special. The
Clifford index of a very special divisor D on Γ is defined by c(D) = deg(D)−
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 Wr
d
parametrizing linear systems gr on a curve. In case Γ has no linear system gr
d d
then wr = −1. Otherwise wr is the maximal integer w ≥ 0 such that for each
d d
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
(cid:48) (cid:48)
caseV (resp. V )isavertexsetofΓ(resp. Γ)thenφiscalledamorphism from
(cid:48) (cid:48)
(Γ,V )to(Γ,V)ifφ(V )⊂V andforeache∈E(Γ,V)thesetφ 1(e)isaunion
(cid:48) (cid:48) (cid:48) −
ofclosuresofedgesof(Γ,V ). Moreoverife ∈E(Γ,V )withe ⊂φ 1(e)then
(cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) −
either φ(e(cid:48)) is a vertex in V(cid:48) or the restriction φe(cid:48) : e(cid:48) → e is a dilation with
some factor de(cid:48)(φ)∈Z+0. In case φ(e(cid:48)) is a vertex then we write de(cid:48)(φ)=0. We
call de(cid:48)(φ) the degree of φ along e(cid:48).
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
(cid:48) (cid:48) (cid:48) (cid:48)
case, for v(cid:48) ∈TP(cid:48)(Γ(cid:48)) and e(cid:48) an edge of (Γ(cid:48),V(cid:48)) such that v(cid:48) is defined by some
connected component of e(cid:48) \{P(cid:48)} we set dv(cid:48)(φ) = de(cid:48)(φ). Such morphism is
called finite in case de(cid:48)(φ) > 0 for all e(cid:48) ∈ E(Γ(cid:48),V(cid:48)). There is a natural map
dφ(P(cid:48)) : TP(cid:48) \{v(cid:48) : dv(cid:48)(φ) = 0} → Tφ(P(cid:48))(Γ) defined as follows. The connected
component of e(cid:48) \{P(cid:48)} defining v(cid:48) ∈ TP(cid:48)(Γ(cid:48)) with dv(cid:48)(φ) (cid:54)= 0 is mapped to a
connected component of φ(e)\{φ(P )} and this defines v ∈ T (Γ), then
(cid:48) (cid:48) φ(P(cid:48))
d (P )(v )=v.
φ (cid:48) (cid:48)
The morphism φ:Γ →Γ of metric graphs is called harmonic at P ∈Γ if
(cid:48) (cid:48) (cid:48)
for each v ∈T (Γ) the number
φ(P(cid:48)
Σ{dv(cid:48)(φ):v(cid:48) ∈TP(cid:48)(Γ(cid:48)) and dφ(P(cid:48))(v(cid:48))=v}
is independent of v. In that case this sum is denoted by dP(cid:48)(φ) and it is called
the degree of φ at P . We say the morphism φ is harmonic if φ is surjective
(cid:48)
and φ is harmonic at each point P ∈ Γ. In this case for P ∈ Γ one has
(cid:48) (cid:48)
Σ(dP(cid:48)(φ) : φ(P(cid:48)) = 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\
(cid:48)
4
V (Γ). A metric graph obtained from Γ as a composition of finitely many
el∞ementary tropical modifications is called a tropical modification of Γ. Two
metric graphs Γ and Γ are called tropically equivalent if there is a common
1 2
tropical modification Γ of Γ and Γ . This terminology can be found in e.g. [3]
1 2
together with some examples.
3 The example
The metric graph G we start with has genus 2 and can be seen in figure 1.
0
Here v and v are two points of valence 3 (all other points have valence 2) and
1 2
m1
e1
q1 q10
v1 e0 v2
m0
q20 q2
e2
m2
Figure 1: the graph G
0
they are connected by three edges e , e and e of mutually different lenghts.
0 1 2
For 0≤i≤2 the point m is the midpoint of e .
i i
Lemma 1. The graph G has a unique g1 given by |v +v |=|2m |=|2m |=
0 2 1 2 0 1
|2m | and in case v ∈ G such that 2v ∈ |v + v | then v = m for some
2 0 1 2 i
0≤i≤2.
