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009 FIRST STEPS IN TROPICAL INTERSECTION THEORY

LARS ALLERMANN AND JOHANNES RAU

ABSTRACT. We establish first parts of a tropical intersection theory.Namely, we definecycles, Cartier divisors and intersection products between these two (without passing torational equivalence) and discuss push-forward and pull-back. We do this first for fansin R

n and then for “abstract” cycles that are fans locally. With regard to applicationsin enumerative geometry, we finally have a look at rational equivalence and intersectionproducts of cycles and cycle classes inR

n.

1. INTRODUCTION

Tropical geometry is a recent development in the field of algebraic geometry that triesto transform algebro-geometric problems into easier, purely combinatorial ones. In thelast few years various authors were able to answer questionsof enumerative algebraicgeometry using these techniques. In order to determine the number of (classical) curvesmeeting given conditions in some ambient space they constructed moduli spaces of tropicalcurves and had to intersect the corresponding tropical conditions in these moduli spaces.Since there is no tropical intersection theory yet the computation of the arising intersectionmultiplicities and the proof of the independence of the choice of the conditions had to berepeated for every single problem without the tools of an elaborated intersection theory(see for example [GM], [KM]).

A first draft of a general tropical intersection theory without proofs has been presentedby Mikhalkin in [M]. The concepts introduced there — if set uprigorously — wouldhelp to unify and solve the above mentioned problems and would provide utilities forfurther applications. Thus in this paper we develop in detail the basics of a general tropicalintersection theory based on Mikhalkin’s ideas.

This paper consists of three parts: In the first part (sections 2 - 4) we firstly introduce affinetropical cycles as balanced weighted fans modulo refinements and affine tropical varietiesas affine cycles with non-negative weights. One would like todefine the intersection oftwo such objects but in general neither is the set-theoreticintersection of two cycles againa cycle nor does the concept of stable intersection as introduced in [RGST] work for arbi-trary ambient spaces as can be seen in example 3.10. Therefore we introduce the notion ofaffine Cartier divisors on tropical cycles as piecewise integer affine linear functions moduloglobally affine linear functions and define a bilinear intersection product of Cartier divi-sors and cycles. We then prove the commutativity of this product and a projection formulafor push-forwards of cycles and pull-backs of Cartier divisors. In the second part (sec-tions 5 - 8) we generalize the theory developed in the first part to abstract cycles whichare abstract polyhedral complexes modulo refinements with affine cycles as local buildingblocks. Again, abstract tropical varieties are just cycleswith non-negative weights. In boththe affine and abstract case a remarkable difference to the classical situation occurs: We

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http://arXiv.org/abs/0709.3705v3

2 LARS ALLERMANN AND JOHANNES RAU

can define the mentioned intersection products on the level of cycles, i.e. we can inter-sect Cartier divisors with cycles and obtain a well-defined cycle — not only a cycle classup to rational equivalence as it is the case in classical algebraic geometry. However, forsimplifying the computations of concrete enumerative numbers we introduce a notion ofrational equivalence of cycles in section 8. In the third part (section 9) we finally use ourtheory to define the intersection product of two cycles with ambient spaceRn. Here againit is remarkable that we can define these intersections — evenfor self-intersections — onthe level of cycles. We suppose this intersection product tobe identical with thestableintersectiondiscussed in [M] and [RGST] though we could not prove it yet.

There are three more articles related to our work that we wantto mention: In [K] the authorstudies the relations between the intersection products oftoric varieties and the tropicalintersection product onRn in the case of transversal intersections. This article is closelyrelated to [FS]: In this work the authors give a description of the Chow cohomology ofa complete toric variety in terms ofMinkowski weights. These objects — representingcocycles in the toric variety — are affine tropical cycles inRn according to our definition.Moreover, there is an intersection product of these Minkowski weights corresponding tothe cup product of the associated cocycles that can be calculated via afan displacementrule. This rule equals the stable intersection of tropical cycles in Rn mentioned above forthe case of affine cycles. But there are also discrepancies between these two interpretationsof Minkowski weights: Morphisms of toric varieties as well as morphisms of affine tropicalcycles are just given by integer linear maps. However, the requirements for the fans arequite different for both kinds of morphisms. Also the functorial behavior is totally differentfor both interpretations: Regarded as toric cocycles, Minkowski weights have pull-backsalong morphisms, whereas interpreted as affine tropical cycles they admit push-forwards.In [ST] the authors study homomorphisms of tori and their induced morphisms of toricvarieties and tropical varieties, respectively. Generically finite morphisms in this contextare closely related to push-forwards of tropical cycles as defined in construction 4.2.

We would like to thank our advisor Andreas Gathmann for numerous helpful discussionsand his inspiring ideas that made this paper possible.

2. AFFINE TROPICAL CYCLES

In this section we will briefly summarize the definitions and some properties of our basicobjects. We refer to [GKM] for more details (but note that we use a slightly more generaldefinition of fan).In the following sectionsΛ will denote a finitely generated free abelian group, i.e. a groupisomorphic toZr for somer ∈ N, andV := Λ ⊗Z R the associated real vector spacecontainingΛ as a lattice. We will denote the dual lattice in the dual vector space byΛ∨ ⊆ V ∨.

Definition 2.1 (Cones). A cone in V is a subsetσ ⊆ V that can be described by finitelymany linear integral equalities and inequalities, i.e. a set of the form

σ = {x ∈ V |f1(x) = 0, . . . , fr(x) = 0, fr+1(x) ≥ 0, . . . , fN(x) ≥ 0}

for some linear formsf1, . . . , fN ∈ Λ∨. We denote byVσ the smallest linear subspaceof V containingσ and byΛσ the latticeVσ ∩ Λ. We define thedimensionof σ to be thedimension ofVσ.

FIRST STEPS IN TROPICAL INTERSECTION THEORY 3

Definition 2.2 (Fans). A fanX in V is a finite set of cones inV satisfying the followingconditions:

(a) The intersection of any two cones inX belongs toX as well,

(b) every coneσ ∈ X is the disjoint unionσ =.⋃τ∈X:τ⊆στ

ri, whereτri denotes therelative interior ofτ , i.e. the interior ofτ in Vτ .

We will denote the set of allk-dimensional cones ofX by X(k). Thedimensionof Xis defined to be the maximum of the dimensions of the cones inX . The fanX is calledpure-dimensionalif each inclusion-maximal cone inX has this dimension. The union ofall cones inX will be denoted|X | ⊆ V . If X is a fan of pure dimensionk then the conesσ ∈ X(k) are calledfacetsof X .

LetX be a fan andσ ∈ X a cone. A coneτ ∈ X with τ ⊆ σ is called afaceof σ. Wewrite this asτ ≤ σ (or τ < σ if in addition τ ( σ holds). Clearly we haveVτ ⊆ Vσ andΛτ ⊆ Λσ in this case. Note thatτ < σ implies thatτ is contained in a proper face (in theusual sense) ofσ.

Construction2.3 (Normal vectors). Let τ < σ be cones of some fanX in V with dim(τ) =dim(σ)− 1. This implies that there is a linear formf ∈ Λ∨σ that is zero onτ , non-negativeonσ and not identically zero onσ. Let uσ ∈ Λσ be a vector generatingΛσ/Λτ ∼= Z withf(uσ) > 0. Note that its classuσ/τ := [uσ] ∈ Λσ/Λτ does not depend on the choice ofuσ. We calluσ/τ the(primitive) normal vectorof σ relative toτ .

Definition 2.4 (Subfans). LetX,Y be fans inV . Y is called asubfanof X if for everyconeσ ∈ Y there exists a coneσ′ ∈ X such thatσ ⊆ σ′. In this case we writeY EX anddefine a mapCY,X : Y → X that maps a coneσ ∈ Y to the unique inclusion-minimalconeσ′ ∈ X with σ ⊆ σ′.

Definition 2.5 (Weighted fans). A weighted fan(X,ωX) of dimensionk in V is a fanXin V of pure dimensionk, together with a mapωX : X(k) → Z. The numberωX(σ) iscalled theweight of the facetσ ∈ X(k). For simplicity we usually writeω(σ) instead ofωX(σ). Moreover, we want to consider theempty fan∅ to be a weighted fan of dimensionk for all k. Furthermore, by abuse of notation we simply writeX for the weighted fan(X,ωX) if the weight functionωX is clear from the context.

Definition 2.6 (Tropical fans). A tropical fan of dimensionk in V is a weighted fan(X,ωX) of dimensionk satisfying the followingbalancing conditionfor everyτ ∈ X(k−1):

∑

σ:τ<σ

ωX(σ) · uσ/τ = 0 ∈ V/Vτ .

Let (X,ωX) be a weighted fan of dimensionk in V andX∗ the fan

X∗ := {τ ∈ X |τ ≤ σ for some facetσ ∈ X with ωX(σ) 6= 0}.

(X∗, ωX∗) := (X∗, ωX |(X∗)(k)) is called thenon-zero partof X and is again a weightedfan of dimensionk in V (note thatX∗ = ∅ is possible). Obviously(X∗, ωX∗) is a tropicalfan if and only if(X,ωX) is one. We call a weighted fan(X,ωX) reducedif all its facetshave non-zero weight, i.e. if(X,ωX) = (X∗, ωX∗) holds.

Remark2.7. Let (X,ωX) be a tropical fan of dimensionk and letτ ∈ X(k−1). Letσ1, . . . , σN be all cones inX with σi > τ . For all i let vσi/τ ∈ Λ be a represen-tative of the primitive normal vectoruσi/τ∈ Λ/Λτ . By the above balancing condition


we have∑Ni=1 ωX(σi) · vσi/τ = λτ for someλτ ∈ Λτ . Obviously we haveλτ =

gcd(ωX(σ1), . . . , ωX(σN )) · λτ for some furtherλτ ∈ Λτ . We can represent the great-est common divisor by a linear combination gcd(ωX(σ1), . . . , ωX(σN )) = α1ωX(σ1) +· · · + αNωX(σN ) with α1, . . . , αN ∈ Z and define

vσi/τ := vσi/τ − αi · λτ

for all i. Note thatvσi/τ is a representative ofuσi/τ , too. Replacing allvσi/τ by vσi/τ we

can always assume that∑Ni=1 ωX(σ) · vσ/τ = 0 ∈ Λ.

Definition 2.8 (Refinements). Let (X,ωX) and(Y, ωY ) be weighted fans inV . We call(Y, ωY ) a refinementof (X,ωX) if the following holds:

(a) Y ∗EX∗,

(b) |Y ∗| = |X∗| and

(c) ωY (σ) = ωX(CY ∗,X∗(σ)) for everyσ ∈ (Y ∗)(dim(Y )).

Note that property (b) implies that eitherX∗ = Y ∗ = ∅ or dim(X) = dim(Y ). We calltwo weighted fans(X,ωX) and(Y, ωY ) in V equivalent(write (X,ωX) ∼ (Y, ωY )) ifthey have a common refinement. Note that(X,ωX)and(X∗, ωX |(X∗)(dim(X))) are alwaysequivalent.

Remark2.9. Note that for a weighted fan(X,ωX) of dimensionk and a refinement(Y, ωY ) we have the following two properties:

(a) |X∗| = |Y ∗|, i.e. the support|X∗| is well-defined on the equivalence class ofX ,

(b) for every coneτ ∈ Y (k−1) there are exactly two cases that can occur: EitherdimCY,X(τ) = k or dimCY,X(τ) = k − 1. In the first case all conesσ ∈ Y (k)

with σ > τ must be contained inCY,X(τ). Thus there are precisely two suchconesσ1 andσ2 with ωY (σ1) = ωY (σ2) anduσ1/τ = −uσ2/τ . In the second casewe have a 1:1 correspondence between conesσ ∈ Y (k) with τ < σ and conesσ′ ∈ X(k) with CY,X(τ) < σ′ preserving weights and normal vectors.

Construction2.10 (Refinements). Let (X,ωX) be a weighted fan andY be any fan inVwith |X | ⊆ |Y |. LetP := {σ ∩ σ′|σ ∈ X,σ′ ∈ Y }. In generalP is not a fan inV as canbe seen in the following example:

X Y

σ′1

σ1

σ3

τ2

τ1

τ3 τ ′1

0σ2 σ′

2

FansX andY such that{σ ∩ σ′|σ ∈ X,σ′ ∈ Y } is not a fan.

HereP containsτ ′1 = σ2 ∩ σ′1, but alsoτ2 = σ1 ∩ σ′2 andτ3 = σ3 ∩ σ′2. Hence property(b) of definition 2.2 is not fulfilled. To resolve this, we define

X ∩ Y := {σ ∈ P |∄ τ ∈ P (dim(σ)) with τ ( σ}.


Note thatX ∩ Y is now a fan inV . We can make it into a weighted fan by settingωX∩Y (σ) := ωX(CX∩Y,X(σ)) for all σ ∈ (X ∩ Y )(dim(X)). Then(X ∩ Y, ωX∩Y ) isa refinement of(X,ωX). Note that if(X,ωX) and(Y, ωY ) are both weighted fans and|X | = |Y | we can form both intersectionsX ∩ Y andY ∩ X . Of course, the underlyingfans are the same in both cases, but the weights may differ since they are always inducedby the first complex.

The following setting is a special case of this construction: Let (X,ωX) be a weighted fanof dimensionk in V and letf ∈ Λ∨ be a non-zero linear form. Then we can construct arefinement of(X,ωX) as follows:

Hf := {{x ∈ V |f(x) ≤ 0}, {x ∈ V |f(x) = 0}, {x ∈ V |f(x) ≥ 0}}

is a fan inV with |Hf | = V . Thus we have|X | ⊆ |Hf | and by our above construction weget a refinement(Xf , ωXf

) := (X ∩Hf , ωX∩Hf) of X .

Obviously we still have to answer the question if the equivalence of weighted fans is indeedan equivalence relation and if this notion of equivalence iswell-defined on tropical fans.We will do this in the following lemma:

Lemma 2.11.

(a) The relation “∼” is an equivalence relation on the set ofk-dimensional weightedfans inV .

(b) If (X,ωX) is a weighted fan of dimensionk and (Y, ωY ) is a refinement then(X,ωX) is a tropical fan if and only if(Y, ωY ) is one.

Proof. Recall that a fan and its non-zero part are always equivalentand that a weighted fanX is tropical if and only if its non-zero partX∗ is. Thus we may assume that all our fansare reduced and the proof is the same as in [GKM, section 2]. �

Having done all these preparations we are now able to introduce the most important objectsfor the succeeding sections:

Definition 2.12(Affine cycles and affine tropical varieties). Let (X,ωX) be a tropical fanof dimensionk in V . We denote by[(X,ωX)] its equivalence class under the equivalencerelation “∼” and byZaff

k (V ) the set of equivalence classes

Zaffk (V ) := {[(X,ωX)]|(X,ωX) tropical fan of dimensionk in V }.

The elements ofZaffk (V ) are calledaffine (tropical)k-cycles in V . A k-cycle [(X,ωX)]

is called anaffine tropical varietyif ωX(σ) ≥ 0 for everyσ ∈ X(k). Note that the lastproperty is independent of the choice of the representativeof [(X,ωX)]. Moreover, notethat0 := [∅] ∈ Zaff

k (V ) for everyk. We define|[(X,ωX)]| := |X∗|. This definition iswell-defined by remark 2.9.

