Tropical intersection theory

Transcription

1 Tropical intersection theory Lars Allermann Dem Fachbereich Mathematik der Technischen Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) vorgelegte Dissertation

2

3 Contents Preface v 1 Foundations of tropical intersection theory Affine tropical cycles Affine Cartier divisors and their associated Weil divisors Push-forward of affine cycles and pull-back of Cartier divisors Abstract tropical cycles Cartier divisors and their associated Weil divisors Push-forward of tropical cycles and pull-back of Cartier divisors Rational equivalence Intersection of cycles in R n Rational equivalence on R n Tropical cycles with real slopes and numerical equivalence Tropical cycles with real slopes Numerical equivalence on fans Chow groups via numerical equivalence Tropical intersection products on smooth varieties Intersection products on tropical linear spaces Intersection products on smooth tropical varieties Pull-backs of cycles on smooth varieties Weil and Cartier divisors under tropical modifications Modifications and contractions Cartier divisors and Weil divisors Chern classes of tropical vector bundles Tropical vector bundles Chern classes Vector bundles on an elliptic curve Bibliography 105 Appendix: Pictures of tropical surfaces 109 iii

4

5 Preface Tropical geometry Tropical geometry is a rather new field within mathematics. Its roots go back to the work of George M. Bergman [B71] as well as Robert Bieri and John R. J. Groves [BG84], but only in the last ten years tropical geometry became a subject on its own. The general idea of the theory is to map objects in algebraic or symplectic geometry to polyhedral objects using some tropicalization process. These latter objects can then be studied by purely combinatorial means, making life much easier in many cases. Nevertheless, in this tropicalization process enough properties of the original objects are preserved such that it is possible to transfer back many tropical results to algebraic or symplectic geometry. Tropical geometry is a useful tool in many different areas of mathematics, such as real enumerative geometry (e.g. [IKS03], [IKS04], [IKS09], [M05]), symplectic geometry (e.g. [A06]), number theory (e.g. [G07a], [G07b]), combinatorics (e.g. [J08]) as well as algebraic statistics and computational biology (e.g. [PS04]). There are a number of ways to approach tropical geometry. In this thesis we choose a purely combinatorial point of view on the topic: We set up the beginnings of an extensive tropical intersection theory on its own, without using the existing theory in algebraic geometry. Nevertheless, our definitions and results are highly inspired by the algebro-geometric theory (cf. [F84]). Results of this thesis In this thesis we set up the beginnings of a tropical intersection theory covering many concepts and tools of its counterpart in algebraic geometry. For instance: We develop notions of tropical varieties and cycles, rational functions and Cartier divisors, intersection products of Cartier divisors with cycles, morphisms of tropical varieties and pull-backs of Cartier divisors and push-forward of cycles as well as rational and numerical equivalence. We prove a projection formula for morphisms of tropical varieties. For the special case that our ambient cycle is R n we prove that the concepts of rational and numerical equivalence agree. Moreover, restricting ourselves to generic cycles we study the numerical equivalence of cycles in more detail. v

6 Preface For the special case that our ambient cycle is a fan we show that every cycle is numerical equivalent to an affine cycle. We define an intersection product of cycles in any smooth tropical variety and prove some basic properties. We use this intersection product to introduce a concept of pull-back of cycles along morphisms of smooth varieties. We prove that under some assumptions the one-to-one correspondence of Weil and Cartier divisors that exists for example on R n is preserved by modifications as introduced in [M06]. We introduce the notions of tropical vector bundles and Chern classes of tropical vector bundles and prove some basic properties. Chapter synopsis This thesis consists of five chapters: Chapter 1 contains the basics of the theory and is essential for the rest of the thesis. Chapters 2 5 are to a large extent independent of each other and can be read separately. Chapter 1: Foundations of tropical intersection theory In section 1.1 we introduce the concept of affine tropical cycles as balanced weighted fans modulo refinements. After that, in section 1.2, we define Cartier divisors to be piecewise integer affine linear functions modulo globally linear functions and set up an intersection product of Cartier divisors and cycles. In section 1.3 we continue with the definitions of morphisms of tropical cycles, of pull-backs of Cartier divisors and push-forwards of cycles and prove a projection formula. In sections 1.4, 1.5 and 1.6 we generalize these concepts to abstract tropical cycles which are abstract polyhedral complexes modulo refinements with affine cycles as local building blocks. In section 1.7 we introduce a concept of rational equivalence. Finally, in sections 1.8 and 1.9, we set up an intersection product of cycles and prove that every cycle is rationally equivalent to some affine cycle in the special case that our ambient cycle is R n. We use this result to show that rational and numerical equivalence agree in this case and prove a tropical Bézout s theorem. Chapter 2: Tropical cycles with real slopes and numerical equivalence In section 2.1 we generalize our definitions of tropical cycles to polyhedral complexes with non-rational slopes. We use these cycles with non-rational slopes in section 2.2 to show that if our ambient cycle is a fan then every subcycle is numerically equivalent to some affine cycle. In section 2.3 we restrict ourselves to cycles in R n that are generic in some sense and study the concept of numerical equivalence in more detail. Chapter 3: Tropical intersection products on smooth varieties In section 3.1 we define an intersection product of tropical cycles on tropical linear spaces L n k and on other, related fans. In section 3.2 we use this result to obtain vi

7 Preface an intersection product of cycles on any smooth tropical variety. Finally, in section 3.3, we use the intersection product to introduce a concept of pull-backs of cycles along morphisms of smooth tropical varieties and prove that this pull-back has all expected properties. Chapter 4: Weil and Cartier divisors under tropical modifications In section 4.1 we introduce modifications and contractions and study their basic properties. In section 4.2 we prove that under some further assumptions a one-to-one correspondence of Weil and Cartier divisors is preserved by modifications. In particular we can prove that on any smooth tropical variety we have a one-to-one correspondence of Weil and Cartier divisors. Moreover, using the result it is possible to prove that there exists a one-to-one correspondence of Weil and Cartier divisors on the moduli space of n-marked abstract tropical curves M 0,n,trop (cf. [H07]). Chapter 5: Chern classes of tropical vector bundles In section 5.1 we give definitions of tropical vector bundles and rational sections of tropical vector bundles. We use these rational sections in section 5.2 to define the Chern classes of such a tropical vector bundle. Moreover, we prove that these Chern classes have all expected properties. In section 5.3 we classify all tropical vector bundles on an elliptic curve up to isomorphisms. Publication of the results This thesis contains material from my articles [AR07], [AR08] and [A09]. In particular, the first chapter is the outcome of joint work with Johannes Rau and it is virtually impossible to specify the contributions each of us made. As far as it can be told, main contributions of Johannes Rau are contained in sections 1.2, 1.5 and 1.7, whereas sections 1.1, 1.3, 1.4, 1.6 and 1.8 are mainly based on my ideas. Section 1.9 contains important contributions of both of us. Moreover, I omit those parts that are to a large extent the work of Johannes Rau. vii

8

9 Preface Danksagung Ich danke......meinem Betreuer Andreas Gathmann für die vielen Ideen, die viele Zeit und überhaupt für die hervorragende Betreuung....meiner Freundin Tanja Berger für viel Motivation und das Pendeln zwischen Frankfurt und Kaiserslautern....Georges François, Marina Franz, Matthias Herold, Michael Kerber, Henning Meyer, Johannes Rau, Stefan Steidel, Oliver Wienand und Christiane Zeck, ohne die es an der Uni ganz schön langweilig gewesen wäre....meinen Eltern für die nicht nur finanzielle Unterstützung während der letzten Jahre. ix

10

11 1 Foundations of tropical intersection theory This chapter consists of three parts: In the first part (sections ) we start with the introduction of affine tropical cycles as balanced weighted fans modulo refinements and affine tropical varieties as affine cycles with non-negative weights. One would like to define the intersection of two such objects, but in general neither is the set-theoretic intersection of two cycles again a cycle, nor does the concept of stable intersection as introduced in [RGST05] work for arbitrary ambient spaces as can be seen in example Therefore we introduce the notion of affine Cartier divisors on tropical cycles as piecewise integer affine linear functions modulo globally affine linear functions and define a bilinear intersection product of Cartier divisors and cycles. We then prove the commutativity of this product and a projection formula for push-forwards of cycles and pull-backs of Cartier divisors. In the second part (sections ) we generalize the theory developed in the first part to abstract cycles which are abstract polyhedral complexes modulo refinements with affine cycles as local building blocks. Again, abstract tropical varieties are just cycles with non-negative weights. In both the affine and abstract case a remarkable difference to the classical situation occurs: We can define the mentioned intersection products on the level of cycles, i.e. we can intersect Cartier divisors with cycles and obtain a well-defined cycle not only a cycle class up to rational equivalence as it is the case in classical algebraic geometry. However, for simplifying the computations of concrete enumerative numbers we introduce a notion of rational equivalence of cycles in section 1.7. In the third part (section ) we finally use our theory to define the intersection product of two cycles with ambient space R n. Here again it is remarkable that we can define these intersections even for self-intersections on the level of cycles. It turns out that this intersection product coincides with the stable intersection discussed in [M06] and [RGST05] (see [K09] and [R08]). Afterwards, we study the special case of rational equivalence in R n in more detail and show that every tropical cycle in R n is equivalent to a uniquely determined affine cycle, called its degree. We use this result to prove a tropical Bézout s theorem. 1.1 Affine tropical cycles In this section we will briefly summarize the definitions and some properties of our basic objects. We refer to [GKM07] for more details (but note that we use a slightly more general definition of fan). Throughout this paper Λ will always denote a finitely generated free abelian group, i.e. 1

12 1.1 Affine tropical cycles a group isomorphic to Z r for some r N, and V := Λ Z R the associated real vector space containing Λ as a lattice. We will denote the dual lattice in the dual vector space by Λ V. Definition (Cones) A cone in V is a subset σ V that can be described by finitely many linear integral equalities and inequalities, i.e. a set of the form σ = {x V f 1 (x) = 0,...,f r (x) = 0,f r+1 (x) 0,...,f N (x) 0} for some linear forms f 1,...,f N Λ. We denote by V σ the smallest linear subspace of V containing σ and by Λ σ the lattice V σ Λ. We define the dimension of σ to be the dimension of V σ. Definition (Fans) A fan X in V is a finite set of cones in V satisfying the following conditions: (a) The intersection of any two cones in X belongs to X as well, (b) every cone σ X is the disjoint union σ =. τ X:τ σ τri, where τ ri denotes the relative interior of τ, i.e. the interior of τ in V τ. We will denote the set of all k-dimensional cones of X by X (k). The dimension of X is defined to be the maximum of the dimensions of the cones in X. The fan X is called pure-dimensional if each inclusion-maximal cone in X has this dimension. The union of all cones in X will be denoted X V. If X is a fan of pure dimension k then the cones σ X (k) are called facets of X. Let X be a fan and σ X a cone. A cone τ X with τ σ is called a face of σ. We write this as τ σ (or τ < σ if in addition τ σ holds). Clearly we have V τ V σ and Λ τ Λ σ in this case. Note that τ < σ implies that τ is contained in a proper face (in the usual sense) of σ. Example The following figure shows three examples of fans of pure dimension two in V = R 2 according to definition Note that the third example is not a fan in the sense of [GKM07, definition 2.4] as for example σ 1 σ 2 is not a face of σ 2 according to that definition. τ 3 τ 2 τ 2 σ 1 τ 1 σ 1 σ 2 0 σ 2 σ 1 σ 2 0 τ 1 τ 4 τ 5 σ3 τ 6 τ 1 τ 3 σ 3 Figure 1.1: Examples of fans in R 2. 2

13 Chapter 1: Foundations of tropical intersection theory X Y τ 1 σ 1 σ 2 0 τ 3 τ τ 2 Figure 1.2: A fan X and a subfan Y X. Construction (Normal vectors) Let τ < σ be cones of some fan X in V with dim(τ) = dim(σ) 1. This implies that there is a linear form f Λ σ that is zero on τ, non-negative on σ and not identically zero on σ. Let u σ Λ σ be a vector generating Λ σ /Λ τ = Z with f(uσ ) > 0. Note that its class u σ/τ := [u σ ] Λ σ /Λ τ does not depend on the choice of u σ. We call u σ/τ the (primitive) normal vector of σ relative to τ. Definition (Subfans) Let X,Y be fans in V. Y is called a subfan of X if for every cone σ Y there exists a cone σ X such that σ σ. In this case we write Y X and define a map C Y,X : Y X that maps a cone σ Y to the unique inclusion-minimal cone σ X with σ σ. Definition (Weighted fans) A weighted fan (X,ω X ) of dimension k in V is a fan X in V of pure dimension k, together with a map ω X : X (k) Z. The number ω X (σ) is called the weight of the facet σ X (k). For simplicity we usually write ω(σ) instead of ω X (σ). Moreover, we want to consider the empty fan to be a weighted fan of dimension k for all k. Furthermore, by abuse of notation we simply write X for the weighted fan (X,ω X ) if the weight function ω X is clear from the context. Definition (Tropical fans) A tropical fan of dimension k in V is a weighted fan (X,ω X ) of dimension k satisfying the following balancing condition for every τ X (k 1) : ω X (σ) u σ/τ = 0 V/V τ. σ:τ<σ Let (X,ω X ) be a weighted fan of dimension k in V and X the fan X := {τ X τ σ for some facet σ X with ω X (σ) 0}. (X,ω X ) := (X,ω X (X )(k)) is called the non-zero part of X and is again a weighted fan of dimension k in V (note that X = is possible). Obviously (X,ω X ) is a tropical fan if and only if (X,ω X ) is one. We call a weighted fan (X,ω X ) reduced if all its facets have non-zero weight, i.e. if (X,ω X ) = (X,ω X ) holds. 3

14 1.1 Affine tropical cycles Example The following figure shows three examples of tropical fans in R 2 and R 3, respectively, according to definition 1.1.7: ω = 1 ω = 1 ω = 2 Figure 1.3: Examples of tropical fans in R 2 and R 3, respectively. The blue cones in the second and third fan are supposed to have weight one, the red cones to have weight minus one. Remark Let (X,ω X ) be a tropical fan of dimension k and let τ X (k 1). Let σ 1,...,σ N be all cones in X with σ i > τ. For all i let v σi /τ Λ be a representative of the primitive normal vector u σi /τ Λ σi /Λ τ. By the above balancing condition we have N i=1 ω X(σ i ) v σi /τ = λ τ for some λ τ Λ τ. Obviously we have λ τ = gcd(ω X (σ 1 ),...,ω X (σ N )) λ τ for some further λ τ Λ τ. We can represent the greatest common divisor by a linear combination gcd(ω X (σ 1 ),...,ω X (σ N )) = α 1 ω X (σ 1 )+ +α N ω X (σ N ) with α 1,...,α N Z and define ṽ σi /τ := v σi /τ α i λ τ for all i. Note that ṽ σi /τ is a representative of u σi /τ, too. Replacing all v σi /τ by ṽ σi /τ we can always assume that N i=1 ω X(σ) v σ/τ = 0 Λ. Definition (Refinements) Let (X,ω X ) and (Y,ω Y ) be weighted fans in V. We call (Y,ω Y ) a refinement of (X,ω X ) if the following holds: (a) Y X, (b) Y = X and (c) ω Y (σ) = ω X (C Y,X (σ)) for every σ (Y ) (dim(y )). Note that property (b) implies that either X = Y = or dim(x) = dim(y ). We call two weighted fans (X,ω X ) and (Y,ω Y ) in V equivalent (write (X,ω X ) (Y,ω Y )) if they have a common refinement. Note that (X,ω X )and (X,ω X (X )(dim(x))) are always equivalent. 4

15 Chapter 1: Foundations of tropical intersection theory X 1 X 2 X ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 ω = 1 Figure 1.4: Two refinements X 1 and X 2 of R 2 and a common refinement X. ω = 1 Remark Note that for a weighted fan (X,ω X ) of dimension k and a refinement (Y,ω Y ) we have the following two properties: (a) X = Y, i.e. the support X is well-defined on the equivalence class of X, (b) for every cone τ Y (k 1) there are exactly two cases that can occur: Either we have dim C Y,X (τ) = k or we have dimc Y,X (τ) = k 1. In the first case all cones σ Y (k) with σ > τ must be contained in C Y,X (τ). Thus there are precisely two such cones σ 1 and σ 2 with ω Y (σ 1 ) = ω Y (σ 2 ) and u σ1 /τ = u σ2 /τ. In the second case we have a 1:1 correspondence between cones σ Y (k) with τ < σ and cones σ X (k) with C Y,X (τ) < σ preserving weights and normal vectors. Construction (Refinements) Let (X,ω X ) be a weighted fan and Y be any fan in V with X Y. Let P := {σ σ σ X,σ Y }. In general P is not a fan in V as can be seen in the following example: X τ 2 Y σ 1 σ 2 σ 2 0 τ 1 σ 1 σ 3 τ 3 τ 1 Figure 1.5: Fans X and Y such that {σ σ σ X,σ Y } is not a fan. Here P contains τ 1 = σ 2 σ 1, but also τ 2 = σ 1 σ 2 and τ 3 = σ 3 σ 2. Hence property (b) of definition is, for instance, not fulfilled for τ 1. To resolve this, we define X Y := {σ P τ P (dim(σ)) with τ σ}. Note that X Y is now a fan in V. We can make it into a weighted fan by setting ω X Y (σ) := ω X (C X Y,X (σ)) for all σ (X Y ) (dim(x)). Then (X Y,ω X Y ) is a refinement of (X,ω X ). Note that if (X,ω X ) and (Y,ω Y ) are both weighted fans and 5