Proof. Clearly2m ∈|v +v |for0≤i≤2andincasev ∈e \{v ,v ,m }then
i 1 2 i 1 2 i
takingv one suchthatthedistanceone fromv tov isequaltothedistance
(cid:48) i i 1
of v to v , then v+v ∈|v +v |. This proves rk(v +v )=1 (it cannot have
(cid:48) 2 (cid:48) 1 2 1 2
rank 2 because g(G ) (cid:54)= 0). It is well-known that a graph of genus at least 2
0
has at most one g1 (see [5]). Indeed, for this graph G , if v (cid:54)=v then v +v is
2 0 2 1
clearly v -reduced, hence |v +v−v | = ∅ and therefore rk(v +v) = 0. This
2 1 2 1
proves the uniqueness of g1 on G .
2 0
Finally for v +v ∈ |v +v | as before (including the possibility v +v =
(cid:48) 1 2 (cid:48)
v +v ),sincev isav-reduceddivisoronehas|v −v|=∅hence|v +v −2v|=∅.
1 2 (cid:48) (cid:48) 1 2
This proves 2v ∈|v +v | implies v =m for some 0≤i≤2.
1 2 i
As indicated in figure 1 we fix q ∈]v ,m [⊂e for i=1,2.
i i i i
5
Lemma 2. There is no g1 on G such that |g1−2m | =(cid:54) ∅ and |g1−2q | =(cid:54) ∅
3 0 3 0 3 i
for i=1,2.
Proof. For i=1,2 we take q ∈e such that the distance on e from v to q is
i(cid:48) i i 1 i
equaltothedistanceone fromv toq . Assumeg1 onG with|g1−2m |=(cid:54) ∅,
i 2 i(cid:48) 3 0 3 0
hence there exists v ∈G such that g1 =|g1+v|.
0 3 2
First assume v ∈ (e ∪e )\{v }. Then q +q +v ∈ g1 and clearly q +v
0 2 1 1 1(cid:48) 3 1(cid:48)
is a q -reduced divisor. This implies |q +v −q | = |g1 −2q | = ∅. In case
1 1(cid:48) 1 3 1
v ∈/ (e ∪e )\{v }thencertainlyv ∈(e ∪e )\{v }andusingsimilararguments
0 2 1 0 1 2
we find |g1−2q |=∅. This finishes the proof of the lemma.
3 2
Now for an integer n ≥ 2 we make a graph G as follows. Fix some more
n
general different points q ,··· ,q on G . Then G is obtained from G by
3 n 0 n 0
attaching a loop γ at m and loops γ at q for each 1 ≤ i ≤ n (we also are
0 0 i i
going to denote m by q ). As an example see a possible picture of G in figure
0 0 6
2. Clearly g(G )=n+3. We prove that the Clifford index of G is at least 2.
n n
m1
e1 γ5
γ1
q5
q1 γ0 q10
v1 q6 e0 v2
m0
γ4 q4 γ6
q20 q3 q2 γ2
e2
γ3 m2
Figure 2: the graph G
6
Proposition 1. Let r be an integer with 1≤r ≤n. Then G has no gr in
n 2r+1
case n≥2.