Construction2.13 (Sums of affine cycles). Let [(X,ωX)] and [(Y, ωY )] be k-cycles inV . We would like to form a fanX + Y by taking the unionX ∪ Y , but obviouslythis collection of cones is in general not a fan. Using appropriate refinements we canresolve this problem: Letf1(x) ≥ 0, . . . , fN1(x) ≥ 0, fN1+1(x) = 0, . . . , fN (x) = 0 andg1(x) ≥ 0, . . . , gM1(x) ≥ 0, gM1+1(x) = 0, . . . , gM (x) = 0 be all different equalities andinequalities occurring in the descriptions of all the conesbelonging toX andY respec-tively. Using construction 2.10 we get refinements

X := X ∩Hf1 ∩ · · · ∩HfN∩Hg1 ∩ · · · ∩HgM


of X andY := Y ∩Hf1 ∩ · · · ∩HfN

∩Hg1 ∩ · · · ∩HgM

of Y (note that the final refinements do not depend on the order of the single refinements).A cone occurring inX or Y is then of the form

σ =

{fi(x) ≤ 0, fj(x) = 0, fk(x) ≥ 0,gi′(x) ≤ 0, gj′(x) = 0, gk′(x) ≥ 0

∣∣∣∣i ∈ I, j ∈ J, k ∈ K,i′ ∈ I ′, j′ ∈ J ′, k′ ∈ K ′

}

for some partitionsI∪J∪K = {1, . . . , N} andI ′∪J ′∪K ′ = {1, . . . ,M}. Now, all theseconesσ belong to the fanHf1∩· · ·∩HfN

∩Hg1∩· · ·∩HgMas well and henceX∪Y fulfills

definition 2.2. Thus, now we can define thesum ofX andY to beX+Y := X∪Y togetherwith weightsωX+Y (σ) := ω

eX(σ) + ωeY (σ) for every facet ofX + Y (we setω�(σ) := 0

if σ does not occur in� ∈ {X, Y }). By construction,(X + Y, ωX+Y ) is again a tropicalfan of dimensionk. Moreover, enlarging the sets{fi}, {gj} by more (in)equalities justcorresponds to refinements ofX andY and only leads to a refinement ofX + Y . Thus,replacing the set of relations by another one that also describes the cones inX andY , orreplacingX orY by refinements keeps the equivalence class[(X+Y, ωX+Y )] unchanged,i.e. taking sums is a well-defined operation on cycles.

This construction immediately leads to the following lemma:

Lemma 2.14. Zaffk (V ) together with the operation “+” from construction 2.13 forms an

abelian group.

Proof. The class of the empty fan0 = [∅] is the neutral element of this operation and[(X,−ωX)] is the inverse element of[(X,ωX)] ∈ Zaff

k (V ). �

Of course we do not want to restrict ourselves to cycles situated in someRn. Therefore wegive the following generalization of definition 2.12:

Definition 2.15. LetX be a fan inV . An affinek-cycle inX is an element[(Y, ωY )] ofZaffk (V ) such that|Y ∗| ⊆ |X |. We denote byZaff

k (X) the set ofk-cycles inX . Note that(Zaffk (X),+

)is a subgroup of(Zaff

k (V ),+). The elements of the groupZaffdimX−1(X) are

calledWeil divisorsonX . If [(X,ωX)] is a cycle inV thenZaffk ([(X,ωX)]) := Zaff

k (X∗).

3. AFFINE CARTIER DIVISORS AND THEIR ASSOCIATEDWEIL DIVISORS

Definition 3.1 (Rational functions). Let C be an affinek-cycle. A (non-zero) rationalfunction onC is a continuous piecewise linear functionϕ : |C| → R, i.e. there existsa representative(X,ωX) of C such that on each coneσ ∈ X , ϕ is the restriction of aninteger affine linear functionϕ|σ = λ+ c, λ ∈ Λ∨σ , c ∈ R. Obviously,c is the same on allfaces byc = ϕ(0) andλ is uniquely determined byϕ and therefore denoted byϕσ := λ.Theset of (non-zero) rational functions ofC is denoted byK∗(C).

Remark3.2 (The zero function and restrictions to subcycles). The “zero” function can bethought of being the constant function−∞, thereforeK(C) := K∗(C) ∪ {−∞}. Withrespect to the operationsmax and+, K(C) is a semifield.Let us note an important difference to the classical case: Let D be an arbitrary subcycle ofC andϕ ∈ K∗(C). Thenϕ||D| ∈ K∗(D), whereas in the classical case it might becomezero. This will be crucial for defining intersection products not only modulo rational equiv-alence. On the other hand, the definition of rational functions given above, requiring the


function to be defined everywhere, seems to be restrictive when compared to the classicalcase, even so “being defined” does not imply “being regular” tropically. In some cases (seeremark 8.6) it would be desirable to generalize our definition while preserving the aboverestriction property.

As in the classical case, each non-zero rational functionϕ onC defines a Weil divisor, i.e.a cycle inZaff

dimC−1(C). The idea of course should be to describe the “zeros” and “poles”of ϕ. A naive approach could be to consider the graph ofϕ in V ×R and “intersect it withV ×{−∞} andV ×{+∞}”. However, our functionϕ takes values only inR, in fact. Onthe other hand, the graph ofϕ is not a tropical object as it is not balanced: Whereϕ is notlinear, our graph gets edges that might violate the balancing condition. So, we first makethe graph balanced by adding new faces in the additional direction (0,−1) ∈ V × R andthen apply our naive approach. Let us make this precise.

Construction3.3 (The associated Weil divisor). LetC be an affinek-cycle inV = Λ ⊗ Randϕ ∈ K∗(C) a rational function onC. Let furthermore(X,ω) be a representativeof C on whose facesϕ is affine linear. Therefore, for each coneσ ∈ X , we get a coneσ := (id×ϕσ)(σ) in V × R of the same dimension. Obviously,Γϕ := {σ|σ ∈ X} formsa fan which we can make into a weighted fan(Γϕ, ω) by ω(σ) := ω(σ). Its support is justthe set-theoretic graph ofϕ− ϕ(0) in |X | × R.For τ < σ with dim(τ) = dim(σ) − 1 let vσ/τ ∈ Λ be a representative of the normalvectoruσ/τ . Then,

(vσ/τ , ϕσ(vσ/τ )

)∈ Λ × Z is a representative of the normal vector

uσ/τ . Therefore, summing around a coneτ with dim τ = dim τ = k − 1, we get

∑

σ∈Γ(k)ϕ

τ<σ

ω(σ)(vσ/τ , ϕσ(vσ/τ )

)=

∑

σ∈X(k)

τ<σ

ω(σ)vσ/τ ,∑

σ∈X(k)

τ<σ

ϕσ(ω(σ)vσ/τ )

.

From the balancing condition for(X,ω) it follows that∑

σ∈X(k):τ<σ ω(σ)vσ/τ ∈ Vτ ,which also means

(∑σ∈X(k):τ<σ ω(σ)vσ/τ , ϕτ

(∑σ∈X(k):τ<σ ω(σ)vσ/τ

))∈ Vτ . There-

fore, moduloVτ , our first sum equals0,

∑

σ∈X(k)

τ<σ

ϕσ(ω(σ)vσ/τ ) − ϕτ

( ∑

σ∈X(k)

τ<σ

ω(σ)vσ/τ

) ∈ V × R.

So, in order to “make(Γϕ, ω) balanced atτ ”, we add the coneϑ := τ + ({0} × R≤0)

with weight ω(ϑ) =∑

σ∈X(k):τ<σ ϕσ(ω(σ)vσ/τ ) − ϕτ

(∑σ∈X(k):τ<σ ω(σ)vσ/τ

). As

obviously [(0,−1)] = uϑ/τ ∈ (V × R)/Vτ , the above calculation shows that then thebalancing condition aroundτ holds. In other words, we build the new fan(Γ′ϕ, ω

′), where

Γ′ϕ := Γϕ ∪{τ + ({0} × R≤0)|τ ∈ Γϕ \ Γ(k)

ϕ

},

ω′|Γ

(k)ϕ

:= ω,

ω′(τ + ({0} × R≤0)) :=∑

σ∈X(k)

τ<σ


( ∑

σ∈X(k)

τ<σ

ω(σ)vσ/τ

)

if dim τ = k − 1.


σ3

σ2

← new edge

σ2

σ3

σ1

σ1

C ⊆ R2

Γϕ ⊆ R2 × R

Construction of a Weil divisor.

This fan is balanced around allτ ∈ Γ(k−1)ϕ . We will show that it is also balanced at all

“new” cones of dimensionk − 1 in proposition 3.7.We now think of intersecting this new fan withV × {−∞} to get our desired Weil divisor(As our weights are allowed to be negative, we can forget about intersecting also withV × {+∞}). This construction leads to the following definition.

Definition 3.4 (Associated Weil divisors). LetC be an affinek-cycle inV = Λ ⊗ R andϕ ∈ K∗(C) a rational function onC. Let furthermore(X,ω) be a representative ofCon whose conesϕ is affine linear. We definediv(ϕ) := ϕ · C := [(

⋃k−1i=0 X

(i), ωϕ)] ∈Zaffk−1(C), where

ωϕ : X(k−1) → Z,

τ 7→∑

σ∈X(k)

τ<σ


( ∑

σ∈X(k)

τ<σ

ω(σ)vσ/τ

)

and thevσ/τ are arbitrary representatives of the normal vectorsuσ/τ .LetD be an arbitrary subcycle ofC. By remark 3.2, we can defineϕ ·D := ϕ||D| ·D.

Remark3.5. Obviously,ωϕ(τ) is independent of the choice of thevσ/τ , as a differentchoice only differs by elements inVτ .Our definition does also not depend on the choice of a representative (X,ω): Let (Y, υ)be a refinement of(X,ω). For τ ∈ Y (k−1), two cases can occur (see also remark 2.9):Let τ ′ := CY,X(τ). If dim τ ′ = k, there are precisely two cones atτ < σ1, σ2 ∈ Y (k),which then fulfillCY,X(σ1) = CY,X(σ2) and thereforeuσ1/τ = −uσ2/τ , υ(σ1) = υ(σ2)andϕσ1 = ϕσ2 . It follows thatυϕ(τ) = 0. If dim τ ′ = k − 1, CY,X gives a one-to-onecorrespondence between{σ ∈ Y (k)|τ < σ} and{σ′ ∈ X(k)|τ ′ < σ′} respecting weightsand normal vectors, and we haveϕσ = ϕCY,X (σ). It follows thatυϕ(τ) = ωϕ(τ ′). So thetwo weighted fans we obtain are equivalent.

Remark3.6 (Affine linear functions and sums). Let ϕ ∈ K∗(C) be globally affine linear,i.e.ϕ = λ||C| + c for someλ ∈ Λ∨, c ∈ R. Then obviouslyϕ · C = 0.Let furthermoreψ ∈ K∗(C) be another rational function onC. Fromϕσ+ψσ = (ϕ+ψ)σit follows that(ϕ+ ψ) · C = ϕ · C + ψ · C.


Proposition 3.7(Balancing Condition and Commutativity).

(a) LetC be an affinek-cycle inV = Λ⊗R andϕ ∈ K∗(C) a rational function onC.Thendiv(ϕ) = ϕ·C is an equivalence class of tropical fans, i.e. its representativesare balanced.

(b) Let ψ ∈ K∗(C) be another rational function onC. Then it holdsψ · (ϕ · C) =ϕ · (ψ · C).

Proof. (a): Let (X,ω) be a representative ofC on whose conesϕ is affine linear. Pick aθ ∈ X(k−2). We choose an elementλ ∈ Λ∨ with λ|Vθ

= ϕθ. By remark 3.6, we can goon withϕ−λ−ϕ(0) ∈ K∗(C) instead ofϕ. By dividing outVθ, we can restrict ourselvesto the situationdimX = 2, θ = {0}.By a further refinement (i.e. by cutting an possibly occuringhalfspace into two piecesalong an additional ray), we can assume that all conesσ ∈ X are simplicial. Thereforeeach two-dimensional coneσ ∈ X(2) is generated by two unique raysτ, τ ′ ∈ X(1), i.e.σ = τ + τ ′. We denote

χ(σ) := [Λσ : Λτ + Λτ ′ ] = [Λσ : Zuτ/{0} + Zuτ ′/{0}],

whereuτ/{0} anduτ ′/{0} denote the primitive normal vectors introduced in construction2.3. Then we get

[uτ ′/{0}] = χ(σ)uσ/τ mod Vτ .

This equation can be shown for example as follows: The linearextension of the followingfunction

index : Λσ \ Λτ → Z,

v 7→ [Λσ : Zuτ/{0} + Zv]

to Λσ is in fact trivial onΛτ . Therefore it can also be considered as a function onΛσ/Λτ .But by definitions we know index(uσ/τ ) = 1 (asuτ/{0} and any representative ofuσ/τform a lattice basis ofΛσ) and index(uτ ′/{0}) = χ(σ), which proves the claim.This means that we can rewrite the balancing condition ofX aroundτ ∈ X(1) only usingthe vectors generating the rays, namely

∑

τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)uτ ′/{0} ∈ Vτ

= λτuτ/{0},

whereλτ is a coefficient inR andσ denotesτ + τ ′ in such sums. Of course, we can alsocompute the weightωϕ(τ) of τ in div(ϕ):

ωϕ(τ) =

∑

τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)ϕ(uτ ′/{0})

− λτϕ(uτ/{0})


Let us now check the balancing condition ofϕ · C around{0} by plugging in these equa-tions. We get

∑

τ∈X(1)

ωϕ(τ)uτ/{0} =∑

τ,τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)ϕ(uτ ′/{0})uτ/{0}

−∑

τ∈X(1)

λτϕ(uτ/{0})uτ/{0}.

By Commutingτ andτ ′ in the first summand we get

∑

τ∈X(1)

ωϕ(τ)uτ/{0} =∑

τ,τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)ϕ(uτ/{0})uτ ′/{0}

−∑

τ∈X(1)

λτϕ(uτ/{0})uτ/{0}

=∑

τ∈X(1)

ϕ(uτ/{0})

∑

τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)uτ ′/{0}

− λτuτ/{0}

︸︷︷︸=0 (balancing condition aroundτ )

= 0.

This finishes the proof of (a).(b): Let (X,ω) be a representative ofC on whose conesϕ andψ are affine linear. Pick aθ ∈ X(k−2). By the same reduction steps as in case (a), we can again restrict ourselves todimX = 2, θ = {0}. With the notations and trick as in (a) we get

ωϕ,ψ({0}) =∑

τ,τ ′∈X(1)

τ+τ ′∈X(2)

ω(σ)

χ(σ)ϕ(uτ ′/{0})ψ(uτ/{0}) = ωψ,ϕ({0}),

which finishes part (b). �

Definition 3.8 (Affine Cartier divisors). LetC be an affinek-cycle. The subgroup of glob-ally affine linear functions inK∗(C) with respect to+ is denoted byO∗(C). We define thegroup of affine Cartier divisors ofC to be the quotient groupDiv(C) := K∗(C)/O∗(C).Let [ϕ] ∈ Div(C) be a Cartier divisor. By remark 3.6, the associated Weil divisordiv([ϕ]) := div(ϕ) is well-defined. We therefore get a bilinear mapping

· : Div(C) × Zaffk (C) → Zaff

k−1(C),

([ϕ], D) 7→ [ϕ] ·D = ϕ ·D,

calledaffine intersection product.

