MODULI SPACES OF RATIONAL TROPICAL CURVES
GRIGORY MIKHALKIN
Abstract. This note is devoted to the definition of moduli spaces
of rational tropical curves with nmarked points. We show that this
space has a structure of a smooth tropical variety of dimension
n − 3. We define the Deligne-Mumford compactification of this
space and tropical ψ-class divisors.
This paper gives a detailed description of the moduli space of tropical
rational curves mentioned in [4]. The survey [4] was prepared under
rather sharp time and volume constraints. As a result the coordinate
presentation of this moduli space from [4] contains a mistake (it was
over-simplified). In this paper we’ll correct the mistake and give a
detailed description on M0,5 as our main example.
1. Introduction: smooth tropical varieties
In this section we follow the definitions of [5] and [4].
The underlying algebra of tropical geometry is given by the semifield
T = R∪{−∞} of tropical numbers. The tropical arithmetic operations
are “a + b” = max{a, b} and “ab” = a + b. The quotation marks are
used to distinguish between the tropical and classical operations. With
respect to addition T is a commutative semigroup with zero “0T” =
−∞. With respect to multiplication T× = T r {0T} ≈ R is an honest
commutative group with the unit “1T” = 0. Furthermore, the addition
and multiplication satisfy to the distribution law “a(b+c)” = “ab+ac”,
a, b, c ∈ T. These operations may be viewed as a result of the so-called
dequantization of the classical arithmetic operations that underlies the
patchworking construction, see [3] and [8].
These tropical operations allow one to define tropical Laurent poly-
nomials. Namely, a tropical Laurent polynomial is a function f : Rn →
f(x) = “
j” = max
(aj + jx),
where jx denotes the scalar product, x ∈ (T×)n ≈ Rn, j ∈ Zn and only
finitely many coefficients aj ∈ T are non-zero (i.e. not −∞).
http://arxiv.org/abs/0704.0839v1
2 GRIGORY MIKHALKIN
Affine-linear functions with integer slopes (for brevity we call them
simply affine functions) form an important subcollection of all Laurent
polynomials. Namely, these are such functions f : Rn → R that both
f and “1T
” = −f are tropical Laurent polynomials.
We equip Tn ≈ [−∞,∞)n with the Euclidean topology. Let U ⊂ Tn
be an open set.
Definition 1.1. A continuous function f : U → T is called regular if
its restriction to U ∩ Rn coincides with a restriction of some tropical
Laurent polynomial to U ∩ Rn.
We denote the sheaf of regular functions on Tn with O (or sometimes
OTn to avoid confusion). Any subset X ⊂ T
n gets an induced regular
sheaf OX by restriction. For our purposes we restrict our attention only
to the case when X is a polyhedral complex, i.e. when X is the closure
of a union of convex polyhedra (possibly unbounded) in Rn such that
the intersection of any number of such polyhedra is their common face.
We say that X is an k-dimensional polyhedral complex if it is obtained
from a union of k-dimensional polyhedra. These polyhedra are called
the facets of X .
Let V ⊂ X be an open set and f ∈ OX(V ) be a regular function
in V . A point x ∈ V is called a “zero point” of f if the restriction of
” = −f toW ⊂ V is not regular for any open neighborhoodW ∋ x.
Note that it may happen that x is a “zero point” for φ : U → T, but
not for φ|X∩U . It is easy to see that if X is a k-dimensional polyhe-
dral complex then the “zero locus” Zf of f is a (k − 1)-dimensional
polyhedral subcomplex.
To each facet of Zf we may associate a natural number, called its
weight (or degree). To do this we choose a “zero point” x inside such
a facet. We say that x is a “simple zero” for f if for any local de-
composition into a sum (i.e. the tropical product) of regular function
f = “gh” = g+h on V near x we have either g or h affine (i.e. without
a “zero”). We say that the weight is l if f can be locally decomposed
into a tropical product of l functions with a simple zero at x.
A regular function f allows us to make the following modification on
its domain V ⊂ X ⊂ Tn. Consider the graph
Γf ⊂ V × T ⊂ T
It is easy to see that the “zero locus”
Γ̄f ⊂ V × T
of the (regular) function “y+ f(x)” (defined on V ×T), where x is the
coordinate on V and y is the coordinate on T, coincides with the union
MODULI SPACES OF RATIONAL TROPICAL CURVES 3
of Γf and the undergraph
UΓf,Z = {(x, y) ∈ V × T | x ∈ Zf , y ≤ f(x)}.
Furthermore, the weight of a facet F ⊂ Γ̄f is 1 if F ∈ Γf (recall that
as V is an unweighted polyhedral complex all the weights of its facets
are equal to one) and is the weight of the corresponding facet of Zf if
F ∈ UΓf,Z . We view Γ̄f as a “tropical closure” of the set-theoretical
graph Γf . Note that we have a map Γ̄f → V . We set Ṽ = Γ̄f to be
the result of the tropical modification µf : Ṽ → V along the regular
function f . The locus Zf is called the center of tropical modification.
The weights of the facets of Ṽ supplies us with some inconvenience
as they should be incorporated in the definition of the regular sheaf OṼ
on Ṽ . Namely, the affine functions defined by OṼ on a facet of weight
w should contain the group of functions that come as restrictions to
this facet of the affine functions on Tn+1 as a subgroup of index w.
Sometimes one can get rid of the weights of Ṽ by a reparameteriza-
tion with the help of a map V̄ → Ṽ that is given by locally linear maps
in the corresponding charts. Indeed, the restriction of µg : V̄ → Ṽ to
a facet is locally given by a linear function between two k-dimensional
affine-linear spaces defined over Z. If its determinant equals to w then
the push-forward of OV̄ supplies an extension of OṼ required by the
weights. Note however that if w > 1 then the converse map is not
defined over Z and thus is not given by elements of OṼ .
Tropical modifications give the basic equivalence relation in Tropical
Geometry. It can be shown that if we start from Tk and do a number of
tropical modifications on it then the result is a k-dimensional polyhe-
dral complex Y ⊂ Tn that satisfies to the following balancing property
(cf. Property 3.3 in [4] where balancing is restated in an equivalent
way).
Property 1.2. Let E ⊂ Y ∩ RN be a (k − 1)-dimensional face and
F1, . . . , Fl be the facets of Y adjacent to F whose weights are w1, . . . , wl.
Let L ⊂ RN be a (N−k)-dimensional affine-linear space with an integer
slope and such that it intersects E. For a generic (real) vector v ∈ RN
the intersection Fj ∩ (L + v) is either empty or a single point. Let
ΛFj ⊂ Z
N be the integer vectors parallel to Fj and ΛL ⊂ Z
N be the
integer vectors parallel to L. Set λj to be the product of wj and the
index of the subgroup ΛFj +ΛL ⊂ Z
N . We say that Y ⊂ Tn is balanced
if for any choice of E, L and a small generic v the sum
j | Fj∩(L+v)6=∅
4 GRIGORY MIKHALKIN
is independent of v. We say that Y is simply balanced if in addition for
every j we can find L and v so that Fj ∩ (L + v) 6= ∅, ιL = 1 and for
every small v there exists an affine hyperplane Hv ⊂ L such that the
intersection Y ∩ (L + v) sits entirely on one side of Hv + v in L + v
while the intersection Y ∩ (Hv + v) is a point.