Proof. First we show G has no linear system g1 in case n ≥ 1 (here G =
n 2 1
G ∪γ ∪γ ). Assume there is a g1 on G . Take 1≤i≤n and let v ∈γ \{q }
0 0 1 2 n i i
and v ∈G such that v+v ∈g1. Let G0 be the closure of G \γ and assume
(cid:48) n (cid:48) 2 n n i
v ∈G0. ItfollowsfromCorollary1in[8]thatrk (v )≥1,becauseg(G0)>0
(cid:48) n G0n (cid:48) n
thisisimpossible. Hencev ∈γ . Onγ thereexistsv”suchthatv+v islinearly
(cid:48) i i (cid:48)
equivalent to q +v” as divisors on γ , hence q +v”∈g1 because of Lemma 1
i i i 2
in[8](thatlemmadoesnotdependontheparticulargraphusedin[8]). Butwe
provedthisimpliesv”=q , hence2q ∈g1. Thisimpliesrk (2q )=1. Indeed,
i i 2 G0 i
takep∈G \{q }andletD bethep-reduceddivisoronG linearlyequivalent
0 i p 0
to 2q . We need to show D −p ≥ 0. The burning algorithm applied to G
i p n
implies D is a p-reduced divisor on G too. Since rk (2q ) = 1 it follows
p n Gn i
6
D −p ≥ 0. So we obtain rk (2q ) = 1, but from Lemma 1 we know this
p G0 i
cannot be true, hence G has no linear system g1.
n 2
Fix some integer r satisfying 1 ≤ r ≤ n−1 and assume G has a linear
n
system gr . For 0 ≤ i ≤ r+1 fix v ∈ γ with v (cid:54)= q . For 0 ≤ i ≤ 2 there
2r+1 i i i i
exists E ∈gr satisfying
i 2r+1
E −(v +v +···v )≥0 .
i i 3 r+1
Forj ∈{3,··· ,r+1}∪{i}letD =E ∩(γ \{q }),henceD −v ≥0. Incase
i,j i j j i,j j
for some j the q -reduced divisor on γ linearly equivalent to D contains a
j j i,j
pointv differentfromq (thenthepointv isunique)thenthereisaneffective
j(cid:48) j j(cid:48)
divisor E on G \γ of degree 2r such that E +v ∈ gr . From Corollary
(cid:48) n j (cid:48) j(cid:48) 2r+1
1 in [8] it follows rk (E )=r, hence G \γ has a linear system gr . Since
Gn\γj (cid:48) n j 2r
g(G \γ )=n+2and2≤2r ≤2n−2thiswouldimplyG \γ isahyperelliptic
n j n j
graph (main result of [7]). But we proved G \γ is not hyperelliptic (it is a
n j
graph G ; the proof of that argument also works on G \γ in case j = 0).
n 1 n 0
−
Since v is not linearly equivalent to q as a divisor on γ it follows D is
j j j i,j
linearly equivalent to m q for some m ≥2 on γ .
i,j j i,j j
So we obtain E ∈gr on G such that
i(cid:48) 2r+1 n
E −(2q +2q +···+2q )≥0
i(cid:48) i 3 r+1
for 0 ≤ i ≤ 2 and E is contained in G . Because of Lemma 2 in [8] those
i(cid:48) 0
divisors E are linearly equivalent as divisors on G . It follows E” = E −
i(cid:48) 0 i i(cid:48)
(2q +···+2q ) with 0 ≤ i ≤ 2 are effective linearly equivalent divisors on
3 r+1
G of degree 3 with E” −2q ≥0. Since g(G )=2 each divisor of degree 3 on
0 i i 0
G defines a g1 and we obtain a contradiction to Lemma 2.
0 3
Finally if G has an gn then because of the Theorem of Riemann-Roch
n 2n+1
|K −gn |=g1, we already excluded this case.
Gn 2n+1 3
Proposition 2. On G we have w1 =1.
n 4
Proof. Fix v ,v ∈ G . We need to prove that there exists a g1 on G such
1 2 n 4 n
that |g1−v −v |=(cid:54) ∅. In case D is an effective divisor of degree 4 on G then
4 1 2 0
from the Riemann-Roch Theorem it follows rk (D)=2. This implies for each
G0
0≤i≤n there is an effective divisor D ≥2q linearly equivalent to D on G .
(cid:48) i 0
Because of Lemma 2 in [8] the divisor D is linearly equivalent to D on G .
(cid:48) n
Moreover for v ∈γ there is an effective divisor on γ linearly equivalent to 2q
i i i
containing v and using the same lemma we obtain the existence of an effective
divisor on G linearly equivalent to D containing v. Similarly, for v ∈ G we
n 0
obtain an effective divisor on G linearly equivalent to D and containing v and
0
againthisdivisorisalsolinearlyequivalenttoDasadivisoronG . Thisproves
n
rk (D)≥1.
