The Riemann–Roch theorem for metric graphs
R. van Dobben de Bruyn
1
Preface
These are the notes of a talk I gave at the graduate student algebraic geometry
seminar at Columbia University. I present a short proof of the Riemann–Roch
theorem on metric graphs. All the arguments are taken from [1], [3] and [4].
We follow the notation from [2], since the main goal of the seminar is to present
the tropical proof of the Brill–Noether theorem.
Baker and Norine [1] gave a proof for Riemann–Roch on (abstract) graphs.
Then Gathmann and Kerber [3] used a limit argument to deduce the result for
metric graphs. At the same time, Mikhalkin and Zharkov [4] gave a different
proof using theta functions and the tropical Jacobian.
The proof we present here is a modification of [1] to the metric case. Most of
the arguments are taken from [4].
2
Notation
Throughout the text, Γ will be a metric graph; that is, a finite one-dimensional
CW-complex with a metric on it. We can also describe Γ with combinatorial
data, by giving a graph G “ pV, Eq with a length function E Ñ Rą0 on the
edges. Such a weighted graph will be called a graph representation of Γ. We will
assume that such a graph representation has no loops (if it does, we add a vertex
somewhere in the loop). All (metric) graphs are assumed to be connected.
A divisor on Γ is a (formal) Z-linear combination of points of Γ. The group
of divisors is denoted DivpΓq. A rational function on Γ is a piecewise linear
function Γ Ñ R, where each of the pieces has integer slope. If f is a rational
function, then for P P Γ we write ordP pf q for the sum of the incoming slopes1 .
Then the principal divisor associated to f is
ÿ
pf q “
ordP pf qP,
P PΓ
where we note that the sum is finite since ordP pf q “ 0 for all points on the
interior of a segment on which f is linear. The subgroup of principal divisors is
denoted PrinpΓq, and the quotient is the Picard group
PicpΓq “ DivpΓq{ PrinpΓq.
A divisor D P DivpΓq is effective if DpP q ě 0 for all P P Γ. Here, DpP q denotes
the coefficient in P of D. The space of effective divisors is denoted Div` pΓq.
1 In
[3] and [4], they instead use the sum of the outgoing slopes.
1
ř
ř
The degree of a divisor D “ P PΓ nP P is the sum P PΓ nP . All principal
divisors have degree 0. The set of divisors of degree d is denoted Divd pΓq, and
the set of effective divisors of degree d is denoted Div`
d pΓq.
3
Linear systems and Riemann–Roch
Definition 3.1. Given a divisor D on Γ, the linear system of D is
(
|D| “ E P Div` pΓq | E „ D .
Remark 3.2. This set is in bijection with the set tf | D`pf q ě 0u{R. However,
the set tf | D ` pf q ě 0u is not a vector space, since we have
ordP pf ` gq “ ordP pf q ` ordP pgq
instead of
ordP pf ` gq ě minpordP pf q, ordP pgqq.
One can prove that |D| is a compact CW-complex, but it does not necessarily
have pure dimension. Instead, we make the following definition.
Definition 3.3. Let D P DivpΓq. The rank of D is the number
ˇ
*
"
ˇ
pΓq
.
rpDq “ max n P Z ˇˇ |D ´ E| ‰ ∅ for all E P Div`
n
Remark 3.4. For n ă 0, the condition is vacuous, hence rpDq ě ´1. We have
rpDq ě 0 if and only if |D| ‰ ∅. On the other hand, rpDq ď degpDq, so the
maximum is well-defined. Note also that if D1 ě 0, then
0 ď rpD1 q ď rpD ` D1 q ´ rpDq ď degpD1 q,
as rpDq can jump by at most one when a point is added to D.
Definition 3.5. The canonical divisor on Γ is
ÿ
K “ KΓ “
pdegpP q ´ 2qP.
P PΓ
Remark 3.6. The sum is finite, since almost all points lie on the interior of an
edge, and hence have degree 2. If g “ dim H1 pΓ, Rq denotes the (topological)
genus of Γ, then the Euler characteristic of Γ is given by
χpΓq “
1
ÿ
p´1qi dim Hi pΓ, Rq “ 1 ´ g.
i“0
If G “ pV, Eq is a graph representation of Γ, then Γ has a CW-complex structure
whose number of 0-cells is |V | and whose number of 1-cells is |E|. Hence,
χpΓq “ |V | ´ |E|.
Hence, |V | ´ |E| “ 1 ´ g, and
ÿ
degpKq “
pdegpP q ´ 2q “ 2|E| ´ 2|V | “ 2g ´ 2.
P PV
2
The main result of these notes is the following theorem.
Theorem 3.7 (Riemann–Roch). Let D P DivpΓq. Then
rpDq ´ rpK ´ Dq “ degpDq ` 1 ´ g.
