rank of divisors on curves and graphs under specialization: the hyperelliptic … · 2014-04-14 ·...

Rank of divisors on curves and graphs under

specialization: the hyperelliptic case and the genus 3

case

(joint work with Kazuhiko Yamaki)

Shu Kawaguchi

Kyoto University

Specialization of Linear Series for Algebraic and Tropical CurvesApril 3, 2014, Banff International Research Station

Shu Kawaguchi (Kyoto) Rank of divisors under specialization April 3, 2014 1 / 34

Plan of the talk

We study, in some sense, an equality condition of the specializationlemma.

.

.

1 Specialization lemma (Baker, Amini-Caporaso)

.

.

2 A question

.

.

3 Hyperelliptic case and genus 3 case (with an outline of proof)

.

.

4 Relation to the algebraic rank (introduced by Caporaso)

References

• Ranks of divisor classes on hyperelliptic curves and graphs underspecialization, preprint arXiv:1304.6979, to appear in IMRN.

• Algebraic rank on hyperelliptic graphs and graphs of genus 3,preprint arXiv:1401.3935.


Specialization lemma (Baker, Amini-Caporaso)

Part 1 Specialization lemma (Baker,

Amini-Caporaso)

Setup

R : a complete discrete valuation ring (cDVR)

K : fractional field of R, K: algebraic closure of K

k : residue field of R, which we assume is algebraically closed

X : regular, generically smooth, semi-stable R-curve

(projective and flat over R, Xk is a reduced nodal curve)

X : generic fiber of X , i.e., X = XK

Xk : special fiber of X



Spec(R)

X

tSpec(k)

Xk

vSpec(K)

X := XK

↓



Reduction graph of X

Xk: the special fiber of X ,a reduced curve with only nodes as singularities

G = (G, w): the (vertex-weighted) dual graph of Xk

irreducible component Cv of Xk ←→ vertex v of G

point in Cv ∩ Cv′ ←→ edge of G connecting v and v′

node in Cv ←→ loop edge at v

V (G): the set of vertices of G

E(G): the set of edges of G

w : V (G)→ Z≥0, w(v) := “geometric genus of Cv”

Γ = (Γ, w): the metric graph associated to G,where each edge of G is assigned length 1

ΓQ := {x ∈ Γ | dist(x, v) ∈ Q for ∀v ∈ V (G)}



Xk

Cv1∼= P1 Cv2

∼= P1

Cv3 elliptic curve

t t

t

v1 v2

v3

0 0

1

G = (G,w)

Specialization mapP ∈ X(K): K-valued point in the generic fiber X

∆P : Zariski closure of P in X , which meets exactly one irreduciblecomponent of the special fiber Xk (by the valuative criterion ofproperness)



Spec(R)

X

tSpec(k)

Xk

vSpec(K)

X := XK

↓

vP∆P

Cv



The assignment

τ : X(K)→ V (G), P 7→ v

gives the specialization map, which extends to

τ∗ : Div(XK)→ Div(ΓQ) :=⊕

x∈ΓQZ[x].

Theorem (Specialization Lemma, Baker, Amini-Caporaso)

For any D ∈ Div(XK), put D := τ∗(D) ∈ Div(ΓQ). Then

rΓ(D) ≥ rX(D).

(Here, let Γ• be the metric graph obtained by adding w(v) loops (of any

length) at v to Γ. Then rΓ(D) = rΓ•(D). )


A question

Part 2 A question

R: a complete discrete valuation ring (cDVR)with Frac(R) = K and algebraically closed residue field k

G = (G, w): a vertex-weighted graphΓ = (Γ, w): associated metric graph of G

QuestionDoes there exist a regular, generically smooth, semi-stable R-curve X

with reduction graph G satisfying the following condition (C)?∀D ∈ Div(ΓQ), ∃D ∈ Div(XK) with D = τ∗(D) s.t.

rΓ(D) = rX(D).


A question

R: a cDVR, G = (G,w): a vertex-weighted graph as before.

Observation

• If such X exists, then the Riemann–Roch formula for G isdeduced from the Riemann–Roch formula for X(:= XK).(This is our original motivation to consider the question.)

• If such X exists, then

τ∗ (W rd (XK)) = W r

d (ΓQ)

where W rd (XK) := {D ∈ Pic(XK) | deg(D) = d, rX(D) ≥ r}, and

W rd (ΓQ) = {D ∈ Pic(ΓQ) | deg(D) = d, rΓ(D) ≥ r}.

• The question is related to the algebraic rank (cf. Len’s talk).


A question

Observation (continued)

• From Jensen’s talk on Monday (if I understand correctly):Γ: a generic rational metric graph of g-loopsX : any regular, generically smooth, strongly semi-stable R-curve

with reduction graph ΓThen Cartwright–Jensen–Payne show∀D ∈ Div(ΓQ), ∃D ∈ Div(XK) with D = τ∗(D) s.t.

rΓ(D) = rX(D).