Gn
Incasev ,v ∈G wecanuseaneffectivedivisorDonG containingv +v .
1 2 0 0 1 2
Then we have rk (D) ≥ 1 and |D−v −v | (cid:54)= ∅. Next assume v ∈ G and
Gn 1 2 1 0
v ∈γ \{q } for some 0≤i≤n. On G take an effective divisor D of degree 4
2 i i 0
7
containing v +2q . Since 2q is linearly equivalent to v +v for some v ∈γ
1 i i 2 2(cid:48) 2(cid:48) i
as a divisor on γ , again using lemma 1 in [8] we find that D−2q +v +v
i i 2 2(cid:48)
is an effective divisor linearly equivalent to D on G . Hence rk (D) ≥ 1 and
n Gn
|D−v −v |=(cid:54) ∅. Assume v ∈γ \{q } and v ∈γ \{q } for some i (cid:54)=i .
1 2 1 i1 i1 2 i2 i2 1 2
Then one makes a similar argument using the divisor 2q +2q on G . Finally
i1 i2 0
ifv ,v ∈γ \{q }forsomeithenoneusesaneffectivedivisorD ofdegree4on
1 2 i i
G containing 3q . Since 3q is linearly equivalent on γ to an effective divisor
0 i i i
D containing v +v again we find rk (D)≥1 and |D−v −v |=(cid:54) ∅.
1 2 Gn 1 2
4 The lifting problem
TheliftingproblemassociatedtoG wearegoingtoconsidernowisthefollow-
n
ing. 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 analyti-
fication of X (as a Berkovich curve). Let R be the valuation ring of K, let
X be a strongly semistable model of X over R 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 dim(W1(X)) = 1? In that case,
n 4
taking into account the result from [10] mentioned in the introduction, this
would give a geometric explanation for w1(G ) = 1. This lifting problem will
4 n
bethemotivationforconsideringtheexistenceofacertainharmonicmorphism
associated to G . Making this motivation we are going to refer to some suited
n
papers for terminology and some definitions. The definitions necessary to un-
derstand the question on the existence of the harmonic morphism are given in
Section2.3. Finallywearegoingtoprovethattheharmonicmorphismdoesnot
exist, proving that the lifting problem has no solution. In particular we obtain
that the classification of metric graphs satisfying w1 = 1 is different from the
4
classification of smooth curves satisfying dim(W1)=1.
4
Assume the lifting problem has a solution. The curve X of that solution
cannot be hyperelliptic since G is not hyperelliptic. From [13] one obtains the
n
followingclassificationofnon-hyperellipticcurvesX ofgenusatleast6satisfying
dim(W1(X))=1: X istrigonal(hasag1), X isasmoothplanecurveofdegree
4 3
5 (hence has genus 6 and has a g2) or X is bi-elliptic (there exists a double
5
covering π : X → E with g(E) = 1). From Proposition 1 we know G has no
n
g1 and no g2, hence the curve X has to be bi-elliptic.
3 5
So assume there exists a morphism π : X → E with g(E) = 1 of degree 2.