Definition 1.3 (cf. [5],[4]). A topological space X enhanced with a
sheaf of tropical functions OX is called a (smooth) tropical variety of
dimension k if for every x ∈ X there exist an open set U ∋ x and an
open set V in a simply balanced polyhedral complex Y ⊂ TN such that
the restrictions OX |U and OY |V are isomorphic.
Tropical varieties are considered up to the equivalence generated by
tropical modifications. It can be shown that a smooth tropical variety
of dimension k can be locally obtained from Tk by a sequence of tropical
modifications centered at smooth tropical varieties of dimension (k−1).
This follows from the following proposition.
Proposition 1.4. Any k-dimensional simply balanced polyhedral com-
plex X ⊂ Rn can be obtained from Tk by a sequence of consecutive trop-
ical modifications whose centers are simply balanced (k−1)-dimensional
polyhedral complexes.
Proof. We prove this proposition inductively by n. Without the loss of
genericity we may assume that X is a fan, i.e. each convex polyhedron
of X is a cone centered at the origin.
The base of the induction, when n = k, is trivial. If n > k let us take
a (n− k)-dimensional affine-linear subspace L ⊂ Rn given by Property
1.2. Choose a linear projection
λ : Rn → Rn−1
defined over Z and such that ker(λ) is a line contained in L.
The image λ(X) ⊂ Rn−1 is a k-dimensional polyhedral complex since
L is transversal to some facets of X . We claim that
λ|X : X → λ(X)
is a tropical modification once we identify Rn and Rn−1×R. The center
of this modification is the locus
Zf = {x ∈ R
n−1 | dim(λ−1(x) ∩X) > 0}.
Here we use the dimension in the usual topological sense. Note that the
(k − 1)-dimensional complex Zf ⊂ R
n−1 is simply balanced, existence
of the needed (n− k)-dimensional affine-linear spaces follows from the
fact that X ⊂ Rn is simply balanced.
MODULI SPACES OF RATIONAL TROPICAL CURVES 5
To justify our claim we note that near any point x ∈ Zf the sub-
complex Y ⊂ X obtained as the (Euclidean topology) closure of X r
λ−1(Zf) is a (set-theoretical) graph of a convex function. This, once
again, follows from the fact that X ⊂ Rn is simply balanced, this time
applied to the points in the facets on X r Y . Thus it gives a regular
tropical function f and it remains only to show that the the weight of
any facet of E ⊂ Zf is 1. But this follows, in turn, from the balancing
condition at λ−1(E) ∩ Y . �
2. Tropical curves and their moduli spaces
The definition of tropical variety is especially easy in dimension 1.
Tropical modifications take a graph into a graph (with arbitrary va-
lence of its vertices) and the tropical structure carried by the sheaf OX
amounts to a complete metric on the complement of the set of 1-valent
vertices of the graph X (cf. [5], [6], [1]). Thus, each 1-valent vertex of
a tropical curve X is adjacent to an edge of infinite length.
A tropical modification allows one to contract such an edge or to
attach it at any point of X other than a 1-valent vertex. If we have a
finite collection of marked points on X then by passing to an equivalent
model if needed we may assume that the set of marked points coincides
with the set of 1-valent vertices. (Of course, if X is a tree then we have
to have at least two marked points to make such assumption.)
The genus of a tropical curve X is dimH1(X). Let Mg,n be the
set of all tropical curves X of genus g with n distinct marked points.
Fixing a combinatorial type of a graph Γ with n marked leaves defines
a subset UΓ ⊂ Mg,n consisting of marked tropical curves with this
combinatorics. A length of any non-leaf edge of Γ defines a real-valued
function on UΓ. Such functions are called edge-length functions. To
avoid difficulties caused by self-automorphisms of X from now on we
restrict our attention to the case g = 0.
Definition 2.1. The combinatorial type of a tropical curve X is its
equivalence class up to homeomorphisms respecting the markings.
Combinatorial types partite the set M0,n into disjoint subsets. The
edge-length functions define the structure of the polyhedral cone RM≥0
in each of those subsets (as the lengths have to be positive). The
number M here is the number of the bounded (non-leaf) edges in X .
By the Euler characteristic reasoning it is equal to n − 3 if X is (1-
and) 3-valent, it is smaller if X has vertices of higher valence.
Furthermore, any face of the polyhedral cone RM≥0 coincides with the
cone corresponding to another combinatorial type, the one where we
6 GRIGORY MIKHALKIN
contract some of the edges of X to points. This gives the adjacency
(fan-like) structure on M0,n, so M0,n is a (non-compact) polyhedral
complex. In particular, it is a topological space.
Theorem 1. The set M0,n for n ≥ 3 admits the structure of an (n−3)-
dimensional tropical variety such that the edge-length functions are reg-
ular within each combinatorial type. Furthermore, the space M0,n can
be tropically embedded in RN for some N (i.e. M0,n can be presented
as a simply balanced complex).
Proof. This theorem is trivial for n = 3 as M0,3 is a point. Otherwise,
any two disjoint ordered pairs of marked points can be used to define
a global regular function on M0,n with values in R = T
×. Namely,
each such ordered pair defines the oriented path on the tropical curve
X connecting the corresponding marked points. These paths can be
embedded.
Since the two pairs of marked points are disjoint the intersection of
the two corresponding paths has to have finite length. We take this
length with the positive sign if the orientations agree and with the
negative sign otherwise. This defines a function on M0,n. We call such
functions the double ratio functions.
Take all possible disjoint pairs of marked points and use them as
coordinates for our embedding
ι : M0,n → R
where N is the number of all possible decompositions of n into two
disjoint pairs. The theorem now follows from the following two lemmas.
Note that, strictly speaking, each coordinate in RN depends not
only on the choice of two disjoint pairs of marked points but also on
the order of points in each pair. However, changing the order in one
of the pairs only reverses the sign of the double ratio. Taking an extra
coordinate for such a change of order would be redundant. Indeed, for
any balanced complex Y ⊂ RN and any affine-linear function λ : RN →
R with an integer slope the graph of λ is a balanced complex in RN+1
isomorphic to the initial complex Y .
Lemma 2.2. The map ι is a topological embedding.
Proof. First, let us prove that ι is an embedding. The combinatorial
type of X is determined by the set of the coordinates that do not
vanish on X . Indeed, any non-leaf edge E of the tree X separates the
leaves (i.e. the set of markings) into two classes corresponding to the
components of XrE. Let us take a coordinate in Rn that corresponds
MODULI SPACES OF RATIONAL TROPICAL CURVES 7
to four marking points (union of the two disjoint pairs) such that two of
these points belong to one class and two to the other class. We call such
a coordinate an E-compatible coordinate. Note that an E-compatible
coordinate vanishes on X if and only if the pairs of markings defined
by the coordinate agree with the pairs defined by the classes.
