Lifting Tropical Curves and Linear Systems on Graphsdzb/conferenceSlides/2010AMSRichmond/Ka… ·...

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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

Transcript of Lifting Tropical Curves and Linear Systems on Graphsdzb/conferenceSlides/2010AMSRichmond/Ka… ·...

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 2 / 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.Tropical graphs are balanced, weighted, integral graphs

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 2 / 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.Tropical graphs are balanced, weighted, integral graphsIntegral: Each edge is a line-segment or a ray parallel to ~u ∈ Zn.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 2 / 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.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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 2 / 31

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 3 / 31

An Elliptic Curve in the Plane

All multiplicities are 1.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 4 / 31

An Elliptic Curve in Space

All multiplicities are 1.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 5 / 31

Statement of Lifting Problem

Lifting Problem: Which tropical graphs are tropicalizations of curves?

Today: necessary conditions.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 6 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 6 / 31

Why?

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

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

Why?

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

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

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

Why?

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

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.

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

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).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 8 / 31

Example of non-liftable curve (cont’d)

This is not liftable to a curve over K because

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 9 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 9 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 9 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 9 / 31

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,

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 9 / 31

Aside for number theorists

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 10 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 10 / 31

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.

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!

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 10 / 31

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 11 / 31

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),

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 11 / 31

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 ).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 11 / 31

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 ).

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 11 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 12 / 31

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.A divisor on Σ is a Z-combination of points of Σ.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 12 / 31

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.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).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 12 / 31

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).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 13 / 31

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).

D = (s) −∑

v

(v)Cv

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

0 ≤ D · Cw = (s) · Cw −∑

E=vw

((w) − (v))

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 13 / 31

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).

D = (s) −∑

v

(v)Cv

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

0 ≤ D · Cw = (s) · Cw −∑

E=vw

((w) − (v))

which can be expressed as

0 ≤ Λ(w) + ∆(w)

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 13 / 31

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).

D = (s) −∑

v

(v)Cv

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

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))

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 14 / 31

Canonical divisor

Σ has 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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 14 / 31

Canonical divisor

Σ has 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.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 14 / 31

Canonical divisor

Σ has 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.

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 )

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 14 / 31

Canonical divisor

Σ has 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.

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 14 / 31

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 15 / 31

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

1 ϕm ∈ L(KΣl),

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 15 / 31

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

1 ϕm ∈ L(KΣl),

2 ϕm = 0 on E with m · E 6= 0,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 15 / 31

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

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 15 / 31

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

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 15 / 31

Cycle-Ampleness Condition

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 16 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 16 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 16 / 31

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,

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.”

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 16 / 31

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).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 17 / 31

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).

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 17 / 31

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).

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).”

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 17 / 31

Elliptic Curve Example

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 18 / 31

Elliptic Curve Example (cont’d)

1 Direct edges towards cycle.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

Elliptic Curve Example (cont’d)

1 Direct edges towards cycle.

2 ϕm must be decreasing on unbounded edges.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

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))

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

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))

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 19 / 31

Elliptic Curve Example (concluded)

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 20 / 31

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!

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 20 / 31

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!

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 20 / 31

A New Example

a c

b

d

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 21 / 31

A New Example

a c

b

d

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 21 / 31

A New Example

a c

b

d

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

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 21 / 31

A New Example

a c

b

d

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

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 21 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

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

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

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

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

4 Slope on edge a is at most 3.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

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

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

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

A New Example (cont’d)

1 Direct edges towards cycle.

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

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

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.

7 deg(Dϕm) ≤ 1 on one cycle.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 22 / 31

Weak well-spacedness condition

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 23 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 23 / 31

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.Let Γ′ be a component of p−1(H) ⊂ Σ with h1(Γ′) > 0.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 23 / 31

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.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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 23 / 31

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 ,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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),

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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,

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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 .

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

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 .Consequently if pe is the node between Cv and Cw then

ordpE(sv ) = (w) − (v).

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 24 / 31

Outline of Proof

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 25 / 31

Outline of Proof

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

1 C extends to a regular semistable model C.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 25 / 31

Outline of Proof

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

1 C extends to a regular semistable model C.

2 (K∗)n extends to a toric scheme P.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 25 / 31

Outline of Proof

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

1 C extends to a regular semistable model C.

2 (K∗)n extends to a toric scheme P.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 25 / 31

Outline of Proof

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

1 C extends to a regular semistable model C.

2 (K∗)n extends to a toric scheme P.

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† .

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 25 / 31

Outline of Proof (cont’d)

Log structures are particularly nice:

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

boundary.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 26 / 31

Outline of Proof (cont’d)

Log structures are particularly nice:

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

boundary.

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

zm where zm is a

character of (K∗)n.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 26 / 31

Outline of Proof (cont’d)

Log structures are particularly nice:

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

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† .

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 26 / 31

Outline of Proof (cont’d)

Let Σ be the dual graph of C0.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 27 / 31

Outline of Proof (cont’d)

Let Σ be the dual graph of C0.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 27 / 31

Outline of Proof (cont’d)

Let Σ be the dual graph of C0.

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

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 27 / 31

Outline of Proof (cont’d)

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (cont’d)

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (cont’d)

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (cont’d)

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (cont’d)

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.

ωm

th=

ds

t−czmwhich is close to being an exact form.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (cont’d)

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.

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)!

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 28 / 31

Outline of Proof (concluded)

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 29 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 29 / 31

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.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 29 / 31

Asides and Future Directions

1 This method is a combinatorial approach to deformation theory.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 30 / 31

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?

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 30 / 31

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.

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 30 / 31

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.

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

Eric Katz (Texas/MSRI) Lifting Tropical Curves November 7, 2010 30 / 31

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

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