• From Cartwright’s talk on Monday (if I understand correctly):Given Γ and an effective divisor D ∈ Div(ΓQ), Cartwright studiesif there exist X and an effective divisor D ∈ Div(XK) withD = τ∗(D) and rΓ(D) = rX(D).


Hyperelliptic case and genus 3 case

Part 3 Hyperelliptic case and genus 3

case

G = (G, w): a vertex-weighted graphΓ = (Γ, w): associated metric graph of G

Definition (genus)

g(G) = g(Γ) := |E(G)| − |V (G)|+ 1g(G) = g(Γ) := g(G) +

∑v∈V (G) w(v)

Definition (hyperelliptic vertex-weighted graph)

G is hyperelliptic ⇐⇒∃D ∈ Div(G) s.t. deg(D) = 2 and rΓ(D) = 1.



Definition (positive-type bridge)

G = (G, w): a vertex-weighted graphAn edge e of G is a bridge ⇐⇒ G \ {e} = ∃G1 q ∃G2

A bridge e is of positive type ⇐⇒ g(G1) ≥ 1 and g(G2) ≥ 1,where Gi = (Gi, w|Gi).

Example

t t

tt

v1 v2

v3v4

0 0

10

← a positive-type bridgenot a positive-typebridge →



Main results

Theorem (hyperelliptic case)

R: cDVR with algebraically closed residue field k

G = (G, w): a hyperelliptic vertex-weighted graphAssume that ch(k) 6= 2. Then the following are equivalent.

(i) There exists a regular, generically smooth, semi-stable R-curve X

with reduction graph G satisfying the condition (C)(i.e. the answer to the question is YES for G).

(ii) For every vertex v of G, there are at most (2w(v) + 2)positive-type bridges emanating from v.

Example ��

��qq YES��

��qq qqQQ��

NO



Theorem (genus 3 case)

R: cDVR with algebraically closed residue field k

G = (G, w): a vertex-weighted non-hyperelliptic graph of genus 3X : any regular, generically smooth, semi-stable R-curve X

with reduction graph G

Then X satisfies the condition (C)(in particular, the answer to the question is YES for G).

Remark

For any vertex-weighted graph G of genus 0 or 1, the answer to thequestion is YES for G. (Note that genus 2 graphs are hyperelliptic.)



Outline of proof of TheoremWe concentrate on the hyperelliptic case.

.

.1 Construction of X (Caporaso, Amini–Baker–Brugalle–Rabinoff,

K-Yamaki)

.

.

2 Computation of rank of reduced divisors on a hyperelliptic graph

.

.

3 Lifting of divisors

Remark

• In the above proof, we use Baker’s specialization lemma. Strictlyspeaking, we don’t use Baker–Norine’s Riemann–Roch formula,but we do use very much the notion of v-reduced divisors.

• The proof of Theorem (genus 3 case), we use Amini–Caporaso’sspecialization lemma and Baker–Norine’s Riemann–Roch formula.



Step 0: Some properties of hyperelliptic graphsHyperelliptic graphs have been studied by Baker–Norine, Chan,Caporaso ...

• G = (G,w): a vertex-weighted graph with g(G) ≥ 2Assume that any vertex v ∈ V (G) of valence 1 satisfies w(v) > 0.Γ: associated metric graph of G

Γ•: metric graph obtained by adding w(v) loops at ∀v to Γ.Then the following are equivalent:(i) G is hyperelliptic;(ii) there exists a unique involution ι on Γ• such that Γ•/ι is a tree.

• G = (G,w): a hyperelliptic vertex-weighted graphΓ = (Γ, w): associated metric graph of G

Γ′: the metric graph that is obtained by contracting all edgeswith endpoint v ∈ V (G) of valence 1 and w(v) = 0. We sayv0 ∈ Γ′

Q ⊆ ΓQ is a fixed point if ι(v0) = v0. (Such v0 always exists.)Shu Kawaguchi (Kyoto) Rank of divisors under specialization April 3, 2014 17 / 34


Step 1: Construction of X

Theorem (Caporaso, Amini–Baker–Brugalle–Rabinoff, K-Yamaki)

R: cDVR with fractional field K and residue field k, ch(k) 6= 2G = (G, w): hyperelliptic vertex-weighted graphAssume that any vertex v ∈ V (G) of valence 1 satisfies w(v) > 0.Then the following are equivalent.

(i) For every vertex v of G, there are at most (2w(v) + 2)positive-type bridges emanating from v.

(ii) There exists a hyperelliptic nodal curve X0 over k with dualgraph G.

(iii) There exists a regular, generically smooth, semi-stable hyperellipticR-curve X with reduction graph G.



