The Riemann Roch theorem for metric graphs

Transcription

1 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 R ą0 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 ord P pfq for the sum of the incoming slopes 1. Then the principal divisor associated to f is pfq ÿ P PΓ ord P pfqp, where we note that the sum is finite since ord P pfq 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

2 The degree of a divisor D ř P PΓ n P P is the sum ř P PΓ n P. All principal divisors have degree 0. The set of divisors of degree d is denoted Div d 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`pfq ě 0u{R. However, the set tf D ` pfq ě 0u is not a vector space, since we have instead of ord P pf ` gq ord P pfq ` ord P pgq ord P pf ` gq ě minpord P pfq, ord P 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 " * rpdq max n P Z ˇ D E for all E P Divǹ pγq. 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 D 1 ě 0, then 0 ď rpd 1 q ď rpd ` D 1 q rpdq ď degpd 1 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 K K Γ ÿ P PΓpdegpP q 2qP. 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 H 1 pγ, Rq denotes the (topological) genus of Γ, then the Euler characteristic of Γ is given by 1ÿ χpγq p 1q i dim H i 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 degpkq ÿ pdegpp q 2q 2 E 2 V 2g 2. P PV 2

3 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 " * U Q P Γ ˇ 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 Q 1 with dpq, Q 1 q ď d is in U. Indeed, let Q 1 Q 1, Q 2,..., Q n be the n distinct points with the same distance to Q as Q 1, where n degpqq. d Q 3 Q 2 Q Q 1 Construct a function f which is constant on tr P Γ dpq, Rq ě dpq, Q 1 qu, and which has slope 1 on each of the segments from Q i to Q. f Γ 3

4 Then pfq Q 1 `... ` Q n nq. Hence, Q 1 ` nq. Since there exists m P Z such that Q ` mp, we conclude that Q 1 ` nmp. Hence, Q 1 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 Q 1 P Γ. Pick an element Q P U sufficiently close to Q 1 so that no vertex of degree different from 2 is closer to Q than Q 1 is. Then the argument above shows that Q 1 P U. Definition 4.2. Let P P Γ. For D P Divǹ pγztp uq, write D P 1 `... ` P n, with dpp, P 1 q ď dpp, P 2 q ď... ď dpp, P n q. Then the multi-distance from D to P is dpd, P q pdpp, P 1 q,..., dpp, P n qq. Then define the preorder ĺ on Divǹ pγztp uq as the pullback of the lexicographic order on R n ą0 along the multi-distance map d: Divǹ pγztp uq ÝÑ R n ą0. Finally, extend ĺ to Div`pΓztP uq by setting D ă D 1 if degpdq ă degpd 1 q. Another way to say the same thing is that if we want to compare dpd, P q and dpd 1, 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 D 1 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 D 1 is d ` m ă d n. Therefore, we automatically have Hence, D cannot be P -reduced. D 1ˇˇΓztP u ă DˇˇΓztP u. 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 D 1 mp for D 1 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, D 1 are both P -reduced, and D ` pfq D 1. Let F min be the subset of Γ where f attains its minimum M. Then the boundary points of F min are the poles of f, hence they are contained in D. 4

5 Now suppose P R F min. Then define the function f ε maxpf, M ` εq. R f ε f M ` ε M P F min For ε sufficiently small, the divisor D 1 pf ε q is still effective: the zeroes of f ε are just those of f, hence are contained in D 1. Moreover, the poles of f ε are ε closer to P than those of f. This contradicts P -reducedness of D. Hence, P P F min. By symmetry, we also have P P F max, hence f is constant, so D D 1. 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 K` ÿ 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 K ÿ pdeg pp q 1qP, 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