The main theorem will be proved in section 6.
4
P -reduced divisors
Lemma 4.1. Let D P DivpΓq, and let P P Γ. Then there exists m P Z such that
|D ` mP | ‰ ∅.
Proof. It suffices do show this for D “ ´Q, for all Q P Γ. We set
ˇ
*
"
ˇ
U “ Q P Γ ˇˇ there exists m P Z such that | ´ Q ` mP | ‰ ∅ .
We will prove that U is both open and closed, which completes the proof since
P P U and Γ is connected.
Note that if Q P U , and d is the distance from Q to the nearest vertex of degree
different from 2, then any point Q1 with dpQ, Q1 q ď d is in U . Indeed, let
Q1 “ Q1 , Q2 , . . . , Qn be the n distinct points with the same distance to Q as
Q1 , where n “ degpQq.
d
Q3
Q1
Q
Q2
Construct a function f which is constant on tR P Γ | dpQ, Rq ě dpQ, Q1 qu, and
which has slope ´1 on each of the segments from Qi to Q.
f
Γ
3
Then pf q “ Q1 ` . . . ` Qn ´ nQ. Hence, | ´ Q1 ` nQ| ‰ ∅. Since there exists
m P Z such that | ´ Q ` mP | ‰ ∅, we conclude that | ´ Q1 ` nmP | ‰ ∅. Hence,
Q1 P U , as claimed.
This already shows that U is open: if Q P U , then any point sufficiently close
to Q is also in U . On the other hand, suppose a sequence of elements in U
converges to some Q1 P Γ. Pick an element Q P U sufficiently close to Q1 so
that no vertex of degree different from 2 is closer to Q than Q1 is. Then the
argument above shows that Q1 P U .
Definition 4.2. Let P P Γ. For D P Div`
n pΓztP uq, write D “ P1 ` . . . ` Pn ,
with
dpP, P1 q ď dpP, P2 q ď . . . ď dpP, Pn q.
Then the multi-distance from D to P is
dpD, P q “ pdpP, P1 q, . . . , dpP, Pn qq.
Then define the preorder ĺ on Div`
n pΓztP uq as the pullback of the lexicographic
order on Rną0 along the multi-distance map
n
d : Div`
n pΓztP uq ÝÑ Rą0 .
Finally, extend ĺ to Div` pΓztP uq by setting D ă D1 if degpDq ă degpD1 q.
Another way to say the same thing is that if we want to compare dpD, P q and
dpD1 , P q, we append zeroes from the front to make them the same length.
Definition 4.3. A divisor D P DivpΓq is P -reduced if its restriction to ΓztP u
is effective, and D is ĺ-minimal among such within its equivalence class.
Remark 4.4. If D is P -reduced, and DpP q “ n, then ´n is the smallest integer
m such that |D ` mP | ‰ ∅. Indeed, D ´ nP is effective since its coefficient at
P is 0 and its restriction to ΓztP u is effective.
Conversely, assume D ` mP „ D1 ě 0 for some m ă ´n, and let d be the
degree of D. Then the degree of D|ΓztP u is d ´ n, whereas the degree of D1 is
d ` m ă d ´ n. Therefore, we automatically have
ˇ
ˇ
D1 ˇΓztP u ă DˇΓztP u .
Hence, D cannot be P -reduced.
Proposition 4.5. For any class rDs P PicpΓq, there exists a unique P -reduced
representative.
Proof. Take the minimal m such that |D ` mP | ‰ ∅. By Remark 4.4, we only
need to consider divisors of the form D1 ´ mP for D1 „ D ` mP effective.
The space |D ` mP | is compact, and we are minimising a bunch of continuous
functions (namely, the various distances to P ). This proves existence.
Now suppose D, D1 are both P -reduced, and D ` pf q “ D1 . Let Fmin be the
subset of Γ where f attains its minimum M . Then the boundary points of Fmin
are the poles of f , hence they are contained in D.
4
Now suppose P R Fmin . Then define the function fε “ maxpf, M ` εq.
R
fε
f
M `ε
M
Fmin
P
For ε sufficiently small, the divisor D1 ´ pfε q is still effective: the zeroes of fε
are just those of f , hence are contained in D1 . Moreover, the poles of fε are ε
closer to P than those of f . This contradicts P -reducedness of D. Hence,
P P Fmin .
By symmetry, we also have P P Fmax , hence f is constant, so D “ D1 .
Remark 4.6. If |D| ‰ ∅, then the P -reduced form is effective. This follows
from the construction in the existence part of the proof.
5
Moderators and the Riemann–Roch axioms
It turns out that we can completely classify the degree g ´ 1 divisors that are
not equivalent to an effective divisor. This leads to the verification of the two
Riemann–Roch axioms (RR1) and (RR2) from Baker–Norine [1].