16 1.1 Affine tropical cycles X = Y we can form both intersections X Y and Y X. Of course, the underlying fans are the same in both cases, but the weights may differ since they are always induced by the first fan. The following setting is a special case of this construction: Let (X,ω X ) be a weighted fan of dimension k in V and let f Λ be a non-zero linear form. Then we can construct a refinement of (X,ω X ) as follows: H f := {{x V f(x) 0}, {x V f(x) = 0}, {x V f(x) 0}} is a fan in V with H f = V. Thus we have X H f and by our above construction we get a refinement (X f,ω Xf ) := (X H f,ω X Hf ) of X. Obviously we still have to answer the question if the equivalence of weighted fans is indeed an equivalence relation and if this notion of equivalence is well-defined on tropical fans. We will do this in the following lemma: Lemma The following two statements hold: (a) The relation is an equivalence relation on the set of k-dimensional weighted fans in V. (b) If (X,ω X ) is a weighted fan of dimension k and (Y,ω Y ) is a refinement then (X,ω X ) is a tropical fan if and only if (Y,ω Y ) is one. Proof. Recall that a fan and its non-zero part are always equivalent and that a weighted fan X is tropical if and only if its non-zero part X is. Thus we may assume that all our fans are reduced and the proof is the same as in [GKM07, section 2]. Having done all these preparations we are now able to introduce the most important objects for the succeeding sections: Definition (Affine cycles and affine tropical varieties) Let (X,ω X ) be a tropical fan of dimension k in V. We denote by [(X,ω X )] its equivalence class under the equivalence relation and by Zk aff (V ) the set of equivalence classes Z aff k (V ) := {[(X,ω X )] (X,ω X ) tropical fan of dimension k in V }. The elements of Zk aff(v ) are called affine (tropical) k-cycles in V. A k-cycle [(X,ω X)] is called an affine tropical variety if ω X (σ) 0 for every σ X (k). Note that the last property is independent of the choice of the representative of [(X,ω X )]. Moreover, note that 0 := [ ] Zk aff(v ) for every k. We define [(X,ω X)] := X. This definition is well-defined by remark Construction (Sums of affine cycles) Let [(X,ω X )] and [(Y,ω Y )] be k-cycles in V. We would like to form a fan X + Y by taking the union X Y, but obviously this collection of cones is in general not a fan. Using appropriate refinements we can resolve this problem: Let f 1 (x) 0,...,f N1 (x) 0, 6

17 Chapter 1: Foundations of tropical intersection theory f N1 +1(x) = 0,...,f N (x) = 0 and g 1 (x) 0,...,g M1 (x) 0, g M1 +1(x) = 0,..., g M (x) = 0 be all different equalities and inequalities occurring in the descriptions of all the cones belonging to X and Y, respectively. Using construction we get refinements X := X H f1 H fn H g1 H gm of X and Ỹ := Y H f1 H fn H g1 H gm of Y (note that the final refinements do not depend on the order of the single refinements). A cone occurring in X or Ỹ is then of the form σ = { fi (x) 0, f j (x) = 0, f k (x) 0, g i (x) 0, g j (x) = 0, g k (x) 0 i I, j J, k K, i I, j J, k K for some partitions I J K = {1,...,N} and I J K = {1,...,M}. Now, all these cones σ belong to the fan H f1 H fn H g1 H gm as well and hence X Ỹ fulfills definition Thus, now we can define the sum of X and Y to be X + Y := X Ỹ together with weights ω X+Y (σ) := ω X(σ) + ωỹ (σ) for every facet of X + Y (we set ω (σ) := 0 if σ does not occur in { X,Ỹ }). By construction, (X + Y,ω X+Y ) is again a tropical fan of dimension k. Moreover, enlarging the sets {f i }, {g j } by more (in)equalities just corresponds to refinements of X and Y and only leads to a refinement of X + Y. Thus, replacing the set of relations by another one that also describes the cones in X and Y, or replacing X or Y by refinements keeps the equivalence class [(X + Y,ω X+Y )] unchanged, i.e. taking sums is a well-defined operation on cycles. This construction immediately leads to the following lemma: Lemma Zk aff (V ) together with the operation + from construction forms an abelian group. Proof. The class of the empty fan 0 = [ ] is the neutral element of this operation and [(X, ω X )] is the inverse element of [(X,ω X )] Zk aff (V ). } Of course we do not want to restrict ourselves to cycles situated in some R n. Therefore we give the following generalization of definition : Definition Let X be a fan in V. An affine k-cycle in X is an element [(Y,ω Y )] of Zk aff (V ) such that Y X. We denote by Zk aff(x) the set of k-cycles in X. Note that ( Zk aff (X), +) is a subgroup of (Zk aff (V ), +). The elements of the group Zaff dim X 1 (X) are called Weil divisors on X. If [(X,ω X )] is a cycle in V then Zk aff([(x,ω X)]) := Zk aff(x ). 7

18 1.2 Affine Cartier divisors and their associated Weil divisors 1.2 Affine Cartier divisors and their associated Weil divisors Definition (Rational functions) Let C be an affine k-cycle. A (non-zero) rational function on C is a continuous piecewise linear function ϕ : C R, i.e. there exists a representative (X,ω X ) of C such that on each cone σ X the map ϕ is the restriction of an integer affine linear function ϕ σ = λ + c, λ Λ σ, c R. Obviously, c is the same on all faces by c = ϕ(0) and λ is uniquely determined by ϕ and therefore denoted by ϕ σ := λ. The set of (non-zero) rational functions of C is denoted by K (C). Remark (The zero function and restrictions to subcycles) The zero function can be thought of being the constant function, therefore K(C) := K (C) { }. With respect to the operations max and +, K(C) is a semifield. Let us note an important difference to the classical case: Let D be an arbitrary subcycle of C and ϕ K (C). Then ϕ D K (D), whereas in the classical case it might become zero. This will be crucial for defining intersection products not only modulo rational equivalence. As in the classical case, each non-zero rational function ϕ on C defines a Weil divisor, i.e. a cycle in Zdim aff C 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 of ϕ in V R and intersect it with V { } and V {+ }. However, our function ϕ takes values only in R, in fact. On the other hand, the graph of ϕ is not a tropical object as it is not balanced: Where ϕ is not linear, our graph gets edges that might violate the balancing condition. So, we first make the graph balanced by adding new faces in the additional direction (0, 1) V R and then apply our naive approach. Let us make this precise. Construction (The associated Weil divisor) Let C be an affine k-cycle in V = Λ R and ϕ K (C) a rational function on C. Let furthermore (X, ω) be a representative of C on whose faces ϕ is affine linear. Therefore, for each cone σ X, we get a cone σ := (id ϕ σ )(σ) in V R of the same dimension. Obviously, Γ ϕ := { σ σ X} forms a fan which we can make into a weighted fan (Γ ϕ, ω) by ω( σ) := ω(σ). Its support is just the set-theoretic graph of ϕ ϕ(0) in X R. For τ < σ with dim(τ) = dim(σ) 1 let v σ/τ Λ be a representative of the normal vector u σ/τ. Then, ( v σ/τ,ϕ σ (v σ/τ ) ) Λ Z is a representative of the normal vector u σ/ τ. Therefore, summing around a cone τ with dim τ = dimτ = k 1, we get σ Γ (k) ϕ τ< σ ω( σ) ( v σ/τ,ϕ σ (v σ/τ ) ) = σ X (k) τ<σ ω(σ)v σ/τ, σ X (k) τ<σ ϕ σ (ω(σ)v σ/τ ). From the balancing condition for (X,ω) it follows that σ X (k) :τ<σ ω(σ)v σ/τ V τ, which also means ( σ X (k) :τ<σ ω(σ)v σ/τ,ϕ τ ( σ X (k) :τ<σ ω(σ)v σ/τ)) V τ. Therefore, 8

19 Chapter 1: Foundations of tropical intersection theory σ 2 σ 3 new edge σ 1 Γ ϕ R 2 R σ 3 σ 2 σ Γ ϕ R 2 1 R new faces σ 2 σ 1 C R 2 σ 3 σ 1 σ 2 C =R 2 σ 3 Figure 1.6: Two examples of the construction of a Weil divisor. modulo V τ, our first sum equals 0, σ X (k) τ<σ ϕ σ (ω(σ)v σ/τ ) ϕ τ ( σ X (k) τ<σ ) ω(σ)v σ/τ V R. So, in order to make (Γ ϕ, ω) balanced at τ, we add the cone ϑ := τ +({0} R 0 ) with weight ω(ϑ) = ( ) σ X (k) :τ<σ ϕ σ(ω(σ)v σ/τ ) ϕ τ σ X (k) :τ<σ ω(σ)v σ/τ. As obviously [(0, 1)] = u ϑ/ τ (V R)/V τ, the above calculation shows that then the balancing condition around τ holds. In other words, we build the new fan (Γ ϕ, ω ), where Γ ϕ ω Γ (k) ϕ ω ( τ + ({0} R 0 )) := := Γ ϕ { τ } + ({0} R 0 ) τ Γ ϕ \ Γ ϕ (k), := ω, ( ) ϕ σ (ω(σ)v σ/τ ) ϕ τ ω(σ)v σ/τ σ X (k) τ<σ if dim τ = k 1. σ X (k) τ<σ This fan is balanced around all τ Γ (k 1) ϕ. We will show that it is also balanced at all new cones of dimension k 1 in proposition We now think of intersecting this new fan with V { } to get our desired Weil divisor (as our weights are allowed to be negative, we can forget about intersecting also with V {+ }). This construction leads to the following definition. Definition (Associated Weil divisors) Let C be an affine k-cycle in V = Λ R and ϕ K (C) a rational function on C. Let furthermore (X,ω) be a representative of C on whose cones ϕ is affine linear. We 9

20 1.2 Affine Cartier divisors and their associated Weil divisors define div(ϕ) := ϕ C := [( k 1 i=0 X(i),ω ϕ )] Zk 1 aff (C), where ω ϕ : X (k 1) Z, τ σ X (k) τ<σ ϕ σ (ω(σ)v σ/τ ) ϕ τ ( σ X (k) τ<σ ω(σ)v σ/τ ) and the v σ/τ are arbitrary representatives of the normal vectors u σ/τ. Let D be an arbitrary subcycle of C. By remark 1.2.2, we can define ϕ D := ϕ D D. Remark Obviously, ω ϕ (τ) is independent of the choice of the v σ/τ, as a different choice only differs by elements in V τ. 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 ): Let τ := C Y,X (τ). If dim τ = k, there are precisely two cones at τ < σ 1,σ 2 Y (k), which then fulfill C Y,X (σ 1 ) = C Y,X (σ 2 ) and therefore u σ1 /τ = u σ2 /τ, υ(σ 1 ) = υ(σ 2 ) and ϕ σ1 = ϕ σ2. It follows that υ ϕ (τ) = 0. If dimτ = k 1, C Y,X gives a one-to-one correspondence between {σ Y (k) τ < σ} and {σ X (k) τ < σ } respecting weights and normal vectors, and we have ϕ σ = ϕ CY,X (σ). It follows that υ ϕ (τ) = ω ϕ (τ ). So the two weighted fans we obtain are equivalent. Remark (Affine linear functions and sums) Let ϕ K (C) be globally affine linear, i.e. ϕ = λ C + c for some λ Λ, c R. Then obviously ϕ C = 0. Let furthermore ψ K (C) be another rational function on C. As ϕ σ + ψ σ = (ϕ + ψ) σ we can conclude that (ϕ + ψ) C = ϕ C + ψ C. Proposition (Balancing Condition and Commutativity) The Weil divisor associated to a Cartier divisor as in definition has the following properties: (a) Let C be an affine k-cycle in V = Λ R and ϕ K (C) a rational function on C. Then div(ϕ) = ϕ C is an equivalence class of tropical fans, i.e. its representatives are balanced. (b) Let ψ K (C) be another rational function on C. Then ψ (ϕ C) = ϕ (ψ C) holds. The proof of this statement is to a large extent the work of Johannes Rau, my coauthor of [AR07] and [AR08]. Hence we skip it here and refer to [AR07, proposition 3.7] instead. Definition (Affine Cartier divisors) Let C be an affine k-cycle. The subgroup of globally affine linear functions in K (C) with respect to + is denoted by O (C). We define the group of affine Cartier divisors of C to be the quotient group Div(C) := K (C)/O (C). 10

21 Chapter 1: Foundations of tropical intersection theory Let [ϕ] Div(C) be a Cartier divisor. By remark 1.2.6, the associated Weil divisor div([ϕ]) := div(ϕ) is well-defined. We therefore get a bilinear mapping called affine intersection product. : Div(C) Zk aff (C) Zk 1(C) aff ([ϕ],d) [ϕ] D = ϕ D, Example (Self-intersection of hyperplanes) Let Λ = Z n (and thus V = R n ), let e 1,...,e n be the standard basis vectors in Z n and e 0 := e 1 e n. By abuse of notation our ambient cycle is R n := [({R n },ω(r n ) = 1)]. Let us consider the linear tropical polynomial h = x 1 x n 0 = max{x 1,...,x n, 0} : R n R. Obviously, h is a rational function in the sense of definition 1.2.1: For each subset I {0, 1,...,n} we denote by σ I the simplicial cone of dimension I generated by the vectors e i for i I. Then h is integer linear on all σ I, namely { 0 if 0 / I, h σi (x 1,...,x n ) = x i if there exists an i {1,...,n} \ I. Let L n k be the k-dimensional fan consisting of all cones σ I with I k and weighted with the trivial weight function ω L n k. Then L n n is a representative of R n fulfilling the conditions of definition We want to show } h {{ h} R n = [L n n k]. ( ) k times This follows inductively from h [L n k+1 ] = [Ln k ], so it remains to compute ω L n k+1,h (σ I ) for all I with I = k < n. Let J := {0, 1,...,n} \ I. Obviously, the (k + 1)-dimensional cones of L n k+1 containing σ I are precisely the cones σ I {j},j J. Moreover, e j is a representative of the normal vector u σi {j} /σ I. Note also that for all i I,I {0, 1,...,n} we have h σi ( e i ) = h σi ( e i ) = h( e i ). Hence we compute ω L n k+1,h(σ I ) = ω L n k+1 (σ I {j} ) h σi {j} ( e j ) }{{} j J =1 ( ) + h σi ω L n k+1 (σ I {j} ) e j }{{} j J =1 }{{} = i I e i = h( e j ) + h( e i ) j J i I = h( e 0 ) + h( e 1 ) + + h( e n ) = = 1 = ω L n k (σ I ), which implies h [L n k+1 ] = [Ln k ] and also equation ( ). We can summarize this example as follows: Firstly, for a tropical polynomial f, the 11

22 1.2 Affine Cartier divisors and their associated Weil divisors Figure 1.7: The rigid curve R in S. associated Weil divisor f R n coincides with the locus of non-differentiability T (f) of f (see [RGST05, section 3]), and secondly, the k-fold self-intersection of a tropical hyperplane in R n is given by its (n k)-skeleton together with trivial weights all equal to 1. Example (A rigid curve) Using notations from example 1.2.9, we consider as ambient cycle the surface S := [L 3 2] = max{x 1,x 2,x 3, 0} R 3. In this surface, we want to show that the curve R := [(R e R,ω R (R e R ) = 1)] Z1 aff (S), where e R := e 1 + e 2, has negative self-intersection in the following sense: We construct a rational function ϕ on S whose associated Weil divisor is R and show that ϕ R = ϕ ϕ S is just the origin with weight 1. This curve and its rigidness were first discussed in [M06, Example 4.11, Example 5.9]. Let us construct ϕ: First we refine L 3 2 to L R by replacing σ {1,2} and σ {0,3} with σ {1,R}, σ {R}, σ {R,2}, σ {0, R}, σ { R} and σ { R,3} (using again the notations from example and e R := e R = e 0 + e 3 ). We define ϕ : S R to be the unique function that is linear on the faces of L R and fulfills 0, e 1, e 2, e 3, e R 0, e 0 1 and e R 1. Analogous to 1.2.9, we can compute for i = 1, 2 for i = 0, 3 and finally ω LR,ϕ(σ {i} ) = ϕ( e 0 ) + ϕ( e 3 ) + ϕ( e R ) = = 0, ω LR,ϕ(σ {i} ) = ϕ( e 1 ) + ϕ( e 2 ) + ϕ( e R ) = = 0, ω LR,ϕ(σ {R} ) = ϕ( e 1 ) + ϕ( e 2 ) ϕ( e R ) = ( 1) = 1, ω LR,ϕ(σ { R} ) = ϕ( e 0 ) + ϕ( e 3 ) ϕ( e R ) = = 1, 12

23 Chapter 1: Foundations of tropical intersection theory which means ϕ S = R. Now we can easily compute ϕ ϕ S = ϕ R on the representative {σ {R},σ { R}, {0}} (with trivial weights) of R: ω R,ϕ ({0}) = ϕ( e R ) + ϕ( e R ) = = 1. Therefore ϕ ϕ S = [({0},ω({0}) = 1)]. Note that we really obtain a cycle with negative weight, not only a cycle class modulo rational equivalence as it is the case in classical algebraic geometry. 1.3 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 Cartier divisors along morphisms of fans and to study the interaction of both constructions. To do this we first of all have to introduce the notion of morphism: Definition (Morphisms of fans) Let X be a fan in V = Λ Z R and Y be a fan in V = Λ Z R. A morphism f : X Y is a Z-linear map, i.e. a map from X V to Y V induced by a Z-linear map f : Λ Λ. By abuse of notation we will usually denote all three maps f, f and f Z id by the same letter f (note that the last two maps are in general not uniquely determined by f : X Y ). A morphism of weighted fans is a morphism of fans. A morphism of affine cycles f : [(X,ω X )] [(Y,ω Y )] is a morphism of fans f : X Y. Note that in this latter case the notion of morphism does not depend on the choice of the representatives by remark Given such a morphism the following construction shows how to build the push-forward fan of a given fan. Afterwards we will show that this construction induces a well-defined operation on cycles. Construction We refer to [GKM07, section 2] for more details on the following construction. Let (X,ω X ) be a weighted fan of pure dimension n in V = Λ Z R, let Y be any fan in V = Λ Z R and let f : X Y be a morphism. Passing to an appropriate refinement of (X,ω X ) the collection of cones f X := {f(σ) σ X contained in a maximal cone of X on which f is injective} is a fan in V of pure dimension n. It can be made into a weighted fan by setting ω f X(σ ) := σ X (n) :f(σ)=σ ω X (σ) Λ σ /f(λ σ) for all σ f X (n). The equivalence class of this weighted fan only depends on the equivalence class of (X,ω X ). 13