This observation suffices to reconstruct the combinatorial type of X .
Furthermore, the length of E equals to the minimal non-zero abso-
lute value of the E-compatible coordinates. This implies that ι is an
embedding. �
Lemma 2.3. The image ι(M0,n) is a simply balanced complex in R
Proof. This is a condition on codimension 1 faces of M0,n. First we
shall check it for the case n = 4. There are three ways to split the
four marking points into two disjoint pairs. Accordingly, there are
three combinatorial types of 3-valent trees with three marked leaves.
Thus our space M0,4 is homeomorphic to the tripod, or the “interior”
of the letter Y , see Figure 1. Each ray of this tripod correspond to
a combinatorial type of a 3-valent tree with 4 leaves while the vertex
correspond to the 4-valent tree.
Figure 1. The tropical moduli space M0,4 and its
points on the corresponding edges.
8 GRIGORY MIKHALKIN
Up to the sign we have the total of three double ratios for n = 4. Let
us e.g. take those defined by the following ordered pairs: {(12), (34)},
{(13), (24)} and {(14), (23)} Each is vanishing on the corresponding
ray of the tripod. Let us parameterize each ray of the tripod by its
only edge-length t ≥ 0 and compute the corresponding map to R3.
We have the following embeddings on the three rays
t 7→ (0, t, t), t 7→ (t, 0,−t), t 7→ (−t,−t, 0).
The sum of the primitive integer vectors parallel to the resulting direc-
tions is 0 and thus ι(M0,4) is balanced.
In the case n > 4 the codimension 1 faces of M0,n correspond to
the combinatorial types of X with a single 4-valent vertex. Near a
point inside of such face F the space M0,n looks like the product of
M0,4 and R
n−4. The factor Rn−4 comes from the edge-lengths on F
(its combinatorial type has n−4 bounded edges) while the factor M0,4
comes from perturbations of the 4-valent vertex (which result in a new
bounded edge in one of the three possible combinatorial types of the
result).
We have a well-defined map from the union U of the F -adjacent
facets to F by contracting the new edge to a point. Note that the
edge-length functions exhibit F as the positive quadrant in Rn−4. Fur-
thermore, in the combinatorial type of F we may choose 4 leaves such
that contracting all other leaves will take place outside of the 4-valent
vertex (see Figure 2). This contraction defines a map U → M0,4.
Figure 2. One of the possible contractions of a tree
with a 4-valent vertex to the tree corresponding to the
origin O ∈ M0,4.
The lemma now follows from the observation that the resulting de-
composition into M0,4 × R
n−4 agrees with the double ratio functions.
Indeed, note that the complement of the 4-valent vertex for a curve
in the combinatorial type F is composed of four components. If the
double ratio is such that its four markings are in one-to-one correspon-
dence with these components then at U it coincides with sum of the
pull-back of the corresponding double ratio in F with the pull-back of
the corresponding double ratio in M0,4. If one of the four components
MODULI SPACES OF RATIONAL TROPICAL CURVES 9
is lacking a marking from the double ratio ρ then ρ|U coincides with
the corresponding pull-back from F . �
Remark 2.4. The functions Zxi,xj from [4] do not define regular func-
tions on M0,n, contrary to what is written in [4]. These functions were
a result of an erroneous simplification of the double ratio functions.
But these functions cannot be regular as they are always positive and
Proposition 5.12 of [4] is not correct. Even the projectivization of the
embedding is not a balanced complex already for M0,5. One should
use the (non-simplified) double ratios instead.
Clearly, the space M0,n is non compact. However it is easy to com-
pactify it by allowing the lengths of bounded edges to assume infinite
values. Let M0,n be the space of connected trees with n (marked)
leaves such that each edge of this tree is assigned a length 0 < l ≤ +∞
so that each leaf has length necessarily equal to +∞.
Corollary 2.5. The space M0,n is a smooth compact tropical variety.
To verify that M0,n is smooth near a point x at the boundary
∂M0,n = M0,n rM0,n
we need to examine those double ratios that are equal to ±∞ at x.
There we use only those signs that result in −∞ do that the map takes
values in TN .
Remark 2.6. Note that the compactification M0,n ⊃ M0,n corresponds
to the Deligne-Mumford compactification in the complex case as under
the 1-parametric family collapse of a Riemannian surface to a tropical
curve the tropical length of an edge corresponds to the rate of growth
of the complex modulus of the holomorphic annulus collapsing to that
edge.
Furthermore, similarly to the complex story the infinite edges de-
compose a tropical curve into components (where the non-leaf edges
are finite). Any tropical map from an infinite edge which is bounded
would have to be constant and thus the image would have to split as
a union of several tropical curves in the target. Such decompositions
were used by Gathmann and Markwig in their deduction of the tropical
WDVV equation in R2, see [1].
3. Tropical ψ-classes
Note that we do have the forgetting maps
ftj : M0,n+1 → M0,n
10 GRIGORY MIKHALKIN
for j = 1, . . . , n + 1 by contracting the leaf with the j-marking. This
map is sometimes called the universal curve. Each marking k 6= j
defines a section σk of ftj. The conormal bundle to σk defines the ψk-
class in complex geometry (to avoid ambiguity we take j = n+1). This
notion can be adapted to our tropical setup.
Recall that so far our choice of tropical models in their equivalence
class was such that the leaves of the tropical curves were in 1-1 cor-
respondence with the markings. For this choice we have the images
σk(M0,n) contained in the boundary part of M0,n+1. This presenta-
tion is compatible with the point of view when we think about line
bundles in tropical geometry to be given by H1(X,O×). Here X is
the base of the bundle and O× is the sheaf of “non-vanishing” tropi-
cal regular functions. Such functions are given in the charts to RN by
affine-linear functions with integer slopes, see [6]. (Recall that T× = R
is an honest group with respect to tropical multiplication, i.e. the
classical addition.)
However, the following alternative construction allows one to obtain
the ψ-classes more geometrically (as we’ll illustrate in an example in
the next section). This approach is based on contracting the leaves
marked by number k.
The canonical class of a tropical curve is supported at its vertices,
namely we take each vertex with the multiplicity equal to its valence
minus 2, cf. [6]. Furthermore, the cotangent bundle near a 3-valent
vertex point can be viewed as a neighborhood of the origin for the line
given by the tropical polynomial “x + y + 1T” in R
2, so the +1 self-
intersection of the line gives the required multiplicity for the canonical
class at any 3-valent vertex. Thus we can use the intersections with the
corresponding codimension 1 faces inM0,n to define the ψ-classes there.
In other words, tropical ψ-classes will be supported on the (n − 4)-
dimensional faces in M0,n.
Namely, for a ψk-class we have to collect those codimension 1 faces
in M0,n whose only 4-valent vertex is adjacent to the leaf marked by
k. After a contraction of this leaf we get a 3-valent vertex, thus the
multiplicity of every face in a ψ-divisor is 1. We arrive to the following
definition.