Definition 5.1. Let ď be a linear order on a graph representation G of Γ. Then
define the moderator associated to ď as
ÿ
K` “
pdeg` pP q ´ 1qP,
P PV
where deg` stands for the number of outgoing edges (i.e. the edges P Q with
Q ą P ). Similarly, define
ÿ
pdeg´ pP q ´ 1qP,
K´ “
P PV
where deg´ “ deg ´ deg` is the number of incoming edges.
Remark 5.2. Note that K ´ is the moderator for the reversed linear order on
G. Moreover, note that
K “ K ` ` K ´,
and degpK ` q “ g ´ 1.
5
Lemma 5.3. If K ` is a moderator, then |K ` | “ ∅.
Proof. Consider K ` ` pf q. Let Fmin be the minimum locus of f , as before. We
may assume without loss of generality that the boundary points of Fmin are in
V (this does not change K ` ).
Let P be the ď-maximal vertex in Fmin , and suppose there are n edges from P
on which f is locally constant near P , and m edges on which f is increasing:
f
P
n edges
m edges.
Fmin
Because P is ď-maximal in Fmin , the edges to P lying in Fmin are all incoming
edges, and thus do not contribute to K ` . Hence,
K ` pP q ď m ´ 1.
On the other hand, each of the edges leaving Fmin contributes a pole of at least
order one to pf q. Hence,
pf qpP q ď ´m.
Thus, K ` ` pf q has a negative coefficient at P .
Lemma 5.4. Let D P DivpΓq be given. Then either |D| ‰ ∅, xor there exists a
moderator K ` such that |K ` ´ D| ‰ ∅.
Proof. They clearly cannot both hold, for this would imply |K ` | ‰ ∅.
Without loss of generality assume D is P -reduced. Consider a graph representation whose vertex set contains the support of D, as well as the point P . We
will inductively construct a linear order on V as follows.
Let P0 “ P ; this will be our maximal element for the linear order. Now suppose
the largest k points
P0 ě P1 ě . . . ě Pk´1
are chosen, for 0 ď k ă #V . Set U “ Uk “ tP0 , . . . , Pk´1 u. Let
ˇ
(
EpU, V zU q “ QR ˇ Q P U, R P V zU ,
and for R P V zU , let EpU, Rq denote the fibre over R.
6
Now suppose DpRq ě #EpU, Rq for all R P V zU . Then define the function f
to be constant on all edges between two points in U or two points in V zU . On
the edges from a point Q P U to a point R P V zU , we let f have slope 1 in an
ε-neighbourhood of R, and constant otherwise.
f
P
U
V zU
Then f has poles at all R P V zU , with multiplicity #EpU, Rq. Hence, D ` pf q
is effective. But its points are closer to P , contradicting P -reducedness. Hence,
there exists a point R P V zU with DpRq ă #EpU, Rq. Let Pk “ R.
Then by construction, for all k ą 0 we have
K ` pPk q “ #EpUk , Pk q ´ 1 ą DpPk q ´ 1,
hence pK ` ´ DqpPk q ě 0. Hence, K ` ´ D is effective on ΓztP u.
Now either DpP q ě 0 and D is effective, or DpP q ă 0 and pK ` ´ DqpP q ě 0,
since K ` pP q “ ´1.
Definition 5.5. Write
"
εpDq “
0
1
if |D| “ ∅,
if |D| ‰ ∅.
Corollary 5.6 (RR1). Let D P DivpΓq. Then there exists a moderator K ` such
that
εpDq ` εpK ` ´ Dq “ 1.
Proof. If εpDq “ 0, then there exists a moderator K ` such that εpK ` ´ Dq “ 1.
If εpDq “ 1, then for any moderator K ` , we have εpK ` ´ Dq “ 0. Pick any.
Corollary 5.7. If deg D “ g ´ 1, and |D| “ ∅, then D „ K ` for some
moderator K ` . If D is P -reduced, then D “ K ` .
Proof. Since |D| “ ∅, there exists a moderator K ` such that |K ` ´ D| ‰ ∅.
But K ` ´ D has degree zero, hence
K ` „ D.
If D is P -reduced, then the proof of Lemma 5.4 shows that K ` ´ D is effective,
hence equal to 0.
7
Corollary 5.8. If degpDq “ g ´ 1, then |D| “ ∅ if and only if D is equivalent
to a moderator.
Proof. Immediate from Lemma 5.3 and Corollary 5.7.
Corollary 5.9 (RR2). If degpDq “ g ´ 1, then εpDq “ εpK ´ Dq.
Proof. Follows from the previous corollary, since D is (linearly equivalent to) a
moderator if and only if K ´ D is.