24 1.3 Push-forward of affine cycles and pull-back of Cartier divisors Example Let X be the fan with cones τ 1,τ 2,τ 3, {0} as shown in the figure R 2 X τ 3 τ 1 {0} f i R τ 2 and let ω X (τ i ) = 1 for i = 1, 2, 3. Moreover, let Y := R be the real line and the morphisms f 1,f 2 : X Y be given by f 1 (x,y) := x + y and f 2 (x,y) := x, respectively. Then (f 1 ) X = (f 2 ) X = {{x 0}, {0}, {x 0}}, but ω (f1 ) X({x 0}) = ω (f1 ) X({x 0}) = 2 and ω (f2 ) X({x 0}) = ω (f2 ) X({x 0}) = 1. Proposition Let (X,ω X ) be a tropical fan of dimension n in V = Λ Z R, let Y be any fan in V = Λ Z R and let f : X Y be a morphism. Then f X is a tropical fan of dimension n. Proof. A proof can be found in [GKM07, section 2]. By construction and proposition the following definition is well-defined: Definition (Push-forward of cycles) Let V = Λ Z R and V = Λ Z R. Moreover, let X Zm aff (V ), Y Zn aff (V ) and f : X Y be a morphism. For [(Z,ω Z )] Zk aff (X) we define f [(Z,ω Z )] := [(f (Z ),ω f (Z ))] Z aff k (Y ). Proposition (Push-forward of cycles) Let V = Λ Z R and V = Λ Z R. Let X Zm aff (V ) and Y Zn aff (V ) be cycles and let f : X Y be a morphism. Then the map is well-defined and Z-linear. Z aff k (X) Z aff k (Y ) : C f C Proof. It remains to prove the linearity: Let (A,ω A ) and (B,ω B ) be two tropical fans of dimension k with A = A, B = B and A, B X. We want to show that f (A + B) f A + f B. Refining A and B as in construction we may assume that A,B A + B. Set à := A + B and { ωa (σ), if σ A ωã(σ) := 0, else for all facets σ Ã. Analogously, set B := A+B with according weights. Then à A and B B. Carrying out a further refinement of A + B like in construction we can reach that f (A + B) = {f(σ) σ A + B contained in a maximal cone of A + B 14

25 Chapter 1: Foundations of tropical intersection theory on which f is injective}. Using à = B = à + B = A + B we get f à = f B = f (à + B) = f (A + B) and it remains to compare the weights: ω f (Ã+ B) (σ ) = ωã+ B(σ) Λ σ /f(λ σ) σ (Ã+ B) (k) :f(σ)=σ [ = ωã(σ) + ω ] B(σ) Λ σ /f(λ σ) = σ (Ã+ B) (k) :f(σ)=σ σ Ã(k) :f(σ)=σ ωã(σ) Λ σ /f(λ σ) + σ B (k) :f(σ)=σ ω B(σ) Λ σ /f(λ σ) = ω f à (σ ) + ω f B(σ ) for all facets σ of f (A +B). Hence f (A +B) f (à + B) = f à +f B f A +f B as weighted fans. Our next step is now to define the pull-back of a Cartier divisor. As promised, our next step will then be to prove a projection formula that describes the interaction between our two constructions. Proposition (Pull-back of Cartier divisors) Let C Zm aff (V ) and D Zn aff (V ) be cycles in V = Λ Z R and V = Λ Z R, respectively, and let f : C D be a morphism. Then there is a well-defined and Z-linear map Div(D) Div(C) : [h] f [h] := [h f]. Proof. The map h h f is obviously Z-linear on rational functions and maps affine linear functions to affine linear functions. Thus it remains to prove that h f is a rational function if h is one: Therefore let (X,ω X ) be any representative of C, let (Y,ω Y ) be a reduced representative of D such that the restriction of h to every cone in Y is affine linear and let f V : V V be a Z-linear map such that f V C = f. Since Z := {f 1 V (σ ) σ Y } is a fan in V and X Z we can construct the refinement X := X Z of X such that h f is affine linear on every cone of X. This finishes the proof. Proposition (Projection formula) Let C Zm aff (V ) and D Zn aff (V ) be cycles in V = Λ Z R and V = Λ Z R respectively and let f : C D be a morphism. Let E Zk aff (C) be a cycle and let ϕ Div(D) be a Cartier divisor. Then the following equation holds: ϕ (f E) = f (f ϕ E) Z aff k 1(D). Proof. Let E = [(Z,ω Z )] and ϕ = [h]. We may assume that Z = Z and h(0) = 0. Replacing Z by a refinement we may additionally assume that f h is linear on every cone of Z (cf. definition 1.2.1) and that f Z = {f(σ) σ Z contained in a maximal cone of Z on which f is injective} 15

26 1.3 Push-forward of affine cycles and pull-back of Cartier divisors (cf. construction 1.3.2). Note that in this case h is linear on the cones of f Z, too. Let σ D be a cone (not necessarily σ f Z) such that h is linear on σ. Then there is a unique linear map h σ : V σ R induced by the restriction h σ. Analogously for f h σ,σ C. For cones τ < σ Z of dimension k 1 and k respectively let v σ/τ Λ be a representative of the primitive normal vector u σ/τ Λ σ /Λ τ of construction Analogously, for τ < σ f Z of dimension k 1 and k respectively let v σ /τ be a representative of u σ /τ Λ σ /Λ τ. Now we want to compare the weighted fans h (f Z) and f (f h Z): Let τ f Z be a cone of dimension k 1. Then we can calculate the weight of τ in h (f Z) as follows: ω h (f Z)(τ ) = = = ω f Z(σ ) h σ (v σ /τ ) σ f Z:σ >τ h τ ω f Z(σ ) v σ /τ σ f Z:σ >τ ω Z (σ) Λ σ /f(λ σ) h σ (v σ /τ ) σ f Z:σ >τ σ Z (k) :f(σ)=σ h τ ω Z (σ) Λ σ /f(λ σ) v σ /τ σ f Z:σ >τ σ Z (k) :f(σ)=σ ω Z (σ) Λ f(σ) /f(λ σ) h f(σ) (v f(σ)/τ ) σ Z (k) :f(σ)>τ h τ ω Z (σ) Λ f(σ) /f(λ σ) v f(σ)/τ σ Z (k) :f(σ)>τ Now let τ f (f h Z) of dimension k 1. The weight of τ in f (f h Z) can be calculated as follows: ω f (f h Z)(τ ) = = τ (f h Z) (k 1) : f(τ)=τ τ (f h Z) (k 1) : f(τ)=τ f h τ ω f h Z(τ) Λ τ /f(λ τ) σ Z (k) :σ>τ σ Z (k) :σ>τ ω Z (σ)f h σ (v σ/τ ) ω Z (σ) v σ/τ Λ τ /f(λ τ) 16

27 Chapter 1: Foundations of tropical intersection theory = ω Z (σ)h f(σ) (f(v σ/τ )) τ (f h Z) (k 1) : f(τ)=τ h f(τ) σ Z (k) :σ>τ σ Z (k) :σ>τ ω Z (σ) f ( ) v σ/τ Λ τ /f(λ τ). Note that f(v σ/τ ) = Λ σ /(Λ τ + Zf(v σ/τ)) v σ /τ +λ σ,τ Λ for some λ σ,τ Λ τ. Since h f(σ) (λ σ,τ ) = h f(τ) (λ σ,τ ) these parts of the corresponding summands in the first and second interior sum cancel using the linearity of h f(τ). Moreover, note that f(v σ/τ ) = λ σ,τ Λ τ for those σ > τ on which f is not injective and that the whole summands cancel in this case. Thus we can conclude that the sum does not change if we restrict the summation to those σ > τ on which f is injective. Using additionally the equation we get ω f (f h Z)(τ ) = Λ σ /f(λ σ) = Λ τ /f(λ τ) Λ σ /(Λ τ + Zf(v σ/τ)) = τ (f h Z) (k 1) : f(τ)=τ h τ σ Z (k) : σ>τ,f(σ)>τ σ Z (k) : σ>τ,f(σ)>τ ω Z (σ) Λ f(σ) /f(λ σ) h f(σ) (v f(σ)/τ ) ω Z (σ) Λ f(σ) /f(λ σ) v f(σ)/τ ω Z (σ) Λ f(σ) /f(λ σ) h f(σ) (v f(σ)/τ ) σ Z (k) :f(σ)>τ h τ ω Z (σ) Λ f(σ) /f(λ σ) v f(σ)/τ. σ Z (k) :f(σ)>τ Note that for the last equation we used again the linearity of h τ. We have checked so far that a cone τ of dimension k 1 occurring in both h (f Z) and f (f h Z) has the same weight in both fans. Thus it remains to examine those cones f(τ),τ Z (k 1) such that f is injective on τ but not on any σ > τ: In this case all vectors v σ/τ are mapped to Λ f(τ). Again, h f(σ) = h f(τ) and by linearity of h f(τ) all summands in the sum cancel as above. Hence the the weight of f(τ) in f (f h Z) is 0 and ϕ (f E) = [h (f Z)] = [f (f h Z)] = f (f ϕ E). 1.4 Abstract tropical cycles In this section we will introduce the notion of abstract tropical cycles as spaces that have tropical fans as local building blocks. Then we will generalize the theory from the previous sections to these spaces. 17

28 1.4 Abstract tropical cycles Definition (Abstract polyhedral complexes) An (abstract) polyhedral complex is a topological space X together with a finite set X of closed subsets of X and an embedding map ϕ σ : σ R nσ for every σ X such that (a) X is closed under taking intersections, i.e. σ σ X for all σ,σ X with σ σ, (b) every image ϕ σ (σ), σ X is a rational polyhedron not contained in a proper affine subspace of R nσ, (c) for every pair σ,σ X the concatenation ϕ σ ϕ 1 σ defined, (d) X =. σ X is integer affine linear where ϕ 1 σ (ϕ σ (σ) ), where ϕ σ (σ) denotes the interior of ϕ σ (σ) in R nσ. For simplicity we will usually drop the embedding maps ϕ σ and denote the polyhedral complex (X, X, {ϕ σ σ X}) by (X, X ) or just by X if no confusion can occur. The closed subsets σ X are called the polyhedra or faces of (X, X ). For σ X the open set σ ri := ϕ 1 σ (ϕ σ (σ) ) is called the relative interior of σ. Like in the case of fans the dimension of (X, X ) is the maximum of the dimensions of its polyhedra. (X, X ) is pure-dimensional if every inclusion-maximal polyhedron has the same dimension. We denote by X (n) the set of polyhedra in (X, X ) of dimension n. Let τ,σ X. Like in the case of fans we write τ σ (or τ < σ) if τ σ (or τ σ respectively). An abstract polyhedral complex (X, X ) of pure dimension n together with a map ω X : X (n) Z is called weighted polyhedral complex of dimension n and ω X (σ) the weight of the polyhedron σ X (n). Like in the case of fans the empty complex is a weighted polyhedral complex of every dimension n. If ((X, X ),ω X ) is a weighted polyhedral complex of dimension n then let X := {τ X τ σ for some σ X (n) with ω X (σ) 0}, X := τ X τ X. With these definitions ( (X, X ),ω X (X ) (n) ) is again a weighted polyhedral complex of dimension n, called the non-zero part of ((X, X ),ω X ). We call a weighted polyhedral complex ((X, X ),ω X ) reduced if ((X, X ),ω X ) = ((X, X ),ω X ) holds. Definition (Subcomplexes and refinements) Let (X, X, {ϕ σ }) and (Y, Y, {ψ τ }) be two polyhedral complexes. (X, X, {ϕ σ }) a subcomplex of (Y, Y, {ψ τ }) if We call (a) X Y, (b) for every σ X exists τ Y with σ τ and (c) the Z-linear structures of X and Y are compatible, i.e. for a pair σ,τ from (b) the maps ϕ σ ψτ 1 and ψ τ ϕ 1 σ are integer affine linear where defined. 18

29 Chapter 1: Foundations of tropical intersection theory We write (X, X, {ϕ σ }) (Y, Y, {ψ τ }) in this case. Analogous to the case of fans we define a map C X,Y : X Y that maps a polyhedron in X to the inclusion-minimal polyhedron in Y containing it. We call a weighted polyhedral complex ((X, X ),ω X ) a refinement of ((Y, Y ),ω Y ) if (a) (X, X ) (Y, Y ), (b) X = Y, (c) ω X (σ) = ω Y (C X,Y (σ)) for all σ (X ) (dim(x)). Definition (Open fans) Let ( F,ω F) be a tropical fan in R n and U R n an open subset containing the origin. The set F := F U := {σ U σ F } together with the induced weight function ω F is called an open (tropical) fan in R n. As in the case of fans let F := σ F σ. Note that the open fan F contains the whole information of the entire fan F as F = {R 0 σ σ F }. Definition (Tropical polyhedral complexes) A tropical polyhedral complex of dimension n is a weighted polyhedral complex ((X, X ),ω X ) of pure dimension n together with the following data: For every polyhedron σ X we are given an open fan F σ in some R nσ and a homeomorphism such that Φ σ : S σ := σ X,σ σ (σ ) ri F σ (a) for all σ X,σ σ holds Φ σ (σ S σ ) F σ and Φ σ is compatible with the Z-linear structure on σ, i.e. Φ σ ϕ 1 σ and ϕ σ Φ 1 σ are integer affine linear where defined, (b) ω X (σ ) = ω Fσ (Φ σ (σ S σ )) for every σ (X ) (n) with σ σ, (c) for every pair σ,τ X there is an integer affine linear map A σ,τ and a commutative diagram Φ τ S σ S τ Φτ (S σ S τ ). Φ σ A σ,τ Φ σ (S σ S τ ) For simplicity of notation we will usually drop the maps Φ σ and write ((X, X ),ω X ) or just X instead of (((X, X ),ω X ), {Φ σ }). A tropical polyhedral complex is called reduced if the underlying weighted polyhedral complex is. Example The following figure shows the topological spaces and the decompositions into polyhedra of two such abstract tropical polyhedral complexes together with the open fan F σ for 19

30 1.4 Abstract tropical cycles every polyhedron σ: R 2 X 1 X2 Remark The above example X 1 is an instance of a more general construction: Let X be a weighted, balanced polyhedral complex in R n as in [M06] or [S05]. Then X can be interpreted as a tropical polyhedral complex in a natural way: For every polyhedron σ X we have trivial embedding maps ϕ σ : σ H σ = R dim(σ), where H σ R n is the smallest affine subspace containing σ. Moreover, for every σ X we have trivial fan charts Φ σ : S σ = σ X,σ σ (σ ) ri Sσ tr, where Sσ tr is a translation of S σ such that σ contains the origin. Then Sσ tr is an open fan sitting in the associated tropical fan S tr σ. Construction (Refinements of tropical polyhedral complexes) Let (((X, X ),ω X ), {Φ σ }) be a tropical polyhedral complex and let ((Y, Y ),ω Y ) be a refinement of its underlying weighted polyhedral complex ((X, X ),ω X ). Then we can make ((Y, Y ),ω Y ) into a tropical polyhedral complex as follows: We may assume that X and Y are reduced as we do not pose any conditions on polyhedra with weight zero. Fix some τ Y and let σ := C Y,X (τ). By definition of refinement, for every τ Y with τ τ there is σ X, σ σ with τ σ. Thus S τ S σ and we have a map Ψ τ := Φ σ Sτ : S τ Ψ τ (S τ ) R nσ. It remains to give Ψ τ (S τ ) the structure of an open fan: We may assume that {0} Ψ τ (τ) (otherwise replace Ψ τ by the concatenation of Ψ τ with an appropriate translation T τ, apply T τ to Fσ X and Φ σ and change the maps A σ,σ and A σ,σ accordingly). Let F σ X := {R 0 σ σ Fσ X } be the tropical fan associated to Fσ X and let F τ Y be the set of cones F τ Y := {R 0 Ψ τ (τ ) τ τ Y }. Note that the conditions on the Z-linear structures on X and Y to be compatible and on Φ σ to be compatible with the Z-linear structure on X assure that F τ Y is a fan in R nσ. In fact, F τ Y with the weights induced by Y is a refinement of ( F σ X,ω F X σ ). Thus the maps Ψ τ together with the open fans { Ψ τ (S τ ) F τ Y }, τ Y fulfill all requirements for a tropical polyhedral complex. Remark If not stated otherwise we will from now on equip every refinement of a tropical polyhedral complex coming from a refinement of the underlying weighted polyhedral complex with the tropical structure constructed in

31 Chapter 1: Foundations of tropical intersection theory Definition (Refinements and equivalence of tropical polyhedral complexes) Let C 1 = ( ((X 1, X 1 ),ω X1 ), {Φ X 1 σ 1 } ) and C 2 = ( ((X 2, X 2 ),ω X2 ), {Φ X 2 σ 2 } ) be tropical polyhedral complexes. We call C 2 a refinement of C 1 if (a) ((X 2, X 2 ),ω X2 ) is a refinement of ((X 1, X 1 ),ω X1 ) and (b) C 2 carries ( the tropical structure) induced by C 1 as in construction 1.4.7, i.e. if C 2 = ((X 2, X 2 ),ω X2 ), { Φ X 2 σ 2 } is the tropical polyhedral complex obtained from C 1 and the refinement ((X 2, X 2 ),ω X2 ) then the maps Φ X 2 σ 2 (Φ X 2 σ 2 ) 1 and Φ X 2 σ 2 ( Φ X 2 σ 2 ) 1 are integer affine linear where defined. We call two tropical polyhedral complexes C 1 and C 2 equivalent (write C 1 C 2 ) if they have a common refinement (as tropical polyhedral complexes). Remark Note that different choices of translation maps T τ in construction only lead to tropical polyhedral complexes carrying the same tropical structure in the sense of definition (b). In particular definition does not depend on the choices we made in construction Note moreover that refinements of (((X, X ),ω X ), {Φ σ }) and ((Y, Y ),ω Y ) in construction only lead to refinements of (((Y, Y ),ω Y ), {Ψ τ }). Construction (Refinements) Let (((X, X, {ϕ σ }),ω X ), {Φ σ }) and (((Y, Y, {ψ τ }),ω Y ), {Ψ τ }) be reduced tropical polyhedral complexes such that (Y, Y ) (X, X ) and the tropical structures on X and Y agree, i.e. for every τ Y and σ := C Y,X (τ) X the maps Ψ τ Φ 1 σ and Φ σ Ψ 1 τ are integer affine linear where defined. Moreover let (((X, X, {ϕ σ }),ω X ), {Φ σ }) be a reduced refinement of (((X, X, ({ϕ σ }),ω X ), {Φ σ }). Like in the case of fans we will construct a refinement ((Y X, Y X, {ψτ Y X }),ω X Y X ), {ΨYτ }) of (((Y, Y, {ψ τ }),ω Y ), {Ψ τ }) such that (Y X, Y X ) (X, X ) and the tropical structures on Y X and X agree: Fix σ X. Note that the compatibility conditions on the Z-linear structures of X, X and Y, X respectively (cf. definition (c)) assure that ϕ σ (σ ), σ X with σ σ as well as ϕ σ (τ), τ Y with τ σ are rational polyhedra in R nσ. Thus in this case ϕ σ (σ τ) = ϕ σ (σ ) ϕ σ (τ) is a rational polyhedron, too. Let H σ,τ = R nτ be the smallest affine subspace of R nσ containing ϕ σ (σ τ). We can consider ϕ σ σ τ to be a map σ τ R nτ. We can hence construct the underlying weighted polyhedral complex of our desired tropical polyhedral complex as follows: Set P := {τ σ τ Y,σ X }, Y X := {τ P τ P (dim(τ)) : τ τ}, Y X := Y and ω Y X (τ) := ω Y (C Y X,Y (τ)) for all τ (Y X ) (dim(y )). It remains to define the maps ψτ Y X X and ΨY τ : For every τ Y X choose a triplet σ X,τ Y,σ X such that σ τ = τ and σ,τ σ and set ψτ Y X := ϕ σ σ τ. With these definitions the weighted polyhedral complex ((Y X, Y X, {ψτ Y X }),ω Y X ) is a refinement of ((Y, Y, {ψ τ }),ω Y ). Thus we can apply construction to obtain maps {Ψ Y τ X } that endow our weighted polyhedral complex with the tropical structure inherited from ((Y, Y, {ψ τ }),ω Y ). Note that the compatibility property between the tropical structures of Y and X is bequeathed to Y X and X, too. 21