Example3.9 (Self-intersection of hyperplanes). Let Λ = Zn (and thusV = Rn), lete1, . . . , en be the standard basis vectors inZn ande0 := −e1 − · · · − en. By abuse ofnotation our ambient cycle isRn := [({Rn}, ω(Rn) = 1)]. Let us consider the “lineartropical polynomial”h = x1⊕· · ·⊕xn⊕0 = max{x1, . . . , xn, 0} : Rn → R. Obviously,h is a rational function in the sense of definition 3.1: For eachsubsetI ( {0, 1, . . . , n} we


denote byσI the simplicial cone of dimension|I| generated by the vectors−ei for i ∈ I.Thenh is integer linear on allσI , namely

h|σI(x1, . . . , xn) =

{0 if 0 /∈ I,xi if there exists ani ∈ {1, . . . , n} \ I.

LetLnk be thek-dimensional fan consisting of all conesσI with |I| ≤ k and weighted withthe trivial weight functionωLn

k. ThenLnn is a representative ofRn fulfilling the conditions

of definition 3.1. We want to show

h · · · · · h︸︷︷︸k times

·Rn = [Lnn−k]. (∗)

This follows inductively fromh · [Lnk+1] = [Lnk ], so it remains to computeωLnk+1,h

(σI) forall I with |I| = k < n. Let J := {0, 1, . . . , n} \ I. Obviously, the(k + 1)-dimensionalcones ofLnk+1 containingσI are precisely the conesσI∪{j}, j ∈ J . Moreover,−ej is a rep-resentative of the normal vectoruσI∪{j}/σI

. Note also that for alli ∈ I ′, I ′ ( {0, 1, . . . , n}

we havehσI′(−ei) = h|σI′

(−ei) = h(−ei). Using this we compute

ωLnk+1,h

(σI) =∑

j∈J

ωLnk+1

(σI∪{j})︸︷︷︸=1

hσI∪{j}(−ej)

+ hσI

(∑

j∈J

ωLnk+1

(σI∪{j})︸︷︷︸=1

ej

︸︷︷︸=−

P

i∈I ei

)

=∑

j∈J

h(−ej) +∑

i∈I

h(−ei)

= h(−e0) + h(−e1) + · · · + h(−en)

= 1 + 0 + · · · + 0 = 1 = ωLnk(σI),

which impliesh · [Lnk+1] = [Lnk ] and also equation(∗).We can summarize this example as follows: Firstly, for a tropical polynomialf , the asso-ciated Weil divisorf · Rn coincides with the locus of non-differentiabilityT (f) of f (see[RGST, section 3]), and secondly, “thek-fold self-intersection of a tropical hyperplane inRn” is given by its(n− k)-skeleton together with trivial weights all equal to1.

Example3.10 (A rigid curve). Using notations from example 3.9, we consider as ambientcycle the surfaceS := [L3

2] = T (x1 ⊕ x2 ⊕ x3 ⊕ 0) in R3. In this surface, we want toshow that the curveR := [(R · eR, ωR(R · eR) = 1)] ∈ Zaff

1 (S), whereeR := e1 + e2,has negative self-intersection in the following sense: We construct a rational functionϕ onS whose associated Weil divisor isR and show thatϕ · R = ϕ · ϕ · S is just the originwith weight−1. This curve and its rigidness were first discussed in [M, Example 4.11.,Example 5.9.].Let us constructϕ. First we refineL3

2 to LR by replacingσ{1,2} andσ{0,3} with σ{1,R},σ{R}, σ{R,2}, σ{0,−R}, σ{−R} andσ{−R,3} (using again the notations from example 3.9ande−R := −eR = e0 + e3). We defineϕ : |S| → R to be the unique function that islinear on the faces ofLR and fulfills

0,−e1,−e2,−e3,−e−R 7→ 0, −e0 7→ 1 and − eR 7→ −1.

Analogous to 3.9, we can compute fori = 1, 2

ωLR,ϕ(σ{i}) = ϕ(−e0) + ϕ(−e3) + ϕ(−eR) = 1 + 0 − 1 = 0,


The rigid curveR in S.

for i = 0, 3

ωLR,ϕ(σ{i}) = ϕ(−e1) + ϕ(−e2) + ϕ(−e−R) = 0 + 0 + 0 = 0,

and finally

ωLR,ϕ(σ{R}) = ϕ(−e1) + ϕ(−e2) − ϕ(−eR) = 0 + 0 − (−1) = 1,

ωLR,ϕ(σ{−R}) = ϕ(−e0) + ϕ(−e3) − ϕ(−e−R) = 1 + 0 + 0 = 1,

which meansϕ ·S = R. Now we can easily computeϕ ·ϕ ·S = ϕ ·R on the representative{σ{R}, σ{−R}, {0}} (with trivial weights) ofR:

ωR,ϕ({0}) = ϕ(−eR) + ϕ(−e−R) = −1 + 0 = −1.

Thereforeϕ · ϕ · S = [({0}, ω({0}) = −1)]. Note that we really obtain a cycle withnegative weight, not only a cycle class modulo rational equivalence as it is the case in“classical” algebraic geometry.

4. PUSH-FORWARD OF AFFINE CYCLES AND PULL-BACK OF CARTIER DIVISORS

The aim of this section is to construct push-forwards of cycles and pull-backs of Cartierdivisors along morphisms of fans and to study the interaction of both constructions. To dothis we first of all have to introduce the notion of morphism:

Definition 4.1 (Morphisms of fans). Let X be a fan inV = Λ ⊗Z R andY be a fan inV ′ = Λ′ ⊗Z R. A morphismf : X → Y is aZ-linear map, i.e. a map from|X | ⊆ V to|Y | ⊆ V ′ induced by aZ-linear mapf : Λ → Λ′. By abuse of notation we will usuallydenote all three mapsf, f and f ⊗Z id by the same letterf (note that the last two mapsare in general not uniquely determined byf : X → Y ). A morphism of weighted fans is amorphism of fans. A morphism of affine cyclesf : [(X,ωX)] → [(Y, ωY )] is a morphismof fansf : X∗ → Y ∗. Note that in this latter case the notion of morphism does notdependon the choice of the representatives by remark 2.9.


Given such a morphism the following construction shows how to build the push-forwardfan of a given fan. Afterwards we will show that this construction induces a well-definedoperation on cycles.

Construction4.2. We refer to [GKM, section 2] for more details on the followingcon-struction. Let(X,ωX) be a weighted fan of pure dimensionn in V = Λ ⊗Z R, let Y beany fan inV ′ = Λ′ ⊗Z R and letf : X → Y be a morphism. Passing to an appropriaterefinement of(X,ωX) the collection of cones

f∗X := {f(σ)|σ ∈ X contained in a maximal cone ofX on whichf is injective}

is a fan inV ′ of pure dimensionn. It can be made into a weighted fan by setting

ωf∗X(σ′) :=∑

σ∈X(n):f(σ)=σ′

ωX(σ) · |Λ′σ′/f(Λσ)|

for all σ′ ∈ f∗X(n). The equivalence class of this weighted fan only depends on the

equivalence class of(X,ωX).

Example4.3. LetX be the fan with conesτ1, τ2, τ3, {0} as shown in the figure

τ1

τ2

τ3

R{0}

R2 ⊃ X

fi

and letωX(τi) = 1 for i = 1, 2, 3. Moreover, letY := R be the real line and themorphismsf1, f2 : X → Y be given byf1(x, y) := x + y andf2(x, y) := x respec-tively. Then(f1)∗X = (f2)∗X = {{x ≤ 0}, {0}, {x ≥ 0}}, butω(f1)∗X({x ≤ 0}) =ω(f1)∗X({x ≥ 0}) = 2 andω(f2)∗X({x ≤ 0}) = ω(f2)∗X({x ≥ 0}) = 1.

Proposition 4.4. Let (X,ωX) be a tropical fan of dimensionn in V = Λ ⊗Z R, let Y beany fan inV ′ = Λ′ ⊗Z R and letf : X → Y be a morphism. Thenf∗X is a tropical fanof dimensionn.

Proof. A proof can be found in [GKM, section 2]. �

By construction 4.2 and proposition 4.4 the following definition is well-defined:

Definition 4.5 (Push-forward of cycles). Let V = Λ ⊗Z R andV ′ = Λ′ ⊗Z R. Moreover,letX ∈ Zaff

m (V ), Y ∈ Zaffn (V ′) andf : X → Y be a morphism. For[(Z, ωZ)] ∈ Zaff

k (X)we define

f∗[(Z, ωZ)] := [(f∗(Z∗), ωf∗(Z∗))] ∈ Zaff

k (Y ).

Proposition 4.6 (Push-forward of cycles). Let V = Λ ⊗Z R andV ′ = Λ′ ⊗Z R. LetX ∈ Zaff

m (V ) andY ∈ Zaffn (V ′) be cycles and letf : X → Y be a morphism. Then the

map

Zaffk (X) −→ Zaff

k (Y ) : C 7−→ f∗C

is well-defined andZ-linear.


Proof. It remains to prove the linearity: Let(A,ωA) and (B,ωB) be two tropical fansof dimensionk with A = A∗, B = B∗ and |A|, |B| ⊆ |X∗|. We want to show thatf∗(A +B) ∼ f∗A+ f∗B. RefiningA andB as in construction 2.13 we may assume thatA,B ⊆ A+B. SetA := A+B and

ωeA(σ) :=

{ωA(σ), if σ ∈ A

0, else

for all facetsσ ∈ A. Analogously, setB := A + B with according weights. ThenA ∼

A and B ∼ B. Carrying out a further refinement ofA + B like in construction 4.2we can reach thatf∗(A + B) = {f(σ)|σ ∈ A+B contained in a maximal cone ofA +

B on whichf is injective}. Using A = B = A + B = A + B we getf∗A = f∗B =

f∗(A+ B) = f∗(A+B) and it remains to compare the weights:

ωf∗( eA+ eB)(σ′) =

∑

σ∈( eA+ eB)(k):f(σ)=σ′

ωeA+ eB(σ) · |Λ′σ′/f(Λσ)|

=∑

σ∈( eA+ eB)(k):f(σ)=σ′

[ω

eA(σ) + ωeB(σ)

]· |Λ′σ′/f(Λσ)|

=∑

σ∈ eA(k):f(σ)=σ′

ωeA(σ) · |Λ′σ′/f(Λσ)| +

∑

σ∈ eB(k):f(σ)=σ′

ωeB(σ) · |Λ′σ′/f(Λσ)|

= ωf∗ eA(σ′) + ωf∗ eB(σ′)

for all facetsσ′ of f∗(A+B). Hencef∗(A+B) ∼ f∗(A+B) = f∗A+f∗B ∼ f∗A+f∗Bas weighted fans. �

Our next step is now to define the pull-back of a Cartier divisor. As promised we will proveafter this a projection formula that describes the interaction between our two constructions.

Proposition 4.7(Pull-back of Cartier divisors). LetC ∈ Zaffm (V ) andD ∈ Zaff

n (V ′) becycles inV = Λ⊗Z R andV ′ = Λ′ ⊗Z R respectively and letf : C → D be a morphism.Then there is a well-defined andZ-linear map

Div(D) −→ Div(C) : [h] 7−→ f∗[h] := [h ◦ f ].

Proof. The maph 7→ h ◦ f is obviouslyZ-linear on rational functions and maps affinelinear functions to affine linear functions. Thus it remainsto prove thath ◦ f is a rationalfunction if h is one: Therefore let(X,ωX) be any representative ofC, let (Y, ωY ) bea reduced representative ofD such that the restriction ofh to every cone inY is affinelinear and letfV : V → V ′ be aZ-linear map such thatfV ||C| = f . SinceZ :=

{f−1V (σ′)|σ′ ∈ Y } is a fan inV and |X | ⊆ |Z| we can construct the refinementX :=

X ∩Z ofX such thath◦f is affine linear on every cone ofX . This finishes the proof. �

Proposition 4.8(Projection formula). LetC ∈ Zaffm (V ) andD ∈ Zaff

n (V ′) be cycles inV = Λ ⊗Z R andV ′ = Λ′ ⊗Z R respectively and letf : C → D be a morphism. LetE ∈ Zaff

k (C) be a cycle and letϕ ∈ Div(D) be a Cartier divisor. Then the followingequation holds:

ϕ · (f∗E) = f∗(f∗ϕ ·E) ∈ Zaff

k−1(D).


Proof. Let E = [(Z, ωZ)] andϕ = [h]. We may assume thatZ = Z∗ andh(0) = 0.ReplacingZ by a refinement we may additionally assume thatf∗h is linear on every coneof Z (cf. definition 3.1) and that

f∗Z = {f(σ)|σ ∈ Z contained in a maximal cone ofZ on whichf is injective}

(cf. construction 4.2). Note that in this caseh is linear on the cones off∗Z, too. Letσ′ ⊆ |D| be a cone (not necessarilyσ′ ∈ f∗Z) such thath is linear onσ′. Then thereis a unique linear maphσ′ : V ′σ′ → R induced by the restrictionh|σ′ . Analogouslyfor f∗hσ, σ ⊆ |C|. For conesτ < σ ∈ Z of dimensionk − 1 andk respectively letvσ/τ ∈ Λ be a representative of the primitive normal vectoruσ/τ ∈ Λ/Λτ of construction2.3. Analogously, forτ ′ < σ′ ∈ f∗Z of dimensionk− 1 andk respectively letvσ′/τ ′ be arepresentative ofuσ′/τ ′ ∈ Λ′/Λ′τ ′. Now we want to compare the weighted fansh · (f∗Z)andf∗(f∗h · Z): Let τ ′ ∈ f∗Z be a cone of dimensionk − 1. Then we can calculate theweight ofτ ′ in h · (f∗Z) as follows:

ωh·(f∗Z)(τ′) =

0

@

X

σ′∈f∗Z:σ′>τ ′

ωf∗Z(σ′) · hσ′(vσ′/τ ′)

1

A

−hτ ′

0

@

X

σ′∈f∗Z:σ′>τ ′

ωf∗Z(σ′) · vσ′/τ ′

1

A

=

0

@

X

σ′∈f∗Z:σ′>τ ′

0

@

X

σ∈Z(k):f(σ)=σ′

ωZ(σ) · |Λ′σ′/f(Λσ)|

1

A · hσ′(vσ′/τ ′)

1

A

−hτ ′

0

@

X

σ′∈f∗Z:σ′>τ ′

0

@

X

σ∈Z(k):f(σ)=σ′

ωZ(σ) · |Λ′σ′/f(Λσ)|

1

A · vσ′/τ ′

1

A

=

0

@

X

σ∈Z(k):f(σ)>τ ′

ωZ(σ) · |Λ′f(σ)/f(Λσ)| · hf(σ)(vf(σ)/τ ′)

1

A

−hτ ′

0

@

X


ωZ(σ) · |Λ′f(σ)/f(Λσ)| · vf(σ)/τ ′

1

A

Now let τ ′ ∈ f∗(f∗h · Z) of dimensionk − 1. The weight ofτ ′ in f∗(f∗h · Z) can be

calculated as follows:

ωf∗(f∗h·Z)(τ′) =

X

τ∈(f∗h·Z)(k−1) :f(τ)=τ ′

ωf∗h·Z(τ ) · |Λ′τ ′/f(Λτ )|

=X

τ∈(f∗h·Z)(k−1) :f(τ)=τ ′

0

@

X

σ∈Z(k):σ>τ

ωZ(σ)f∗hσ(vσ/τ )

− f∗hτ

0

@

X

σ∈Z(k):σ>τ

ωZ(σ) · vσ/τ

1

A

1

A · |Λ′τ ′/f(Λτ )|


=X

τ∈(f∗h·Z)(k−1) :f(τ)=τ ′

0

@

X

σ∈Z(k):σ>τ

ωZ(σ)hf(σ)(f(vσ/τ ))

− hf(τ)

0

@

X

σ∈Z(k):σ>τ

ωZ(σ) · f`

vσ/τ´

1

A

1

A · |Λ′τ ′/f(Λτ )|.