Definition 3.1. The tropical ψk-divisor Ψk ⊂ M0,n is the union of
those (n−4)-dimensional faces that correspond to tropical curves with
a 4-valent vertex adjacent to the leaf marked by k, k = 1, . . . , n. Each
such face is taken with the multiplicity 1.
Proposition 3.2. The subcomplex Ψk is a divisor, i.e. satisfies the
balancing condition.
MODULI SPACES OF RATIONAL TROPICAL CURVES 11
Proof. Recall that the balancing condition is a condition at (n − 5)-
dimensional faces. In M0,n there are two types of such faces, one
corresponding to tropical curves with two 4-valent vertices and one
corresponding to a tropical curve with a 5-valent vertex.
Near the faces of the first type the moduli space M0,n is locally a
product of two copies of M0,4 and R
n−5. The Ψ-divisor is a product
of Rn−5, one copy of M0,4 and the central (3-valent) point in the other
copy of M0,4 (this is the point corresponding to the 4-valent vertex
adjacent to the leaf marked by k). Thus the balancing condition holds
trivially in this case.
Near the faces of the second type the moduli space M0,n is locally a
product of M0,5 and R
n−5. As in the proof of Theorem 1 each double
ratio decomposes to the sum of the corresponding double ration inM0,5
(perhaps trivial if two of the markings for the double ratio correspond
to the same edge adjacent to the 5-valent vertex) and an affine-linear
function in Rn−5. Thus it suffices to check only the balancing condition
for the Ψ-divisors in M0,5. This example is considered in details in the
next section. The balancing condition there follows from Proposition
4.1. �
Conjecturally, the tropical Ψ-divisors are limits of some natural rep-
resentatives of the divisors for the complex ψ-classes under the collapse
of the complex moduli space onto the corresponding tropical moduli
space M0,n. Note that our choice for the tropical Ψ-divisor is not con-
tained in the boundary ∂M0,n ⊂ M0,n (cf. the calculus of the complex
boundary classes in [2]), but comes as a closure of a divisor in M0,n.
4. The space M0,5
We have already described the moduli space M0,4 as the tripod of
Figure 1. It has only one 0-dimensional face O ∈ M0,4. This point
(considered as a divisor) coincides with the divisors Ψ1 = Ψ2 = Ψ3 =
Ψ4. The description of M0,5 is somewhat more interesting.
There are 15 combinatorial types of 3-valent trees with 5 marked
leaves. If we forget about the markings there is only one homeomor-
phism class for such a curve (see Figure 3). To get the number of
non-isomorphic markings we take the number all possible reordering of
vertices (equal to 5! = 120) and divide by 23 = 8 as there is an 8-fold
symmetry of reordering. Indeed there is one symmetry interchanging
the left two leaves, one interchanging the right two leaves and the cen-
tral symmetry around the central leave of the 3-valent tree on top of
Figure 3.
12 GRIGORY MIKHALKIN
1 15 5
Figure 3. Adjunction of combinatorial types corre-
sponding to the quadrant connecting the rays (45) and
(12).
(25) (13)
(15) (23)
Figure 4. The link of the origin in M0,5.
MODULI SPACES OF RATIONAL TROPICAL CURVES 13
Thus the space M0,5 is a union of 15 quadrants R
≥0. These quad-
rants are attached along the rays which correspond to the combinato-
rial types of curves with one 4-valent vertex. Such curves also have one
3-valent vertex which is adjacent to two leaves and the only bounded
edge of the curve, see the bottom of Figure 3. Such combinatorial types
are determined by the markings of the two leaves emanating from the
3-valent vertex. Thus we have a total of
= 10 of such rays.
The two boundary edges of the quadrant correspond to contractions
of the bounded edges of the combinatorial type as shown on Figure
3. The global picture of adjacency of quadrants and rays is shown
on Figure 4 where the reader may recognize the well-known Petersen
graph, cf. the related tropical Grassmannian picture in [7]. Vertices of
this graph correspond to the rays of M0,5 while the edges correspond
to the quadrants. Thus the whole picture may be interpreted as the
link of the only vertex O ∈ M0,5 (the point O corresponds to the tree
with a 5-valent vertex adjacent to all the leaves).
To locate the Ψk-divisor we recall that the kth leaf has to be adjacent
to a 4-valent vertex if it appears in Ψk. This means that Ψk consists
of 6 rays that are marked by pairs not containing k.
Proposition 4.1. The subcomplex Ψk ⊂ M0,5 is a divisor.
Proof. Since the whole M0,5 is S5-symmetric it suffices to check the
balancing condition only for Ψ1. The embedding M0,5 ⊂ R
N is given
by the double ratios, so it suffices to check that for each double ratio
function the sum of its gradients on the six rays of Ψ1 vanishes.
If the double ratio is determined by two pairs disjoint from the mark-
ing 1, e.g. by {(23), (45)} then its restriction onto the six rays of Ψ1 is
the same as its restriction to the three rays M0,4 taken twice and thus
balanced. Namely its gradient is 1 on the rays (24) and (35); −1 on
the rays (25) and (34); and 0 on the rays (23) and (45).
If the four markings of the double ratio contain the marking 1 then
thanks to the symmetry we may assume that the double ratio is given
by {(12), (34)}. It vanishes on the rays (34), (35), (45) and (25); it has
gradient +1 on the ray (24) and the gradient −1 on the ray (23). Once
again, the balancing condition holds. �
As our final example of the paper we would like to describe explicitly
the universal curve
ft5 : M0,5 → M0,4.
This is presented on Figure 5. Once again, we interpret the Peterson
graph as the link L of the vertex O ∈ M0,5. Similarly, the link of the
14 GRIGORY MIKHALKIN
Figure 5. The three fibers and four sections of the
universal curve ft5 : M0,5 → M0,4.
origin in M0,4 consists of three points. Thus L is the union of the fibers
of ft5 (away from a neighborhood of infinity) over these three points
and four copies of a neighborhood of the origin in M0,4 corresponding
to the four sections σ1, σ2, σ3 and σ4 of the universal curve. Figure
5 depicts the fibers in L with solid lines and the sections with dashed
lines.
Acknowledgements. I am thankful to Valery Alexeev and Kristin
Shaw for discussions related to geometry of tropical moduli spaces.
My research is supported in part by NSERC.
References
[1] Gathmann, A., Markwig, H., Kontsevich’s formula and the WDVV equations
in tropical geometry, http://arxiv.org/abs/math.AG/0509628.
[2] Keel, S., Intersection theory of moduli space of stable N -pointed curves of
genus zero, Transactions of the AMS 330 (1992), 545–574.
[3] Litvinov, G. L., The Maslov dequantization, idempotent and tropical mathe-
matics: a very brief introduction. In Idempotent mathematics and mathemat-
ical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005,
1–17.
[4] Mikhalkin, G., Tropical Geometry and its application, to appear in the Pro-
ceedings on the ICM-2006, Madrid; http://arxiv.org/abs/math/0601041.
[5] Mikhalkin, G., Tropical Geometry, book in preparation.
http://arxiv.org/abs/math.AG/0509628
http://arxiv.org/abs/math/0601041
MODULI SPACES OF RATIONAL TROPICAL CURVES 15
[6] Mikhalkin, G., Zharkov, I., Tropical curves, their Jacobians and Theta func-
tions, http://arxiv.org/abs/math/0612267.