32 1.4 Abstract tropical cycles Lemma The equivalence of tropical polyhedral complexes is an equivalence relation. Proof. Let C 1 = ( ((X 1, X 1 ),ω X1 ), {Φ X 1 σ 1 } ), C 2 = ( ((X 2, X 2 ),ω X2 ), {Φ X 2 σ 2 } ) ( and C 3 = ((X3, X 3 ),ω X3 ), {Φ X 3 σ 3 } ) be tropical polyhedral complexes such that C 1 C 2 via a common refinement D 1 = ( ((Y 1, Y 1 ),ω Y1 ), {Φ Y 1 σ 1 } ) and C 2 C 3 via a common refinement D 2 = ( ((Y 2, Y 2 ),ω Y2 ), {Φ Y 2 σ 2 } ). We have to construct a common refinement of C 1 and C 3 : First of all we may assume that D 1 and D 2 are reduced. Using construction we get a refinement D 3 := ( ((Y 1 Y 2, Y 1 Y 2 ),ω Y1 Y 2 ), {Φ Y 1 Y 2 τ } ) of D 1 with (Y 1 Y 2, Y 1 Y 2 ) (Y 2, Y 2 ) and a tropical structure that is compatible with the tropical structure on D 2. It is easily checked that D 3 is a refinement of D 2, too. Definition (Abstract tropical cycles) Let ((X, X ),ω X ) be an n-dimensional tropical polyhedral complex. Its equivalence class [((X, X ),ω X )] is called an (abstract) tropical n-cycle. The set of n-cycles is denoted by Z n. Since the underlying topological space X of a tropical polyhedral complex ((X, X ),ω X ) is by definition invariant under refinements we define [((X, X ),ω X )] := X. Like in the affine case, an n-cycle [((X, X ),ω X )] is called an (abstract) tropical variety if ω X (σ) 0 for all σ X (n). Let C Z n and D Z k be two tropical cycles. D is called an (abstract) tropical cycle in C or a subcycle of C if there exists a representative (((Z, Z ),ω Z ), {Ψ τ }) of D and a reduced representative (((X, X ),ω X ), {Φ σ }) of C such that (a) (Z, Z ) (X, X ), (b) the tropical structures on Z and X agree, i.e. for every τ Z the maps Ψ τ Φ 1 C Z,X (τ) and Φ C Z,X (τ) Ψ 1 τ are integer affine linear where defined. The set of tropical k-cycles in C is denoted by Z k (C). Remark and Definition (a) Let X be a finite set of rational polyhedra in R n, f Hom(Z n, Z) a linear form and b R. Then let H f,b := { {x R n f(x) b}, {x R n f(x) = b}, {x R n f(x) b} }. Like in the case of fans (cf. construction ) we can form sets P := {σ σ σ X,σ H f,b } and X H f,b := {σ P τ P (dim(σ)) with τ σ}. (b) Again let X be a finite set of rational polyhedra in R n. Let {f i b i i = 1,...,N} be all (integral) inequalities occurring in the description of all polyhedra in X. Then we can construct the set X H f1,b 1 H fn,b N. Note that for every collection of polyhedra X this set X H f1,b 1 H fn,b N is a (usual) rational polyhedral complex (i.e. for every polyhedron τ X every face (in the usual sense) of σ is contained in X and the intersection of every two polyhedra in X is a common face of each). Moreover note that the result is independent of the order of the f i and if {g i c i i = 1,...,M} is a different set of inequalities describing the polyhedra in X 22

33 Chapter 1: Foundations of tropical intersection theory then X H f1,b 1 H fn,b N and X H g1,c 1 H gm,c M have a common refinement, namely X H f1,b 1 H fn,b N H g1,c 1 H gm,c M. Construction (Sums of tropical cycles) Let C Z n be a tropical cycle. Like in the affine case the set of tropical k-cycles in C can be made into an abelian group by defining the sum of two such k-cycles as follows: Let D 1 and D 2 Z k (C) be the two cycles whose sum we want to construct. By definition there are reduced representatives ( ((X 1, X 1 ),ω X1 ), {Φ X 1 τ } ) ( and ((X2, X 2 ),ω X2 ), {Φ X 2 τ } ) of C and reduced representatives ( ((Y, Y ),ω Y ), {Φ Y τ } ) of D 1 and ( ((Z, Z ),ω Z ), {Φ Z τ } ) of D 2 such that (Y, Y ) (X 1, X 1 ) and the tropical structures on Y and X 1 agree and (Z, Z ) (X 2, X 2 ) and the tropical structures on Z and X 2 agree. As is an equivalence relation there is a common refinement ( ((X, X, {ϕ τ }),ω X ), {Φ X τ } ) of X 1 and X 2 which we may assume to be reduced. Applying construction to Y and X we obtain the tropical polyhedral complex ( ((Y X, Y X ),ω Y X ), {Φ Y τ X } ) which is a refinement of Y, has a tropical structure that is compatible with the tropical structure on X and fulfils (Y X, Y X ) (X, X ). If we further apply construction to Z and X we get a refinement ( ((Z X, Z X ),ω Z X ), {Φ Z X τ } ) of Z with analogous properties. Now fix some polyhedron σ X and let τ 1,...,τ r Y X and τ r+1,...,τ s Z X be all polyhedra of Y X and Z X respectively that are contained in σ. Note that property (a) of definition implies that for all i = 1,...,r the image ϕ σ (τ i ) is a rational polyhedron in R nσ. Like in remark and definition let {f i b i i = 1,...,N} be the set of all integral inequalities occurring in the description of all polyhedra ϕ σ (τ i ),i = 1,...,s and let RY σ X := {ϕ σ(τ i ) i = 1,...,r} H f1,b 1 H fn,b N and RZ X σ := := {ϕ 1 σ (τ) τ RY σ X } and {ϕ σ (τ i ) i = r +1,...,s} H f1,b 1 H fn,b N. Then PY σ X := {ϕ 1 σ (τ) τ RZ X σ } are a kind of local refinement of Y X and Z X, P σ Z X respectively, but taking the union over all maximal polyhedra σ X (n) does in general not lead to global refinements as there may be overlaps between polyhedra coming from different σ. We resolve this as follows: For σ X (n), τ n 1 i=0 X(i) let PY,τ σ := { P Y σ X τ is the inclusion-minimal polyhedron of X containing } and P Y,n := σ X { P σ (n) Y X τ X(n 1) : τ}. Analogously for PZ,τ σ and P Z,n. Then let Ỹ := P Y,n { τ σ τ σ PY,τ} σ and Z := P Z,n τ X (i) :i<n τ X (i) :i<n { σ X (n) :τ σ σ X (n) :τ σ τ σ τ σ PZ,τ} σ. Moreover for every τ Ỹ Z choose some σ X (n) with τ σ and let ψ τ := ϕ σ τ. Note that by construction (Ỹ, Y X ) and ( Z, Z X ) with structure maps ψ τ,τ X or τ Z respectively and weight functions ωỹ and ω Z induced by Y X and Z X are refinements of Y X and Z X (we need here that R σ Y X and Rσ Z X were usual polyhedral complexes in R nσ ). Thus we can endow them with the tropical structures inherited from Y X and Z X respectively (cf. construction 1.4.7). As 23

34 1.4 Abstract tropical cycles X Z X Y X σ 1 σ 2 Construct local refinements Look at faces in the boundary Look at faces in the boundary σ 1 σ 2 Make the refinements fit together in the boundary σ 1 σ 2 σ 1 σ 2 Figure 1.8: An illustration of the process described in construction

35 Chapter 1: Foundations of tropical intersection theory ( X Ỹ, Y X Z X ) is a polyhedral complex now, we can form ((P, P ),ω P ) := (( X Ỹ, Y X Z X ),ω P), where ω P (σ) := ωỹ (σ) + ω Z(σ) for all σ P (k) (we set ω (σ) := 0 for σ /, {Ỹ, Z}). Recall that the tropical structures on Ỹ and Z are inherited from Y X and Z X and are thus compatible with the tropical structure on X. Thus Φ X σ (Sσ P ) Fσ X with weights induced from P is an open fan (the corresponding complete tropical fan is just the sum of the fans coming from Ỹ and Z). Thus we can set Φ σ := Φ X σ S P σ : Sσ P Φ X σ (Sσ P ) and can hence define the sum D 1 + D 2 to be [( )] D 1 + D 2 := ((P, P ),ω P ), { Φ σ }. Note that the class [(((P, P ),ω P ), { Φ σ })] is independent of the choices we made, i.e. the sum D 1 + D 2 is well-defined. Lemma Let C Z n be a tropical cycle. The set Z k (C) together with the operation + from construction forms an abelian group. Proof. The class of the empty complex 0 = [ ] is the neutral element of this operation and [((Y, Y ), ω Y )] is the inverse element of [((Y, Y ),ω Y )] Z k (C). Definition Let (((X, X, {ϕ σ }),ω X ), {Φ σ }) and (((Y, Y, {ψ τ }),ω Y ), {Ψ τ }) be tropical polyhedral complexes. We denote by their cartesian product where (((X, X, {ϕ σ }),ω X ), {Φ σ }) (((Y, Y, {ψ τ }),ω Y ), {Ψ τ }) (((X Y, X Y, {ϑ σ τ }),ω X Y ), {Θ σ τ }), X Y := {σ τ σ X,τ Y }, ϑ σ τ := ϕ σ ψ τ : σ τ R nσ R nτ, ω X Y (σ τ) := ω X (σ) ω Y (τ), Θ σ τ := Φ σ Ψ τ : S X σ S Y τ F X σ F Y τ. Let F σ X and F τ Y be the entire fans associated with Fσ X and Fτ Y from above. Obviously, the product F σ X F τ Y := {α β α F σ X,β F τ Y } with weight function ω F X σ F (α β) := ω τ Y F X σ (α) ω F Y τ (β) is again a tropical fan and thus its intersection with Fσ X Fτ Y yields an open fan (cf. definition 1.4.3). Hence the cartesian product (((X Y, X Y, {ϑ σ τ }),ω X Y ), {Θ σ τ }) 25

36 1.5 Cartier divisors and their associated Weil divisors ψ 1 on S α1 ψ 2 on S α2 ψ 1 ψ 2 on the overlaps C C C α 1 α 1 α 1 α 2 α 2 α 2 Figure 1.9: The Cartier divisor ϕ defined in example is again a tropical polyhedral complex. If C = [(X,ω X )] and D = [(Y,ω Y )] are tropical cycles we define C D := [(X,ω X ) (Y,ω Y )] for (X,ω X ) (Y,ω Y ) as defined above. Note that C D does not depend on the choice of the representatives X and Y. 1.5 Cartier divisors and their associated Weil divisors Definition (Rational functions and Cartier divisors) Let C be an abstract k-cycle and let U be an open set in C. A (non-zero) rational function on U is a continuous function ϕ : U R such that there exists a representative (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) of C such that for each face σ X the map ϕ m 1 σ is locally integer affine linear (where defined). The set of all non-zero rational functions on U is denoted by KC (U) or just K (U). If additionally for each face σ X the map ϕ Mσ 1 is locally integer affine linear (where defined), ϕ is called regular invertible. The set of all regular invertible functions on U is denoted by OC (U) or just O (U). A representative of a Cartier divisor on C is a finite set {(U 1,ϕ 1 ),...,(U l,ϕ l )}, where {U i } is an open covering of C and ϕ i K (U i ) are rational functions on U i that only differ in regular invertible functions on the overlaps, in other words, for all i j we have ϕ i Ui U j ϕ j Ui U j O (U i U j ). We define the sum of two representatives by {(U i,ϕ i )}+{(V j,ψ j )} = {(U i V j,ϕ i +ψ j )}, which obviously fulfills again the condition on the overlaps. We call two representatives {(U i,ϕ i )}, {(V j,ψ j )} equivalent if ϕ i ψ j is regular invertible (where defined) for all i,j, i.e. {(U i,ϕ i )} {(V j,ψ j )} = {(W k,γ k )} with γ k O (W k ). Obviously, + induces a group structure on the set of equivalence classes of representatives with the neutral element {( C,c 0 )}, where c 0 is the constant zero function. This group is denoted by Div(C) and its elements are called Cartier divisors on C. Example Let us give an example of a Cartier divisor which is not globally defined by a rational function: As abstract cycle C we take the elliptic curve [X 2 ] from example (the brackets resemble the fact that, to be precise, we take the equivalence class of the polyhedral complex X 2 with respect to refinements). By α 1,α 2 we denote the two 26

37 Chapter 1: Foundations of tropical intersection theory vertices in X 2. W.l.o.g. we can assume that the maps M αi map the points α i exactly to 0 R. Of course, the stars S α1,s α2 cover our whole space C = X 2. So we can define the Cartier divisor ϕ := [{(S α1,ψ 1 ), (S α2,ψ 2 )}], where ψ 1 := max(0,x) M α1 and ψ 2 := c 0 M α2 with c 0 the constant zero function. Let us check the condition on the overlaps: On one open half of our curve the two functions coincide, whereas on the other open half they differ by a linear function. So we constructed an Cartier divisor which can not be globally defined by one rational function (as ψ 1 can not be completed to a continuous function on C ). Remark (Restrictions to subcycles) Note that, as in the affine case (see remark 1.2.2), we can restrict a non-zero rational function ϕ KC (U) to an arbitrary subcycle D C, i.e. ϕ U D KD (U D ). It is also true that a regular invertible function ϕ OC (U) restricted to D is again regular invertible, i.e. ϕ U D OD (U D ). Hence we can also restrict a Cartier divisor [{(U i,ϕ i )}] Div(C) to D by setting [{(U i,ϕ i )}] D := [{(U i D,ϕ i Ui D )}] Div(D). Construction (Intersection products) Let C be an abstract k-cycle and ϕ = [{(U i,ϕ i )}] Div(C) a Cartier divisor on C. By definition and lemma , there exists a representative (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) of C such that for all i and σ X the map ϕ i m 1 σ is locally integer affine linear (where defined). We can also assume that X = X, as our functions are defined on C = X at the most. We would like to define the intersection product ϕ C to be [( ((Y, ) ) Y, {mσ } σ Y,ωX,ϕ, { M σ S Y σ : Sσ Y Fσ Y } )], σ Y where Y := k 1 i=0 X (i), Y := σ Y σ, S Y σ = (σ σ Y σ σ ) ri, F Y σ := and ω X,ϕ is an appropriate weight function. So it remains to construct ω X,ϕ (τ) for τ X (k 1). First, we do this pointwise, i.e. we construct ω X,ϕ (p) for p (τ) ri. Given a p (τ) ri, we pick an i with p U i. Let V be the connected component of M τ (U i S τ ) containing M τ (p). Then the function ϕ i Mτ 1 V can be uniquely extended to a rational function ϕ i K ([( F τ )]), where ( F,ω Fτ τ ) is the tropical fan generated by the open fan,ω Fτ (F τ,ω Fτ ). So, in the affine case, we can compute ω Fτ, ϕ i (R M τ (τ)) (see construction and definition 1.2.4) and define ω X,ϕ (p) := ω Fτ, ϕ i (R M τ (τ)). This definition is well-defined, namely if we pick another j with p U j and denote by V the connected component of M τ (U j S τ ) containing M τ (p), we know by definition of a Cartier divisor that ϕ i Mτ 1 V V ϕ j Mτ 1 V V is affine linear, hence ϕ i ϕ j is affine linear. By remark we get ω Fτ, ϕ i (R M τ (τ)) = ω Fτ, ϕ j (R M τ (τ)). The same argument shows that our definition does not depend on the choice of a representative {(U i,ϕ i )} of ϕ. But as (τ) ri is connected, the continuous function ω X,ϕ : (τ) ri Z must be constant. k 1 i=0 F (i) σ 27

38 1.6 Cartier divisors and their associated Weil divisors Hence, we define ω X,ϕ (τ) := ω X,ϕ (p) for some p (τ) ri. With this weight function ( ((Y, Y, {mσ } σ Y ),ωx,ϕ ), {M σ SYσ } σ Y ) is a tropical polyhedral complex. Let us now check if the equivalence class of this complex is independent of the choice of representatives of C. Let therefore (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) be a refinement of (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) (we can again assume X = (X ) ). Then, for each σ X, the map M CX,X (σ ) M 1 σ embeds F σ into a refinement of F CX,X (σ ). Applying the affine statement here (see remark 1.2.5), we deduce that for each τ X (k 1) we have ω X,ϕ(τ ) = 0 (if dimc X,X(τ ) = k) or ω X,ϕ(τ ) = ω X,ϕ (C X,X(τ )) (if dimc X,X(τ ) = k 1). Definition (Intersection products) Let C be an abstract k-cycle and ϕ = [{(U i,ϕ i )}] Div(C) a Cartier divisor on C. Let furthermore (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) be a reduced representative of C such that for all i and σ X the map ϕ i m 1 σ is locally integer affine linear (where defined). The associated Weil divisor div(ϕ) = ϕ C is defined to be [( ((Y k 1 := i=0 X (i), σ Y σ, {m σ } σ Y ),ωx,ϕ ), {M σ S Y σ } σ Y )] Z k 1 (C), where S Y σ = σ Y σ σ (σ ) ri and ω X,ϕ is the weight function defined in construction Let D Z l (C) be an arbitrary subcycle of C of dimension l. We define the intersection product of ϕ with D to be ϕ D := ϕ D D Z l 1 (C). Example Let us compute the Weil divisor associated to our Cartier divisor ϕ on the elliptic curve C constructed in example In fact, there is nothing to compute: One can see immediately from the picture that div(ϕ) is just the vertex α 1 with multiplicity 1 (the multiplicity of α 2 is 0 as in order to compute it, one has to use the constant function ψ 2 ). Let us stress that this single point can not be obtained as the Weil divisor of a (global) rational function, as all such divisors must have degree 0 (this is defined precisely and proven in remark 1.7.3). Proposition (Commutativity) Let ϕ,ψ Div(C) be two Cartier divisors on C. Then ψ (ϕ C) = ϕ (ψ C). Proof. Say ϕ = [{(U i,ϕ i )}] and ψ = [{(V j,ψ j )}]. Using lemma we find a reduced representative (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) of C such that for all i,j and all σ X the maps ϕ i m 1 σ and ψ j m 1 σ are locally integer affine linear (where defined). For θ X (k 2), p (θ) ri and i,j with p U i V j we get (using notations from construction 1.5.4) ω X,ϕ,ψ (θ) = ω X,ϕ,ψ (p) = ω Fθ, ϕ i, ψ j (R M θ (θ)) and similarily ω X,ψ,ϕ (θ) = ω Fθ, ψ j, ϕ i (R M θ (θ)). Using the corresponding statement in the affine case now (see proposition (b)), we deduce that the two weight functions are equal, which proves the claim. 28