Note thatf(vσ/τ ) = |Λ′σ′/(Λ′τ ′ + Zf(vσ/τ ))| · vσ′/τ ′ + λσ,τ ∈ Λ′ for someλσ,τ ∈ Λ′τ ′ .Sincehf(σ)(λσ,τ ) = hf(τ)(λσ,τ ) these parts of the corresponding summands in the firstand second interior sum cancel using the linearity ofhf(τ). Moreover, note thatf(vσ/τ ) =λσ,τ ∈ Λ′τ ′ for thoseσ > τ on whichf is not injective and that the whole summandscancel in this case. Thus we can conclude that the sum does notchange if we restrict thesummation to thoseσ > τ on whichf is injective. Using additionally the equation

|Λ′σ′/f(Λσ)| = |Λ′τ ′/f(Λτ )| · |Λ′σ′/(Λ′τ ′ + Zf(vσ/τ ))|

we get

ωf∗(f∗h·Z)(τ′) =

X

τ∈(f∗h·Z)(k−1):f(τ)=τ ′

0

B

B

B

@

X

σ∈Z(k):σ>τ,f(σ)>τ ′


− hτ ′

0

B

B

B

@

X

σ∈Z(k):σ>τ,f(σ)>τ ′

ωZ(σ) · |Λ′f(σ)/f(Λσ)| · vf(σ)/τ ′

1

C

C

C

A

1

C

C

C

A

=

0

@

X



1

A

−hτ ′

0

@

X


ωZ(σ) · |Λ′f(σ)/f(Λσ)| · vf(σ)/τ ′

1

A .

Note that for the last equation we used again the linearity ofhτ ′ . We have checked sofar that a coneτ ′ of dimensionk − 1 occurring in bothh · (f∗Z) andf∗(f∗h · Z) has thesame weight in both fans. Thus it remains to examine those conesf(τ), τ ∈ Z(k−1) suchthatf is injective onτ but not on anyσ > τ : In this case all vectorsvσ/τ are mapped toΛ′f(τ). Again,hf(σ) = hf(τ) and by linearity ofhf(τ) all summands in the sum cancel asabove. Hence the the weight off(τ) in f∗(f∗h · Z) is 0 andϕ · (f∗E) = [h · (f∗Z)] =[f∗(f

∗h · Z)] = f∗(f∗ϕ ·E). �

5. ABSTRACT TROPICAL CYCLES

In this section we will introduce the notion of abstract tropical cycles as spaces that havetropical fans as local building blocks. Then we will generalize the theory from the previoussections to these spaces.

Definition 5.1 (Abstract polyhedral complexes). An (abstract) polyhedral complexis atopological space|X | together with a finite setX of closed subsets of|X | and an embed-ding mapϕσ : σ → Rnσ for everyσ ∈ X such that


(a) X is closed under taking intersections, i.e.σ ∩ σ′ ∈ X for all σ, σ′ ∈ X withσ ∩ σ′ 6= ∅,

(b) every imageϕσ(σ), σ ∈ X is a rational polyhedron not contained in a proper affinesubspace ofRnσ ,

(c) for every pairσ, σ′ ∈ X the concatenationϕσ ◦ ϕ−1σ′ is integer affine linear where

defined,

(d) |X | =.⋃

σ∈X

ϕ−1σ (ϕσ(σ)◦), whereϕσ(σ)◦ denotes the interior ofϕσ(σ) in Rnσ .

For simplicity we will usually drop the embedding mapsϕσ and denote the polyhedralcomplex(X, |X |, {ϕσ|σ ∈ X}) by (X, |X |) or just byX if no confusion can occur. Theclosed subsetsσ ∈ X are called thepolyhedraor faces of(X, |X |). For σ ∈ X theopen setσri := ϕ−1

σ (ϕσ(σ)◦) is called therelative interior ofσ. Like in the case of fansthedimensionof (X, |X |) is the maximum of the dimensions of its polyhedra.(X, |X |)is pure-dimensionalif every inclusion-maximal polyhedron has the same dimension. Wedenote byX(n) the set of polyhedra in(X, |X |) of dimensionn. Letτ, σ ∈ X . Like in thecase of fans we writeτ ≤ σ (or τ < σ) if τ ⊆ σ (or τ ( σ respectively).

An abstract polyhedral complex(X, |X |) of pure dimensionn together with a mapωX :X(n) → Z is calledweighted polyhedral complexof dimensionn andωX(σ) theweightof the polyhedronσ ∈ X(n). Like in the case of fans the empty complex∅ is a weightedpolyhedral complex of every dimensionn. If ((X, |X |), ωX) is a weighted polyhedralcomplex of dimensionn then let

X∗ := {τ ∈ X |τ ⊆ σ for someσ ∈ X(n) with ωX(σ) 6= 0}, |X∗| :=⋃

τ∈X∗

τ ⊆ |X |.

With these definitions((X∗, |X∗|), ωX |(X∗)(n)

)is again a weighted polyhedral complex

of dimensionn, called thenon-zero partof ((X, |X |), ωX). We call a weighted polyhedralcomplex((X, |X |), ωX) reducedif ((X, |X |), ωX) = ((X∗, |X∗|), ωX∗) holds.

Definition 5.2 (Subcomplexes and refinements). Let (X, |X |, {ϕσ}) and(Y, |Y |, {ψτ})be two polyhedral complexes. We call(X, |X |, {ϕσ}) asubcomplexof (Y, |Y |, {ψτ}) if

(a) |X | ⊆ |Y |,

(b) for everyσ ∈ X existsτ ∈ Y with σ ⊆ τ and

(c) theZ-linear structures ofX andY are compatible, i.e. for a pairσ, τ from (b) themapsϕσ ◦ ψ−1

τ andψτ ◦ ϕ−1σ are integer affine linear where defined.

We write (X, |X |, {ϕσ})E (Y, |Y |, {ψτ}) in this case. Analogous to the case of fans wedefine a mapCX,Y : X → Y that maps a polyhedron inX to the inclusion-minimalpolyhedron inY containing it.We call a weighted polyhedral complex((X, |X |), ωX) a refinementof ((Y, |Y |), ωY ) if

(a) (X∗, |X∗|)E (Y ∗, |Y ∗|),

(b) |X∗| = |Y ∗|,

(c) ωX(σ) = ωY (CX∗,Y ∗(σ)) for all σ ∈ (X∗)(dim(X)).

Definition 5.3 (Open fans). Let (F , ωeF ) be a tropical fan inRn andU ⊆ Rn an open

subset containing the origin. The setF := F ∩ U := {σ ∩ U |σ ∈ F} together with the


induced weight functionωF is called anopen (tropical) fanin Rn. Like in the case of fanslet |F | :=

⋃σ′∈F σ

′. Note that the open fanF contains the whole information of the entire

fan F asF = {R≥0 · σ′|σ′ ∈ F}.

Definition 5.4 (Tropical polyhedral complexes). A tropical polyhedral complexof dimen-sionn is a weighted polyhedral complex((X, |X |), ωX) of pure dimensionn together withthe following data: For every polyhedronσ ∈ X∗ we are given an open fanFσ in someRnσ and a homeomorphism

Φσ : Sσ :=⋃

σ′∈X∗,σ′⊇σ

(σ′)ri∼−→ |Fσ|

such that

(a) for all σ′ ∈ X∗, σ′ ⊇ σ holdsΦσ(σ′ ∩ Sσ) ∈ Fσ andΦσ is compatible with the

Z-linear structure onσ′, i.e.Φσ ◦ϕ−1σ′ andϕσ′ ◦Φ−1

σ are integer affine linear wheredefined,

(b) ωX(σ′) = ωFσ(Φσ(σ

′ ∩ Sσ)) for everyσ′ ∈ (X∗)(n) with σ′ ⊇ σ,

(c) for every pairσ, τ ∈ X∗ there is an integer affine linear mapAσ,τ and a commuta-tive diagram

Sσ ∩ Sτ

∼Φσ

��

∼

Φτ // Φτ (Sσ ∩ Sτ )

Φσ(Sσ ∩ Sτ )

Aσ,τ

77n

nn

nn

nn

nn

nn

n

.

For simplicity of notation we will usually drop the mapsΦσ and write((X, |X |), ωX) orjustX instead of(((X, |X |), ωX), {Φσ}). A tropical polyhedral complex is calledreducedif the underlying weighted polyhedral complex is.

Example5.5. The following figure shows the topological spaces and the decompositionsinto polyhedra of two such abstract tropical polyhedral complexes together with the openfanFσ for every polyhedronσ:

X1 X2

R2

Construction 5.6 (Refinements of tropical polyhedral complexes). Let(((X, |X |), ωX), {Φσ}) be a tropical polyhedral complex and let((Y, |Y |), ωY ) be a re-finement of its underlying weighted polyhedral complex((X, |X |), ωX). Then we can


make((Y, |Y |), ωY ) into a tropical polyhedral complex as follows: We may assumethatX andY are reduced as we do not pose any conditions on polyhedra withweight zero.Fix someτ ∈ Y and letσ := CY,X(τ). By definition of refinement, for everyτ ′ ∈ Ywith τ ′ ≥ τ there isσ′ ∈ X , σ′ ≥ σ with τ ′ ⊆ σ′. ThusSτ ⊆ Sσ and we have a mapΨτ := Φσ|Sτ

: Sτ∼→ Ψτ (Sτ ) ⊆ Rnσ . It remains to giveΨτ (Sτ ) the structure of an

open fan: We may assume that{0} ⊆ Ψτ (τ) (otherwise replaceΨτ by the concatenatingof Ψτ with an appropriate translationTτ , applyTτ to FXσ andΦσ and change the mapsAσ,σ′ andAσ′,σ accordingly). LetFXσ := {R≥0 · σ′|σ′ ∈ FXσ } be the tropical fan associ-ated toFXσ and letFYτ be the set of conesFYτ := {R≥0 · Ψτ (τ

′)|τ ≤ τ ′ ∈ Y }. Note thatthe conditions on theZ-linear structures onX andY to be compatible and onΦσ to becompatible with theZ-linear structure onX assure thatFYτ is a fan inRnσ . In fact, FYτwith the weights induced byY is a refinement of(FXσ , ω eFX

σ). Thus the mapsΨτ together

with the open fans{ ∩ Ψτ (Sτ )| ∈ FYτ }, τ ∈ Y fulfill all requirements for a tropicalpolyhedral complex.

Remark5.7. If not stated otherwise we will from now on equip every refinement of a trop-ical polyhedral complex coming from a refinement of the underlying weighted polyhedralcomplex with the tropical structure constructed in 5.6.

Definition 5.8 (Refinements and equivalence of tropical polyhedral complexes). LetC1 =(((X1, |X1|), ωX1), {Φ

X1σ1

})

andC2 =(((X2, |X2|), ωX2), {Φ

X2σ2

})

be tropical polyhe-dral complexes. We callC2 a refinementof C1 if

(a) ((X2, |X2|), ωX2) is a refinement of((X1, |X1|), ωX1) and

(b) C2 carries the tropical structure induced byC1 like in construction 5.6, i.e. if

C′2 =(((X2, |X2|), ωX2), {Φ

X2σ2

})

is the tropical polyhedral complex obtained

fromC1 and the refinement((X2, |X2|), ωX2) then the mapsΦX2σ2

◦ (ΦX2σ2

)−1 and

ΦX2σ2

◦ (ΦX2σ2

)−1 are integer affine linear where defined.

We call two tropical polyhedral complexesC1 andC2 equivalent(writeC1 ∼ C2) if theyhave a common refinement (as tropical polyhedral complexes).

Remark5.9. Note that different choices of translation mapsTτ in construction 5.6 onlylead to tropical polyhedral complexes carrying the same tropical structure in the senseof definition 5.8 (b). In particular definition 5.8 does not depend on the choices wemade in construction 5.6. Note moreover that refinements of(((X, |X |), ωX), {Φσ}) and((Y, |Y |), ωY ) in construction 5.6 only lead to refinements of(((Y, |Y |), ωY ), {Ψτ}).

Construction 5.10. (Refinements) Let (((X, |X |, {ϕσ}), ωX), {Φσ}) and(((Y, |Y |, {ψτ}), ωY ), {Ψτ}) be reduced tropical polyhedral complexes such that(Y, |Y |)E (X, |X |) and the tropical structures onX andY agree, i.e. for everyτ ∈ Yandσ := CY,X(τ) ∈ X the mapsΨτ◦Φ−1

σ andΦσ◦Ψ−1τ are integer affine linear where de-

fined. Moreover let (((X ′, |X ′|, {ϕ′σ′}), ωX′), {Φ′σ′}) be a reducedrefinement of (((X, |X |, {ϕσ}), ωX), {Φσ}). Like in the case of fans we will

construct a refinement(((Y ∩X ′, |Y ∩X ′|, {ψY ∩X

′

τ ′ }), ωY ∩X′), {ΨY ∩X′

τ ′ })

of

(((Y, |Y |, {ψτ}), ωY ), {Ψτ}) such that(Y ∩X ′, |Y ∩X ′|)E (X ′, |X ′|) and the tropicalstructures onY ∩X ′ andX ′ agree:Fix σ ∈ X . Note that the compatibility conditions on theZ-linear structures ofX ′,X andY , X respectively (cf. 5.2 (c)) assure thatϕσ(σ′), σ′ ∈ X ′ with σ′ ⊆ σ as


well asϕσ(τ), τ ∈ Y with τ ⊆ σ are rational polyhedra inRnσ . Thus in this caseϕσ(σ

′ ∩ τ) = ϕσ(σ′) ∩ ϕσ(τ) is a rational polyhedron, too. LetHσ′,τ

∼= Rnτ be thesmallest affine subspace ofRnσ containingϕσ(σ′ ∩ τ). We can considerϕσ|σ′∩τ to be amapσ′ ∩ τ → Rnτ . We can hence construct the underlying weighted polyhedralcomplexof our desired tropical polyhedral complex as follows: SetP := {τ ∩ σ′|τ ∈ Y, σ′ ∈ X ′},Y ∩ X ′ := {τ ∈ P |∄τ ∈ P (dim(τ)) : τ ( τ}, |Y ∩ X ′| := |Y | andωY ∩X′(τ) :=

ωY (CY ∩X′,Y (τ)) for all τ ∈ (Y ∩ X ′)(dim(Y )). It remains to define the mapsψY ∩X′

τ ′

andΨY ∩X′

τ ′ : For everyτ ′ ∈ Y ∩ X ′ choose a tripletσ′ ∈ X ′, τ ∈ Y, σ ∈ X suchthat σ′ ∩ τ = τ ′ andσ′, τ ⊆ σ and setψY ∩X

′

τ ′ := ϕσ|σ′∩τ . With these definitionsthe weighted polyhedral complex((Y ∩X ′, |Y ∩X ′|, {ψY ∩X

′

τ ′ }), ωY∩X′) is a refinementof ((Y, |Y |, {ψτ}), ωY ). Thus we can apply construction 5.6 to obtain maps{ΨY ∩X′

τ ′ }that endow our weighted polyhedral complex with the tropical structure inherited from((Y, |Y |, {ψτ}), ωY ). Note that the compatibility property between the tropicalstructuresof Y andX is bequeathed toY ∩X ′ andX ′, too.