6
Proof of the Riemann–Roch theorem
Baker and Norine [1] proved that the Riemann–Roch theorem is equivalent to
the Riemann–Roch axioms (RR1) and (RR2) above. We will only prove the
implication that we need. We follow the treatment of [1].
„
Lemma 6.1. Let ψ : A ÝÑ A1 be a bijection, and let f : A Ñ Z, f 1 : A1 Ñ Z be
bounded below functions. Suppose there exists c P Z such that
f paq ´ f 1 pψpaqq “ c
for all a P A. Then
˙
ˆ
˙ ˆ
1 1
f pa q “ c.
min f paq ´ min
1
1
aPA
a PA
Proof. This is straightforward, and left as an exercise to the reader.
Definition 6.2. Let D be a divisor. Then the zero divisor of D is
ÿ
Dp0q “
DpP qP.
P PΓ
DpP qě0
The polar divisor of D is
Dp8q “
ÿ
´DpP qP.
P PΓ
DpP qď0
We write
deg` pDq “ degpDp0q q,
deg´ pDq “ degpDp8q q.
Remark 6.3. Note that D “ Dp0q ´ Dp8q . Hence,
degpDq “ deg` pDq ´ deg´ pDq.
8
Lemma 6.4. Let D P DivpΓq. Then
¨
˛
rpDq “ ˝ min
deg` pD1 ´ K ` q‚´ 1.
1
D „D
K`
Proof. Denote the right hand side by r1 pDq. Now suppose r1 pDq ą rpDq. Then
there exists an effective divisor E of degree r1 pDq such that |D ´ E| “ ∅. Then
by (RR1) there exists a moderator K ` such that
|K ` ´ D ` E| ‰ ∅.
Hence, there exists D1 „ D and an effective E 1 such that
K ` ´ D1 ` E “ E 1 .
Then D1 ´ K ` “ E ´ E 1 , so
deg` pD1 ´ K ` q ´ 1 ď degpEq ´ 1 “ r1 pDq ´ 1,
contradicting the definition of r1 pDq. Hence,
r1 pDq ď rpDq.
Conversely, choose D1 „ D and K ` attaining the minimum. Then
deg` pD1 ´ K ` q “ r1 pDq ` 1,
so there exist effective divisors E, E 1 with degpEq “ r1 pDq ` 1 such that
D1 ´ K ` “ E ´ E 1 .
Then D ´ K ` „ E ´ E 1 . By (RR1), we have εpK ` ´ E 1 q “ 0, as E 1 is effective.
Hence,
|D ´ E| “ |K ` ´ E 1 | “ ∅.
Hence,
rpDq ď degpEq ´ 1 “ r1 pDq,
which proves the other inequality.
Proof of Riemann–Roch. Let D P DivpΓq. For any D1 „ D, we get
deg` pD1 ´ K ` q ´ deg` ppK ´ D1 q ´ K ´ q “ deg` pD1 ´ K ` q ´ deg` pK ` ´ D1 q
“ degpD1 ´ K ` q
“ degpDq ` 1 ´ g.
(6.1)
We set A “ tpD1 , K ` q | D1 „ Du and A1 “ tpD2 , K ` q | D2 „ K ´ Du, and let
„
ψ : A ÝÑ A1
pD1 , K ` q ÞÝÑ pK ´ D1 , K ´ q.
9
Finally, set
f : A ÝÑ Z
pD , K ` q ÞÝÑ deg` pD1 ´ K ` q ´ 1,
1
and
f 1 : A1 ÝÑ Z
pD2 , K ` q ÞÝÑ deg` pD2 ´ K ` q ´ 1.
Then (6.1) shows that
f paq ´ f 1 pψpaqq “ degpDq ` 1 ´ g,
for all a P A. By Lemma 6.1, we get
ˆ
˙ ˆ
˙
1 1
min f paq ´ min
f pa q “ degpDq ` 1 ´ g.
1
1
aPA
a PA
By Lemma 6.4, this translates to
rpDq ´ rpK ´ Dq “ degpDq ` 1 ´ g,
which is the Riemann–Roch formula.
Remark 6.5. We of course also get the usual corollaries, e.g. Clifford’s theorem,
etc.
References
[1] M. Baker, S. Norine, Riemann–Roch and Abel–Jacobi theory on a finite
graph. Adv. Math. 215–2 (2007) p. 766–788.
[2] F. Cools, J. Draisma, S. Payne, E. Robeva, A tropical proof of the Brill–
Noether theorem. Adv. Math. 230–2 (2012) p. 759–776
[3] A. Gathmann, M. Kerber, A Riemann–Roch theorem in tropical geometry.
Math. Z. 259–1 (2008) p. 217–230.
[4] G. Mikhalkin, I. Zharkov, Tropical curves, their Jacobians and theta functions. Contemp. Math. 465 (2008) p. 203–231.
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