Remark

• The equivalence between (i) (graph with 2w(v) + 2 condition) and(ii) (existence of hyperelliptic nodal curve X0) is due to Caporaso.

• A version of Theorem (where K is not a discrete valuation field,but an algebraically closed valuation field) is shown byAmini–Baker–Brugalle–Rabinoff.

• The equivalence between (ii) (existence of hyperelliptic nodalcurve X0) and (iii) (existence of hyperelliptic R-curve) is shown bya deformation argument.



Corollary

R: cDVR with fractional field K and residue field k, ch(k) 6= 2G = (G, w): hyperelliptic vertex-weighted graphThen the following are equivalent.

(i) For every vertex v of G, there are at most (2w(v) + 2)positive-type bridges emanating from v.

(ii) There exists a regular, generically smooth, semi-stable R-curve X

with reduction graph G such that the generic fiber X ishyperelliptic.

We are going to show that this X has the desired property.



Step 2: Computation of rank of reduced divisors on ahyperelliptic graph

G = (G, w): a hyperelliptic vertex-weighted graphΓ = (Γ, w): associated metric graph of G

v0 ∈ ΓQ: a fixed point

Theorem (rank on hyperelliptic vertex-weighted graph)

For any v0-reduced effective divisor E ∈ Div(Γ),we put r = bE(v0)

2 c, where E(v0) is the coefficient of E at v0. Then

rΓ(E) =

r (if deg(E)− r ≤ g(Γ))

deg(E)− g(Γ) (if deg(E)− r ≥ g(Γ) + 1)



There is a corresponding formula on rank on a hyperelliptic curve.

Proposition (rank on hyperelliptic curve)

X: a hyperelliptic curve over K (smooth, connected)ιX : hyperelliptic involution on X

For any effective divisor E ∈ Div(XK), we express E as

E = P1 + · · ·+ Pr + ιX(P1) + · · ·+ ιX(Pr) + Q1 + · · ·+ Qs,

where P1, . . . , Pr, Q1, . . . , Qs ∈ X(K) and ιX(Qi) 6= Qj for any i 6= j

with 1 ≤ i, j ≤ s. Then

rX(E) =

r (if deg(E)− r ≤ g(X)),

deg(E)− g(X) (if deg(E)− r ≥ g(X) + 1).



Comparison of rank on a hyperelliptic vertex-weighted graph and rankon hyperelliptic curve, and Baker’s specialization lemma imply thefollowing:

G = (G, w): a hyperelliptic vertex-weighted graphX : regular, generically smooth, semi-stable R-curve with

reduction graph G with hyperelliptic generic fiber X

If E ∈ Div(ΓQ) is a v0-reduced effective divisor, then there existsE ∈ Div(XK) with τ∗(E) = E s.t.

rΓ(E) = rX(E).



Step 3: Lifting of divisors

G = (G, w): a hyperelliptic vertex-weighted graph with metric graphΓX : a regular, generically smooth, semi-stable R-curve with

reduction graph G with hyperelliptic generic fiber X

We want to show:∀D ∈ Div(ΓQ), ∃D ∈ Div(XK) with D = τ∗(D) s.t. rΓ(D) = rX(D).

Have shown: OK if D is v0-reduced.

In general, for D ∈ Div(ΓQ), let E ∈ Div(ΓQ) be the v0-reduced divisorsuch that N := D − E ∈ Prin(ΓQ). Let E ∈ Div(XK) be a desired lift.Since τ∗ : Prin(XK)→ Prin(ΓQ) is surjective, there existsN ∈ Prin(XK) such that τ∗(N) = N . We put D := E + N ∈ Div(XK).Then D is a desired lift. 2


Relation to the algebraic rank

Part 4 Relation to the algebraic rank

k: an algebraically closed fieldG = (G, w): a vertex-weighted graph with associated metric graph ΓD ∈ Div(G)

As in Len’s talk,

Definition (algebraic rank, Caporaso)

ralg,kG

(D) := maxX0

{min

E

{max

E0

rX0(E0)}}

,

where X0 runs through all connected reduced nodal curves over k withdual graph G, E ∈ Div(G) runs through all divisors with E ∼ D, andE0 ∈ Div(X0) runs through all Cartier divisors on X0 withdeg(E0|Cv) = E(v) for any v ∈ V (G).



Result

Theorem (algebraic rank for hyperelliptic or genus 3 graphs)

Assume one of the following.

(i) G is hyperelliptic and ch(k) 6= 2;

(ii) G is non-hyperelliptic and of genus 3.

Then, for any D ∈ Div(G), we have ralg,kG

(D) ≥ rG(D).

Remark

• Here rG(D) := rΓ(D), where Γ is the associated metric graph of G.