Lemma 5.11. The equivalence of tropical polyhedral complexes is an equivalence rela-tion.

Proof. Let C1 =(((X1, |X1|), ωX1), {Φ

X1σ1

}), C2 =

(((X2, |X2|), ωX2), {Φ

X2σ2

})

andC3 =

(((X3, |X3|), ωX3), {Φ

X3σ3

})

be tropical polyhedral complexes such thatC1 ∼ C2

via a common refinementD1 =(((Y1, |Y1|), ωY1), {Φ

Y1σ1})

andC2 ∼ C3 via a commonrefinementD2 =

(((Y2, |Y2|), ωY2), {Φ

Y2σ2}). We have to construct a common refinement

of C1 andC3: First of all we may assume thatD1 andD2 are reduced. Using construction5.10 we get a refinementD3 :=

(((Y1 ∩ Y2, |Y1 ∩ Y2|), ωY1∩Y2), {Φ

Y1∩Y2τ }

)of D1 with

(Y1 ∩Y2, |Y1 ∩Y2|)E (Y2, |Y2|) and a tropical structure that is compatible with the tropicalstructure onD2. It is easily checked thatD3 is a refinement ofD2, too. �

Definition 5.12(Abstract tropical cycles). Let ((X, |X |), ωX) be ann-dimensional tropi-cal polyhedral complex. Its equivalence class[((X, |X |), ωX)] is called an(abstract) trop-ical n-cycle. The set ofn-cycles is denoted byZn. Since the topological space|X∗| ofa tropical polyhedral complex((X, |X |), ωX) is by definition invariant under refinementswe define

∣∣ [((X, |X |), ωX)]∣∣ := |X∗|. Like in the affine case, ann-cycle((X, |X |), ωX)

is called an(abstract) tropical varietyif ωX(σ) ≥ 0 for all σ ∈ X(n).

Let C ∈ Zn andD ∈ Zk be two tropical cycles.D is called an(abstract) tropical cyclein C or asubcycle ofC if there exists a representative(((Z, |Z|), ωZ), {Ψτ}) of D and areduced representative(((X, |X |), ωX), {Φσ}) of C such that

(a) (Z, |Z|)E (X, |X |),

(b) the tropical structures onZ and X agree, i.e. for everyτ ∈ Z the mapsΨτ ◦ Φ−1

CZ,X(τ) andΦCZ,X(τ) ◦ Ψ−1τ are integer affine linear where defined.

The set of tropicalk-cycles inC is denoted byZk(C).

Remark and Definition5.13. (a) LetX be a finite set of rational polyhedra inRn, f ∈Hom(Zn,Z) a linear form andb ∈ R. Then let

Hf,b :={{x ∈ Rn|f(x) ≤ b}, {x ∈ Rn|f(x) = b}, {x ∈ Rn|f(x) ≥ b}

}.

Like in the case of fans (cf. construction 2.10) we can form sets P :={σ ∩ σ′|σ ∈ X,σ′ ∈ Hf,b} andX ∩Hf,b := {σ ∈ P |∄ τ ∈ P (dim(σ)) with τ ( σ}.


(b) Again letX be a finite set of rational polyhedra inRn. Let {fi ≤ bi|i = 1, . . . , N} beall (integral) inequalities occurring in the description of all polyhedra inX . Then we canconstruct the setX ∩Hf1,b1 ∩ · · · ∩HfN ,bN

. Note that for every collection of polyhedraX this setX ∩ Hf1,b1 ∩ · · · ∩ HfN ,bN

is a (usual) rational polyhedral complex (i.e. forevery polyhedronτ ∈ X every face (in the usual sense) ofσ is contained inX and theintersection of every two polyhedra inX is a common face of each). Moreover note thatthe result is independent of the order of thefi and if {gi ≤ ci|i = 1, . . . ,M} is a dif-ferent set of inequalities describing the polyhedra inX thenX ∩Hf1,b1 ∩ · · · ∩HfN ,bN

andX ∩Hg1,c1 ∩ · · · ∩HgM ,cMhave a common refinement, namelyX ∩Hf1,b1 ∩ · · · ∩

HfN ,bN∩Hg1,c1 ∩ · · · ∩HgM ,cM

.

Construction5.14 (Sums of tropical cycles). Let C ∈ Zn be a tropical cycle. Likein the affine case the set of tropicalk-cycles inC can be made into an abelian groupby defining the sum of two suchk-cycles as follows: LetD1 andD2 ∈ Zk(C) bethe two cycles whose sum we want to construct. By definition there are reduced rep-resentatives

(((X1, |X1|), ωX1), {Φ

X1τ }

)and

(((X2, |X2|), ωX2), {Φ

X2τ }

)of C and re-

duced representatives(((Y, |Y |), ωY ), {ΦYτ }

)of D1 and

(((Z, |Z|), ωZ), {ΦZτ }

)of D2

such that(Y, |Y |)E(X1, |X1|) and the tropical structures onY andX1 agree and(Z, |Z|)E(X2, |X2|) and the tropical structures onZ andX2 agree. As “∼” is an equivalence re-lation there is a common refinement

(((X, |X |, {ϕτ}), ωX), {ΦXτ }

)of X1 andX2 which

we may assume to be reduced. Applying construction 5.10 toY andX we obtain the trop-ical polyhedral complex

(((Y ∩X, |Y ∩X |), ωY ∩X), {ΦY ∩Xτ }

)which is a refinement of

Y , has a tropical structure that is compatible with the tropical structure onX and fulfils(Y ∩X, |Y ∩X |)E (X, |X |). If we further apply construction 5.10 toZ andX we get arefinement

(((Z ∩X, |Z ∩X |), ωZ∩X), {ΦZ∩Xτ }

)of Z with analogous properties. Now

fix some polyhedronσ ∈ X and letτ1, . . . , τr ∈ Y ∩ X andτr+1, . . . , τs ∈ Z ∩ X beall polyhedra ofY ∩ X andZ ∩ X respectively that are contained inσ. Note that prop-erty (a) of definition 5.12 implies that for alli = 1, . . . , r the imageϕσ(τi) is a rationalpolyhedron inRnσ . Like in remark and definition 5.13 let{fi ≤ bi|i = 1, . . . , N} bethe set of all integral inequalities occurring in the description of all polyhedraϕσ(τi), i =1, . . . , s and letRσY ∩X := {ϕσ(τi)|i = 1, . . . , r} ∩Hf1,b1 ∩ · · · ∩HfN ,bN

andRσZ∩X :={ϕσ(τi)|i = r+1, . . . , s}∩Hf1,b1 ∩· · ·∩HfN ,bN

. ThenP σY ∩X := {ϕ−1σ (τ)|τ ∈ RσY ∩X}

andP σZ∩X := {ϕ−1σ (τ)|τ ∈ RσZ∩X} are a kind of local refinement ofY ∩X andZ ∩X

respectively, but taking the union over all maximal polyhedra σ ∈ X(n) does in gen-eral not lead to global refinements as there may be overlaps between polyhedra com-ing from differentσ. We resolve this as follows: Forσ ∈ X(n), τ ∈

⋃n−1i=0 X

(i) letP σY,τ := { ∈ P σY ∩X |τ is the inclusion-minimal polyhedron ofX containing } and

PY,n :=⋃σ∈X(n){ ∈ P σY ∩X |∄τ ∈ X(n−1) : ⊆ τ}. Analogously forP σZ,τ andPZ,n.

Then let Y := PY,n ∪(⋃

τ∈X(i):i<n{⋂σ∈X(n):τ⊆σ τσ|τσ ∈ P σY,τ}

)and Z :=

PZ,n ∪(⋃

τ∈X(i):i<n{⋂σ∈X(n):τ⊆σ τσ|τσ ∈ P σZ,τ}

). Moreover for everyτ ∈ Y ∪ Z

choose someσ ∈ X(n) with τ ⊆ σ and letψτ := ϕσ|τ . Note that by construction(Y , |Y ∩X |) and(Z, |Z ∩X |) with structure mapsψτ , τ ∈ X or τ ∈ Z respectively andweight functionsω

eY andωeZ induced byY ∩X andZ ∩X are refinements ofY ∩X and

Z ∩ X (we need here thatRσY ∩X andRσZ∩X were usual polyhedral complexes inRnσ ).Thus we can endow them with the tropical structures inherited from Y ∩ X andZ ∩ X

respectively (cf. construction 5.6). As(X∪ Y , |Y ∩X |∪ |Z∩X |) is a polyhedral complex


in theboundary

σ2

refinements

σ1

σ1

σ1

Make the

fit together

σ1 σ2

X

boundarythe

faces inLook at

boundarythe

faces inLook at

σ2

σ2

Y ∩X

Construct local refinements

Z ∩X

An illustration of the process described in construction 5.14.


now, we can form

((P, |P |), ωP ) := ((X ∪ Y , |Y ∩X | ∪ |Z ∩X |), ωP ),

whereωP (σ) := ωeY (σ) + ω

eZ(σ) for all σ ∈ P (k) (we setω�(σ) := 0 for σ /∈ �,

� ∈ {Y , Z}). Recall that the tropical structures onY andZ are inherited fromY ∩X andZ ∩ X and are thus compatible with the tropical structure onX . ThusΦXσ (SPσ ) ⊆ |FXσ |with weights induced fromP is an open fan (the corresponding complete tropical fan isjust the sum of the fans coming fromY andZ). Thus we can setΦσ := ΦXσ |SP

σ: SPσ

∼→

ΦXσ (SPσ ) and can hence define the sumD1 +D2 to be

D1 +D2 :=[(

((P, |P |), ωP ), {Φσ})].

Note that the class[(((P, |P |), ωP ), {Φσ})] is independent of the choices we made, i.e. thesumD1 +D2 is well-defined.

Lemma 5.15. LetC ∈ Zn be a tropical cycle. The setZk(C) together with the operation“+” from construction 5.14 forms an abelian group.

Proof. The class of the empty complex0 = [∅] is the neutral element of this operation and[((Y, |Y |),−ωY )] is the inverse element of[((Y, |Y |), ωY )] ∈ Zk(C). �

6. CARTIER DIVISORS AND THEIR ASSOCIATEDWEIL DIVISORS

Definition 6.1 (Rational functions and Cartier divisors). LetC be an abstractk-cycle andlet U be an open set in|C|. A (non-zero) rational function onU is a continuous functionϕ : U → R such that there exists a representative(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X)of C such that for each faceσ ∈ X the mapϕ ◦m−1

σ is locally integer affine linear (wheredefined). Theset of all non-zero rational functions onU is denoted byK∗C(U) or justK∗(U).If additionally for each faceσ ∈ X the mapϕ◦M−1

σ is locally integer affine linear (wheredefined),ϕ is calledregular invertible. Theset of all regular invertible functions onU isdenoted byO∗C(U) or justO∗(U).A representative of a Cartier divisor onC is a finite set{(U1, ϕ1), . . . , (Ul, ϕl)}, where{Ui} is an open covering of|C| andϕi ∈ K∗(Ui) are rational functions onUi that onlydiffer in regular invertible functions on the overlaps, in other words, for alli 6= j we haveϕi|Ui∩Uj

− ϕj |Ui∩Uj∈ O∗(Ui ∩ Uj).

We define thesumof two representatives by{(Ui, ϕi)} + {(Vj , ψj)} = {(Ui ∩ Vj , ϕi +ψj)}, which obviously fulfills again the condition on the overlaps.We call two representatives{(Ui, ϕi)}, {(Vj , ψj)} equivalentif ϕi−ψj is regular invertible(where defined) for alli, j, i.e. {(Ui, ϕi)}−{(Vj, ψj)} = {(Wk, γk)} with γk ∈ O∗(Wk).Obviously, “+” induces a group structure on the set of equivalence classes of representa-tives with the neutral element{(|C|, c0)}, wherec0 is the constant zero function. Thisgroup is denoted byDiv(C) and its elements are calledCartier divisors onC.

Example6.2. Let us give an example of a Cartier divisor which is not globally definedby a rational function: As abstract cycleC we take the elliptic curve[X2] from example5.5 (the brackets resemble the fact that, to be precise, we take the equivalence class of thepolyhedral complexX2 with respect to refinements). Byα1, α2 we denote the two verticesin X2. W.l.o.g. we can assume that the mapsMαi

map the pointsαi exactly to0 ∈ R. Ofcourse, the starsSα1 , Sα2 cover our whole space|C| = |X2|. So we can define the Cartier


α2

α1

α2

α1

α2

α1

ψ1 onSα1 ψ2 onSα2 ψ1 − ψ2 on the overlaps

C C C

The Cartier divisorϕ defined in example 6.2.

divisorϕ := [{(Sα1 , ψ1), (Sα2 , ψ2)}], whereψ1 := max(0, x)◦Mα1 andψ2 := c0 ◦Mα2

with c0 the constant zero function. Let us check the condition on theoverlaps: On oneopen half of our curve the two functions coincide, whereas onthe other open half theydiffer by a linear function. So we constructed an Cartier divisor which can not be globallydefined by one rational function (asψ1 can not be completed to a continuous function on|C|).

Remark6.3 (Restrictions to subcycles). Note that, as in the affine case (see remark 3.2),we can restrict a non-zero rational functionϕ ∈ K∗C(U) to an arbitrary subcycleD ⊆ C,i.e. ϕ|U∩|D| ∈ K∗D(U ∩ |D|). It is also true that a regular invertible functionϕ ∈ O∗C(U)restricted toD is again regular invertible, i.e.ϕ|U∩|D| ∈ O∗D(U ∩ |D|). Hence we canalso restrict a Cartier divisor[{(Ui, ϕi)}] ∈ Div(C) to D by setting[{(Ui, ϕi)}] |D :=[{(Ui ∩ |D|, ϕi|Ui∩|D|)}] ∈ Div(D). Let us also stress again that we still require ourobjects to be defined everywhere (on a given open subsetU ). This causes problems likefor example in remark 8.6.