[7] Speyer, D., Sturmfels, B., The tropical Grassmannian. Adv. Geom. 4 (2004),
no. 3, 389–411.
[8] Viro, O. Ya., Dequantization of real algebraic geometry on logarithmic paper.
In European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math.,
201, Birkhäuser, Basel, 2001, 135–146.
Department of Mathematics, University of Toronto, 40 St George
St, Toronto ON M5S 2E4 Canada
http://arxiv.org/abs/math/0612267
	1. Introduction: smooth tropical varieties
	2. Tropical curves and their moduli spaces
	3. Tropical -classes
	4. The space M0,5
	References

  This note is devoted to the definition of moduli spaces of rational tropical
curves with n marked points. We show that this space has a structure of a
smooth tropical variety of dimension n-3. We define the Deligne-Mumford
compactification of this space and tropical $\psi$-class divisors.

Introduction: smooth tropical varieties
In this section we follow the definitions of [5] and [4].
The underlying algebra of tropical geometry is given by the semifield
T = R∪{−∞} of tropical numbers. The tropical arithmetic operations
are “a + b” = max{a, b} and “ab” = a + b. The quotation marks are
used to distinguish between the tropical and classical operations. With
respect to addition T is a commutative semigroup with zero “0T” =
−∞. With respect to multiplication T× = T r {0T} ≈ R is an honest
commutative group with the unit “1T” = 0. Furthermore, the addition
and multiplication satisfy to the distribution law “a(b+c)” = “ab+ac”,
a, b, c ∈ T. These operations may be viewed as a result of the so-called
dequantization of the classical arithmetic operations that underlies the
patchworking construction, see [3] and [8].
These tropical operations allow one to define tropical Laurent poly-
nomials. Namely, a tropical Laurent polynomial is a function f : Rn →
f(x) = “
j” = max
(aj + jx),
where jx denotes the scalar product, x ∈ (T×)n ≈ Rn, j ∈ Zn and only
finitely many coefficients aj ∈ T are non-zero (i.e. not −∞).
http://arxiv.org/abs/0704.0839v1
2 GRIGORY MIKHALKIN
Affine-linear functions with integer slopes (for brevity we call them
simply affine functions) form an important subcollection of all Laurent
polynomials. Namely, these are such functions f : Rn → R that both
f and “1T
” = −f are tropical Laurent polynomials.
We equip Tn ≈ [−∞,∞)n with the Euclidean topology. Let U ⊂ Tn
be an open set.
Definition 1.1. A continuous function f : U → T is called regular if
its restriction to U ∩ Rn coincides with a restriction of some tropical
Laurent polynomial to U ∩ Rn.
We denote the sheaf of regular functions on Tn with O (or sometimes
OTn to avoid confusion). Any subset X ⊂ T
n gets an induced regular
sheaf OX by restriction. For our purposes we restrict our attention only
to the case when X is a polyhedral complex, i.e. when X is the closure
of a union of convex polyhedra (possibly unbounded) in Rn such that
the intersection of any number of such polyhedra is their common face.
We say that X is an k-dimensional polyhedral complex if it is obtained
from a union of k-dimensional polyhedra. These polyhedra are called
the facets of X .
Let V ⊂ X be an open set and f ∈ OX(V ) be a regular function
in V . A point x ∈ V is called a “zero point” of f if the restriction of
” = −f toW ⊂ V is not regular for any open neighborhoodW ∋ x.
Note that it may happen that x is a “zero point” for φ : U → T, but
not for φ|X∩U . It is easy to see that if X is a k-dimensional polyhe-
dral complex then the “zero locus” Zf of f is a (k − 1)-dimensional
polyhedral subcomplex.
To each facet of Zf we may associate a natural number, called its
weight (or degree). To do this we choose a “zero point” x inside such
a facet. We say that x is a “simple zero” for f if for any local de-
composition into a sum (i.e. the tropical product) of regular function
f = “gh” = g+h on V near x we have either g or h affine (i.e. without
a “zero”). We say that the weight is l if f can be locally decomposed
into a tropical product of l functions with a simple zero at x.
A regular function f allows us to make the following modification on
its domain V ⊂ X ⊂ Tn. Consider the graph
Γf ⊂ V × T ⊂ T
It is easy to see that the “zero locus”
Γ̄f ⊂ V × T
of the (regular) function “y+ f(x)” (defined on V ×T), where x is the
coordinate on V and y is the coordinate on T, coincides with the union
MODULI SPACES OF RATIONAL TROPICAL CURVES 3
of Γf and the undergraph
UΓf,Z = {(x, y) ∈ V × T | x ∈ Zf , y ≤ f(x)}.
Furthermore, the weight of a facet F ⊂ Γ̄f is 1 if F ∈ Γf (recall that
as V is an unweighted polyhedral complex all the weights of its facets
are equal to one) and is the weight of the corresponding facet of Zf if
F ∈ UΓf,Z . We view Γ̄f as a “tropical closure” of the set-theoretical
graph Γf . Note that we have a map Γ̄f → V . We set Ṽ = Γ̄f to be
the result of the tropical modification µf : Ṽ → V along the regular
function f . The locus Zf is called the center of tropical modification.
The weights of the facets of Ṽ supplies us with some inconvenience
as they should be incorporated in the definition of the regular sheaf OṼ
on Ṽ . Namely, the affine functions defined by OṼ on a facet of weight
w should contain the group of functions that come as restrictions to
this facet of the affine functions on Tn+1 as a subgroup of index w.
Sometimes one can get rid of the weights of Ṽ by a reparameteriza-
tion with the help of a map V̄ → Ṽ that is given by locally linear maps
in the corresponding charts. Indeed, the restriction of µg : V̄ → Ṽ to
a facet is locally given by a linear function between two k-dimensional
affine-linear spaces defined over Z. If its determinant equals to w then
the push-forward of OV̄ supplies an extension of OṼ required by the
weights. Note however that if w > 1 then the converse map is not
defined over Z and thus is not given by elements of OṼ .
Tropical modifications give the basic equivalence relation in Tropical
Geometry. It can be shown that if we start from Tk and do a number of
tropical modifications on it then the result is a k-dimensional polyhe-
dral complex Y ⊂ Tn that satisfies to the following balancing property
(cf. Property 3.3 in [4] where balancing is restated in an equivalent
way).
Property 1.2. Let E ⊂ Y ∩ RN be a (k − 1)-dimensional face and
F1, . . . , Fl be the facets of Y adjacent to F whose weights are w1, . . . , wl.
