Lifting Tropical Curves and Linear Systems on Graphsdzb/conferenceSlides/2010AMSRichmond/Ka… · Lifting Tropical Curves and Linear Systems on Graphs Eric Katz (Texas/MSRI) November

Lifting Tropical Curves and Linear Systems on Graphs

Eric Katz (Texas/MSRI)

November 7, 2010

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 1 / 31

Tropicalization

Let K = Ct = C((t)), the field of Puiseux series. K hasnon-Archimedean valuation v : K∗ → Q ⊂ R. Tropicalization map:

Trop : curves C ⊂ (K∗)n → tropical graphs Σ = Trop(C ) ⊂ Rn

Trop(C ) is defined as v(C ) ⊂ Rn where v : (K∗)n → Rn is the product ofvaluation maps and closure is topological.


Tropicalization



Trop(C ) is defined as v(C ) ⊂ Rn where v : (K∗)n → Rn is the product ofvaluation maps and closure is topological.Tropical graphs are balanced, weighted, integral graphs


Tropicalization



Trop(C ) is defined as v(C ) ⊂ Rn where v : (K∗)n → Rn is the product ofvaluation maps and closure is topological.Tropical graphs are balanced, weighted, integral graphsIntegral: Each edge is a line-segment or a ray parallel to ~u ∈ Zn.


Tropicalization



Trop(C ) is defined as v(C ) ⊂ Rn where v : (K∗)n → Rn is the product ofvaluation maps and closure is topological.Tropical graphs are balanced, weighted, integral graphsIntegral: Each edge is a line-segment or a ray parallel to ~u ∈ Zn.Weighted: Each edge has a weight (multiplicity) m(σ) ∈ N.


Tropicalization

Balanced: For v , a vertex of Σ and adjacent edges E1, . . . ,Ek in primitiveZn directions, ~u1, . . . , ~uk then

∑

m(Ei )~ui = ~0.

Example:

m = 2

m = 1

m = 1


An Elliptic Curve in the Plane

All multiplicities are 1.


An Elliptic Curve in Space

All multiplicities are 1.


Statement of Lifting Problem

Lifting Problem: Which tropical graphs are tropicalizations of curves?

Today: necessary conditions.


Statement of Lifting Problem

Lifting Problem: Which tropical graphs are tropicalizations of curves?

Today: necessary conditions.

Speyer: Elliptic Curves, necessary and sufficient conditions in genus 1.

Nishinou and Brugalle-Mikhalkin: Generalization of Speyer’s result inone-bouquet case.


Why?

There are tropical curves that are not tropicalizations, telling thedifference is subtle.


Why?


The problem is combinatorial, but what kind of combinatorics evenencodes this?


Why?


The problem is combinatorial, but what kind of combinatorics evenencodes this?

Closely tied to deformation theory which is often grungy, maybethere’s a combinatorial approach.


Example of non-liftable curve

Change the length of a bounded edge in the spatial elliptic curve so that itdoes not lie on the tropicalization of any plane (possible by dimensioncounting).


Example of non-liftable curve (cont’d)

This is not liftable to a curve over K because




1 three unbounded edges in each direction in the curve shows that itmust be a cubic,





2 the loop in the curve shows that any lift must have genus at least 1,






3 any classical cubic is either genus 0 and spatial or genus 1 and planar,






3 any classical cubic is either genus 0 and spatial or genus 1 and planar,

no lift of the curve can be planar or genus 0, so the curve does not lift.


Aside for number theorists

Instead of considering C ⊂ (K∗)n, we may consider C ⊂ A where A is anAbelian variety.




If C is a curve over K with maximally degenerate reduction, way considerthe Abel-Jacobi map, i : C → J(C ). By uniformization, J(C ) = (K∗)n/Λ.




If C is a curve over K with maximally degenerate reduction, way considerthe Abel-Jacobi map, i : C → J(C ). By uniformization, J(C ) = (K∗)n/Λ.

We can tropicalize this to get Trop(i) : Σ → Rn/v(Λ). Here Rn/v(Λ) is areal torus. Our lifting method involves the combinatorics of pulling back1-forms from ambient space to C and restricting to closed fiber ofsemistable model. More later!


Parameterized Tropical Graphs

A tropical parameterization of a tropical graph Σ is a map p : Σ → Σ(maps vertices to vertices but may contract edges) such that




1 Σ is a tropical graph (balanced where each edge is given the directionof its image),





2∑

E∈p−1(E)

m(E ) = m(E ).





