Constructing the moduli space of parametrized

Constructing the moduli space of tropical curves

Motivation

Constructing the moduli space of parametrized tropical curves

Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N

Hannah Markwig joint with Andreas Gathmann and Michael Kerber IMA Minneapolis

June 2007

Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Motivation

Motivation

Give the moduli space of parametrized tropical curves, Mtrop,0,n (Pr , d), the structure of a tropical variety (respectively, a tropical fan). This space can be used to count tropical curves (e.g. Kontsevich’s formula). r

Only a local understanding of Mtrop,0,n (P , d): know the cones and how they are glued, but 1 2

how to embed the space globally into some RN ? what kind of map is the evaluation map?

Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Contents

Constructing the moduli space of tropical curves

Motivation

1

Tropical fans

Tropical fans Constructing M 0,n (Pr , d)

2

Constructing M 0,n (Pr , d)

3

Tropical Mtrop,0,N

Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

4

Parametrized tropical curves

5

The evaluation map

Tropical fans Λ denotes a lattice (ZN ) and V = Λ ⊗Z R the corresponding real vector space.

Definition A tropical fan in V is a fan such that 1

cones σ are cut out by integral inequalities and equations (Λσ is the smallest sublattice of Λ that contains σ ∩ Λ),

2

full-dimensional cones σ are equipped with a weight ω(σ) ∈ N>0 , and

3

for each cone τ of codimension 1 the balancing condition is fulfilled.

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

For a fan of dimension 1 the balancing condition means: ` 1´

Constructing the moduli space of tropical curves

1

ω=2 `−1´ 0

Motivation Tropical fans

`

1 ´ −1

For a higher-dimensional fan, project the cone τ and its full-dimensional neighboring cones along τ :

Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Two tropical fans are called equivalent, if they have a common refinement. (The weights have to agree, too.)

Motivation Tropical fans

A tropical fan is called irreducible, if there is no tropical fan of the same dimension which is strictly contained.

Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Lemma Let X and Y be tropical fans of dimension n, X irreducible, and Y contained in X. Then Y = λ · X for some λ in Q>0 . Idea of the proof: suitable refinement λ := minσ∈X ωY (σ)/ωX (σ) α such that α · λ ∈ Z new weight function: ω(σ) = α · (ωY (σ) − λωX (σ)) new fan containing only σ | ω(σ) > 0. It has to be empty.

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Definition A morphism of fans f : X → Y is a Z-linear map, that is, a map induced by a linear map from Λ to Λ0 . construct the image fan f (X) in Y : roughly: cones f (σ), where σ is contained in a full-dimensional cone on which f is injective. the cones f (σ) might overlap → suitable refinement 0

define the weight of σ in f (X) to be X ωf (X) (σ 0 ) = ωX (σ) · |Λ0σ0 /f (Λσ )|. σ∈X|f (σ)=σ 0

f (X) is a tropical fan, too.

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Corollary X, Y tropical fans of the same dimension n, f : X → Y . Y irreducible. For Q such that Q∈

0 σQ ,

0 dim(σQ )

Tropical fans Constructing M 0,n (Pr , d)

=n

inverse images P ∈ σP , dim(σP ) = n define the multiplicity

Tropical Mtrop,0,N Parametrized tropical curves

multP f := Then the sum

Motivation

ωX (σP ) 0 0 /f (ΛσP )|. 0 ) · |ΛσQ ωY (σQ X

multP f

P | f (P )=Q

does not depend on Q(“degree of f ”).

The evaluation map

Constructing the moduli space of tropical curves

Idea of the proof:

Motivation

suitable refinements

Tropical fans

Y irreducible =⇒ f (X) = λ · Y for some λ.

Constructing M 0,n (Pr , d)

Therefore X

multP f =

P | f (P )=Q 0

ωf (X) (σ ) =λ = ωY (σ 0 ) does not depend on Q.

X σ | f (σ)=σ 0

Tropical Mtrop,0,N

ωX (σ) · |Λ0σ0 /f (Λσ )| ωY (σ 0 )

Parametrized tropical curves The evaluation map

Constructing M 0,n (Pr , d)

Constructing the moduli space of tropical curves

Motivation

U open subset of M 0,n (Pr , d) such that f ∗ {xi = 0} consists of d distinct nonspecial points. To (C, pi , f ) associate (C, pi , qi,j ) ∈ M 0,N (where N := n + (r + 1)d) by marking the points f ∗ {xi = 0}.

Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves

q3,3

q1,3 p3

The evaluation map

q2,3

q1,1 q1,2 q2,1

p3

p1

p2 q2,2 q3,1 q3,2

p1

p2

Constructing the moduli space of tropical curves

B ⊂ M 0,N open subset s. th. divisors si := linearly equivalent. si

r + 1 sections of a line bundle on P1

P

j qi,j

are

Motivation Tropical fans

map to Pr

(C, pi , Φ ◦ f ) and (C, pi , f ) yield the same (C, pi , qi,j ), where '

Φ : [x0 : . . . : xr ] 7→ [x0 : λ1 x1 : . . . : λr xr ]. The map [s0 : . . . : sr ] is as good as [s0 : λ1 s1 : . . . : λr sr ]. Let Y be the corresponding (C∗ )r -bundle over B. Then Y /(Sr+1 d ) = U.

Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Tropical Mtrop,0,N

Constructing the moduli space of tropical curves

Definition An abstract tropical curve with n markings is a tree, such that

Motivation Tropical fans

each vertex is at least 3-valent, n of the leaves (ends, unbounded edges) are marked and the bounded edges are equipped with a (positive) length. Mtrop,0,N = space of abst. trop. curves with N ends, all marked x2 l1 = 4

x1 l2 = 2.5

x3

x4

Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

N Mtrop,0,N can be embedded as a tropical fan in R( 2 )−N : ij-th coordinate given by the distance of leaf i and leaf j.

Constructing the moduli space of tropical curves

Motivation

1 2

1 3 4

12 13 14 23 24 34

1 0 1 1 0 1

Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves

Divide out W spanned by trees with a 2-valent and an N -valent vertex: 2

1

1 3

4

12 13 14 23 24 34

0 1 0 1 0 1

The evaluation map

Mtrop,0,4 can be embedded into R2 = R6 /W . It consists of three rays generated by (0, 1, 1, 1, 1, 0) (1, 0, 1, 1, 0, 1) (1, 1, 0, 0, 1, 1) Their sum = the sum of the 4 generators of W =⇒ Mtrop,0,4 is a tropical fan.

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Parametrized tropical curves

Constructing the moduli space of tropical curves

Definition (Γ, xi , h) is an n-marked parametrized tropical curve of degree d to Pr if: (Γ, xi ) abstract tropical curve with n + (r + 1)d ends, h : Γ → Rr a continuous map satisfying: On each edge E, h is of the form h|E : [0, l(E)] → Rr : t 7→ a + v(E) · t (a ∈ Rr , v(E) ∈ Zr “direction” of E). (v(E) = product of the weight ω(E) and the primitive integral vector.) For every vertex the balancing condition holds. Marked ends xi contracted to a point by h (i.e. v(xi ) = 0). P d of the other ends mapped to −ei , and d to ei .

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves

The length of an image edge h(E) is determined by the length of E in the abstract tropical curve and the direction v(E) (if v(E) 6= 0).

The evaluation map

How associate an abstract tropical curve to (Γ, pi , h)? Mark the other ends by qi,j . All directions are determined by the directions of the ends:

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

=⇒ the map h is determined by the position of one point (e.g. h(x1 )) in Rr , the abstract curve (Γ, pi , qi,j ) and the directions v(qi,j ).

Get a map Mtrop,0,N × Rr → Mtrop,0,n (Pr , d). Cover, number of inverse images varies for the different cones. Example: Mtrop,0,0 (P2 , 2).

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

The picture encodes both graph and map. Avoid this by labelling the ends (for enumerative statements, we have to divide by |Sr+1 d | later.)

Constructing the moduli space of tropical curves

Motivation

lab Mtrop,0,n (Pr , d)

=

Mtrop,0,N × Rr

(Γ, xi , qi,j , h) 7→ ((Γ, xi , qi,j ), h(x1 )) in the sense: bijection between the two sets of cones, each cone is equal to its image cone under this bijection, gluing coincides. The right hand side is a tropical fan.

Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Constructing the moduli space of tropical curves

Definition

Motivation

The map

Tropical fans

evi :

lab Mtrop,0,n (Pr , d)

→

R

r

(Γ, x1 , . . . xn , h) 7−→ h(xi ) is called the i-th evaluation map.

Lemma The i-th evaluation map evi is a morphism of fans.

Constructing M 0,n (Pr , d) Tropical Mtrop,0,N Parametrized tropical curves The evaluation map

Idea of the proof: lab Identify M0,n,trop (Pr , d) with M0,N,trop × Rr .

Define

Constructing the moduli space of tropical curves

Motivation N

ev0i : R( 2 ) × Rr

Tropical fans

−→ Rr

(a1,2 , . . . aN −1,N , b) 7−→ b +

N 1X

2

Constructing M 0,n (Pr , d)

(a1,k − ai,k ) vk

k

Parametrized tropical curves

ev0i is 0 on W and it is a linear map. x1

The evaluation map

l(E) = 1 xi

I v(E)

Tropical Mtrop,0,N

J

For all ends k ∈ I, a1,k = 0, ai,k = 1 =⇒ count negatively. For all ends l ∈ J, a1,l = 1, ai,l = 0 =⇒ count positively. P P Balancing condition =⇒ − k∈I vk = l∈J vl = v(E).

Now the statement ”the degree of the evaluation map is constant” (or:“the number of tropical curves through given points does not depend on the points”) is an easy consequence of the fact that this space is a tropical fan and ev is a morphism of fans.

Constructing the moduli space of tropical curves

Motivation Tropical fans Constructing M 0,n (Pr , d) Tropical Mtrop,0,N

References: A. Gathmann, M. Kerber, H. Markwig, Tropical fans and the moduli space of rational tropical curves, preprint. G. Mikhalkin, Moduli spaces of rational tropical curves, preprint. G. Mikhalkin, Tropical Geometry and its applications, Proceedings of the ICM, Madrid Spain (2006), 827–852.

Parametrized tropical curves The evaluation map