Construction6.4 (Intersection products). LetC be an abstractk-cycle andϕ = [{(Ui, ϕi)}]∈ Div(C) a Cartier divisor onC. By definition 6.1 and lemma 5.11, there exists a repre-sentative(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X) of C such that for alli andσ ∈ X themapϕi ◦ m−1

σ is locally integer affine linear (where defined). We can also assume thatX = X∗, as our functions are defined on|C| = |X∗| at the most. We would like to definethe intersection productϕ · C to be

[(((Y, |Y |, {mσ}σ∈Y

), ωX,ϕ

),{Mσ|SY

σ: SYσ → |FYσ |

}σ∈Y

)],

where

Y :=

k−1⋃

i=0

X(i), |Y | :=⋃

σ∈Y

σ, SYσ =⋃

σ′∈Yσ⊆σ′

(σ′)ri, FYσ :=

k−1⋃

i=0

F (i)σ

andωX,ϕ is an appropriate weight function. So it remains to construct ωX,ϕ(τ) for τ ∈X(k−1).First, we do this pointwise, i.e. we constructωX,ϕ(p) for p ∈ (τ)ri. Given ap ∈ (τ)ri,we pick ani with p ∈ Ui. Let V be the connected component ofMτ (Ui ∩ Sτ ) con-tainingMτ (p). Then the functionϕi ◦ M−1

τ |V can be uniquely extended to a rationalfunctionϕi ∈ K∗([(Fτ , ωFτ

)]), where(Fτ , ωFτ) is the tropical fan generated by the open

fan(Fτ , ωFτ). So, in the affine case, we can computeωFτ ,ϕi

(R ·Mτ (τ)) (see construction3.3 and definition 3.4) and defineωX,ϕ(p) := ωFτ ,ϕi

(R ·Mτ (τ)).This definition is well-defined, namely if we pick anotherj with p ∈ Uj and denote byV ′

the connected component ofMτ (Uj ∩ Sτ ) containingMτ (p), we know by definition of aCartier divisor thatϕi ◦M−1

τ |V ∩V ′ − ϕj ◦M−1τ |V ∩V ′ is affine linear, henceϕi − ϕj is


affine linear. By remark 3.6 we getωFτ ,ϕi(R ·Mτ (τ)) = ωFτ ,ϕj

(R ·Mτ (τ)).The same argument shows that our definition does not depend onthe choice of a represen-tative{(Ui, ϕi)} of ϕ.But as(τ)ri is connected, the continuous functionωX,ϕ : (τ)ri → Z must be constant.Hence, we defineωX,ϕ(τ) := ωX,ϕ(p) for somep ∈ (τ)ri. With this weight function

(((Y, |Y |, {mσ}σ∈Y

), ωX,ϕ

), {Mσ|SY

σ}σ∈Y

)

is a tropical polyhedral complex.Let us now check if the equivalence class of this complex is independent of the choice ofrepresentatives ofC. Let therefore(((X ′, |X ′|, {mσ′}σ′∈X′), ωX′), {Mσ′}σ′∈X′) be a re-finement of(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X) (we can again assumeX ′ = X ′∗).Then, for eachσ′ ∈ X ′, the mapMCX′,X (σ′) ◦ M−1

σ′ embedsFσ′ into a refinementof FCX′,X (σ′). Applying the affine statement here (see remark 3.5), we deduce that for

eachτ ′ ∈ X ′(k−1) it holds ωX′,ϕ(τ ′) = 0 (if dimCX′,X(τ ′) = k) or ωX′,ϕ(τ ′)) =ωX,ϕ(CX′,X(τ ′)) (if dimCX′,X(τ ′) = k − 1).

Definition 6.5 (Intersection products). LetC be an abstractk-cycle andϕ = [{(Ui, ϕi)}] ∈Div(C) a Cartier divisor onC. Let furthermore(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X)be a representative ofC such that|X | = |C| and for alli andσ ∈ X the mapϕi ◦m−1

σ islocally integer affine linear (where defined). Theassociated Weil divisordiv(ϕ) = ϕ · Cis defined to be

[(((Y :=

k−1⋃

i=0

X(i),⋃

σ∈Y

σ, {mσ}σ∈Y), ωX,ϕ

), {Mσ|SY

σ}σ∈Y

)]∈ Zk−1(C),

whereSYσ =⋃σ′∈Yσ⊆σ′

(σ′)ri andωX,ϕ is the weight function constructed in construction 6.4.

LetD ∈ Zl(C) be an arbitrary subcycle ofC of dimensionl. We define theintersectionproduct ofϕ with D to beϕ ·D := ϕ|D ·D ∈ Zl−1(C).

Example6.6. Let us compute the Weil divisor associated to our Cartier divisorϕ on theelliptic curveC constructed in example 6.2. In fact, there is nothing to compute: One cansee immediately from the picture thatdiv(ϕ) is just the vertexα1 with multiplicity 1 (themultiplicity of α2 is 0 as in order to compute it, one has to use the constant function ψ2).Let us stress that this single point can not be obtained as theWeil divisor of a (global)rational function, as all such divisors must have “degree 0”(this is defined precisely andproven in remark 8.4 and lemma 8.3).

Proposition 6.7(Commutativity). Letϕ, ψ ∈ Div(C) be two Cartier divisors onC. Thenψ · (ϕ · C) = ϕ · (ψ · C).

Proof. Sayϕ = [{(Ui, ϕi)}] andψ = [{(Vj , ψj)}]. Using lemma 5.11 we find a repre-sentative(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X) of C such that|X | = |C| and for alli, j andσ ∈ X the mapsϕi ◦m−1

σ andψj ◦m−1σ are locally integer affine linear (where

defined). Forθ ∈ X(k−2), p ∈ (θ)ri andi, j with p ∈ Ui ∩ Vj we get (using notationsfrom construction 6.4)ωX,ϕ,ψ(θ) = ωX,ϕ,ψ(p) = ωFθ,ϕi,ψj

(R · Mθ(θ)) and similarilyωX,ψ,ϕ(θ) = ωFθ,ψj,ϕi

(R ·Mθ(θ)). Using the corresponding statement in the affine casenow (see proposition 3.7 (b)), we deduce that the two weight functions are equal whichproves the claim. �


7. PUSH-FORWARD OF TROPICAL CYCLES AND PULL-BACK OF CARTIER DIVISORS

Definition 7.1 (Morphisms of tropical cycles). LetC ∈ Zn andD ∈ Zm be two tropicalcycles. Amorphismf : C → D of tropical cycles is a continuous mapf : |C| → |D|with the following property: There exist reduced representatives(((X, |X |), ωX), {Φσ})of C and(((Y, |Y |), ωY ), {Ψτ}) of D such that

(a) for every polyhedronσ ∈ X there exists a polyhedronσ ∈ Y with f(σ) ⊆ σ,

(b) for every pairσ, σ from (a) the mapΨeσ ◦ f ◦ Φ−1

σ : |FXσ | → |FYeσ | induces a

morphism of fansFXσ → FYeσ (cf. definition 4.1), whereFXσ and FY

eσ are thetropical fans associated toFXσ andFY

eσ respectively (cf. definition 5.3).

First of all we want to show that the restriction of a morphismto a subcycle is again amorphism:

Lemma 7.2. Let C ∈ Zn andD ∈ Zm be two cycles,f : C → D a morphism andE ∈ Zk(C) a subcycle ofC. Then the mapf ||E| : |E| → |D| induces a morphism oftropical cyclesf |E : E → D.

Proof. By definition of morphism there exist reduced representatives((X1, |X1|), ωX1)of C and ((Y, |Y |), ωY ) of D such that properties (a) and (b) in definition 7.1 are ful-filled. By definition of subcycle there exist reduced representatives((Z1, |Z1|), ωZ1) ofE and((X2, |X2|), ωX2) of C such that properties (a) and (b) in definition 5.12 are ful-filled, i.e. such that(Z1, |Z1|)E (X2, |X2|) and the tropical structures onZ1 andX2 agree.As “∼” is an equivalence relation there exists a common refinement((X, |X |), ωX) of((X1, |X1|), ωX1) and((X2, |X2|), ωX2) which we may assume to be reduced. Applyingconstruction 5.10 toZ1 and X we obtain a refinement((Z, |Z|), ωZ) :=((Z1 ∩X, |Z1 ∩X |), ωZ1∩X) of ((Z1, |Z1|), ωZ1) such that(Z, |Z|)E (X, |X |) and thetropical structures onZ andX agree. Thus properties (a) and (b) of definition 7.1 arefulfilled by Z andY and the restricted mapf ||E| : |E| → |D| gives us a morphismf |E : E → D. �

If we are given a morphism and a tropical cycle the following construction shows how tobuild the push-forward cycle of the given one along our morphism:

Construction7.3 (Push-forward of tropical cycles). Let C ∈ Zn andD ∈ Zm be twocycles and letf : C → D be a morphism. Let(((X, |X |, {ϕσ}), ωX), {Φσ}) and(((Y, |Y |, {ψσ}), ωY ), {Ψτ}) be representatives ofC andD fulfilling properties (a) and(b) of definition 7.1. Consider the collection of polyhedra

Z := {f(σ)|σ ∈ X contained in a maximal polyhedron ofX on whichf is injective}.

In generalZ is not a polyhedral complex. We resolve this by subdividing the polyhedra inZ and refiningX accordingly:Fix some polyhedronσ ∈ Y (m) and letτ1, . . . , τr ∈ Z be all polyhedra that are containedin σ. Property (b) of definition 7.1 implies that{ψ

eσ(τi)|i = 1, . . . , r} is a set of rationalpolyhedra inRneσ . Like in remark and definition 5.13 let{gi(x) ≤ bi|i = 1, . . . , N},gi ∈ Hom(Zneσ ,Z), bi ∈ R be all inequalities occurring in the description of all polyhedrain {ψ

eσ(τi)|i = 1, . . . , r} and let

Reσ := {ψ

eσ(τi)|i = 1, . . . , r} ∩HG1,b1 ∩ · · · ∩HGN ,bN,

Peσ := {ψ−1

eσ (τ)|τ ∈ Rσi}.


Like in construction 5.14Peσ can be seen as a kind of local refinement ofZ. But here

again taking the union over all maximal polyhedraσ ∈ Y (m) does in general not leadto a global refinement as there may be overlaps between polyhedra coming from dif-ferent σ. We fix this as follows (cf. 5.14): Forσ ∈ Y (m) and τ ∈

⋃m−1i=0 Y (i) let

P eσZ,eτ := { ∈ P

eσ|τ is the inclusion minimal polyhedron ofY containing } and

PZ,m :=⋃

eσ∈Y (m){ ∈ Peσ|∄τ ∈ Y (m−1) : ⊆ τ}. Then Z :=

PZ,m ∪(⋃

eτ∈Y (i):i<m{⋂

eσ∈Y (m):eτ⊆eσ τeσ|τeσ ∈ P eσZ,eτ}

)is the set of polyhedra (without any

overlaps now) that shall induce our wanted refinement ofX : Let T :={σ ∈ X(n)|f is injective onσ}, Q0 := {τ ∈ X |∄σ ∈ T : τ ⊆ σ} andQ1 :=(⋃

σ∈T {(f |σ)−1(τ)|τ ∈ Z, τ ⊆ f(σ)}

). Then defineX := Q0 ∪Q1.

Let τ ∈ Q1 and chooseσ ∈ T with τ ⊆ σ. Property (b) of definition 7.1 implies thatψ

eσ ◦ f ◦ ϕ−1σ is integer affine linear where defined. Henceϕσ(τ) is a rational polyhedron

in Rnσ . Denote byHσ,τ the smallest affine subspace ofRnσ containingϕσ(τ). We canconsider τ := ϕσ|τ to be a map τ : τ → Hσ,τ

∼= Rnτ . Note that by construction(X, |X |, {τ}) is a polyhedral complex. We endow it with the weight functionω

eX and

tropical structure{Φ eXτ } induced byX . Now we are able to define

f∗X := {f(σ)|σ ∈ X contained in a maximal polyhedron ofX on whichf is injective}

and|f∗X | :=⋃τ∈f∗X

τ . For every polyhedronτ ∈ f∗X let στ ∈ Y be the inclusion-minimal polyhedron containingτ . Then defineϑτ := ψστ

|τ : τ → Hστ ,τ∼= Rnτ , where

Hστ ,τ ⊆ Rnστ is the smallest affine subspace containing the rational polyhedronψστ(τ) ∈

Z. Note that this makes(f∗X, |f∗X |, {ϑτ}) into a polyhedral complex. Moreover notethat property (b) of definition 7.1 still holds forX andY . Hence we can assign weightsand tropical fans tof∗X as follows: Letσ ∈ f∗X , let σ ∈ Y be the inclusion-minimalpolyhedron containing it and letτ1, . . . , τr ∈ X be all polyhedra withf(τi) = σ that arecontained in a maximal polyhedron ofX on whichf is injective. Then letΨ

eσ(Seσ) = FYeσ

andΦeXτi

(Sτi) = F

eXτi

respectively be the corresponding open fans andFYeσ , F eX

τibe the

associated tropical fans. Property (b) of definition 7.1 implies thatf∗FeXτi

⊆ |FYeσ | is again

a tropical fan (note that we do not need to refineFeXτi

to construct this push-forward). Thuswe can define(F f∗Xσ , ω

eF f∗Xσ

):=

(r⋃

i=1

f∗FeXτi,

r∑

i=1

ωf∗ eF fX

τi

)and F f∗Xσ := F f∗Xσ ∩ Ψ

eσ(Sσ)

(here again we assume thatωf∗ eF fXτi

(τ) = 0 if τ /∈ f∗FeXτi

). Moreover we define

Θσ := Ψeσ|Sσ

: Sσ → |F f∗Xσ |.

Then the mapΘσ, σ ∈ f∗X is 1:1 on polyhedra and we can endow the maximal polyhedraof f∗X with weightsωf∗X(·) coming fromF f∗Xσ in this way. These weights are obviouslywell-defined by property (c) of the tropical polyhedral complex Y (cf. definition 5.4) andthe mapsΘσ for differentσ ∈ f∗X are obviously compatible. Hence we can define

f∗C :=

[(((f∗X, |f∗X |, {ϑτ}), ωf∗X

), {Θτ}

)]∈ Zn(D).

Note that the class[(((f∗X, |f∗X |, {ϑτ}), ωf∗X), {Θτ})] is independent of the choices wemade. Thus construction 7.3 immediately leads to the following


Corollary 7.4 (Push-forward of tropical cycles). LetC ∈ Zn andD ∈ Zm be two cyclesand letf : C → D be a morphism. Then for allk there is a well-defined andZ-linear map

Zk(C) −→ Zk(D) : E 7−→ f∗E := (f |E)∗E.