2∑

E∈p−1(E)

m(E ) = m(E ).

Note: If all the multiplicities of Σ are 1 and all vertices are trivalent, thenthe only parameterization of Σ is the identity.


Specialization of line bundles

Quick Review of Baker’s Specialization:Let C be a curve over K. C is defined over some field C((t)) after possibly

replacing t1N by t. Pick a regular semistable model C over O = C[[t]]. Let

Σ be the dual graph of C0. For v ∈ V (Σ), let Cv be the correspondingcomponent. For E , a bounded edge, let pE be the corresponding node.





Σ be the dual graph of C0. For v ∈ V (Σ), let Cv be the correspondingcomponent. For E , a bounded edge, let pE be the corresponding node.A divisor on Σ is a Z-combination of points of Σ.





Σ be the dual graph of C0. For v ∈ V (Σ), let Cv be the correspondingcomponent. For E , a bounded edge, let pE be the corresponding node.A divisor on Σ is a Z-combination of points of Σ.Let L be a line bundle on C. The divisor associated to L is

Λ =∑

v

deg(L|Cv)(v).


Specialization of sections

If s ∈ Γ(C,L), write

(s) = D +∑

v

(v)Cv

where D is a horizontal divisor. is called the vanishing functions (orderof vanishing of s on generic points of components of central fiber).




(s) = D +∑

v

(v)Cv


D = (s) −∑

v

(v)Cv

so for w ∈ V (Σ), (since D intersects Cw properly)

0 ≤ D · Cw = (s) · Cw −∑

E=vw

((w) − (v))




(s) = D +∑

v

(v)Cv


D = (s) −∑

v

(v)Cv


0 ≤ D · Cw = (s) · Cw −∑

E=vw

((w) − (v))

which can be expressed as

0 ≤ Λ(w) + ∆(w)




(s) = D +∑

v

(v)Cv


D = (s) −∑

v

(v)Cv


0 ≤ D · Cw = (s) · Cw −∑

E=vw

((w) − (v))

which can be expressed as

0 ≤ Λ(w) + ∆(w)

or as ∈ L(Λ).Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 13 / 31

Canonical divisor

Σ has canonical divisor:

KΣ =∑

v

(2g(v) − 2 + deg(v))


Canonical divisor


KΣ =∑

v

(2g(v) − 2 + deg(v))

We will need to make sense of the condition that ∈ L(KΣ). Interpolate as a piecewise-linear function on Σ. Assumption:g(v) = 0.


Canonical divisor


KΣ =∑

v

(2g(v) − 2 + deg(v))


if v ∈ Σ, E ∋ v , write s(v ,E ) for the slope of on E coming from v .


Canonical divisor


KΣ =∑

v

(2g(v) − 2 + deg(v))


if v ∈ Σ, E ∋ v , write s(v ,E ) for the slope of on E coming from v .The Laplacian of ϕ is

∆()(v) = −∑

E∋v

s(v ,E )


Canonical divisor


KΣ =∑

v

(2g(v) − 2 + deg(v))


if v ∈ Σ, E ∋ v , write s(v ,E ) for the slope of on E coming from v .The Laplacian of ϕ is

∆()(v) = −∑

E∋v

s(v ,E )

Note: For ∈ L(KΣ), at bivalent points v , KΣ(v) = 0 and the condition∆()(v) = ∆()(v) + KΣ(v) ≥ 0 means that slopes only decrease.


Main Theorem

Theorem: If Σ ⊂ Rn is a tropicalization then there exists p : Σ → Σ andfor all m ∈ Zn, there is a piecewise-linear function ϕm : Σl → R≥0 (Σl isthe l -fold subdivision of Σ) with Z-slopes such that


Main Theorem


1 ϕm ∈ L(KΣl),


Main Theorem


1 ϕm ∈ L(KΣl),

2 ϕm = 0 on E with m · E 6= 0,


Main Theorem


1 ϕm ∈ L(KΣl),

2 ϕm = 0 on E with m · E 6= 0,

3 ϕm never has slope 0 on edges E with m · E = 0,


Main Theorem


1 ϕm ∈ L(KΣl),

2 ϕm = 0 on E with m · E 6= 0,

3 ϕm never has slope 0 on edges E with m · E = 0,

4 ϕm obeys the cycle-ampleness condition.


Cycle-Ampleness Condition

Let H be a hyperplane given by H = x |x · m = c.




Let Γ be a cycle in the interior of p−1(H) ⊂ Σ.