39 Chapter 1: Foundations of tropical intersection theory 1.6 Push-forward of tropical cycles and pull-back of Cartier divisors Definition (Morphisms of tropical cycles) Let C Z n and D Z m be two tropical cycles. A morphism f : C D of tropical cycles is a continuous map f : C D with the following property: There exist reduced representatives (((X, X ),ω X ), {Φ σ }) of C and (((Y, Y ),ω Y ), {Ψ τ }) of D such that (a) for every polyhedron σ X there exists a polyhedron σ Y with f(σ) σ, (b) for every pair σ, σ from (a) the map Ψ σ f Φ 1 σ : Fσ X Fσ Ỹ induces a morphism of fans F σ X F σ Ỹ (cf. definition 1.3.1), where F σ X and F σ Ỹ are the tropical fans associated to Fσ X and Fσ Ỹ respectively (cf. definition 1.4.3). First of all we want to show that the restriction of a morphism to a subcycle is again a morphism: Lemma Let C Z n and D Z m be two cycles, f : C D a morphism and E Z k (C) a subcycle of C. Then the map f E : E D induces a morphism of tropical cycles f E : E D. Proof. By the definition of a morphism there exist reduced representatives ((X 1, X 1 ),ω X1 ) of C and ((Y, Y ),ω Y ) of D such that properties (a) and (b) in definition are fulfilled. By the definition of a subcycle there exist reduced representatives ((Z 1, Z 1 ),ω Z1 ) of E and ((X 2, X 2 ),ω X2 ) of C such that properties (a) and (b) in definition are fulfilled, i.e. such that (Z 1, Z 1 ) (X 2, X 2 ) and the tropical structures on Z 1 and X 2 agree. As is an equivalence relation there exists a common refinement ((X, X ),ω X ) of ((X 1, X 1 ),ω X1 ) and ((X 2, X 2 ),ω X2 ) which we may assume to be reduced. Applying construction to Z 1 and X we obtain a refinement ((Z, Z ),ω Z ) := ((Z 1 X, Z 1 X ),ω Z1 X) of ((Z 1, Z 1 ),ω Z1 ) such that (Z, Z ) (X, X ) and the tropical structures on Z and X agree. Thus properties (a) and (b) of definition are fulfilled by Z and Y and the restricted map f E : E D gives us a morphism f E : E D. If we are given a morphism and a tropical cycle the following construction shows how to build the push-forward cycle of the given one along our morphism: Construction (Push-forward of tropical cycles) Let C Z n and D Z m be two cycles and let f : C D be a morphism. Let (((X, X, {ϕ σ }),ω X ), {Φ σ }) and (((Y, Y, {ψ σ }),ω Y ), {Ψ τ }) be representatives of C and D fulfilling properties (a) and (b) of definition Consider the collection of polyhedra Z := {f(σ) σ X contained in a maximal polyhedron of X on which f is injective}. In general Z is not a polyhedral complex. We resolve this by subdividing the polyhedra in Z and refining X accordingly: 29

40 1.6 Push-forward of tropical cycles and pull-back of Cartier divisors Fix some polyhedron σ Y (m) and let τ 1,...,τ r Z be all polyhedra that are contained in σ. Property (b) of definition implies that {ψ σ (τ i ) i = 1,...,r} is a set of rational polyhedra in R n σ. As in remark and definition let {gi (x) b i i = 1,...,N}, g i Hom(Z n σ, Z), bi R be all inequalities occurring in the description of all polyhedra in {ψ σ (τ i ) i = 1,...,r} and let R σ := {ψ σ (τ i ) i = 1,...,r} H G1,b 1 H GN,b N, P σ := {ψ 1 σ (τ) τ R σ i }. As in construction P σ can be seen as a kind of local refinement of Z. But here again taking the union over all maximal polyhedra σ Y (m) does in general not lead to a global refinement as there may be overlaps between polyhedra coming from different σ. We fix this as follows (cf. construction ): For σ Y (m) and τ m 1 i=0 Y (i) let and Then P σ Z, τ := { P σ τ is the inclusion minimal polyhedron of Y containing } P Z,m := Z := P Z,m σ Y (m) { P σ τ Y (m 1) : τ}. τ Y (i) :i<m { σ Y (m) : τ σ τ σ τ σ PZ, τ} σ is the set of polyhedra (without any overlaps now) that shall induce our wanted refinement of X: Let T := {σ X (n) f is injective on σ}, and Q 0 := {τ X σ T : τ σ} ( ) Q 1 := {(f σ ) 1 (τ) τ Z,τ f(σ)}. σ T Then define X := Q 0 Q 1. Let τ Q 1 and choose σ T with τ σ. Property (b) of definition implies that ψ σ f ϕ 1 σ is integer affine linear where defined. Hence ϕ σ (τ) is a rational polyhedron in R nσ. Denote by H σ,τ the smallest affine subspace of R nσ containing ϕ σ (τ). We can consider τ := ϕ σ τ to be a map τ : τ H σ,τ = R n τ. Note that by construction ( X, X, { τ}) is a polyhedral complex. We endow it with the weight function ω X and tropical structure {Φ X τ } induced by X. Now we are able to define f X := {f(σ) σ X contained in a maximal polyhedron of X on which f is injective} and f X := τ f τ. For every polyhedron τ f X X let σ τ Y be the inclusionminimal polyhedron containing τ. Then define ϑ τ := ψ στ τ : τ H στ,τ = R n τ, where H στ,τ R nστ is the smallest affine subspace containing the rational polyhedron ψ στ (τ) Z. Note that this makes (f X, f X, {ϑ τ }) into a polyhedral complex. Moreover, note that property (b) of definition still holds for X and Y. Hence we 30

41 Chapter 1: Foundations of tropical intersection theory can assign weights and tropical fans to f X as follows: Let σ f X, let σ Y be the inclusion-minimal polyhedron containing it and let τ 1,...,τ r X be all polyhedra with f(τ i ) = σ that are contained in a maximal polyhedron of X on which f is injective. Then let Ψ σ (S σ ) = Fσ Ỹ and Φ X τi (S τi ) = F X τi respectively be the corresponding open fans and F σ Ỹ, F Xτi be the associated tropical fans. Property (b) of definition implies that f F Xτi F σ Ỹ is again a tropical fan (note that we do not need to refine F X τi to construct this push-forward). Thus we can define ( F f X σ,ω F f X σ ( ) r := f F Xτi, i=1 ) r ω f F Xτi i=1 and F f X σ := F f X σ Ψ σ (S σ ) (here again we assume that ω f F Xτi (τ) = 0 if τ / f F Xτi ). Moreover we define Θ σ := Ψ σ Sσ : S σ F f X σ. Then the map Θ σ, σ f X is 1:1 on polyhedra and we can endow the maximal polyhedra of f X with weights ω f X( ) coming from Fσ f X in this way. These weights are obviously well-defined by property (c) of the tropical polyhedral complex Y (cf. definition 1.4.4) and the maps Θ σ for different σ f X are obviously compatible. Hence we can define [ (((f ) f C := X, f X, {ϑ τ }),ω f X, {Θτ } )] Z n (D). Note that the class [(((f X, f X, {ϑ τ }),ω f X), {Θ τ })] is independent of the choices we made. Thus construction immediately leads to the following Corollary (Push-forward of tropical cycles) Let C Z n and D Z m be two cycles and let f : C D be a morphism. Then for all k there is a well-defined and Z-linear map Z k (C) Z k (D) : E f E := (f E ) E. Proof. The linearity can be proven similar to the affine case (cf. proposition 1.3.6). Our next aim is to define the pull-back of Cartier divisors. But first we need the following Lemma Let C Z n and D Z m be two tropical cycles and let f : C D be a morphism. By definition there exist reduced representatives (((X, X, {ϕ σ }),ω X ), {Φ σ }) of C and (((Y, Y, {ψ τ }),ω Y ), {Ψ τ }) of D such that properties (a) and (b) in definition are fulfilled. Let (((Y 1, Y 1, {ψ τ }),ω Y 1 ), {Ψ τ }) be a refinement of Y. Then there is a refinement (((X 1, X 1, {ϕ σ }),ω X 1 ), {Φ σ}) of X such that properties (a) and (b) of definition are fulfilled for X 1 and Y 1. 31

42 1.7 Rational equivalence Proof. Let X 1 := {σ f 1 (τ) σ X,τ Y 1 }. By property (b) of definition all ϕ σ (σ f 1 (τ)) are rational polyhedra in R nσ. For every σ X 1 choose σ X such that σ = σ f 1 (τ) for some τ Y 1. Then we can define ϕ σ := ϕ σ σ : σ H σ,σ = R n σ, where H σ,σ is the smallest affine subspace of R nσ containing ϕ σ (σ ). Moreover let X 1 := X. Note that with these settings (X 1, X 1, {ϕ σ }) is a polyhedral complex. We can endow it with the weight function ω X1 and the tropical structure {Φ σ } induced by X. Together with Y 1 the tropical polyhedral complex (((X 1, X 1, {ϕ σ }),ω X 1 ), {Φ σ }) fulfills the requirements (a) and (b) of definition Proposition (Pull-back of Cartier divisors) Let C Z n and D Z m be tropical cycles and let f : C D be a morphism. Then there is a well-defined and Z-linear map Div(D) Div(C) : [{(U i,h i )}] f [{(U i,h i )}] := [{(f 1 (U i ),h i f)}]. Proof. We have to show that h f KC (f 1 (U)) for h KD (U) and that h f OC (f 1 (U)) for h OD (U). Then the rest is obvious. So let h KD (U). Then there exists a representative (((Y, Y, {ψ σ}),ω Y ), {Ψ τ }) of D such that for every polyhedron σ Y the map h ψσ 1 is locally integer affine linear. Moreover, since f is a morphism there exist representatives (((X, X, {ϕ σ }),ω X ), {Φ τ }) of C and (((Y, Y, {ψ σ }),ω Y ), {Ψ τ }) of D such that properties (a) and (b) of definition are fulfilled, i.e. f(σ) σ Y for all σ X and the maps Ψ σ f Φ 1 σ induce morphisms of fans. By lemma we may assume that Y = Y. Now let σ X and choose some σ Y such that f(σ) σ. Property (b) of definition implies that ψ σ f ϕ 1 σ and Ψ σ f Φ 1 σ are integer affine linear. Thus h f ϕ 1 σ = (h ψ 1 σ ) (ψ σ f ϕ 1 σ ) is locally integer affine linear and h f KC (f 1 (U)). If additionally h Ψ 1 σ is locally integer affine linear then so is h f Φ 1 σ = (h Ψ 1 σ ) (Ψ σ f Φ 1 σ ). Hence h f OC (f 1 (U)) for h OD (U). Our last step in this chapter is to state the analogon of the projection formula from 1.3.8: Proposition (Projection formula) Let C Z n and D Z m be two cycles and f : C D be a morphism. Let E Z k (C) be a subcycle of C and d Div(D) be a Cartier divisor. Then the following holds: d (f C) = f (f d C) Z k 1 (D). Proof. The claim follows from the constructions of f C and f d, from definition and proposition Rational equivalence In this section we will introduce a concept of rational equivalence and prove that this equivalence is compatible with push-forwards of cycles and pull-backs of Cartier divisors. 32

43 Chapter 1: Foundations of tropical intersection theory Definition (Rational equivalence) Let C be a cycle. We denote by R(C) := {( C,h) h bounded} Div(C) the subgroup of all Cartier divisors globally given by a bounded rational function. Then we define the Picard group of C to be Pic(C) := Div(C)/R(C). Moreover, let D be a subcycle of C. We call D rationally equivalent to zero on C if there exist a tropical cycle C of dimension dim(d) + 1, a morphism f : C C and a bounded rational function h on C such that f (h C ) = D. Let D be another subcycle of C. Then we call D and D rationally equivalent, denoted by D D, if D D is rationally equivalent to zero. We define the k-th Chow group of C to be A k (C) := Z k (C)/. To prove that this equivalence is not too strong for applications in enumerative geometry we need the following lemma: Lemma Let C be a one-dimensional abstract tropical cycle, ϕ R(C) a bounded rational function on C and (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) a reduced representative of C such that for all σ X the map ϕ m 1 σ i.e. ϕ C is of degree zero. =: ϕ σ is integer affine linear. Then ω ϕ ({p}) = 0, {p} X (0) The proof of this statement is to a large extent the work of Johannes Rau, my coauthor of [AR07] and [AR08]. Hence we skip it here and refer to [AR07, lemma 8.3] instead. Remark Note that this property {p} X (0) ω ϕ ({p}) = 0 is preserved by pushing forward. As a consequence, for any tropical cycle C there is a well-defined morphism deg : A 0 (C) Z : [λ 1 P λ r P r ] λ λ r. For D A 0 (C) the number deg(d) is called the degree of D. Lemma Let D be a cycle in C that is rationally equivalent to zero. Then the following holds: (a) Let E be another cycle. Then D E is also rationally equivalent to zero. (b) Let ϕ Div(C) be a Cartier divisor on C. Then ϕ D is also rationally equivalent to zero. 33

44 1.8 Intersection of cycles in R n (c) Let g : C C be a morphism. Then g (D) is also rationally equivalent to zero. Proof. Let f : C C be a morphism and h a bounded function on C such that f (h C ) = D. Then f id : C E C E provides (a), restricting f to f : f (ϕ) C C provides (b) and composing f with g provides (c). Remark In particular, the above lemma implies that there is a well-defined map Pic(C) d A d (C) Z : (([ϕ 1 ],...,[ϕ d ]),D) deg([ϕ 1... ϕ d D]), where C is our ambient cycle. This map is of particular interest when dealing with enumerative questions. Of course, our notion of rational equivalence should also be compatible with the pullback of Cartier divisors. This is indeed the case: Lemma (Pull-back of rational equivalence) Let C,D be tropical cycles and let f : C D be a morphism between them. Then the pull-back map Div(D) Div(C),ϕ f ϕ induces a well-defined map on the quotients Pic(D) Pic(C) : [ϕ] [f ϕ]. Proof. We only have to show that for each element ( D,ψ) R(D) the pull-back Cartier divisor f ( D,ψ) lies in R(C). But this follows from the trivial fact that the composition ψ f of a bounded function ψ and an arbitrary map f is again bounded. 1.8 Intersection of cycles in R n So far we are only able to intersect Cartier divisors with cycles. Our aim in this section is now to define the intersection of two cycles with ambient cycle R n (with trivial structure maps). We do this as follows: Remark We can express the diagonal in R n R n [(, 1)] = [({(x,x) x R n }, 1)] Z n (R n R n ) as a product of Cartier divisors, namely [(, 1)] = ψ 1 ψ n (R n R n ), where ψ i = [{(R n R n, max{0,x i y i })}] Div(R n R n ), i = 1,...,n. We will use this fact to define the intersection product of any two cycles in R n. Definition Let π : R n R n R n : (x,y) x. Then we define the intersection product of cycles in R n by Z n k (R n ) Z n l (R n ) Z n k l (R n ) (C,D) C D := π ( (C D)), 34

45 Chapter 1: Foundations of tropical intersection theory where π denotes the push-forward as defined in and (C D) := ψ 1 ψ n (C D) with ψ 1,...,ψ n as defined in remark Having defined this intersection product of arbitrary cycles in R n we will prove now some basic properties. But as a start we need the following lemmas: Lemma Let C Z k (R n ) be a cycle with representative (X,ω X ) and let ψ 1,...,ψ n be the Cartier divisors defined in remark Then (X j,ω Xj ) with X j := { (R n σ) {(x,y) R n R n x i = y i for i = j,...,n} σ X }, ω Xj ( (R n σ) {(x,y) R n R n x i = y i for i = j,...,n} ) := ω X (σ) is a representative of ψ j ψ n (R n C). Proof. We use induction on j. For j = n + 1 there is nothing to show. Now let the above representative be correct for some j + 1. We have to show that X j is a tropical polyhedral complex and that it represents ψ j ψ n (R n C): Note that dim ((R n σ) {(x,y) R n R n x i = y i for i = j,...,n}) < dim ((R n σ) {(x,y) R n R n x i = y i for i = j + 1,...,n}) ( ) for all σ X. Hence X j is a tropical polyhedral complex. Moreover note that X j+1 := {σ {x j y j = 0},σ {x j y j 0},σ {x j y j 0} σ X j+1 } with weights induced by X j+1 is a refinement of X j+1 such that max{0,x j y j } is linear on every face of X j+1. By ( ) there are exactly two types of faces of codimension one in X j+1 : (i) (R n σ) {x i y i = 0 for i = j,...,n} with σ X, codim(σ) = 0, (ii) (R n σ) {x i y i = 0 for i = j + 1,...,n; x j y j 0} or (R n σ) {x i y i = 0 for i = j + 1,...,n; x j y j 0} with σ X, codim(σ) = 1, where the faces of the second type are not contained in {(x,y) R n R n x j = y j }. Hence max{0,x j y j } is linear on a neighborhood of every face of type (ii) and thus these faces get weight zero in max{0,x j y j } X j+1. The faces of type (i) are weighted by ω Xj+1 ((R n σ) {x i y i = 0 for i = j + 1,...,n}) in max{0,x j y j } X j+1 since x 1 y 1,...,x n y n are part of a lattice basis of (Z n Z n ). Thus max{0,x j y j } X j+1 = X j and X j is a representative of ψ j ψ n (R n C). Corollary Let C Z k (R n ) be a cycle. Then we have the equation: R n C = C. 35