Proof. The linearity can be proven similar to the affine case (cf. proposition 4.6). �

Our next aim is to define the pull-back of Cartier divisors. But first we need the following

Lemma 7.5. LetC ∈ Zn andD ∈ Zm be two tropical cycles and letf : C → D be amorphism. By definition there exist reduced representatives(((X, |X |, {ϕσ}), ωX), {Φσ})ofC and(((Y, |Y |, {ψτ}), ωY ), {Ψτ}) ofD such that properties (a) and (b) in definition7.1 are fulfilled. Let(((Y1, |Y1|, {ψ′τ ′}), ωY1), {Ψ

′τ ′}) be a refinement ofY . Then there

is a refinement(((X1, |X1|, {ϕ′σ′}), ωX1), {Φ′σ}) ofX such that properties (a) and (b) of

definition 7.1 are fulfilled forX1 andY1.

Proof. Let X1 := {σ ∩ f−1(τ)|σ ∈ X, τ ∈ Y1}. By property (b) of definition 7.1 allϕσ(σ ∩ f−1(τ)) are rational polyhedra inRnσ . For everyσ′ ∈ X1 chooseσ ∈ X suchthatσ′ = σ ∩ f−1(τ) for someτ ∈ Y1. Then we can defineϕ′σ′ := ϕσ|σ′ : σ′ → Hσ,σ′ ∼=Rnσ′ , whereHσ,σ′ is the smallest affine subspace ofRnσ containingϕσ(σ′). Moreoverlet |X1| := |X |. Note that with these settings(X1, |X1|, {ϕ′σ′}) is a polyhedral complex.We can endow it with the weight functionωX1 and the tropical structure{Φ′σ′} induced byX . Together withY1 the tropical polyhedral complex(((X1, |X1|, {ϕ′σ′}), ωX1), {Φ

′σ′})

fulfills the requirements (a) and (b) of definition 7.1. �

Proposition 7.6 (Pull-back of Cartier divisors). Let C ∈ Zn andD ∈ Zm be tropicalcycles and letf : C → D be a morphism. Then there is a well-defined andZ-linear map

Div(D) −→ Div(C) : [{(Ui, hi)}] 7−→ f∗[{(Ui, hi)}] := [{(f−1(Ui), hi ◦ f)}].

Proof. We have to show thath ◦ f ∈ K∗C(f−1(U)) for h ∈ K∗D(U) and thath ◦ f ∈O∗C(f−1(U)) for h ∈ O∗D(U). Then the rest is obvious.So leth ∈ K∗D(U). Then there exists a representative(((Y, |Y |, {ψσ}), ωY ), {Ψτ}) of Dsuch that for every polyhedronσ ∈ Y the maph◦ψ−1

σ is locally integer affine linear. More-over, sincef is a morphism there exist representatives(((X, |X |, {ϕσ}), ωX), {Φτ}) of Cand(((Y ′, |Y ′|, {ψ′σ′}), ωY ′), {Ψ′τ ′}) of D such that properties (a) and (b) of definition7.1 are fulfilled, i.e.f(σ) ⊆ σ ∈ Y ′ for all σ ∈ X and the mapsΨ

eσ ◦ f ◦Φ−1σ induce mor-

phisms of fans. By lemma 7.5 we may assume thatY = Y ′. Now letσ ∈ X and choosesomeσ ∈ Y such thatf(σ) ⊆ σ. Property (b) of definition 7.1 implies thatψ

eσ ◦ f ◦ ϕ−1σ

andΨeσ ◦ f ◦Φ−1

σ are integer affine linear. Thush ◦ f ◦ϕ−1σ = (h ◦ψ−1

eσ ) ◦ (ψeσ ◦ f ◦ϕ

−1σ )

is locally integer affine linear andh ◦ f ∈ K∗C(f−1(U)). If additionallyh ◦ Ψ−1eσ is lo-

cally integer affine linear then so ish ◦ f ◦ Φ−1σ = (h ◦ Ψ−1

eσ ) ◦ (Ψeσ ◦ f ◦ Φ−1

σ ). Henceh ◦ f ∈ O∗C(f−1(U)) for h ∈ O∗D(U). �

Our last step in this chapter is to state the analogon of the projection formula from 4.8:

Proposition 7.7 (Projection formula). LetC ∈ Zn andD ∈ Zm be two cycles andf :C → D be a morphism. LetE ∈ Zk(C) be a subcycle ofC andd ∈ Div(D) be a Cartierdivisor. Then the following holds:

d · (f∗C) = f∗(f∗d · C) ∈ Zk−1(D).


Proof. The claim follows from the constructions off∗C andf∗d, from definition 6.5 andproposition 4.8. �

8. RATIONAL EQUIVALENCE

We will now make some first steps in establishing a concept of rational equivalence.

We fix an abstract tropical cycleA as ambient space and an arbitrary subgroupR ⊆Div(A) of the group of Cartier divisors onA. We define thePicard groupas the quo-tient groupPic(A) := Div(A)/R. Let Rk ⊆ Zk(A) denote the group generated by{ϕ · C|ϕ ∈ R,C ∈ Zk+1(A)}, i.e. by allk-dimensional cycles obtained by intersectinga Cartier divisor fromR with an arbitrary(k + 1)-dimensional cycle. We define thek-thChow groupto beAk(A) := Zk(A)/Rk.

Corollary 8.1 (Intersection products modulo rational equivalence). The map

· : Pic(A) ×Ak(A) → Ak−1(A),

([ϕ], [D]) 7→ [ϕ ·D]

is well-defined and bilinear.

Proof. By definition, for eachϕ ∈ R,D ∈ Zk(A) we haveϕ ·D ∈ Rk−1. Let furthermoreϕ · C be an element inRk (whereϕ ∈ R,C ∈ Zk(A)). Then it follows from proposition3.7 b) that for arbitraryψ ∈ Div(A) we getψ · (ϕ · C) = ϕ · (ψ · C) ∈ Rk−1. The claimfollows from the bilinearity of the intersection product. �

So far, our intersection theory takes place (at least locally) in Rn, which can be consid-ered as then-dimensional tropical algebraic torus. Especially, if we generated rationalequivalence by all rational functions onA, the resulting Chow groups and intersectionproducts would be useless in enumerative geometry: As in theclassical case, the divisorof a rational function might have components in the “boundary” of some compactificationof the “affine” varietyRn. Therefore, in the following we restrict the functions thatgen-erate rational equivalence to those “whose divisor in any torical compactification has nocomponents in the boundary”.

Definition 8.2 (Rational equivalence generated by bounded functions). Let A be an ab-stract tropical cycle andR(A) := {[(|A|, ϕ)]|ϕ bounded} be the group of all Cartier divi-sors globally given by a bounded rational function. We definethePicard groupPic(A) :=Div(A)/R(A) and theChow groupsAk(A) as above. We call two Cartier divisors (twok-dimensional subcycles resp.)rationally equivalent, if their classes inPic(A) (Ak(A)resp.) are the same.

Let us prove that we do not divide out too much for applications in enumerative geometry.

Lemma 8.3. LetC be an one-dimensional abstract tropical cycle,ϕ ∈ R(C) a boundedrational function onC and(((X, |X |, {mσ}σ∈X), ωX), {Mσ}σ∈X) a representative ofCsuch that|X | = |C| and for all σ ∈ X the mapϕ ◦m−1

σ =: ϕσ is integer affine linear.Then ∑

{p}∈X(0)

ωϕ({p}) = 0,

i.e.ϕ · C is of degree zero.


Proof. By definition, for all{p} ∈ X(0) we have

ωϕ({p}) =∑

σ∈X(1)

p∈σ

ω(σ)ϕσ(uσ/{p}).

Note that ifσ ∈ X(1) contains two different vertices, say∂1σ and∂2σ, we haveuσ/{∂1σ} =−uσ/{∂2σ}. If, otherwise,σ contains less than two vertices,mσ(σ) is a non-compact poly-hedron and thereforeϕ can only be bounded if it is constant onσ. Together we get

∑

{p}∈X(0)

ωϕ({p}) =∑

{p}∈X(0)

∑

σ∈X(1)

p∈σ

ω(σ)ϕσ(uσ/{p})

=∑

σ∈X(1)

∃! ∂σ ∈σ

ω(σ)ϕσ(uσ/{∂σ})

+∑

σ∈X(1)

∃! ∂1σ,∂2σ ∈σ

ω(σ)ϕσ(uσ/{∂1σ}) + ω(σ)ϕσ(uσ/{∂2σ})

=∑

σ∈X(1)

∃! ∂σ ∈σ

ω(σ) · 0

+∑

σ∈X(1)

∃! ∂1σ,∂2σ ∈σ

ω(σ)(ϕσ(uσ/{∂1σ}) − ϕσ(uσ/{∂1σ})︸︷︷︸

=0

)

= 0.

�

Remark8.4. As a consequence, for any cycleC ∈ Z∗(A) there is a well-defined morphism

deg: A0(C) −→ Z : [λ1P1 + . . .+ λrPr] 7−→ λ1 + . . .+ λr.

ForD ∈ A0(C) the number deg(D) is called thedegreeof D.Moreover, by corollary 8.1 there is a well-defined map of top products

Pic(A)d −→ Z : ([ϕ1], . . . , [ϕdim(C)]) 7−→ deg([ϕ1 · . . . · ϕdim(C) · C]),

whereA is our ambient cycle andd is the dimension ofC. Of course, this map is ofparticular interest when dealing with enumerative questions.

Of course, our chosen rational equivalenceR(A) := {[(|A|, ϕ)]|ϕ bounded} should alsobe compatible with pull-back and push-forward. However, inthe push-forward case weface problems due to our definition of rational functions. Let us first state the positiveresult in the pull-back case.

Lemma 8.5 (Pull-back of rational equivalence). LetC,D be tropical cycles and letf :C → D be a morphism between them. Then the pull-back mapDiv(D) → Div(C), ϕ 7→f∗ϕ induces a well-defined map on the quotientsPic(D) → Pic(C), [ϕ] 7→ [f∗ϕ].

Proof. We only have to show that for each element(|D|, ψ) ∈ R(D) the pull-back Cartierdivisorf∗(|D|, ψ) lies inR(C). But this follows from the trivial fact that the compositionψ ◦ f of a bounded functionψ and an arbitrary mapf is again bounded. �


Remark8.6 (Push-forward of rational equivalence). The corresponding statement for push-forwards is false! Let us again consider the elliptic curveC from Example 6.2. On thiscurve, the Weil divisor associated to the bounded rational functionψ illustrated in thepicture below equalsdiv(ψ) = α1 + α2 − α3 − α4.

α1

α2 α3

α4C

+1

+1 −1

−1

div(ψ) = α1 + α2 − α3 − α4

ψ

Let us now consider the cycleD obtained by identifyingα1 with α3 and the canonicalprojection mapf : C → D.

α2 α3

α1 α4C D

α2 eα α4

fglue

The push-forward ofdiv(ψ) under this morphism isf∗ div(ψ) = α2 − α4. But this Weildivisor can obviously not be obtained by a rational functiononD. This problem is due toour restrictive definition of rational functions (see remark 3.2). We are currently workingon a refined version of the related definitions.

9. INTERSECTION OF CYCLES INRn

So far we are only able to intersect Cartier divisors with cycles. Our aim in this section isnow to define the intersection of two cycles with ambient cycle Rn (with trivial structuremaps). But first we need some preparations:

Definition 9.1. Let (((X, |X |, {ϕσ}), ωX), {Φσ}) and (((Y, |Y |, {ψτ}), ωY ), {Ψτ}) betropical polyhedral complexes. We denote by

(((X, |X |, {ϕσ}), ωX), {Φσ}) × (((Y, |Y |, {ψτ}), ωY ), {Ψτ})

theircartesian product

(((X × Y, |X | × |Y |, {ϑσ×τ}), ωX×Y ), {Θσ×τ}),

where

X × Y := {σ × τ |σ ∈ X, τ ∈ Y } ,

ϑσ×τ := ϕσ × ψτ : σ × τ −→ Rnσ × Rnτ ,

ωX×Y (σ × τ) := ωX(σ) · ωY (τ),

Θσ×τ := Φσ × Ψτ : SXσ × SYτ −→ |FXσ | × |FYτ |.

Let FXσ and FYτ be the entire fans associated withFXσ andFYτ from above. Obvi-ously, the productFXσ × FYτ := {α × β|α ∈ FXσ , β ∈ FYτ } with weight function


ωeFX

σ ×eFY

τ(α× β) := ω

eFXσ

(α) · ωeFY

τ(β) is again a tropical fan and thus its intersection

with |FXσ | × |FYτ | yields an open fan (cf. definition 5.3). Hence the cartesian product(((X × Y, |X | × |Y |, {ϑσ×τ}), ωX×Y ), {Θσ×τ}) is again a tropical polyhedral complex.

If C = [(X,ωX)] andD = [(Y, ωY )] are tropical cycles we define

C ×D := [(X,ωX) × (Y, ωY )]

for (X,ωX) × (Y, ωY ) as defined above. Note thatC ×D does not depend on the choiceof the representativesX andY .

Remark9.2. We can express the diagonal inRn × Rn

[(△, 1)] = [({(x, x)|x ∈ Rn}, 1)] ∈ Zn(Rn × Rn)

as a product of Cartier divisors, namely

[(△, 1)] = ψ1 · · ·ψn · Rn × Rn,

whereψi = [{(Rn,max{0, xi − yi})}] ∈ Div(Rn × Rn), i = 1, . . . , n. We will use thisability to define the intersection product of any two cycles in Rn.

Definition 9.3. Let π : Rn × Rn → Rn : (x, y) 7→ x. Then we define the intersectionproduct of cycles inRn by

Zn−k(Rn) × Zn−l(R

n) −→ Zn−k−l(Rn)

(C,D) 7−→ C ·D := π∗(△ · (C ×D)),

whereπ∗ denotes the push-forward as defined in 7.4 and△·(C×D) := ψ1 · · ·ψn ·(C×D)with ψ1, . . . , ψn as defined in remark 9.2.

Having defined this intersection product of arbitrary cycles inRn we will prove now somebasic properties. But as a start we need the following lemmas:

Lemma 9.4. LetC ∈ Zk(Rn) be a cycle with representative(X,ωX) and letψ1, . . . , ψnbe the Cartier divisors defined in remark 9.2. Then(Xj , ωXj

) with

Xj :={(Rn × σ) ∩ {(x, y) ∈ Rn × Rn|xi = yi for i = j, . . . , n}|σ ∈ X

},

ωXj

((Rn × σ) ∩ {(x, y) ∈ Rn × Rn|xi = yi for i = j, . . . , n}

):= ωX(σ)

is a representative ofψj · · ·ψn · Rn × C.

