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write Div(Γ) for the additive group of all divisors. A general elementof Div(Γ) is written as

D = a1v1 + · · · + arvr,

where ai ∈ Z and vi ∈ Γ. We define the degree of D by

deg(D) = a1 + · · · + ar.

A rational function on Γ is a piecewise linear function on Γ withintegral slopes. If f is a rational function of Γ and v ∈ Γ, define ordv(f )by the sum of incoming slopes of f at v. Then define div(f ) ∈ Div(Γ)by

div(f ) =v∈Γ

ordv(f ) · v

and call it the divisor of f . The divisors of rational functions on Γcompose a subgroup Prin(Γ) of Div(Γ), the subgroup of principal di-visors.

The Picard group of Γ is defined by

P ic(Γ) = Div(Γ)/Prin(Γ).

One sees every principal divisor has degree 0, so there is a well-definedmap

deg : P ic(Γ) → Z.

An element of P ic(Γ) is called a divisor class. Denote by P icd(Γ) thesubset deg−1(d) of P ic(Γ). In particular, P ic0(Γ) is a subgroup of P ic(Γ).

2.2. Linear system of divisors. A divisor D = a1v1 + · · · + arvr ∈Div(Γ) is effective if each coefficient ai is nonnegative. A divisor D

islinearly equivalent to D if D − D ∈ P rin(Γ).

The rank r(D) of an effective divisor D is the largest integer r suchthat, for every effective divisor E of degree r, D −E is linearly equiva-lent to an effective divisor. If D is not linearly equivalent to an effectivedivisor, then set r(D) = −1.

For nonnegetive integers r and d, the Brill-Noether locus

W rd (Γ) ⊂ P icd(Γ)

is the set of divisor classes of degree d and rank at least r.In classical theory of divisors on Riemann surfaces, for an Riemann

surface of genus g, the Brill-Noether locus W rd (C ) is similarly defined.Let

ρ(g,r,d) = g − (r + 1)(g − d + r)

be the Brill-Noether number. Then it is known that for a general curve,if ρ(g,r,d) is nonnegetive, the dimension of W rd (C ) is equal to ρ(g,r,d),

and if ρ(g,r,d) is negative, then W rd (C ) is an empty set.2


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For graphs however, there are open sets in the moduli space of metricgraphs of genus g, where dim W rd (Γ) is strictly larger than ρ(g,r,d) (see

[3]). One of the conjectures of Caporaso is the following.

Conjecture 2.1. Assume g ≥ 2 and ρ(g,r,d) < 0. Then there exists a 3-regular graph Γ of genus g for which W rd (Γ) = ∅.

Theorem 2.2. Caporaso’s conjecture holds for the following graphs.

l0 l1

l3

l4

l5

l6

l7 l3g-5 l3g-2 l3g-1

l3g-4

l3g-3

l2

Figure 1. The graph Γg

Here the edge lengths of the edges li, 1 ≤ i ≤ 3g − 3 are given by

length(li) = εi,

with ε small positive constant.

Remark 2.3. The edge length need not strictly take these values. If we perturb the lengths of edges slightly (compared to its original length, for example, we can change the length of the edge li to ε

i + O(εi+1)), then the resulting graph still satisfies the conclusion of Theorem 2.2. Thus,we actually have the open subset of the moduli space of the metric graphs where the conclusion of Theorem 2.2 holds.

Remark 2.4. The same line of argument proves that in fact the full Brill-Noether theorem is true for these graphs. That is, when the Brill-Noether number ρ is nonnegative, then the corresponding Brill-Noether locus has dimension ρ. In this note, we concentrate on the case ρ


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An important tool for the proof is the chip-firing deformation whichwe introduce below. It gives a convenient way to construct linearly

equivalent divisors of a given divisor.Recall the chip-firing move on a graph [1, 2]. Let Γ be a graph and

D a divisor on it. Let v be a k-valent vertex of Γ and v1, · · · , vk be theneighboring vertices of v. Then applying a chip-firing move at v to Dgives the new divisor D

D = D + kv − v1 − v2 − · · · − vk (’borrowing’ move), or

D = D − kv + v1 + v2 + · · · + vk (’giving’ move).