46 1.8 Intersection of cycles in R n Proof. Let (X,ω X ) be a representative of C, let π : R n R n R n : (x,y) x and let ψ 1,...,ψ n be the Cartier divisors defined in remark By lemma we know that X 1 = {{(x,x) x σ} σ X} with ω X1 ({(x,x) x σ}) = ω X (σ) is a representative of ψ 1 ψ n (R n C). Hence R n C = π (ψ 1 ψ n R n C) = [π (X 1,ω X1 )] = [(X,ω X )] = C. Lemma Let C Z k (R n ) and D Z l (R m ) be abstract cycles, ϕ Div(R n ) a Cartier divisor and π : R n R m R n :(x,y) x. Then: (ϕ C) D = π ϕ (C D). Proof. We prove the statement for affine cycles C,D and an affine Cartier divisor ϕ. The general case then follows by applying the statement locally. Choose arbitrary representatives Y of D and h of ϕ and choose a representative X of C such that h is linear on every face of X. This implies that π h is linear on every face of X Y, too. In X 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 all σ τ with τ > τ. We get ω h (σ τ) = 0 in h (X Y ) as π h is linear on σ Y. For the first type the adjacent facets are exactly all σ τ with σ > σ and the weights can be calculated exactly like for h X. This finishes the proof. Let C and D be cycles in R n. Assume that C can be expressed as a product of Cartier divisors, i.e. there are ϕ 1,...,ϕ r Div(R n ) such that C = ϕ r ϕ 1 R n. The obvious questions are now how C D relates to ϕ r ϕ 1 D and whether ϕ r ϕ 1 D depends on the choice of the Cartier divisors ϕ i. To answer this question we first prove a somewhat stronger statement: Lemma Let C Z k (R n ) and D Z l (R n ) be cycles and ϕ Div(R n ) a Cartier divisor. Then we have the equality: (ϕ C) D = ϕ (C D). Proof. Let π : R n R n R n : (x,y) x be like above. The following holds: (ϕ C) D = π ( ((ϕ C) D)) = π (π ϕ (C D)) = ϕ π ( (C D)) = ϕ (C D). 36

47 Chapter 1: Foundations of tropical intersection theory Corollary Let C Z k (R n ) be a cycle such that there are Cartier divisors ϕ 1,...,ϕ r Div(R n ) with ϕ r ϕ 1 R n = C and let D Z l (R n ) be any cycle. Then ϕ r ϕ 1 D = C D. Proof. Applying lemma and lemma we obtain C D = (ϕ r ϕ 1 R n ) D = ϕ r ϕ 1 (R n D) = ϕ r ϕ 1 D. Remark Note that corollary in particular implies that our definition of the intersection product on R n (cf. definition 1.8.2) is independent of the choice of the Cartier divisors describing the diagonal. Theorem Let C,C Z k (R n ), D Z l (R n ) and E Z m (R n ) be cycles. Then the following equations hold: (a) C D = D C, (b) (C + C ) D = C D + C D, (c) (C D) E = C (D E). Proof. (a): Let π i : R n R n R n : (x 1,x 2 ) x i and π 12 : (R n R n ) 2 R n R n : (x 1,x 2,x 3,x 4 ) (x 1,x 2 ). Moreover, let R n be the diagonal in R n R n and R n R n be the diagonal in (R n R n ) 2. Then we can conclude by corollary that and hence ψ 1 ψ n (C D) = R n (C D) = π 12 ( R n R n ( R n (C D))) R n C D = π 1 (ψ 1 ψ n (C D)) = π 1 ( R n (C D)) = π 2 ( R n (C D)) = π 1 ( R n (D C)) = π 1 (ψ 1 ψ n (D C)) = D C. (b): Follows immediately by bilinearity of the intersection product Div(R n R n ) Z p (R n R n ) Z p 1 (R n R n ), 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 π 12 : (R n ) 3 (R n ) 2 : (x,y,z) (x,y), π 13 : (R n ) 3 (R n ) 2 : (x,y,z) (x,z) and π 23 : (R n ) 3 (R n ) 2 : (x,y,z) (y,z). An easy calculation shows that (C π ( (D E))) = π 12 (C ( (D E))) (1.1) 37

48 1.9 Intersection of cycles in R n and (π ( (C D)) E) = π 13 (( (C D)) E). (1.2) Now let ψ 1,...,ψ n be the Cartier divisors defined in remark We label these Cartier divisors with pairs of letters ψ xy i to point out the coordinates they are acting on. We obtain (C π ( (D E))) (1.1) = π 12 (C ( (D E))) = ψ xy 1 ψn xy π 12 (C (ψ yz 1 ψn yz (D E))) = π 12 ((π 12 ) ψ xy 1 (π 12 ) ψn xy (C (ψ yz 1 ψn yz (D E)))) = π 12 ((π 23 ) ψ yz 1 (π 23 ) ψn yz (π 12 ) ψ xy 1 (π 12 ) ψn xy (C D E)) = π 13 ((π 12 ) ψ xy 1 (π 12 ) ψn xy (π 13 ) ψ1 xz (π 13 ) ψn xz (C D E)) = π 13 ((π 13 ) ψ1 xz (π 13 ) ψn xz (ψ xy 1 ψn xy (C D)) E) = ψ1 xz ψn xz π 13 ((ψ xy 1 ψn xy (C D)) E) = π 13 (( (C D)) E) (1.2) = (π ( (C D)) E). This proves (c). It remains to show that our intersection product is well-defined modulo rational equivalence. If this is the case the intersection product induced on A (R n ) clearly inherits the properties of the intersection product on Z (R n ) we have proven in this section. Proposition The intersection product Z n k (R n ) Z n l (R n ) Z n k l (R n ) induces a well-defined and bilinear map A n k (R n ) A n l (R n ) Proof. The intersection product C D is defined to be A n k l (R n ) : ([C], [D]) [C] [D] := [C D]. π (max{x 1,y 1 } max{x r,y r } (C D)), where the x i (resp. y i ) are the coordinates of the first (resp. second) factor of R n R n and π : R n R n R n is the projection onto the first factor (cf. definition 1.8.2). Thus we can apply lemma (a) (c) and the claim follows. 38

49 Chapter 1: Foundations of tropical intersection theory 1.9 Rational equivalence on R n In this section we will prove that every tropical cycle in R n is rationally equivalent to a uniquely determined affine cycle, called its degree. We will use this equivalence to prove a tropical version of Bézout s theorem. We start the section with the easiest example of rationally equivalent cycles in R n, namely translations: Definition Let X be a tropical polyhedral complex in R n and let v R n. We denote by X(v) the translation X(v) := {σ + v σ X} of X along v. If [X] = C Z k (R n ) then C(v) := [X(v)]. Note that the class C(v) is independent of the representative X. Lemma Let C be a subcycle of R n and let v R n be an arbitrary vector. Then the equation holds. C(v) C Proof. Consider the cycle C R in R n R with morphism f : R n R R n, (x,t) x + t e i, where e i is the i-th unit vector in R n. For µ R let h µ be the bounded function 0 t 0 h µ (x,t) = t 0 t µ µ t µ. Then f (h µ C R) = C C(µ e i ), which proves the claim. Definition Let C be a cycle in R n of codimension k. Then we define d C to be the map d C : Z k (R n ) Z, D deg(c D). Lemma Let C = [(X,ω X )] be a d-dimensional affine cycle in R n. Then there always exists a representative (X,ω X ) of C and a complete simplicial fan Θ such that X Θ. Proof. Let X 0 := X = {σ 1,...,σ N } and let σ i = {x R n f σ i 1 (x) 0,...,f σ i k σi (x) 0}. Moreover, let Y 0 := {R n } and for f Λ let H f := {{x f(x) 0}, {x f(x) = 0}, {x f(x) 0}}. 39

50 1.9 Rational equivalence on R n For all i = 1,...,N we construct refinements X i := X i 1 H f σ i 1... H f σ i kσ i and Y i := Y i 1 H f σ i 1... H f σ i kσ i as described in [GKM07, 2.5(e)]. This construction yields fans X N and Y N with X (k) N for all k and X = X N. Moreover, Y N is a complete fan in R n. We can make Y (k) N Y N into a simplicial fan by further subdividing its cones: Let Θ := Y N. If σ Θ (p) is generated by vectors v 1,...,v q then remove σ and add all cones R 0 v i R 0 v ik for 1 k p and 1 i 1 <... < i k q to Θ. Finally, we take (X,ω X ) := (X,ω X ) Θ as described in [GKM07, 2.11(b)]. Lemma Let C 1 and C 2 be affine cycles in R n with d C1 = d C2. Then C 1 = C 2. The proof of this statement is to a large extent the work of Johannes Rau, my coauthor of [AR07] and [AR08]. Hence we skip it here and refer to [AR08, lemma 6] instead. Combining this statement with proposition we can immediately conclude the following: Corollary Let C 1 and C 2 be affine cycles in R n with C 1 C 2. Then C 1 = C 2. Theorem Let C be a cycle in R n. Then there exists an affine cycle δ(x) in R n with X δ(x). Proof. Let (X 1,ω X1 ) be a representative of C 1 := C. Refining (X 1,ω X1 ) we may assume that every polyhedron σ X 1 is the convex hull of its 1-skeleton (see for example [Z95, 1.2 and 2.2]) and that every polyhedron σ X 1 contains at least one vertex σ P σ X (0) 1. The 1-skeleton of X 1 is a finite graph Γ with edges X (1) 1 = {ε 1,...,ε N } and vertices X (0) 1 = {P 0,...,P M }. By lemma we may assume that P 0 is the origin. On every edge ε i of this graph we choose an orientation and a primitive direction vector v i Λ εi respecting this orientation (see figure 1.10 (a)). Then for i = 1,...,N let l i v i be the length of the edge ε i (we set l i = if ε i is unbounded). Adjacency of the bounded edges in the graph Γ yields a system of linear equations in the variables l i having the entries of the vectors v i as coefficients (see figure 1.10 (b)). As the system is solved by the given lengths l i R >0 and all vectors v i are integral there exists a positive and integral solution l 1,...,l N. Using these numbers l i we construct a polyhedral complex X1, t t R as follows: We keep the position of the point P 0 fixed and for all i = 1,...,N with l i < we change the length of the edge ε i to l i + t l i. For all unbounded edges ε i we just keep the directions and the lengths unchanged. For a given polyhedron σ X 1 this process yields a deformation σ t of 40

51 Chapter 1: Foundations of tropical intersection theory l 3 v 3 l 4 v 4 l 2 v 2 l 5 v 5 l 2 v 2 l 1 v 1 P 0 l 6 v 6 l 1 v 1 P0 l N v N (a) The oriented graph constructed from Γ. l 1 v 1 +l 2 v 2 +l 3 v 3 l 4 v 4 l 5 v 5 l 6 v 6 = 0 (b) Adjacency in this graph yields linear equations. σ t 2 σ t 3 σ1 t P 0 P 0 (c) Two examples of the shrinking process. Figure 1.10: Constructions described in the proof of theorem σ which is not necessarily a polyhedron, but that can be decomposed into polyhedra σ t 1,...,σ t p t (see figure 1.10 (c)). If such a polyhedron σ t j is of dimension dim(c), then we define its weight to be ω X t 1 (σ t j) := ( 1) δ(σt j ) ω X1 (σ), where δ(σ t j) is defined as follows: δ(σ t j) := d(σ t j ), where the sum is taken over all values t R between 0 and t such that at least one of the lengths l iq +t l i q occurring in the boundary of σj t is zero and d(σj t ) := dim(σj) t dim(σj t ). We denote by X 1 t the set of all polyhedra σj t for σ X 1 and by ω X t 1 the weight function on the polyhedra of maximal dimension. Refining and possibly merging some of the σj t (we have to add up the weights of all merged polyhedra) yields a tropical polyhedral complex (X1,ω t X t 1 ). Note that (X1,ω 0 X 0 1 ) = (X 1,ω X1 ). Furthermore, for σ X 1 we can consider the set σ := ( pt ) σj t {t} R n R. t R j=1 This set naturally splits up into polyhedra σ 1,..., σ sσ. If a polyhedron σ i is of maximal dimension we associate the weight ω X t 1 (σ t j) to it, where σ t j is a polyhedron containing a point in the relative interior of σ i (this weight is obviously well-defined). We denote by Z the set { σ 1,..., σ sσ σ X 1 } and by ω Z the weight function on the polyhedra of maximal dimension. The choice of the weights ω Z ( σ i ) ensures that refining some of the σ i yields a tropical polyhedral complex (Z,ω Z ). Now, for µ R let ϕ µ be the rational function defined by ϕ µ : R n R R : (x,t) max{0,t} max{µ,t}. Let σ i Z be a polyhedron of maximal dimension and let (possibly after a refinement of X1) t σj t σ i be a polyhedron of X1 t of maximal dimension. As every polyhedron in X 1 contains at least one vertex, this property also holds for X1 t and we can choose a vertex P σ t j σj. t Let P σ t+1 be the translation of P j σ t j in X1 t+1. We have P σ t+1 j P σ t j = k ±l i j v ij j=1 41

52 1.9 Rational equivalence on R n Z ω = 1 X 0 1 X 1 1 X 2 1 Figure 1.11: An example of a cycle Z as constructed in theorem for some i j {1,...,N}. Hence ( k j=1 ±l i j v ij 1 ) Λ Z is a generator of (Λ Z) σi /(Λ Z) σ t and we can deduce that ϕ µ [(Z,ω Z )] = [(X 0 1,ω X 0 1 )] [(X µ 1,ω X µ 1 )]. Now let t 0 R <0 be the largest value such that there exists an edge that has been shrunk to length 0, i.e. an edge ε i (X t 0 1 ) (1) with length l i + t 0 l i = 0. We conclude that ϕ t0 [(Z,ω Z )] = C 1 C 2, where C 2 := [(X t 0 1,ω X t 0 1 )] can be seen as the cycle C = C 1 with at least one bounded polyhedron shrunk to one dimension less. We repeat the whole process until all bounded polyhedra are shrunk to a point, i.e. until we obtain an affine cycle C p. By construction we have which proves the claim. C = C 1 C 2... C p, Definition Let C be a cycle in R n. We define the recession cycle or degree of C, denoted by δ(c), to be the affine cycle equivalent to C. This affine cycle exists by theorem and is unique by lemma Remark Let σ be a polyhedron in R n. We define the recession cone of σ to be rc(σ) := {v R n x + R 0 v σ x σ} = {v R n x σ s.t. x + R 0 v σ}. 42

53 Chapter 1: Foundations of tropical intersection theory The two sets coincide as σ is closed and convex. Let C be a d-dimensional cycle in R n with representative (X,ω X ) and let R(C) := {rc(σ) σ X}. By removing all cones of R(C) that are not contained in a d-dimensional cone and by subdividing the remaining cones we can make this set into a fan R(C) of pure dimension d. To every cone σ R(C) (d) we associate the weight ω R(C) (σ) := σ X σ rc(σ ) The proof of theorem indeed shows that holds. ω X (σ ). δ(c) = [(R(C),ω R(C) )] Theorem Let C,D be two tropical cycles in R n. Then the following are equivalent: i) C D ii) d C = d D iii) δ(c) = δ(d) Proof. i) ii) follows from proposition iii) i) is an immediate consequence of theorem ii) iii) follows from theorem 1.9.7, i) ii) and lemma Remark In other words, the above theorem says: The notions of rational equivalence, numerical equivalence and having the same degree coincide. Theorem (General Bézout s theorem) Let C,D be two tropical cycles in R n. Then δ(c D) = δ(c) δ(d). Proof. We apply theorem and proposition and get δ(c D) C D δ(c) δ(d). By corollary two rationally equivalent affine cycles are equal. Our last step in this section is to prove a version of Bézout s theorem for a special class of tropical cycles in R n called P n -generic cycles. But first we need some further definitions: 43

54 1.9 Rational equivalence on R n Definition Let C Z k (R n ) be a tropical cycle and let L n k be the tropical fan defined in example Then we define the degree of C to be the number deg(c) := deg(c L n codim X), where the second map deg : Z 0 (R n ) Z : λ 1 P λ r P r λ λ r is the usual degree map. Then the map deg : Z k (R n ) Z is obviously linear by definition. Moreover, we define the degree of [C] A k (R n ) to be deg([c]) := deg(c). Note that deg([c]) is well-defined by remark and proposition Definition (P n -generic cycles) Let C Z k (R n ) be a tropical cycle. C is called P n -generic if for one (and thus for every) representative X of C holds: For every face σ X (k) there exists a polytope P σ R n of some dimension r {0,...,k} and a cone σ (L n k )(k r) such that σ P σ + σ. Remark Note that that the above definition of P n -generic cycles is equivalent to the condition that the recession cycle δ(c) of our cycle C Z k (R n ) is contained in the tropical fan L n k. By [GKM07, example 2.18] this implies that C is Pn -generic if and only if δ(c) = λ L n k holds for some integer λ. In this case we can easily conclude that deg(c) = deg(δ(c)) = λ. Y X Figure 1.12: The intersection of two P n -generic cycles of degrees 2 and 3. Theorem (Bézout s theorem) Let C Z k (R n ) and D Z n k (R n ) be two tropical cycles of complementary dimensions. Moreover, assume that C is P n -generic. Then the following equation holds: deg(c D) = deg(c) deg(d). Proof. By proposition , theorem and remark we can conclude that deg(c D) = deg(δ(c) D) = deg((deg(c) L n k ) D) = deg(c) deg(l n k D) = deg(c) deg(d). 44

55 2 Tropical cycles with real slopes and numerical equivalence In this chapter we introduce a generalized definition of tropical fans and tropical cycles that allows non-rational slopes. Obviously, these objects cannot be realized as images of algebraic varieties over the field of Puiseux series under the valuation, but they are quite useful for some theoretical aspects of tropical geometry. For example, using tropical cycles with real slopes we give a proof that in any tropical fan any cycle is numerically equivalent to its degree cycle. 2.1 Tropical cycles with real slopes We start this section giving the necessary generalizations of the first definitions in [GKM07] and chapter 1. In the following, let V always be a real vector space of finite dimension. Definition A polyhedron in V is a non-empty subset σ V of the form σ = {x V f 1 (x) = b 1,...,f r (x) = b r,f r+1 (x) b r+1,...,f N (x) b N } for some linear forms f 1,...,f N V and numbers b 1,...,b N R. A cone in V is a polyhedron for which all the numbers b 1,...,b N are zero. Definition A polyhedral complex X in V is a finite set of polyhedra in V such that the following properties are fulfilled: (a) All faces of the polyhedra in X are again in X and (b) the intersection of two polyhedra in X is either a face of each or empty. A fan in V is a polyhedral complex consisting of cones only. As before we denote by X the union of all polyhedra in X. Definition A marked fan in V is a pure-dimensional simplicial fan X in V together with a generating vector v σ for every edge in X, i.e. a vector v σ σ \ {0} for every σ X (1). As in [GKM07, construction 2.13], for all pairs σ X (dim(x)) and τ X (dim(x) 1) with 45