6 Lemma 5.3. If K` is a moderator, then K`. Proof. Consider K` ` pfq. Let F min be the minimum locus of f, as before. We may assume without loss of generality that the boundary points of F min are in V (this does not change K`). Let P be the ď-maximal vertex in F min, 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 n edges P m edges. F min Because P is ď-maximal in F min, the edges to P lying in F min 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 F min contributes a pole of at least order one to pfq. Hence, pfqpp q ď m. Thus, K` ` pfq 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 P 0 P ; this will be our maximal element for the linear order. Now suppose the largest k points P 0 ě P 1 ě... ě P k 1 are chosen, for 0 ď k ă #V. Set U U k tp 0,..., P k 1 u. Let EpU, V zuq QR ˇˇ Q P U, R P V zu (, and for R P V zu, let EpU, Rq denote the fibre over R. 6

7 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 ` pfq 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 P k R. Then by construction, for all k ą 0 we have K`pP k q #EpU k, P k q 1 ą DpP k q 1, hence pk` DqpP k 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 if D, 1 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` moderator K`. If D is P -reduced, then D K`. for some 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

8 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 A 1 be a bijection, and let f : A Ñ Z, f 1 : A 1 Ñ Z be bounded below functions. Suppose there exists c P Z such that fpaq f 1 pψpaqq c for all a P A. Then ˆ ˆ min fpaq min f 1 pa 1 q c. apa a 1 PA 1 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 D p0q ÿ P PΓ DpP qě0 DpP qp. The polar divisor of D is We write D p8q ÿ DpP qp. P PΓ DpP qď0 deg`pdq degpd p0q q, deg pdq degpd p8q q. Remark 6.3. Note that D D p0q D p8q. Hence, degpdq deg`pdq deg pdq. 8

9 Lemma 6.4. Let D P DivpΓq. Then rpdq min deg`pd 1 K`q 1. D 1 D K` Proof. Denote the right hand side by r 1 pdq. Now suppose r 1 pdq ą rpdq. Then there exists an effective divisor E of degree r 1 pdq such that D E. Then by (RR1) there exists a moderator K` such that K` D ` E. Hence, there exists D 1 D and an effective E 1 such that Then D 1 K` E E 1, so K` D 1 ` E E 1. deg`pd 1 K`q 1 ď degpeq 1 r 1 pdq 1, contradicting the definition of r 1 pdq. Hence, r 1 pdq ď rpdq. Conversely, choose D 1 D and K` attaining the minimum. Then deg`pd 1 K`q r 1 pdq ` 1, so there exist effective divisors E, E 1 with degpeq r 1 pdq ` 1 such that D 1 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, which proves the other inequality. rpdq ď degpeq 1 r 1 pdq, Proof of Riemann Roch. Let D P DivpΓq. For any D 1 D, we get deg`pd 1 K`q deg`ppk D 1 q K q deg`pd 1 K`q deg`pk` D 1 q degpd 1 K`q degpdq ` 1 g. (6.1) We set A tpd 1, K`q D 1 Du and A 1 tpd 2, K`q D 2 K Du, and let ψ : A ÝÑ A 1 pd 1, K`q ÞÝÑ pk D 1, K q. 9

10 Finally, set f : A ÝÑ Z pd 1, K`q ÞÝÑ deg`pd 1 K`q 1, and f 1 : A 1 ÝÑ Z pd 2, K`q ÞÝÑ deg`pd 2 K`q 1. Then (6.1) shows that fpaq f 1 pψpaqq degpdq ` 1 g, for all a P A. By Lemma 6.1, we get ˆ ˆ min fpaq min f 1 pa 1 q degpdq ` 1 g. apa a 1 PA 1 By Lemma 6.4, this translates to which is the Riemann Roch formula. rpdq rpk Dq degpdq ` 1 g, 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 (2007) p [2] F. Cools, J. Draisma, S. Payne, E. Robeva, A tropical proof of the Brill Noether theorem. Adv. Math (2012) p [3] A. Gathmann, M. Kerber, A Riemann Roch theorem in tropical geometry. Math. Z (2008) p [4] G. Mikhalkin, I. Zharkov, Tropical curves, their Jacobians and theta functions. Contemp. Math. 465 (2008) p