Let L ⊂ RN be a (N−k)-dimensional affine-linear space with an integer
slope and such that it intersects E. For a generic (real) vector v ∈ RN
the intersection Fj ∩ (L + v) is either empty or a single point. Let
ΛFj ⊂ Z
N be the integer vectors parallel to Fj and ΛL ⊂ Z
N be the
integer vectors parallel to L. Set λj to be the product of wj and the
index of the subgroup ΛFj +ΛL ⊂ Z
N . We say that Y ⊂ Tn is balanced
if for any choice of E, L and a small generic v the sum
j | Fj∩(L+v)6=∅
4 GRIGORY MIKHALKIN
is independent of v. We say that Y is simply balanced if in addition for
every j we can find L and v so that Fj ∩ (L + v) 6= ∅, ιL = 1 and for
every small v there exists an affine hyperplane Hv ⊂ L such that the
intersection Y ∩ (L + v) sits entirely on one side of Hv + v in L + v
while the intersection Y ∩ (Hv + v) is a point.
Definition 1.3 (cf. [5],[4]). A topological space X enhanced with a
sheaf of tropical functions OX is called a (smooth) tropical variety of
dimension k if for every x ∈ X there exist an open set U ∋ x and an
open set V in a simply balanced polyhedral complex Y ⊂ TN such that
the restrictions OX |U and OY |V are isomorphic.
Tropical varieties are considered up to the equivalence generated by
tropical modifications. It can be shown that a smooth tropical variety
of dimension k can be locally obtained from Tk by a sequence of tropical
modifications centered at smooth tropical varieties of dimension (k−1).
This follows from the following proposition.
Proposition 1.4. Any k-dimensional simply balanced polyhedral com-
plex X ⊂ Rn can be obtained from Tk by a sequence of consecutive trop-
ical modifications whose centers are simply balanced (k−1)-dimensional
polyhedral complexes.
Proof. We prove this proposition inductively by n. Without the loss of
genericity we may assume that X is a fan, i.e. each convex polyhedron
of X is a cone centered at the origin.
The base of the induction, when n = k, is trivial. If n > k let us take
a (n− k)-dimensional affine-linear subspace L ⊂ Rn given by Property
1.2. Choose a linear projection
λ : Rn → Rn−1
defined over Z and such that ker(λ) is a line contained in L.
The image λ(X) ⊂ Rn−1 is a k-dimensional polyhedral complex since
L is transversal to some facets of X . We claim that
λ|X : X → λ(X)
is a tropical modification once we identify Rn and Rn−1×R. The center
of this modification is the locus
Zf = {x ∈ R
n−1 | dim(λ−1(x) ∩X) > 0}.
Here we use the dimension in the usual topological sense. Note that the
(k − 1)-dimensional complex Zf ⊂ R
n−1 is simply balanced, existence
of the needed (n− k)-dimensional affine-linear spaces follows from the
fact that X ⊂ Rn is simply balanced.
MODULI SPACES OF RATIONAL TROPICAL CURVES 5
To justify our claim we note that near any point x ∈ Zf the sub-
complex Y ⊂ X obtained as the (Euclidean topology) closure of X r
λ−1(Zf) is a (set-theoretical) graph of a convex function. This, once
again, follows from the fact that X ⊂ Rn is simply balanced, this time
applied to the points in the facets on X r Y . Thus it gives a regular
tropical function f and it remains only to show that the the weight of
any facet of E ⊂ Zf is 1. But this follows, in turn, from the balancing
condition at λ−1(E) ∩ Y . �
2. Tropical curves and their moduli spaces
The definition of tropical variety is especially easy in dimension 1.
Tropical modifications take a graph into a graph (with arbitrary va-
lence of its vertices) and the tropical structure carried by the sheaf OX
amounts to a complete metric on the complement of the set of 1-valent
vertices of the graph X (cf. [5], [6], [1]). Thus, each 1-valent vertex of
a tropical curve X is adjacent to an edge of infinite length.
A tropical modification allows one to contract such an edge or to
attach it at any point of X other than a 1-valent vertex. If we have a
finite collection of marked points on X then by passing to an equivalent
model if needed we may assume that the set of marked points coincides
with the set of 1-valent vertices. (Of course, if X is a tree then we have
to have at least two marked points to make such assumption.)
The genus of a tropical curve X is dimH1(X). Let Mg,n be the
set of all tropical curves X of genus g with n distinct marked points.
Fixing a combinatorial type of a graph Γ with n marked leaves defines
a subset UΓ ⊂ Mg,n consisting of marked tropical curves with this
combinatorics. A length of any non-leaf edge of Γ defines a real-valued
function on UΓ. Such functions are called edge-length functions. To
avoid difficulties caused by self-automorphisms of X from now on we
restrict our attention to the case g = 0.
Definition 2.1. The combinatorial type of a tropical curve X is its
equivalence class up to homeomorphisms respecting the markings.
Combinatorial types partite the set M0,n into disjoint subsets. The
edge-length functions define the structure of the polyhedral cone RM≥0
in each of those subsets (as the lengths have to be positive). The
number M here is the number of the bounded (non-leaf) edges in X .
By the Euler characteristic reasoning it is equal to n − 3 if X is (1-
and) 3-valent, it is smaller if X has vertices of higher valence.
Furthermore, any face of the polyhedral cone RM≥0 coincides with the
cone corresponding to another combinatorial type, the one where we
6 GRIGORY MIKHALKIN
contract some of the edges of X to points. This gives the adjacency
(fan-like) structure on M0,n, so M0,n is a (non-compact) polyhedral
complex. In particular, it is a topological space.
Theorem 1. The set M0,n for n ≥ 3 admits the structure of an (n−3)-
dimensional tropical variety such that the edge-length functions are reg-
ular within each combinatorial type. Furthermore, the space M0,n can
be tropically embedded in RN for some N (i.e. M0,n can be presented
as a simply balanced complex).
Proof. This theorem is trivial for n = 3 as M0,3 is a point. Otherwise,
any two disjoint ordered pairs of marked points can be used to define
a global regular function on M0,n with values in R = T
×. Namely,
each such ordered pair defines the oriented path on the tropical curve
X connecting the corresponding marked points. These paths can be
embedded.
Since the two pairs of marked points are disjoint the intersection of
the two corresponding paths has to have finite length. We take this
length with the positive sign if the orientations agree and with the
negative sign otherwise. This defines a function on M0,n. We call such
functions the double ratio functions.
Take all possible disjoint pairs of marked points and use them as
coordinates for our embedding
ι : M0,n → R
where N is the number of all possible decompositions of n into two
disjoint pairs. The theorem now follows from the following two lemmas.
Note that, strictly speaking, each coordinate in RN depends not
only on the choice of two disjoint pairs of marked points but also on
the order of points in each pair. However, changing the order in one
of the pairs only reverses the sign of the double ratio. Taking an extra
coordinate for such a change of order would be redundant. Indeed, for
any balanced complex Y ⊂ RN and any affine-linear function λ : RN →
R with an integer slope the graph of λ is a balanced complex in RN+1
isomorphic to the initial complex Y .
Lemma 2.2. The map ι is a topological embedding.