Any divisor linearly equivalent to D can be obtained as a result of asequence of chip-firing moves.

In the case of a metric graph, a straightforward generalization of chip-firing move can be defined, which we call chip-firing deformation .To define it, we prepare some notation. Let Γ be a compact connectedmetric graph. Let v be a k-valent point of Γ (a point on the interior of an edge is a divalent point. The valency at the vertices of Γ is definedas usual). Consider the open subset Γ \ {v} of Γ. This is a graph withk open ends. Add one valent vertices to each of these ends. This givesa closed graph Γ which is not necessarily connected.

Definition 3.1. We call the graph Γ the graph obtained from Γ bycutting Γ at v.

Let D be a divisor on Γ whose summand at the point v is lv withl > 0.

Definition 3.2. A divisor D on Γ is obtained from D by cutting the pair (Γ, D) at v if

D = D − lv +k

i=1

civi

with ci ≥ 0 and∑k

i=1 ci = l.

Note that D − lv does not have a summand at the point v, so wecan think of it as a divisor on the graph Γ. Thus, the expression of D

in Definition 3.2 makes sense.Now we define the chip-firing deformation. For simplicity, we only

define it for the case of positive ends. Let Γ be a metric graph and Dan effective divisor on Γ. Let

v1, · · · , va

be points of Γ and Γ be the graph obtained from Γ be cutting it ateach of the points v1, · · · , va. Let

Γ1, · · · , Γ

b

be the connected components of Γ. Assume that by cutting the pair

(Γ, D) suitably at each of the points v1, · · · , va, we obtain a component4


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If the statement of the lemma is not true, there is an effective divisorD linearly equivalent to D such that

f i(D) ≡ 0 mod εi+1.

Consider a chip-firing deformation which changes the value of f i.Obviously the relevant graph Γ1 must contain a part of the edge li.Then there are two cases:

(1) There are two vertices of Γ 1 contained in li.(2) There is only one vertex of Γ

1 contained in li.

In the case (1), the graph Γ1

is a subsegment of li, or contains thecomplement of a subsegment of li. In each case, it is clear that thecorresponding chip-firing deformation does not change f i. In the case

(2), the graph Γ

1 contains a part of some edge l j with j > i. Then,the corresponding chip-firing deformation can change the value of f i atmost O(εi+1). Since the degree of the divisor D is bounded by 3g, itfollows that we can change the value of f i at most O(ε

i+1), proving thelemma.

We name the vertices of the graph Γg as follows:

l0 l1

l3

l4

l5

l6

l7 l3g-8 l3g-5 l3g-4

l3g-7

l3g-3

l2v1

v2

v3

v4

v5

v6

v2g-5

v2g-4

v2g-3

v2g-2

Figure 2

Consider the following open graph γ k+1 of the graph Γg:

l0 l1

l3

l4

l5

l6

l7 l3k-2

l3k-1

l3k

l2v1

v2

v3

v4

v5

v6

v2k-1

v2k

v2k+1

v2k+2

Figure 3

It has k + 1 vertical edges l0, l1, l4, · · · , l3k−2. Let a0, a1, a4, · · · , a3k−2be the middle points of these edges. Let S k+1 be the set of orderedsequences of k + 1 nonnegetive integers (z 0, z 1, z 4, · · · , z 3k−2) satisfying

z 0 + z 1 + z 4 + · · · + z 3k−2 = k + 1.7


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proves Theorem 2.2 for the case p = 1 (ρ = −1). The other cases withρ = −1 can be proved by similar argument.

References

[1] Biggs, N., Chip-firing and the critical group of a graph. J. Algebraic Combi-natorics., 9 (1999), 25-45.

[2] Baker, M. and Norine, S., Riemann-Roch and Abel-Jacobi Theory on a Finite Graph., Advances in Mathematics 215 (2007), 766–788.

[3] Caporaso,L., Algebraic and combinatorial Brill-Noether theory.ArXiv:1106.1140.

[4] Cools, F., Draisma, J, Payne, S. and Robeva, E. A tropical proof of the Brill-Noether Theorem. Advances in Mathematics. 230 (2012), 759-776.

Department of Mathematics, Rikkyo University, Tokyo, Japan

E-mail address : [email protected]

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