• Caporaso–Len–Melo prove that the other directionralg,kG

(D) ≤ rG(D) holds for any vertex-weighted graph G andD ∈ Div(G). Thus for a hyperelliptic or a genus 3 graph, theequality ralg,k

G(D) = rG(D) holds.



Remark (continued)

• Caporaso–Len–Melo give many other examples of graphs with theequality ralg,k

G(D) = rG(D).

• They also show that there exist a graph G and a divisorD ∈ Div(G) such that ralg,k

G(D) < rG(D).

Outline of the proof of Theorem

.

.

1 A variant of the question on lifting of divisors

.

.

2 Relation between lifting divisors and algebraic rank

.

.

3 Decomposition of graphs with bridges



Step 1: A variant of the questionR: cDVR with Frac(R) = K and residue field k

G = (G, w): a vertex-weighted graph

Question (a variant)Does there exist a regular, generically smooth, semi-stable R-curve X

with reduction graph G satisfying the following condition (F)?∀D ∈ Div(G), ∃D ∈ Div(X) with D = ρ∗(D) s.t.

rG(D) = rX(D).

Here, ρ∗ : Div(X)→ Div(G) is the specialization map.

(Comparison to the original question: D in Div(G) not in Div(ΓQ); D in

Div(X) not in Div(XK); and the specialization map is ρ∗ not

τ∗ : Div(XK)→ Div(ΓQ). )



Step 2: Relation between Question (a variant) and algebraicrankCaporaso kindly told us relation between Question (a variant) andalgebraic rank as follows.

Proposition

R: cDVR with Frac(R) = K and residue field k

G = (G, w): a vertex-weighted graphX : a regular, generically smooth, semi-stable R-curve with

reduction graph G with generic fiber X

Assume that X satisfies the condition (F).Then, for any divisor D ∈ Div(G), we have

ralg,kG

(D) ≥ rG(D).



Proof (Caporaso)

Recall

ralg,kG

(D) := maxX0

{min

E

{max

E0

rX0(E0)}}

.

If such X exists, we let X0 = Xk.For any E ∈ Div(G) with D ∼ E, we take a lift E ∈ Div(X) preservingthe rank. Let E be the Zariski closure of E in X , and we putE0 := E |X0 . Then

ralg,kG

(D) ≥ rX0(E0)(∗)≥ rX(E)

(∗∗)= rG(E) = rG(D),

where (*) is deduced from the upper-semicontinuity of the cohomology,and (**) is the condition (F). 2



R: cDVR with Frac(R) = K and residue field k

G = (G, w): a vertex-weighted graphAs in the original question, we have the following theorem.

Theorem

Assume one of the following.

(i) G is hyperelliptic and ch(k) 6= 2; Further, for each v ∈ V (G), thereare at most (2w(v) + 2) positive-type bridges emanating from v.


Then there exists a regular, generically smooth, semi-stable R-curve X

with reduction graph G satisfying the condition (F) (i.e. the answer toQuestion (a variant) is YES.)



Thus we have, for any D ∈ Div(G),

ralg,kG

(D) ≥ rG(D)

provided that G is one of the following.

(i) G is hyperelliptic and ch(k) 6= 2; Further, for each v ∈ V (G), thereare at most (2w(v) + 2) positive-type bridges emanating from v.


We want to delete the (2w(v) + 2) condition for the hyperelliptic case.



Step 3: Decomposition of graphs with bridgesG = (G, w): a vertex-weighted graph having a bridge e

with endpoints v1, v2

Gi (i = 1, 2): connected components of G \ {e} with vi ∈ V (Gi)Gi := (Gi, w|V (Gi)

)

Proposition

For any D ∈ Div(G), let Di ∈ Div(Gi) be the restriction of D to Gi.Then

rG(D) ≤

rG1(D1) + rG2

(D2) + 1 (if vi ∈ Bs(|Di|) for ∀i = 1, 2),

rG1(D1) + rG2

(D2) (otherwise).

Remark

If G is hyperelliptic, then Gi is hyperelliptic (or of genus 0 or 1).



There is a corresponding formula on nodal curves.

Lemma

Let X be a nodal curve. We assume that X has a decomposition asX = X1 ∪X2 into two nodal curves so that X1 and X2 meet at exactlyone point p. Let D be a Cartier divisor on X, and we setDi = D

∣∣∣Xi

∈ Div(Xi) for i = 1, 2. Then

rX(D) =

rX1(D1) + rX2(D2) + 1 (if p ∈ Bs(|Di|) for ∀i = 1, 2),

rX1(D1) + rX2(D2) (otherwise).

Combining Proposition and Lemma, we obtain ralg,kG

(D) ≥ rG(D) forany hyperelliptic graph G (without the (2w(v) + 2) condition). 2


rank of divisors on curves and graphs under specialization: the hyperelliptic … · 2014-04-14 ·...

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