56 2.1 Tropical cycles with real slopes τ < σ there is exactly one edge σ X (1) that is a face of σ but not of τ. We denote the associated vector v σ by v σ/τ and call it the normal vector of σ relative to τ. Definition A tropical fan (with real slopes) in V is a pair (X,ω X ) where X is a marked fan and ω X : X (dim(x)) R is a weight function on the cones of maximal dimension such that the balancing condition ω X (σ) v σ/τ = 0 in V/V τ σ:τ<σ is fulfilled for all τ X (dim(x) 1). As in definition we denote by (X,ω X ) := ({τ X τ σ for some σ X (dim(x)) with ω X (σ) 0},ω X (X )(dim(x))) the non-zero part of X. Remark and Definition Let (X,ω X ) be a tropical fan with real slopes, let σ 1 X (1) be an edge of X and let λ R >0 be a positive number. Then let X := X and { ωx (σ), if σ 1 σ ω X (σ) := 1 ω λ X(σ), if σ 1 σ. Moreover, replace in X the vector v σ1 by λ v σ1. We would like to call (X,ω X ) and (X,ω X ) the same tropical fan, but according to definition this is not the case. Hence, in the following we will always identify tropical fans that arise one from the other by operations as above. We also have to take care of this ambiguity when defining refinements: Definition Let (X,ω X ) and (Y,ω Y ) be tropical fans with real slopes in V. Analogously to definition we say that Y is a subfan of X if for every cone σ Y there exists a cone σ X such that σ σ. In this case we define a map C Y,X : Y X that maps a cone σ Y to the inclusion-minimal cone σ X with σ σ. We say that (Y,ω Y ) is a refinement of (X,ω X ) if (a) Y is a subfan of X, (b) Y = X and (c) ω X (C Y,X (σ)) det(a σ,cy,x (σ)) = ω Y (σ) for every σ (Y ) (dim(y )), where A σ,cy,x (σ) is a matrix expressing the generating vectors of C Y,X (σ) in terms of the generating vectors of σ. Remark Using the above definitions we can carry over the other basic definitions from section 1.1, in particular the definition of affine tropical cycles, to our context. Note that in some cases, e.g. for taking sums of cycles, it is necessary to further refine the tropical fans our constructions in section 1.1 yield to obtain simplicial fans. Moreover, we can 46

57 Chapter 2: Tropical cycles with real slopes and numerical equivalence define open marked fans and open tropical fans in analogy to definition Using rational functions with arbitrary slopes it is obviously possible to define intersection products of Cartier divisors and tropical cycles with real slopes in exactly the same way as we did in section 1.2. What is missing so far is the notion of morphisms: Definition Let (X,ω X ) and (Y,ω Y ) be tropical fans with real slopes in finite-dimensional real vector spaces V and V, respectively. A morphism f : X Y is just the restriction of an R-linear map f : V V to X. As before we can refine X in such a way that f X := {f(σ) σ X is contained in a maximal cone of X on which f is injective} is a fan in V. We can make f X into a tropical fans as follows: For every cone σ f X (1) we choose a generating vector v σ σ \ {0}. Moreover, for all cones σ f X (dim(x)) we set ω f X(σ ) := σ X:f(σ)=σ ω X (σ) det(a σ,f(σ)), where A σ,f(σ) is a matrix expressing the images f(v τi ) of the generating vectors v τi of σ in terms of the chosen generating vectors of σ. In the following we will define marked polyhedral complexes and tropical polyhedral complexes with real slopes in analogy to marked and tropical fans. This will lead us to (embedded) tropical cycles with real slopes as spaces looking locally like affine cycles. Definition A marked polyhedral complex in V is a pure-dimensional polyhedral complex X in V fulfilling the following conditions: (a) For every polyhedron σ X there exists a vertex P σ X (0) with P σ σ, (b) for every vertex P X (0) and every edge σ X (1) with P σ we are given a vector v P σ V such that the set {σ X P σ} together with the set of vectors {v P σ P σ X (1) } is an open marked fan in V (after a suitable translation), (c) for all vertices P 1,P 2 X (0) with P 1 P 2 and every edge σ X (1) with P 1,P 2 σ holds v P 1 σ = v P 2 σ and (d) for every pair of vertices P 1,P 2 X (0) and all polyhedra σ X with P 1,P 2 σ the base change from v P 1 τ 1,...,v P 1 τ r to v P 2,...,v P 2 has determinant ±1, where 1 r τ 1,...,τ r X (1) and 1,..., r X (1) are all the edges contained in σ that contain P 1 respectively P 2. Definition A tropical polyhedral complex (with real slopes) in V is a pair (X,ω X ) where X is a marked polyhedral complex and ω X : X (dim(x)) R is a weight function on the polyhedra of maximal dimension such that for all vertices P X (0) these weights make the open marked fan {σ X P σ} into a tropical fan. As before, we denote by (X,ω X ) := ({τ X τ σ for some σ X (dim(x)) with ω X (σ) 0},ω X (X ) (dim(x))) the non-zero part of X. 47

58 2.1 Tropical cycles with real slopes Definition Let (X,ω X ) and (Y,ω Y ) be tropical polyhedral complexes with real slopes in V. Exactly as in definition we say that Y is a subcomplex of X if for every polyhedron σ Y there exists a polyhedron σ X such that σ σ. Again, we can define a map C Y,X : Y X that maps a polyhedron σ Y to the inclusion-minimal polyhedron σ X with σ σ. We say that (Y,ω Y ) is a refinement of (X,ω X ) if (a) Y is a subcomplex of X, (b) Y = X and (c) ω X (C Y,X (σ)) det(a σ,cy,x (σ)) = ω Y (σ) for every σ (Y ) (dim(y )), where the matrix A σ,cy,x (σ) arises as follows: Pick vertices P σ and Q C Y,X (σ). Then A σ,cy,x (σ) is a matrix expressing the generating vectors vτ Q i of C Y,X (σ) in the open marked fan { σ X Q σ} in terms of the generating vectors v P i of σ in the open marked fan { σ Y P σ}. By property (d) of definition condition (c) is independent of the choice of the vertices P,Q. Remark As before, using these new definitions we can carry over the definitions and results from sections , in particular the definition of a tropical cycle and the notion of a morphism of cycles, to our context. Here again it is necessary in some cases to further refine the polyhedral complexes we obtain by our constructions in chapter 1 to end up with simplicial complexes. Note that properties (c) and (d) of definition make sure that we can define the intersection of a Cartier divisor with a cycle by taking local intersection products as we did in definition Example We want to consider tropical curves in R 2, i.e. tropical cycles of dimension one according to remark , and their moduli space in analogy to [GKM07, section 4]. Therefore we fix a degree 0 := {v 1,v 2,v 3 },v i R 2, 3 i=1 v i = 0 and say a curve with real slopes has degree d if its degree (in the sense of [GKM07, definition 4.1]) is {v 1,...,v 1,v 2,...,v 2,v 3,...,v 3 }, where each v i occurs exactly d times. Given n := 3d 1 points P 1,...,P n R 2 we want to calculate the number of tropical curves with real slopes of degree d passing through P 1,...,P n (counted with multiplicities) as an intersection product on the corresponding moduli space M lab 0,n,trop(R 2,d) = M 0,3d+n,trop R 2 : Let the matrix A be given by A = ( v 1 v 2 ) 1. Hence we have maps M 0,3d+n,trop R 2 ev i R 2 A R 2. 48

59 Chapter 2: Tropical cycles with real slopes and numerical equivalence v 2 v 2 Av 3 v 3 Av 1 Av 3 A Av 1 v 3 v 1 v 1 Av 2 Av 2 Figure 2.1: A curve with real slopes mapped to an ordinary degree-2-curve. Note that for a curve C of degree d in R 2 we have that A(C) is an ordinary degreed-curve, i.e. A(C) has degree d in the sense of [GKM07, definition 4.1]. Let P i = (a (i) 1,a (i) 2 ). We can write P i as P i = max{a (i) 1,x} max{a (i) 2,y} R 2, where x,y are the coordinates on R 2. Moreover, we have A max{a (i) 1,x} A max{a (i) 2,y} R 2 = det(a) P i = det(a) max{a (i) 1,x} max{a (i) 2,y} R 2. Using this equation and the projection formula we can conclude that ( n ) N d = deg ev i A max{a (i) 1,x} ev i A max{a (i) 2,y} M 0,n+3d,trop R 2 ( i=1 n ) = deg det(a) ev i max{a (i) 1,x} ev i max{a (i) 2,y} M 0,n+3d,trop R 2 i=1 ( n ) = det(a) n deg ev i max{a (i) 1,x} ev i max{a (i) 2,y} M 0,n+3d,trop R 2 i=1 = det(a) n Ñd, where N d is the number of ordinary degree-d-curves passing through P 1,...,P n (counted with multiplicities) and Ñd is exactly the number we want to calculate. Hence we obtain the equation 1 Ñ d = det(a) N n d = det (v 1 v 2 ) n N d. Remark In the following, the notions of tropical fan, tropical polyhedral complex and tropical cycle will always refer to the objects introduced in chapter 1. We will state explicitly if a fan or cycle is a tropical fan or tropical cycle with real slopes according to definition or remark , respectively. 49

60 2.2 Numerical equivalence on fans v 2 u 1 v 1 v 3 τ τ u 2 u 3 v 2 v 1 v 3 ω Z ( τ) = ω X (τ) v 3 v 1 0 ω Z ( τ ) = ( 1) 2 ω X (τ) v 2 Figure 2.2: Cones τ, τ with normal vectors v i over a polyhedron τ. 2.2 Numerical equivalence on fans Definition Let Y be a tropical fan in some vector space V and let C,D Z k (Y ) be tropical cycles. We call C and D numerically equivalent if deg(ϕ 1 ϕ k C) = deg(ϕ 1 ϕ k D) holds for all Cartier divisors ϕ 1,...,ϕ k on Y. Proposition Let Y be a tropical fan in R n and let C Z k (Y ) be a tropical cycle. Then C and its degree cycle δ(c) Z k (Y ) are numerically equivalent (see definition for the definition of δ(c)). Proof. Let X be a representative of C. We perform the same construction as in the proof of theorem 1.9.7, but with some modifications: Like before, we translate our chosen vertex P 0 to the point P 0 := 0, but instead of replacing the lengths l i by l i +t l i we replace l i by t l i. Note that we do not change the directions and lengths of unbounded edges. After a suitable refinement of X the cones arising from the polyhedra τ X for t 0 in this way are simplicial. Moreover, for t 0 a polyhedron τ X in general yields a union of simplicial cones. We denote this set of cones by Z. We can make Z into a tropical polyhedral complex by further refining it and equipping it with weights and normal vectors as follows: Let first τ X be a bounded polyhedron. Performing operations as described in remark and definition we may assume that every normal vector associated to an edge E of τ with endpoints P E and Q E is just P E Q E (cf. figure 2.2). If τ (for t 0) and τ (for t 0) are the cones in Z arising from τ and P 1,...,P r X (0) are all vertices in τ we choose ±(P 1, 1),...,±(P r, 1) R n R as generating vectors of τ and τ, respectively. If τ X is an unbounded polyhedron we have all vectors (v, 0) R n R, where v is the primitive normal vector of an unbounded edge in τ in our original complex X, as additional primitive normal vectors generating τ. In both cases we set ω Z ( τ) := ω X (τ) if τ is a cone of maximal dimension. Moreover, if τ is unbounded, τ is in general not a cone but a union of cones that contains the primitive normal vectors (v, 0) R n R, too, but due to 50

61 Chapter 2: Tropical cycles with real slopes and numerical equivalence δ(c) C Figure 2.3: A curve C and its degree cycle δ(c). the necessary refinements we need some additional generating vectors in this case. Let τ i be one such cone (or τ i = τ if τ is bounded). Then we set ω Z ( τ i) := ( 1) d(τ) ω X (τ), where d(τ) := dim(( τ i) t ) dim(rc(( τ i) t )) and ( τ i) t is the intersection of ( τ i) with R n {t} for some t < 0 (cf. theorem 1.9.7, remark and figure 2.2). Performing a refinement of overlapping cones and adding up their weights as in theorem the above choices make Z into a tropical polyhedral complex with real slopes (Z,ω Z ) according to definition in R n R such that moreover max{1,t} [(Z,ω Z )] = C( P 0 ) holds, where t is the coordinate of the additional factor R and C( P 0 ) is the translation of C by the vector P 0. Additionally we have max{0,t} [(Z,ω Z )] = δ(c) in this case. Let f : R n R R n be the map given by (x,t) x + t P 0. Then f maps Z {0 t 1} to Y. We can conclude the following: deg (ϕ 1 ϕ k (C δ(c))) = deg (ϕ 1 ϕ k f (max{1,t} max{0,t} [(Z,ω Z )])) = deg (f (f ϕ 1 f ϕ k (max{1,t} max{0,t}) [(Z,ω Z )])) = deg (f ϕ 1 f ϕ k (max{1,t} max{0,t}) [(Z,ω Z )]) ) = deg ( f ϕ 1 f ϕ k (max{1,t} max{0,t}) [(Z,ω Z )] ( ) = deg (max{1,t} max{0,t}) f ϕ 1 f ϕ k [(Z,ω Z )] = 0, where f ϕ i is any continuation of f ϕ i : Z {0 t 1} R on the whole cycle Z. Hence, using the linearity of deg and the intersection product, we can conclude that deg (ϕ 1 ϕ k C) = deg (ϕ 1 ϕ k δ(c)) for all Cartier divisors ϕ 1,...,ϕ k on Y and thus that C and δ(c) are numerically equivalent. 2.3 Chow groups via numerical equivalence In analogy to P n -generic cycles (cf. definition ) we can define generic cycles corresponding to other toric compactifications of the algebraic torus: 51

62 2.3 Chow groups via numerical equivalence Definition (F-generic cycles) Let F be a complete fan in R n corresponding to a compact and smooth toric variety, i.e. F = R n and every cone of F is generated by part of a lattice basis. Let C Z k (R n ) be a tropical cycle. C is called F-generic if for one (and thus for every) representative X of C holds: For every face σ X (k) there exists a polytope P σ R n of some dimension r {0,...,k} and a cone σ F (k r) such that σ P σ + σ. We denote by Z F k (Rn ) the group of all k-dimensional and F-generic cycles in R n. Remark The above definition of F-generic cycles is equivalent to the condition that the recession cycle δ(c) of our cycle C Z k (R n ) is contained in the k-skeleton of our given fan F. Hence, for every F-generic cylce C there is a representative Y of δ(c) such that Y k i=0 F (i). Example Let F 1 and F 2 be the complete fans in R 2 and R 3, respectively, defining P 1 P 1 and P 1 P 1 P 1, respectively, as a toric variety. The following figure shows examples of F 1 - and F 2 -generic cycles: C D Figure 2.4: An F 1 -generic curve C and an F 2 -generic cycle D. For the rest of this section F will always denote a complete fan in R n corresponding to a compact and smooth toric variety as in definition Definition (F-numerical equivalence) Let C,C Z k (R n ) be arbitrary tropical cycles in R n. We say that C and C are F-numerical equivalent, denoted by C num C, if the maps d C : Zk F(Rn ) Z : D deg(c D) and d C : Zk F(Rn ) Z : D deg(c D) coincide (cf. definition 1.9.3). We define the k-th tropical Chow group of R n induced by numerical equivalence to be the group A F k (R n ) := Z k (R n )/ num. Theorem Let X(F) be the toric variety associated to F and A k (X(F)) and A k (X(F)) its k-th Chow group and k-th Chow cohomology group, respectively. Then we have an isomorphism: A F k (R n ) = A k (X(F)), 52

63 Chapter 2: Tropical cycles with real slopes and numerical equivalence i.e. the classical Chow cohomology groups and the tropical Chow groups are naturally isomorphic. Hence we also have A F k (R n ) = A k (X(F)), but there is no canonical map giving this last isomorphism. Proof. As X(F) is a smooth toric variety we can deduce that A k (X(F)) is free. Hence, by [FS97, theorem 2.1] there are Minkowski weights c 1,...,c r on F (n k) freely generating A k (X(F)). We define X(c i ) to be the tropical fan with cones X(c i ) := n k i=0 F (i) and weight function c i. Moreover, [F84, corollary 17.4] and [FS97, proposition 1.4] imply that the map I : A k (X(F)) Hom(A k (X(F)), Z) Hom(A n k (X(F)), Z) c deg((c ) [X(F)]) is an isomorphism. As A n k (X(F)) is a free group, we can conclude that A k (X(F)) = A n k (X(F)). Thus, there are again r Minkowski weights d 1,...,d r on F (k) freely generating the group A n k (X(F)) and associated tropical fans X(d i ) := k i=0 F (i) with weight functions d i. The isomorphism I is then given by where I : A k (X(F)) Hom(A n k (X(F)), Z) Z r Z r (α 1,...,α r ) ( r j=1 γ 1jα j,..., r j=1 γ rjα j ), γ ij := deg((d i c j ) [X(F)]) = (d i c j )({0}). The last equation follows from [FS97, proposition 3.1]. By [R08, theorem 1.9] we can deduce that moreover γ ij = deg(x(d i ) X(c j )) holds. Now, let C Z n k (R n ) be a tropical cycle and let δ i := deg(x(d i ) C) Z. As the map I is an isomorphism, the matrix (γ ij ) ij is invertible over Z and there are α 1,...,α r Z with r j=1 γ ijα j = δ i for all i = 1,...,r. We claim that C num α 1 X(c 1 )+...+α r X(c r ): Let D Z F k (Rn ) be an F-generic cycle. By remark there exists a representative Y of δ(d) such that Y k i=0 F (i). The weight function ω Y defines a Minkowski weight c(y ) on F n k. Hence, there are coefficients β 1,...,β r Z such that c(y ) = β 1 d β r d r and equivalently Y = β 1 X(d 1 ) β r X(d r ). Thus we obtain deg(y C) = deg((β 1 X(d 1 ) β r X(d r )) C) = β 1 δ β r δ r. Moreover, we can deduce that deg(y (α 1 X(c 1 ) α r X(c r ))) = deg((β 1 X(d 1 ) β r X(d r )) (α 1 X(c 1 ) α r X(c r ))) = ( )) r i=1 (β i rj=1 α jγ ij = r i=1 β iδ i. 53