Thisinducesamapπan :Xan →EanbetweentheBerkovichanalytifications. In
caseE isnotaTatecurvetheneachstrongsemistablereductionofE containsa
componentsofgenus1initsspecialfiber,inparticulartheaugmentationmapof
theassociatedskeletonhasauniquepointwithvalue1. Otherwisesuchskeleton
can be considered as a metric graph of genus 1. Each skeleton associated to a
semistablereductionofX istropicallyequivalenttothegraphG ,inparticular
n
it can be considered as a metric graph. From the results in [2] it follows that
there exist skeletons Γ˜ (resp. Γ) of X (resp. E) such that π induces a finite
harmonicmorphismΓ˜ →Γofdegree2. SinceΓ˜ isametricgraph(augmentation
8
map identically zero) this is also the case for Γ hence Γ is a metric graph. So
in case G is liftable to smooth curve X satisfying dim(W1(X))=1 then there
n 4
exist a tropical modification Γ˜ of G and a metric graph Γ of genus 1 such that
n
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 3. Let φ:(Γ ,V )→(Γ ,V ) be a finite harmonic morphism between
1 1 2 2
metric graphs with vertex sets. Let (T ,V ) ⊂ (Γ ,V ) be a subgraph such that
(cid:48) (cid:48) 1 1
T isatree, Γ \T ⊂Γ isconnectedandΓ \T ∩T consistsofauniquepoint
(cid:48) 1 (cid:48) 1 1 (cid:48) (cid:48)
t (in particular t ∈ V ). There is no subtree (T,V) of (T ,V ) different from a
1 (cid:48) (cid:48)
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
(cid:48)
contained in a loop Γ of Γ . Let l(T) (resp. l(Γ)) be the sum of the lenghts of
2
all the edges of T (resp. Γ). By definition one has l(T) ≤ deg(φ)l(Γ). 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 (cid:54)= t and let f be
the edge of T having q as a vertex point. This edge f defines v ∈ T (Γ ),
q 1
let w = d (q)(v), hence φ(q) ∈ Γ and w ∈ T (Γ). Since Γ is a loop there
φ φ(q)
is a unique w ∈ T (Γ) with w (cid:54)= w and since φ is harmonic there exists
(cid:48) φ(q) (cid:48)
v ∈T (Γ ) with d (q)(v )=w . Let f be the edge in Γ having q as a vertex
(cid:48) q 1 φ (cid:48) (cid:48) (cid:48) 1
point and defining v . Since f (cid:54)= f and q (cid:54)= t one has f is an edge of T \T.
(cid:48) (cid:48) (cid:48) (cid:48)
Since T is a tree, also T ∪f is a tree and one has φ(T ∪f ) ⊂ Γ. Moreover
(cid:48) (cid:48) (cid:48) (cid:48)
l(T ∪f )=l(T)+l(f ) and l(f ) has as a fixed lower bound the minimal length
(cid:48) (cid:48) (cid:48)
of an edge contained in Γ .
1
Theorem 1. There does not exist a tropical modification G of G such that
(cid:48)n n
there exists a graph E with g(E)=1 and a finite harmonic morphism φ:G →
(cid:48)n
E of degree 2.
Proof. Assume G is a tropical modification of G and φ : G → E is a finite
(cid:48)n n (cid:48)n
harmonic morphism of degree 2 of metric graphs with g(E)=1.
Step 1: g(φ(G ))=1.
0
Assumeg(φ(G ))=0. Wearegoingtoprovethatφ(G )looksasinfigure3
0 0
(cid:12) (cid:12)
withφ(cid:12)v1miv2 :v1miv2 →[φ(v1),φ(mi)]hasdegree2and(cid:93)((φ(cid:12)v1miv2)−1(q))=2
for all q ∈[φ(v ),φ(m )[ for 0≤i≤2.
1 i
Consider the loop e = v m v m v . Since φ(G ) has genus 0 it follows
1 1 1 2 0 1 0
φ(e ) is a subtree T of φ(G ). Since T is the image of a loop and deg(φ)=2,
1 1 0 1
(cid:12)
it follows that (cid:93)((φ(cid:12)e1)−1(q)) = 1 for some q ∈ φ(e1) if and only if q is a point
of valence 1 of φ(e ). Since deg(φ)=2 it follows d (φ)=2 for such point q and
1 q
in particular d (φ) = 1 for q ∈ e if φ(q) does not have valence 1 on φ(e ). In
q 1 1
caseφ(e )wouldhaveapointq ofvalence3thenthereexistatleast3different
1 (cid:48)
points q on e with φ(q) = q , contradicting deg(φ) = 2. Hence φ(e ) can be
1 (cid:48) 1
9
φ(m1)
φ(m0) φ(v1)=φ(v2)
φ(m2)
Figure 3: In case g(φ(G )=0
0
consideredasafiniteedgewithtwovertices. Alsoforeachq ∈e andv ∈T (e )
1 q 1
one has d (φ)=1.