Proof. We use induction onj. Forj = n+ 1 there is nothing to show. Now let the aboverepresentative be correct for somej + 1. We have to show thatXj is a tropical polyhedralcomplex and that it representsψj · · ·ψn · Rn × C: Note that

dim ((Rn × σ) ∩ {(x, y) ∈ Rn × Rn|xi = yi for i = j, . . . , n})< dim ((Rn × σ) ∩ {(x, y) ∈ Rn × Rn|xi = yi for i = j + 1, . . . , n})

(∗)

for all σ ∈ X . HenceXj is a tropical polyhedral complex. Moreover note that

Xj+1 := {σ ∩ {xj − yj = 0}, σ ∩ {xj − yj ≤ 0}, σ ∩ {xj − yj ≥ 0}|σ ∈ Xj+1}

with weights induced byXj+1 is a refinement ofXj+1 such thatmax{0, xj−yj} is linearon every face ofXj+1. By (∗) there are exactly two types of faces of codimension one inXj+1:

(i) (Rn × σ) ∩ {xi − yi = 0 for i = j, . . . , n} with σ ∈ X , codim(σ) = 0,


(ii) (Rn × σ) ∩ {xi − yi = 0 for i = j + 1, . . . , n; xj − yj ≤ 0} or(Rn × σ) ∩ {xi − yi = 0 for i = j + 1, . . . , n; xj − yj ≥ 0}with σ ∈ X ,codim(σ) = 1,

where the faces of the second type are not contained in{(x, y) ∈ Rn × Rn|xj = yj}.Hencemax{0, xj − yj} is linear on a neighborhood of every face of type (ii) and thusthese faces get weight zero inmax{0, xj − yj} · Xj+1. The faces of type (i) are weightedbyωXj+1 ((R

n×σ)∩{xi−yi = 0 for i = j+1, . . . , n}) in max{0, xj−yj} · Xj+1 sincex1 − y1, . . . , xn − yn are part of a lattice basis of(Zn × Zn)∨. Thusmax{0, xj − yj} ·

Xj+1 = Xj andXj is a representative ofψj · · ·ψn · Rn × C. �

Corollary 9.5. LetC ∈ Zk(Rn) be a cycle. Then we have the equation:

Rn · C = C.

Proof. Let (X,ωX) be a representative ofC, let π : Rn × Rn → Rn : (x, y) 7→ x andlet ψ1, . . . , ψn be the Cartier divisors defined in remark 9.2. By lemma 9.4 we know thatX1 = {{(x, x)|x ∈ σ}|σ ∈ X} with ωX1({(x, x)|x ∈ σ}) = ωX(σ) is a representative ofψ1 · · ·ψn · Rn × C. Hence

Rn · C = π∗(ψ1 · · ·ψn · Rn × C) = [π∗(X1, ωX1)] = [(X,ωX)] = C.

�

Lemma 9.6. Let C ∈ Zk(Rn) andD ∈ Zl(Rm) be abstract cycles,ϕ ∈ Div(Rn) aCartier divisor andπ : Rn × Rm → Rn :(x, y) 7→ x. Then:

(ϕ · C) ×D = π∗ϕ · (C ×D).

Proof. We prove the statement for affine cyclesC,D and an affine Cartier divisorϕ. Thegeneral case then follows by applying the statement locally.Choose arbitrary representativesY of D andh of ϕ and choose a representativeX of Csuch thath is linear on every face ofX . This implies thatπ∗h is linear on every face ofX × Y , too. InX × Y we have two types of faces of codimension one:

(i) σ × τ with σ ∈ X, τ ∈ Y, codim(σ) = 1, codim(τ) = 0,

(ii) σ × τ with σ ∈ X, τ ∈ Y, codim(σ) = 0, codim(τ) = 1.

For the second type the adjacent facets are exactly allσ×τ with τ > τ . We getωh(σ×τ) =0 in h ·X ×Y asπ∗h is linear onσ× |Y |. For the first type the adjacent facets are exactlyall σ× τ with σ > σ and the weights can be calculated exactly like forh ·X . This finishesthe proof. �

Let C andD be cycles inRn. Assume thatC can be expressed as a product of Cartierdivisors, i.e. there areϕ1, . . . , ϕr ∈ Div(Rn) such thatC = ϕr · · ·ϕ1 · Rn. The obviousquestions are now howC · D relates toϕr · · ·ϕ1 · D and whetherϕr · · ·ϕ1 · D dependson the choice of the Cartier divisorsϕi. To answer this question we first prove a somewhatstronger statement:

Lemma 9.7. LetC ∈ Zk(Rn) andD ∈ Zl(Rn) be cycles andϕ ∈ Div(Rn) a Cartierdivisor. Then we have the equality:

(ϕ · C) ·D = ϕ · (C ·D).


Proof. Let π : Rn × Rn → Rn : (x, y) 7→ x be like above. It holds:

(ϕ · C) ·D = π∗(△ · (ϕ · C) ×D)

9.6= π∗(π

∗ϕ · △ · C ×D)

7.7= ϕ · π∗(△ · C ×D)

= ϕ · (C ·D).

�

Corollary 9.8. Let C ∈ Zk(Rn) be a cycle such that there are Cartier divisorsϕ1, . . . , ϕr ∈ Div(Rn) with ϕr · · ·ϕ1 · Rn = C and letD ∈ Zl(Rn) be any cycle.Then

ϕr · · ·ϕ1 ·D = C ·D.

Proof. Applying lemma 9.7 and lemma 9.4 we obtain

C ·D = (ϕr · · ·ϕ1 · Rn) ·D = ϕr · · ·ϕ1 · (Rn ·D) = ϕr · · ·ϕ1 ·D.

�

Remark9.9. Note that corollary 9.8 in particular implies that our definition of the intersec-tion product onRn (cf. 9.3) is independent of the choice of the Cartier divisors describingthe diagonal△.

Theorem 9.10. LetC,C′ ∈ Zk(Rn),D ∈ Zl(Rn) andE ∈ Zm(Rn) be cycles. Then thefollowing equations hold:

(a) C ·D = D · C,

(b) (C + C′) ·D = C ·D + C′ ·D,

(c) (C ·D) ·E = C · (D · E).

Proof. (a): Letψ1, . . . , ψn ∈ Div(Rn × Rn) be like defined in remark 9.2. Note that forevery i ∈ {1, . . . , n} the mapsmax{0, xi − yi} andmax{0, yi − xi} only differ by aglobally linear map and hence define the same Cartier divisor. Thus we get

π∗(ψ1 · · ·ψn · C ×D) = π∗(ψ1 · · ·ψn ·D × C).

(b): Follows immediately by bilinearity of the intersection product

Div(Rn × Rn) × Zp(Rn × Rn)

·−→ Zp−1(R

n × Rn),

linearity of the push-forward and the fact that(C + C′) ×D = C ×D + C′ ×D.(c): We will show that△ · C × (π∗(△ ·D × E)) = △ · (π∗(△ · C ×D) × E) :Let π12 : (Rn)3 → (Rn)2 : (x, y, z) 7→ (x, y), π13 : (Rn)3 → (Rn)2 : (x, y, z) 7→ (x, z)andπ23 : (Rn)3 → (Rn)2 : (x, y, z) 7→ (y, z). An easy calculation shows that

△ · C × (π∗(△ ·D × E)) = △ · π12∗ (C × (△ ·D × E)) (1)

and

△ · (π∗(△ · C ×D) × E) = △ · π13∗ ((△ · C ×D) × E). (2)


Now letψ1, . . . , ψn be the Cartier divisors defined in remark 9.2. We label these Cartierdivisors with pairs of lettersψxyi to point out the coordinates they are acting on. We obtain

△ · C × (π∗(△ ·D × E))

(1)= △ · π12

∗ (C × (△ ·D × E))

= ψxy1 · · ·ψxyn · π12∗ (C × (ψyz1 · · ·ψyzn ·D × E))

7.7= π12

∗ ((π12)∗ψxy1 · · · (π12)∗ψxyn · C × (ψyz1 · · ·ψyzn ·D × E))

9.6= π12

∗ ((π23)∗ψyz1 · · · (π23)∗ψyzn · (π12)∗ψxy1 · · · (π12)∗ψxyn · C ×D × E)

9.8= π13

∗ ((π12)∗ψxy1 · · · (π12)∗ψxyn · (π13)∗ψxz1 · · · (π13)∗ψxzn · C ×D × E)

9.6= π13

∗ ((π13)∗ψxz1 · · · (π13)∗ψxzn · (ψxy1 · · ·ψxyn · C ×D) × E)

7.7= ψxz1 · · ·ψxzn · π13

∗ ((ψxy1 · · ·ψxyn · C ×D) × E)

= △ · π13∗ ((△ · C ×D) × E)

(2)= △ · (π∗(△ · C ×D) × E).

This proves (d). �

It remains to show that our intersection product is well-defined modulo rational equiva-lence. If this is the case the intersection product induced on A∗(Rn) clearly inherits theproperties of the intersection product onZ∗(Rn) we have proven in this section.

Proposition 9.11. The intersection productZn−k(Rn) × Zn−l(Rn)·

−→ Zn−k−l(Rn)induces a well-defined and bilinear map

An−k(Rn) ×An−l(R

n)·

−→ An−k−l(Rn) : ([C], [D]) 7−→ [C] · [D] := [C ·D].

Proof. Let h · C ∈ Rn−k (cf. section 8) andD ∈ Zn−l(Rn). Using lemma 9.7 we canconclude that(h · C) ·D = h · (C ·D) ∈ Rn−k−l. �

Our last step in this section is to prove a Bezout-style theorem for a special class of tropicalcycles inRn calledPn-generic cycles. But first we need some further definitions:

Definition 9.12. LetX be a tropical polyhedral complex inRn and letv ∈ Rn. We denotebyX(v) the translation

X(v) := {σ + v|σ ∈ X}

of X alongv. If [X ] = C ∈ Zk(Rn) thenC(v) := [X(v)]. Note that the classC(v) isindependent of the representativeX .

Definition 9.13. LetC ∈ Zk(Rn) be a tropical cycle and letLnk be the tropical fan definedin example 3.9. Then we define thedegree ofC to be the number

deg(C) := deg(C · [LncodimX ]),

where the second mapdeg : Z0(Rn) → Z : λ1P1 + . . . + λrPr 7→ λ1 + . . . + λr is theusual degree map. Then the mapdeg : Zk(Rn) → Z is obviously linear by definition.Moreover, we define the degree of[C] ∈ Ak(Rn) to bedeg([C]) := deg(C). Note thatdeg([C]) is well-defined by remark 8.4.


Lemma 9.14. LetC ∈ Zk(Rn) andD ∈ Zn−k(Rn) be two tropical cycles of complemen-tary dimensions. Then

deg(C ·D) = deg(C(v1) ·D(v2))

for all vectorsv1, v2 ∈ Rn. In particulardeg(C) = deg(C(v)) for all v ∈ Rn.

Proof. Let π : Rn × Rn → Rn : (x, y) 7→ x be the projection map as above and foru = (u1, . . . , un) ∈ Rn let

△(u) · (C ×D) := ψ1(u1) · · ·ψn(un) · (C ×D)

withψi(ui) := [{(Rn,max{0, xi−yi+ui})}] ∈ Div(Rn×Rn) be the intersection with thetranslated diagonal (cf. definition 9.3). Note that the rational functionmax{0, xi − yi} −max{0, xi − yi + ui} is bounded and that hence[ψi] = [ψi(ui)] ∈ Pic(Rn×Rn) for all i.It follows that

[△ · (C ×D)] = [△(u) · (C ×D)] ∈ A0(Rn × Rn)

and thus we get

deg(C ·D) = deg(π∗(△ · (C ×D)))

= deg(△ · (C ×D))

8.4= deg(△(v1 − v2) · (C ×D))

= deg(△ · (C(v1) ×D(v2)))

= deg(π∗(△ · (C(v1) ×D(v2))))

= deg(C(v1) ·D(v2)).

�

Definition 9.15 (Pn-generic cycles). Let C ∈ Zk(Rn) be a tropical cycle.C is calledPn-genericif for one (and thus for every) representativeX of C holds: For every faceσ ∈ X(k) there exists a polytopePσ ⊆ Rn of some dimensionr ∈ {0, . . . , k} and a coneσ ∈ (Lnk )

(k−r) such thatσ ⊆ Pσ + σ.

Theorem 9.16(Bezout’s theorem). LetC ∈ Zk(Rn) andD ∈ Zn−k(Rn) be two tropicalcycles of complementary dimensions. Moreover, assume thatC andD are Pn-generic.Then:

deg(C ·D) = deg(C) · deg(D).

Proof. Let (X,ωX) be a representative ofC and(Y, ωY ) be a representative ofD. MovingX along a (generic) direction vectora = (a1, . . . , an) ∈ Rk≫0 × Rn−k≪0 we can reach that|X(a)| and|Y | intersect in points in the interior of maximal faces only, namely |X(a)| ∩|Y | = {Pij |i = 1, . . . , r; j = 1, . . . , s} with Pij = σi ∩ σ′j for facets (we use the notationintroduced in example 3.9 for the cones ofLnk here)

• σi ∈ X(a)(k) with σi ⊆ σ{1,...,k} + ui ∈ Lnk(ui) and

• σ′j ∈ Y (n−k) with σ′j ⊆ σ{k+1,...,n} + vj ∈ Lnn−k(vj).

Hence we can conclude thatX(a) · Y =∑r

i=1

∑sj=1 ωX(σi)ωY (σ′j)Pij and thus by

lemma 9.14

deg(X · Y ) = deg(X(a) · Y ) =

r∑

i=1

s∑

j=1

ωX(σi)ωY (σ′j).


σ3

Y

X

σ′1 σ′

2

σ1

σ2

The intersection ofX(a) andY as described in 9.16.

Moreover we can deduce that|X(a)| ∩ |Lnn−k(v1)| = {P11, . . . , Pr1}. HenceX(a) ·Lnn−k(v1) =

∑ri=1 ωX(σi)Pi1 and again by lemma 9.14

deg(X) = deg(X(a) · Lnn−k(v1)) =r∑

i=1

ωX(σi).

Analogously we obtain

deg(Y ) = deg(Y · Lnk(u1)) =

s∑

j=1

ωY (σ′j).

Thus the claim follows. �

REFERENCES

[FS] W. Fulton, B. Sturmfels,Intersection Theory on Toric Varieties, Topology, Volume 36, Number 2,March 1997 , pp. 335–353(19).

[GKM] A. Gathmann, M.Kerber and H. Markwig,Tropical fans and the moduli spaces of tropical curves,Compositio Mathematica (to appear), preprint math.AG/0708.2268.

[GM] A. Gathmann, H. Markwig,Kontsevich’s formula and the WDVV equations in tropical geometry, Ad-vances in Mathematics, Volume 217, Issue 2, January 2008, pp. 537–560(24).

[K] E. Katz, A Tropical Toolkit, preprint math.AG/0610878.[KM] M.Kerber, H. Markwig, Counting tropical elliptic plane curves with fixed j-invariant, Commentarii

Mathematici Helvetici (to appear), preprint math.AG/0608472.[M] G. Mikhalkin, Tropical Geometry and its applications, Proceedings of the ICM, Madrid, Spain (2006),

827–852(26).[RGST] J. Richter-Gebert, B. Sturmfels and T. Theobald,First steps in tropical geometry, Idempotent Math-

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LARS ALLERMANN , FACHBEREICH MATHEMATIK , TU KAISERSLAUTERN, POSTFACH3049, 67653 KAI -SERSLAUTERN, GERMANY

E-mail address: [email protected]

JOHANNESRAU , FACHBEREICHMATHEMATIK , TU KAISERSLAUTERN, POSTFACH3049, 67653 KAISERS-LAUTERN, GERMANY

E-mail address: [email protected]

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