Proof. First, let us prove that ι is an embedding. The combinatorial
type of X is determined by the set of the coordinates that do not
vanish on X . Indeed, any non-leaf edge E of the tree X separates the
leaves (i.e. the set of markings) into two classes corresponding to the
components of XrE. Let us take a coordinate in Rn that corresponds
MODULI SPACES OF RATIONAL TROPICAL CURVES 7
to four marking points (union of the two disjoint pairs) such that two of
these points belong to one class and two to the other class. We call such
a coordinate an E-compatible coordinate. Note that an E-compatible
coordinate vanishes on X if and only if the pairs of markings defined
by the coordinate agree with the pairs defined by the classes.
This observation suffices to reconstruct the combinatorial type of X .
Furthermore, the length of E equals to the minimal non-zero abso-
lute value of the E-compatible coordinates. This implies that ι is an
embedding. �
Lemma 2.3. The image ι(M0,n) is a simply balanced complex in R
Proof. This is a condition on codimension 1 faces of M0,n. First we
shall check it for the case n = 4. There are three ways to split the
four marking points into two disjoint pairs. Accordingly, there are
three combinatorial types of 3-valent trees with three marked leaves.
Thus our space M0,4 is homeomorphic to the tripod, or the “interior”
of the letter Y , see Figure 1. Each ray of this tripod correspond to
a combinatorial type of a 3-valent tree with 4 leaves while the vertex
correspond to the 4-valent tree.
Figure 1. The tropical moduli space M0,4 and its
points on the corresponding edges.
8 GRIGORY MIKHALKIN
Up to the sign we have the total of three double ratios for n = 4. Let
us e.g. take those defined by the following ordered pairs: {(12), (34)},
{(13), (24)} and {(14), (23)} Each is vanishing on the corresponding
ray of the tripod. Let us parameterize each ray of the tripod by its
only edge-length t ≥ 0 and compute the corresponding map to R3.
We have the following embeddings on the three rays
t 7→ (0, t, t), t 7→ (t, 0,−t), t 7→ (−t,−t, 0).
The sum of the primitive integer vectors parallel to the resulting direc-
tions is 0 and thus ι(M0,4) is balanced.
In the case n > 4 the codimension 1 faces of M0,n correspond to
the combinatorial types of X with a single 4-valent vertex. Near a
point inside of such face F the space M0,n looks like the product of
M0,4 and R
n−4. The factor Rn−4 comes from the edge-lengths on F
(its combinatorial type has n−4 bounded edges) while the factor M0,4
comes from perturbations of the 4-valent vertex (which result in a new
bounded edge in one of the three possible combinatorial types of the
result).
We have a well-defined map from the union U of the F -adjacent
facets to F by contracting the new edge to a point. Note that the
edge-length functions exhibit F as the positive quadrant in Rn−4. Fur-
thermore, in the combinatorial type of F we may choose 4 leaves such
that contracting all other leaves will take place outside of the 4-valent
vertex (see Figure 2). This contraction defines a map U → M0,4.
Figure 2. One of the possible contractions of a tree
with a 4-valent vertex to the tree corresponding to the
origin O ∈ M0,4.
The lemma now follows from the observation that the resulting de-
composition into M0,4 × R
n−4 agrees with the double ratio functions.
Indeed, note that the complement of the 4-valent vertex for a curve
in the combinatorial type F is composed of four components. If the
double ratio is such that its four markings are in one-to-one correspon-
dence with these components then at U it coincides with sum of the
pull-back of the corresponding double ratio in F with the pull-back of
the corresponding double ratio in M0,4. If one of the four components
MODULI SPACES OF RATIONAL TROPICAL CURVES 9
is lacking a marking from the double ratio ρ then ρ|U coincides with
the corresponding pull-back from F . �
Remark 2.4. The functions Zxi,xj from [4] do not define regular func-
tions on M0,n, contrary to what is written in [4]. These functions were
a result of an erroneous simplification of the double ratio functions.
But these functions cannot be regular as they are always positive and
Proposition 5.12 of [4] is not correct. Even the projectivization of the
embedding is not a balanced complex already for M0,5. One should
use the (non-simplified) double ratios instead.
Clearly, the space M0,n is non compact. However it is easy to com-
pactify it by allowing the lengths of bounded edges to assume infinite
values. Let M0,n be the space of connected trees with n (marked)
leaves such that each edge of this tree is assigned a length 0 < l ≤ +∞
so that each leaf has length necessarily equal to +∞.
Corollary 2.5. The space M0,n is a smooth compact tropical variety.
To verify that M0,n is smooth near a point x at the boundary
∂M0,n = M0,n rM0,n
we need to examine those double ratios that are equal to ±∞ at x.
There we use only those signs that result in −∞ do that the map takes
values in TN .
Remark 2.6. Note that the compactification M0,n ⊃ M0,n corresponds
to the Deligne-Mumford compactification in the complex case as under
the 1-parametric family collapse of a Riemannian surface to a tropical
curve the tropical length of an edge corresponds to the rate of growth
of the complex modulus of the holomorphic annulus collapsing to that
edge.
Furthermore, similarly to the complex story the infinite edges de-
compose a tropical curve into components (where the non-leaf edges
are finite). Any tropical map from an infinite edge which is bounded
would have to be constant and thus the image would have to split as
a union of several tropical curves in the target. Such decompositions
were used by Gathmann and Markwig in their deduction of the tropical
WDVV equation in R2, see [1].
3. Tropical ψ-classes
Note that we do have the forgetting maps
ftj : M0,n+1 → M0,n
10 GRIGORY MIKHALKIN
for j = 1, . . . , n + 1 by contracting the leaf with the j-marking. This
map is sometimes called the universal curve. Each marking k 6= j
defines a section σk of ftj. The conormal bundle to σk defines the ψk-
class in complex geometry (to avoid ambiguity we take j = n+1). This
notion can be adapted to our tropical setup.
Recall that so far our choice of tropical models in their equivalence
class was such that the leaves of the tropical curves were in 1-1 cor-
respondence with the markings. For this choice we have the images
σk(M0,n) contained in the boundary part of M0,n+1. This presenta-
tion is compatible with the point of view when we think about line
bundles in tropical geometry to be given by H1(X,O×). Here X is
the base of the bundle and O× is the sheaf of “non-vanishing” tropi-
cal regular functions. Such functions are given in the charts to RN by
affine-linear functions with integer slopes, see [6]. (Recall that T× = R
is an honest group with respect to tropical multiplication, i.e. the
classical addition.)
However, the following alternative construction allows one to obtain
the ψ-classes more geometrically (as we’ll illustrate in an example in
the next section). This approach is based on contracting the leaves
marked by number k.
The canonical class of a tropical curve is supported at its vertices,
namely we take each vertex with the multiplicity equal to its valence
minus 2, cf. [6]. Furthermore, the cotangent bundle near a 3-valent
vertex point can be viewed as a neighborhood of the origin for the line
given by the tropical polynomial “x + y + 1T” in R
2, so the +1 self-
intersection of the line gives the required multiplicity for the canonical
class at any 3-valent vertex. Thus we can use the intersections with the
corresponding codimension 1 faces inM0,n to define the ψ-classes there.
In other words, tropical ψ-classes will be supported on the (n − 4)-
dimensional faces in M0,n.