Set h = minv∈Γ (ϕm(v)) then,





Set h = minv∈Γ (ϕm(v)) then,

Dϕm ≡∑

v∈Γ|ϕm(v)=h

∑

E 6∈Γ|s(v ,E)<0

(−s(v ,E ))

≥ 2.

“sum of positive slopes coming into the cycle at min’s of ϕm must be atleast 2.”


Sections of Canonical Bundle

Before we use these conditions, we need the following observation:ϕm ∈ L(KΣl

) translates into

∆(ϕm)(v) = −∑

E∋v

s(v ,E ) ≥ 2 − deg(v).




) translates into

∆(ϕm)(v) = −∑

E∋v

s(v ,E ) ≥ 2 − deg(v).

If v ∈ Γ is a vertex with edges E1, . . . ,Ek ,F1, . . . ,Fl (partitioned in anyway). By hypothesis s(v ,Ei ), s(v ,Fj ) 6= 0.




) translates into

∆(ϕm)(v) = −∑

E∋v

s(v ,E ) ≥ 2 − deg(v).

If v ∈ Γ is a vertex with edges E1, . . . ,Ek ,F1, . . . ,Fl (partitioned in anyway). By hypothesis s(v ,Ei ), s(v ,Fj ) 6= 0.

∑

s(v ,Fj ) ≤(

∑

−s(v ,Ei ))

+ (deg(v) − 2))

“At v, sum of outgoing slope along edges Fj is less than sum of incomingslopes along edges Ei plus (deg(v) − 2).”


Elliptic Curve Example

Note: This is p−1(H) where H is the plane of the screen.


Elliptic Curve Example (cont’d)

1 Direct edges towards cycle.




2 ϕm must be decreasing on unbounded edges.





3 ϕm is equal to 0 on ∂(p−1(H)) and has slope at most 1 there.






4 Slopes of ϕm only decrease along edge as we move towards cycle.







5 Slope of ϕm is at most 1 as it turns the corner and heads to cycle.








6 There is positive incoming slope at ≥ 3 points on the cycle. At thosepoints, ϕm is equal to distance to ∂(p−1(H))








6 There is positive incoming slope at ≥ 3 points on the cycle. At thosepoints, ϕm is equal to distance to ∂(p−1(H))

7 For deg(Dϕm) ≥ 2, the minimum distance must be achieved at leasttwice.


Elliptic Curve Example (concluded)

In summary, minimum distance from Γ to Σ \ p−1(H) must be achieved bytwo paths.




This is Speyer’s well-spacedness condition!




This is Speyer’s well-spacedness condition!

Also get generalization to higher genus as given by Nishinou andBrugalle-Mikhalkin. This requires strong conditions on combinatorics of Σ.


A New Example

a c

b

d


A New Example

a c

b

d

Embed in the plane so that it is balanced in the plane.


A New Example

a c

b

d


Add unbounded edges pointing out of the plane to ensure that is globallybalanced. Give every edge multiplicity 1. Can ensure that onlyparameterization is the identity.


A New Example

a c

b

d


Add unbounded edges pointing out of the plane to ensure that is globallybalanced. Give every edge multiplicity 1. Can ensure that onlyparameterization is the identity.

There does not exist the desired ϕm, so it does not lift.


A New Example (cont’d)
















4 Slope on edge a is at most 3.







5 Slopes on edges b, c , d sum to at most 5, so they contribute at mostone point to Dϕm on one of the cycles.

6 Long edges are too long for ϕm to have positive slope and to alsointersect a cycle in a minimum of ϕm.







5 Slopes on edges b, c , d sum to at most 5, so they contribute at mostone point to Dϕm on one of the cycles.

6 Long edges are too long for ϕm to have positive slope and to alsointersect a cycle in a minimum of ϕm.

7 deg(Dϕm) ≤ 1 on one cycle.


Weak well-spacedness condition

There’s a weak version of Speyer’s condition in higher genera that holdsfor curves of complicated combinatorial type.




Theorem: Let H be a hyperplane with normal vector m ∈ Zn.




Theorem: Let H be a hyperplane with normal vector m ∈ Zn.Let Γ′ be a component of p−1(H) ⊂ Σ with h1(Γ′) > 0.




Theorem: Let H be a hyperplane with normal vector m ∈ Zn.Let Γ′ be a component of p−1(H) ⊂ Σ with h1(Γ′) > 0.Then ∂Γ′ does not consist of a single trivalent vertex of Σ.


Observation: controls poles of sections!