v
Assumeφ(v )(cid:54)=φ(v ). Thenφ(v m v )isapathfromφ(v )toφ(v )outside
1 2 1 2 2 1 2
of φ(e ). This would imply g(φ(G )) ≥ 1, contradicting g(φ(G )) = 0, hence
1 0 0
φ(v )=φ(v ). Repeatingthepreviousargumentsfortheloope =v m v m v
1 2 2 1 2 2 0 1
(cid:12)
one obtains the given description for φ(cid:12)G0 :G0 →φ(G0).
Consider φ(q )∈φ(v m v ) and q (cid:54)=q on v m v with φ(q )=φ(q ). We
1 1 1 2 1(cid:48) 1 1 1 2 1 1(cid:48)
obtain d (φ) = 1. In case φ(γ ) would be a tree then it would imply φ(q )
q1 1 1
cannot be a point of valence 1 on φ(γ ) hence there exists q ∈ γ \{q } with
1 1(cid:48)(cid:48) 1 1
φ(q ) = φ(q ). Hence (cid:93)(φ 1(φ(q))) ≥ 3, contradicting deg(φ) = 2. It follows
1 1(cid:48)(cid:48) −
g(φ(γ ))=1, hence φ(γ ) contains a loop e in E. Using the same argument as
1 1 (cid:48)1
before we obtain φ(q )∈e . Also e ∩φ(G )={φ(q )}.
1 (cid:48)1 (cid:48)1 0 1
Repeatingtheargumentsusingq andγ weobtainaloope inE suchthat
2 2 (cid:48)2
e ∩φ(G )={φ(q )}, hence e ∩e =∅. Since g(E)=1 this is impossible. As
(cid:48)2 0 2 (cid:48)1 (cid:48)2
a conclusion we obtain g(φ(G ))=1.
0
In case φ(e ) would have genus 0, from the previous arguments it follows
1
that for q ∈ v m v \{v ,v } one has φ(q) ∈/ φ(e ). In case φ(v ) (cid:54)= φ(v ) it
1 2 2 1 2 1 1 2
implies φ(e ) has genus 1 with e = v m v m v . In case φ(v ) = φ(v ) and
2 2 1 0 2 1 1 1 2
g(φ(e ))=0 too, it would imply g(φ(G ))=0. Hence we can assume φ(e ) has
2 0 1
genus 1 (but then φ(e ) could have genus 0).
2
(cid:12)
Step 2: φ(cid:12)e1 : e1 → φ(e1) is an isomorphism (meaning it is finite harmonic of
degree 1)
Since g(φ(e ))=1 it follows there is a loop e in φ(e ), finitely many points
1 1
r ,··· ,r on e and finitely many trees T inside φ(e ) such that
1 t i 1
T ∩e={r }
i i
T ∩T =∅ in case i(cid:54)=j
i j
φ(e )=e∪T ∪···∪T
1 1 t
(of course t = 0, hence φ(e ) is a loop, is also possible; we are going to prove
1
that t=0).
In case val (r ) > 3 for some 1 ≤ i ≤ t then φ 1(r ) contain at least 3
φ(e1) i − i
different points on e , contradicting deg(φ) = 2. So we obtain a situation like
1
in Figure 4. Let r ∈ {r ,··· ,r } and let T be the associated subtree of φ(e ).
1 t 1
Thenφ 1(r)={r ,r }⊂e withr (cid:54)=r . ThetangentspaceT (φ(e ))consists
− (cid:48) (cid:48)(cid:48) 1 (cid:48) (cid:48)(cid:48) r 1
of 3 elements (see Figure 5). Hence there exists w ∈ Tr(cid:48)(G0)\Tr(cid:48)(e1) such
10