Namely, for a ψk-class we have to collect those codimension 1 faces
in M0,n whose only 4-valent vertex is adjacent to the leaf marked by
k. After a contraction of this leaf we get a 3-valent vertex, thus the
multiplicity of every face in a ψ-divisor is 1. We arrive to the following
definition.
Definition 3.1. The tropical ψk-divisor Ψk ⊂ M0,n is the union of
those (n−4)-dimensional faces that correspond to tropical curves with
a 4-valent vertex adjacent to the leaf marked by k, k = 1, . . . , n. Each
such face is taken with the multiplicity 1.
Proposition 3.2. The subcomplex Ψk is a divisor, i.e. satisfies the
balancing condition.
MODULI SPACES OF RATIONAL TROPICAL CURVES 11
Proof. Recall that the balancing condition is a condition at (n − 5)-
dimensional faces. In M0,n there are two types of such faces, one
corresponding to tropical curves with two 4-valent vertices and one
corresponding to a tropical curve with a 5-valent vertex.
Near the faces of the first type the moduli space M0,n is locally a
product of two copies of M0,4 and R
n−5. The Ψ-divisor is a product
of Rn−5, one copy of M0,4 and the central (3-valent) point in the other
copy of M0,4 (this is the point corresponding to the 4-valent vertex
adjacent to the leaf marked by k). Thus the balancing condition holds
trivially in this case.
Near the faces of the second type the moduli space M0,n is locally a
product of M0,5 and R
n−5. As in the proof of Theorem 1 each double
ratio decomposes to the sum of the corresponding double ration inM0,5
(perhaps trivial if two of the markings for the double ratio correspond
to the same edge adjacent to the 5-valent vertex) and an affine-linear
function in Rn−5. Thus it suffices to check only the balancing condition
for the Ψ-divisors in M0,5. This example is considered in details in the
next section. The balancing condition there follows from Proposition
4.1. �
Conjecturally, the tropical Ψ-divisors are limits of some natural rep-
resentatives of the divisors for the complex ψ-classes under the collapse
of the complex moduli space onto the corresponding tropical moduli
space M0,n. Note that our choice for the tropical Ψ-divisor is not con-
tained in the boundary ∂M0,n ⊂ M0,n (cf. the calculus of the complex
boundary classes in [2]), but comes as a closure of a divisor in M0,n.
4. The space M0,5
We have already described the moduli space M0,4 as the tripod of
Figure 1. It has only one 0-dimensional face O ∈ M0,4. This point
(considered as a divisor) coincides with the divisors Ψ1 = Ψ2 = Ψ3 =
Ψ4. The description of M0,5 is somewhat more interesting.
There are 15 combinatorial types of 3-valent trees with 5 marked
leaves. If we forget about the markings there is only one homeomor-
phism class for such a curve (see Figure 3). To get the number of
non-isomorphic markings we take the number all possible reordering of
vertices (equal to 5! = 120) and divide by 23 = 8 as there is an 8-fold
symmetry of reordering. Indeed there is one symmetry interchanging
the left two leaves, one interchanging the right two leaves and the cen-
tral symmetry around the central leave of the 3-valent tree on top of
Figure 3.
12 GRIGORY MIKHALKIN
1 15 5
Figure 3. Adjunction of combinatorial types corre-
sponding to the quadrant connecting the rays (45) and
(12).
(25) (13)
(15) (23)
Figure 4. The link of the origin in M0,5.
MODULI SPACES OF RATIONAL TROPICAL CURVES 13
Thus the space M0,5 is a union of 15 quadrants R
≥0. These quad-
rants are attached along the rays which correspond to the combinato-
rial types of curves with one 4-valent vertex. Such curves also have one
3-valent vertex which is adjacent to two leaves and the only bounded
edge of the curve, see the bottom of Figure 3. Such combinatorial types
are determined by the markings of the two leaves emanating from the
3-valent vertex. Thus we have a total of
= 10 of such rays.
The two boundary edges of the quadrant correspond to contractions
of the bounded edges of the combinatorial type as shown on Figure
3. The global picture of adjacency of quadrants and rays is shown
on Figure 4 where the reader may recognize the well-known Petersen
graph, cf. the related tropical Grassmannian picture in [7]. Vertices of
this graph correspond to the rays of M0,5 while the edges correspond
to the quadrants. Thus the whole picture may be interpreted as the
link of the only vertex O ∈ M0,5 (the point O corresponds to the tree
with a 5-valent vertex adjacent to all the leaves).
To locate the Ψk-divisor we recall that the kth leaf has to be adjacent
to a 4-valent vertex if it appears in Ψk. This means that Ψk consists
of 6 rays that are marked by pairs not containing k.
Proposition 4.1. The subcomplex Ψk ⊂ M0,5 is a divisor.
Proof. Since the whole M0,5 is S5-symmetric it suffices to check the
balancing condition only for Ψ1. The embedding M0,5 ⊂ R
N is given
by the double ratios, so it suffices to check that for each double ratio
function the sum of its gradients on the six rays of Ψ1 vanishes.
If the double ratio is determined by two pairs disjoint from the mark-
ing 1, e.g. by {(23), (45)} then its restriction onto the six rays of Ψ1 is
the same as its restriction to the three rays M0,4 taken twice and thus
balanced. Namely its gradient is 1 on the rays (24) and (35); −1 on
the rays (25) and (34); and 0 on the rays (23) and (45).
If the four markings of the double ratio contain the marking 1 then
thanks to the symmetry we may assume that the double ratio is given
by {(12), (34)}. It vanishes on the rays (34), (35), (45) and (25); it has
gradient +1 on the ray (24) and the gradient −1 on the ray (23). Once
again, the balancing condition holds. �
As our final example of the paper we would like to describe explicitly
the universal curve
ft5 : M0,5 → M0,4.
This is presented on Figure 5. Once again, we interpret the Peterson
graph as the link L of the vertex O ∈ M0,5. Similarly, the link of the
14 GRIGORY MIKHALKIN
Figure 5. The three fibers and four sections of the
universal curve ft5 : M0,5 → M0,4.
origin in M0,4 consists of three points. Thus L is the union of the fibers
of ft5 (away from a neighborhood of infinity) over these three points
and four copies of a neighborhood of the origin in M0,4 corresponding
to the four sections σ1, σ2, σ3 and σ4 of the universal curve. Figure
5 depicts the fibers in L with solid lines and the sections with dashed
lines.
Acknowledgements. I am thankful to Valery Alexeev and Kristin
Shaw for discussions related to geometry of tropical moduli spaces.
My research is supported in part by NSERC.
References
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http://arxiv.org/abs/math/0601041
MODULI SPACES OF RATIONAL TROPICAL CURVES 15
[6] Mikhalkin, G., Zharkov, I., Tropical curves, their Jacobians and Theta func-
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Department of Mathematics, University of Toronto, 40 St George
St, Toronto ON M5S 2E4 Canada
http://arxiv.org/abs/math/0612267
	1. Introduction: smooth tropical varieties
	2. Tropical curves and their moduli spaces
	3. Tropical -classes
	4. The space M0,5
	References