Let O be the valuation ring of K = C((t)), suppose everything over K.

C is a family of curves over O extending C ,





let Σ be its log dual graph (infinite ends correspond to markedpoints),






let L be a line bundle on C and s, a section,







after base-extension, the zeroes of sK on CK consist of K-rationalpoints and therefore, specialize to smooth points on the special fiber,








let be the vanishing function of s,









for v ∈ V (Σ), let sv = st(v) |Cv

, a nonvanishing rational function onCv











If E = vw is an edge, then s vanishes generically to order (v) on Cv andto order (w) on Cv .











If E = vw is an edge, then s vanishes generically to order (v) on Cv andto order (w) on Cv .Consequently if pe is the node between Cv and Cw then

ordpE(sv ) = (w) − (v).


Outline of Proof

Try to extend relevant objects over O. (Nishinou-Siebert stable reductiontheorem)


Outline of Proof


1 C extends to a regular semistable model C.


Outline of Proof



2 (K∗)n extends to a toric scheme P.


Outline of Proof




3 C → (K∗)n extends to a stable map f : C → P.


Outline of Proof




3 C → (K∗)n extends to a stable map f : C → P.

4 C and P have log structures (magic dust) giving them locally free logcotangent sheaves Ω1

C†/O† ,Ω1P†/O† .


Outline of Proof (cont’d)

Log structures are particularly nice:

1 Ω1C†/O† is relative dualizing sheaf twisted by points mapped to toric

boundary.





boundary.

2 Ω1P†/O† is trivial bundle. Sections of the form dzm

zm where zm is a

character of (K∗)n.





boundary.

2 Ω1P†/O† is trivial bundle. Sections of the form dzm

zm where zm is a

character of (K∗)n.

3 f is a log morphism. ωm = f ∗ dzm

zm is a section of log cotangent bundleΩ1C†/O† .



Let Σ be the dual graph of C0.




The cotangent sheaf Ω1C†/O† specializes to KΣ.




The cotangent sheaf Ω1C†/O† specializes to KΣ.

Specializing ωm (almost) gives ϕm ∈ L(KΣ).



Cycle-ampleness condition is the only hard part. Look at ωm in a formalneighborhood UΣ of components of C0 corresponding to cycle Σ.



Cycle-ampleness condition is the only hard part. Look at ωm in a formalneighborhood UΣ of components of C0 corresponding to cycle Σ.Since Γ is mapped into a hyperplane orthogonal to m, t−c f ∗zm is aregular function on UΣ.




If h = minv∈Σ (ϕm(v)), ωm = t−cd(f ∗zm)t−c f ∗zm vanishes to order h.

Consequently, t−c f ∗zm is equal to a constant L up to order h.






ωm is equal to an exact 1-form up to low orders: let

s =f ∗(t−czm) − L

th.








th.

ωm

th=

ds

t−czmwhich is close to being an exact form.








th.

ωm

th=

ds

t−czmwhich is close to being an exact form.

Upshot: at components of the central fiber Cv with ϕm(v) = h, s|Cvand

ωm

th have the same poles (where ωm

th is considered as a log 1-form)!


Outline of Proof (concluded)

The components of C0 corresponding to Σ are a cycle, (C0)Σ of rationalcurves. This is a degenerate elliptic curve.




Punchline: A rational function on an elliptic curve needs at least two poles.




Punchline: A rational function on an elliptic curve needs at least two poles.

Dϕm picks up the poles of s|(C0)Σ . This immediately gives cycle-amplenesscondition.


Asides and Future Directions

1 This method is a combinatorial approach to deformation theory.




2 Gives an additional combinatorial structure on tropicalizations ofcurves. Higher dimensions?





3 Method works in finite residue characteristic as long as you excludewild phenomena.





3 Method works in finite residue characteristic as long as you excludewild phenomena.

4 Abelian varieties also have lots of 1-forms. What does 1-form ineffective Chabauty give?


Thanks!

Lifting Tropical Curves in Space and Linear Systems on Graphs,arXiv:1009.1783

Speyer, David. Uniformizing Tropical Curves I: Genus Zero and One,arXiv:0711.2677

Nishinou, Takeo. Correspondence Theorems for Tropical Curves,arXiv:0912.5090


Lifting Tropical Curves and Linear Systems on Graphsdzb/conferenceSlides/2010AMSRichmond/Ka… · Lifting Tropical Curves and Linear Systems on Graphs Eric Katz (Texas/MSRI) November

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