64 2.3 Chow groups via numerical equivalence Thus we have C num α 1 X(c 1 ) α r X(c r ). Now, let α 1,...,α r Z be given such that α 1 X(c 1 ) α r X(c r ) num 0. In particular, this implies that deg((α 1 X(c 1 ) α r X(c r )) X(d i )) = r α j γ ij = 0 j=1 for all i = 1,...,r. As (γ ij ) ij is invertible we can deduce that α 1 =... = α r = 0. Altogether we can conclude that the map [ r r ] X(c 1 ),...,X(c r ) Z A F k (R n ) : α i X(c i ) α i X(c i ) i=1 i=1 num is an isomorphism. Hence we obtain A F k (R n ) = X(c 1 ),...,X(c r ) Z = c1,...,c r Z = A k (X(F)) = A k (X(F)), which finishes the proof. 54

65 3 Tropical intersection products on smooth varieties In this chapter we define an intersection product of tropical cycles on tropical linear spaces L n k, i.e. on tropical fans of the type max{0,x 1,...,x n } n k R n, in analogy to section 1.8. Afterwards we use this result to obtain an intersection product of cycles on every smooth tropical variety, i.e. on every tropical variety that arises from gluing such tropical linear spaces. In contrast to classical algebraic geometry these products always yield well-defined cycles, not cycle classes only. Using these intersection products we are able to define the pull-back of a tropical cycle along a morphism between smooth tropical varieties. 3.1 Intersection products on tropical linear spaces In this section we will give a proof that tropical linear spaces L n k admit an intersection product. Therefore we show at first that the diagonal in the Cartesian product L n k Ln k of such a linear space with itself is a sum of products of Cartier divisors. Given two cycles C and D we can then intersect the diagonal with C D and define the product C D to be the projection thereof. Throughout the section e 1,...,e n will always be the standard basis vectors in R n and e 0 := e 1... e n. We begin the section with our basic definitions: Definition (Tropical linear spaces) For I {0, 1,...,n} let σ I be the cone generated by the vectors e i, i I. We denote by L n k the tropical fan consisting of all cones σ I with I {0, 1,...,n} and I k, whose maximal cones all have weight one (cf. example 1.2.9). This fan L n k is a representative of the tropical linear space max{0,x 1,...,x n } n k R n. Definition Let C Z k (R n ) be a tropical cycle and let the map i : R n R n R n be given by x (x, x). Then the push-forward cycle is called the diagonal of C C. C := i (C) Z k (R n R n ) In order to express the diagonal in L n k Ln k by means of Cartier divisors we first have to refine L n k Ln k in such a way that the diagonal is a subfan of this refinement: 55

66 3.1 Intersection products on tropical linear spaces Definition Let Fk n be the refinement of Ln k Ln k that arises recursively from Ln k Ln k as follows: Let M := (L n k Ln k )(2k) be the set of maximal cones in L n k Ln k. If a cone σ M is generated by ( ) ( ) ei 0,,v 0 e 3,...,v 2k i for some i and vectors {( eµ v j e µ ) ( eµ, 0 ) ( 0, e µ then replace the cone σ by the two cones spanned by ( ) ( ) ei ei,,v e i 0 3,...,v 2k and ( ei e i ) ( 0, e i ),v 3,...,v 2k, ) µ = 0,...,n} respectively. Repeat this process until there are no more cones in M that can be replaced. The fan Fk n is then the set of all faces of all cones in M. The next lemma and the following corollary provide technical tools needed in the proofs of the subsequent theorems: Lemma Let F be a complete and smooth fan in R n (in the sense of toric geometry) and let h 1,...,h r, r n, be rational functions on R n that are linear on every cone of F. Then the intersection product h 1 h r F is given by ( n r ) h 1 h r F = F (i),ω h1 h r with some weight function ω h1 h r on the cones of dimension n r. For a cone σ F (n r) such that all functions h 1,...,h r are identically zero on σ the weight ω h1 h r (σ) is given by ω h1 h r (σ) = h 1 (u 1 ) h r (u r ), i=0 r u1,...,r ur F (1) : r u1,...,r ur,σ span a cone in F (n) where r ui denotes the ray generated by the primitive lattice vector u i. Proof. We prove the claim by induction on r: For r = 1 the above formula is just the definition of the intersection product (see definition 1.2.4). For r > 1 we have the equation ω h1 h r (σ) = ω h2 hr( σ) h 1 (u 1 ). r u1 F (1) : r u1,σ span a cone σ F (n r+1) 56

67 Chapter 3: Tropical intersection products on smooth varieties Using the induction hypothesis we can conclude that = = ω h1 h r (σ) r u1 F (1) : r u1,σ span a cone σ F (n r+1) h 1 (u 1 ) h r (u r ). r u1,...,r ur F (1) : r u1,...,r ur,σ span a cone in F (n) h 2 (u 2 ) h r (u r ) r u2,...,r ur F (1) : r u2,...,r ur, σ span a cone in F (n) h 1(u 1 ) Corollary Under the above assumptions the weight of the cone σ can equivalently be written as ω h1 h r (σ) = perm ((h i (u j ) i,j=1,...,r ), F (n) : is generated by r u1,...,r ur,σ where perm(a) denotes the permanent of the matrix A. Proof. Using lemma we can conclude that ω h1 h r (σ) = h 1 (u 1 ) h r (u r ) u 1,...,u r F (1) : u 1,...,u r,σ span a cone in F (n) ( = h1 (u p(1) ) h r (u p(r) ) ) p S r = F (n), is generated by u 1,...,u r,σ F (n) : is generated by u 1,...,u r,σ perm ((h i (u j ) i,j=1,...,r ). Remark Note that the above assumption that all rational functions are identically zero on σ is not really a restriction: We can always achieve this by adding suitable globally linear functions to the rational functions h i which does not change the intersection product and in particular not the weight ω h1 h r (σ). Notation Let F be a simplicial fan in R n and let u be a generator of a ray r u in F. By abuse of notation we also denote by u the unique rational function on F that is linear on every cone in F, that has the value one on u and that is identically zero on all rays of F other than r u. If not stated otherwise, vectors considered as Cartier divisors will from now on always denote rational functions on the complete fan F n n. 57

68 3.1 Intersection products on tropical linear spaces Notation Let C be a tropical cycle and let h 1,...,h r Div(C) be Cartier divisors on C. If P(x 1,...,x r ) = α i1,...,i r x i 1 1 x ir r i i r d is a polynomial in variables x 1,...,x r we denote by P(h 1,...,h r ) C the intersection product P(h 1,...,h r ) C := ( αi1,...,i r h i 1 1 h ir r C ). i i r d In the following theorem we give a description of the diagonal L n n k Cartier divisors on our fan Fn n : by means of Theorem The fan (( ) ( e e 0 )) (( en... 0 ) ( 0 + e 0 )) (( e0 0 ) ( e0 + e 0 )) k F n n is a representative of the diagonal L n n k. Proof. First of all, note that ( e0 0 ) ( e0 + e 0 ) is a representation of the tropical polynomial max{0,x 1,...,x n }, where x 1,...,x n are the coordinates of the first factor of R n R n. Applying lemma we obtain [ (( ) ( )) k ] e0 e0 + Fn n = [L n n k R n ]. 0 e 0 By lemma we can conclude that R n [L n n k Rn ] = i ([L n n k ]) = L n and n k hence it suffices to show that [X] = R n for (( ) ( )) (( ) ( )) e1 0 en 0 X := Fn n 0 e 0 0 e 0 to prove the claim. Therefore, let σ = r 1,...,r n R 0 X (n) be a cone not contained in R n. We will show that the weight of σ in X has to be zero. W.l.o.g. we assume that {( ) ( )} e1 en r 1 D :=,...,. e 1 e n Moreover, let {( e1 T := 0 ) ( en,..., 0 We distinguish between two cases: )} {( 0 and B := e 1 ) ( 0,..., e n )}. 58

69 Chapter 3: Tropical intersection products on smooth varieties 1. First, we assume that r i {( e0 0 ) ( 0, e 0 )},i = 1,...,n. Changing the given rational functions by globally linear functions we can rewrite the above intersection product as X = ϕ 1 ϕ n Fn n, where ( ) ( ) ( ) ei 0 ei +, if {r 0 e 0 0 1,...,r n } ϕ i = ( ) ( ) 0 e0 +, else. e i 0 All occurring rational functions are identically zero on σ now and we can apply corollary 3.1.5: If the weight of σ in X is non-zero there must be at least one cone σ = r 1,...,r n,v 1,...,v n R 0 Fn n such that perm ((ϕ i (v j )) i,j ) 0. We study three subcases: ( ) ( ) ei ej a) More than one vector r i is contained in T (or in B): If, 0 0 ( ) ( ) 0 e0 {r 1,...,r n } for some i j then ϕ i = + and ϕ e i 0 j = ( ) ( ) ( ) ( ) 0 e Hence we need two vectors out of,, e j 0 e i e ( ) j e0 among the v 0 µ such that perm ((ϕ i (v j )) i,j ) can be non-zero. But ( ) ( ) ( ) there is no cone in Fn n ei 0 ej containing and or and 0 e i 0 ( ) 0. (Analogously for B.) e j ( ) e0 b) There are vectors r i T and r j B: Then we need both vectors 0 ( ) 0 and among the v e µ such that perm ((ϕ i (v j )) i,j ) can be non-zero. 0 But again there is no cone in Fn n containing these two vectors. ( ) e0 c) r 1 T (or r 1 B) and r j D for j 1: Like in (a) we need v 1 = 0 {( ) ( )} e2 en and v 2,...,v n,..., such that perm ((ϕ 0 0 i (v j )) i,j ) can ( ) ( ) be non-zero. But there is no cone in Fn n e1 en containing,..., 0 0 ( ) e0 and. (Analogously for B.) 0 2. Now we assume that r 1 = ( e0 0 ) ( ( 0 or r 1 = e 0 )). 59

70 3.1 Intersection products on tropical linear spaces Again, by changing the given rational functions by globally linear functions we can rewrite the intersection product as X = ϕ 1 ϕ n F n n, where ϕ i = ( ei 0 ) ( 0 + e 0 ), if some equivalent rational function not involving r 1,...,r n, else. ( ei 0 ) {r 1,...,r n } Again we reached that all ( rational ) functions are identically zero ( on) σ and ( we can ) ei ei 0 apply corollary 3.1.5: If {r 0 1,...,r n } then ϕ i = + 0 e ( ) ( ) 0 ei 0 and we need or among the v 0 e µ such that perm ((ϕ i (v j )) i,j ) can 0 ( ) ( ) 0 be non-zero. But as there is no cone in Fn n e0 containing and we e 0 0 ( ) ( ) ( ) ei e1 en must have {v 0 1,...,v n }. Hence all the vectors,..., 0 0 ( ) e0 and must be contained in {r 0 1,...,r n,v 1,...,v n }, but there is no such ( ) 0 cone in Fn n. (Analogously for r 1 =.) e 0 Our last step in this proof is to show that at least one cone in the diagonal of R n R n occurs with weight one in X. As the diagonal is irreducible we can then conclude by [GKM07, lemma 2.21] that the whole diagonal occurs with weight one. We have a look at the cone As all the rational functions ( e1 σ = r 1,...,r n R 0 = e 1 ϕ i = ( ei 0 ) ( en,..., e n ) ( ) 0 + e 0 ). R 0 are already zero on σ we can immediately apply corollary 3.1.5: There is exactly one cone σ = r 1,...,r n,v 1,...,v n R 0 = ( e1 e 1 ),..., ( en e n ), ( e1 0 ) ( en,..., 0 in F n n containing σ such that the permanent perm ((ϕ i (v j )) i,j ) is non-zero and for this cone we have perm ((ϕ i (v j )) i,j ) = 1. This finishes the proof. Our next step is to derive a description of the diagonal L n n k on L n n k Ln n k from our description on F n n : ) R 0 60

71 Chapter 3: Tropical intersection products on smooth varieties Theorem The intersection product in theorem can be rewritten as ( r ) (( ) ( )) k 0 e0 h i,1...h i,n k + e 0 e 0 i=1 (( e0 0 ) ( e0 + e 0 )) k F n n for some Cartier divisors h i,j on F n n. We have to prepare the proof of the theorem by the following lemma: Lemma Let C Z l (L n n k ) be a subcycle of Ln n k. Then the following intersection products are zero: ( ) ( ) e0 0 (a) (C R 0 e n ), 0 (b) v i1 v in k+r (C R n ), (c) ( 0 e 0 ) ( e0 e 0 ) s v i1 v in k s+r (C Rn ), where r, s > 0 and the vectors v ij are pairwise distinct. {( e0 0 ) ( en,..., 0 ) ( e0, e 0 ) ( en,..., e n Proof. (a) and (b): In both cases, a cone that can occur in the intersection product with non-zero weight has to be contained ( in ) a ( cone of ) Fn n that is contained in L n n k Rn e0 0 and that contains the vectors, or v 0 e i1,...,v in k+r, respectively. But 0 there are no such cones. (c): By (a) and lemma we can rewrite the intersection product as = = = )} ( ) ( ) s 0 e0 v e 0 e i1 v (C in k s+r Rn ) ( ) (( 0 ) ( )) s 0 e0 e0 + v e 0 0 e i1 v (C in k s+r Rn ) ( ) [(( 0 ) ( )) s 0 e0 e0 v e i1 v in k s+r + C] R 0 0 e n ( ) 0 0 v e i1 v in k s+r [max{0,x 1,...,x n } s C] R n, 0 which is zero by (b) as max{0,x 1,...,x n } s C is contained in L n n k s. 61

72 3.1 Intersection products on tropical linear spaces Proof of theorem By theorem we have the representation L n n k = (( e 1 0 = (( e 1 0 ) ( + 0 )) (( e 0... en 0 ) ( en 0 ) ( + 0 )) (( e 0 e0 ) ( 0 + e0 )) k e 0 [F n n ] }{{} ) ( ) n ) e 0 [L n n k R n ]. =[L n n k Rn ] ( ) s 0 By lemma (b) all the summands containing with a power s < k are e 0 zero. Hence we can rewrite the intersection product as [ ( ) ( ) ( ) n k (( ) ( )) k e1 ek 0 0 e0 L n n k = A] 0 0 e 0 e 0 e 0 [L n n k Rn ], where A contains all the summands we added too much. Thus all the summands of A are of the form ( ) s ( ) t 0 e0 α v 1 v n s r e 0 e 0 {( ) ( )} e1 en for some integer α, vectors v i,..., and powers 1 t k, s n. By lemma (b) and (c) such a summand applied to [L n n k Rn ] is zero if s < k and only those summands remain in A that have t 1,s k. Let ( 0 S := α v 1 v n s r e 0 ) s ( e0 e 0 ) t be one of the remaining summands. By lemma (a) we obtain the equation = = = α v 1 v n s t ( ) 0 s ( e 0 e0 ) t e 0 [L n ( n k R n ] t ( t j) α v1 v n s t ( ) ) 0 s ( e 0 e0 ) j ( e 0 e0 ) t j 0 [L n n k Rn ] j=0 ( α v 1 v n s t ( ) 0 s (( e 0 e0 ) ( 0 + e0 )) ) t e 0 [L n n k Rn ] [ (( 0 ) ( e 0 + e0 )) k e ( 0 α v 1 v n s t ( ) 0 s k (( e 0 e0 ) ( 0 + e0 )) ) ] t e 0 B S [L n n k Rn ], where B S contains again all the summands we added too much. Thus all the summands of B S are of the form ( ) ( ) s s ( ) s ( ) t ( ) t t t 0 S e0 e0 e0 := β v 1 v n s t j e 0 e 0 0 e 0 for some integer β and powers 1 s k, 0 t t. If s s < k we group all corresponding summands together as ( ) s s ( ) s (( ) ( )) t 0 e0 e0 e0 β v 1 v n s t +. e 0 e 0 0 e 0 62

73 Chapter 3: Tropical intersection products on smooth varieties This product applied to [L n n k Rn ] is zero by lemma (b) and (c). Moreover, all summands S with s s k and t > 0 yield zero on [L n n k Rn ] by lemma (a). Thus only those summands S are left in B S that are of the form ( ) s s ( ) t+s 0 S = β e0 v 1 v n s t e 0 e 0 with s s k and s 1. Applying this process inductively to all summands with (( ) ( )) k 0 e0 t = 1,...,n k 1 in which we could not factor out +, yet, we e 0 e ( ) 0 e0 can by and by increase the power of in all remaining summands until finally e 0 only one summand ( ) k ( ) n k 0 e0 γ e 0 e 0 is left. But ( ) k ( ) n k 0 e0 γ [L e 0 e n n k Rn ] (( ) ( 0 )) k (( ) ( )) n k 0 e0 e0 e0 = γ + + [L e 0 e 0 0 e n n k Rn ] 0 as ( ) i ( ) k i (( ) ( )) n k 0 e0 e0 e0 + [L e 0 e 0 0 e n n k Rn ] ( ) 0 i ( ) k i 0 e0 = [L e 0 e n 0 R n ] 0 = 0 for all i < k by lemma (b) and ( ) k ( ) j ( 0 e0 e0 e 0 0 e 0 for all j > 0 by lemma (a). This proves the claim. ) n k j [L n n k R n ] = 0 Corollary The Cartier divisors h i,j from theorem provide the following description of the diagonal L n n k : r L n n k = h i,1...h i,n k [L n n k L n n k]. i=1 Proof. Let x 1,...,x n be the coordinates of the first and y 1,...,y n be coordinates of the second factor of R n R n. Applying lemma we can conclude that [ (( ) ( )) k (( ) ( )) k ] 0 e0 e0 e0 + + Fn e 0 e 0 0 e n 0 = [ ] max{0,x 1,...,x n } k max{0,y 1,...,y n } k Fn n = [L n n k Ln n k ] 63

74 3.1 Intersection products on tropical linear spaces and hence by theorem and theorem that r h i,1...h i,n k [L n n k L n n k] = L n n k. i=1 Remark As lemma does not only hold on L n n k Rn but also on any C R n with C a subcycle of L n n k, the proof of theorem indeed shows that ( r ) (( ) ( )) k 0 e0 h i,1...h i,n k + (C R i=1 e 0 e n ) 0 (( ) ( )) (( ) ( )) e1 0 en 0 = (C R 0 e 0 0 e n ) 0 for all cycles C Z l (L n n k ). Using corollary we can conclude that ( r ) (( ) ( )) k 0 e0 h i,1...h i,n k + (C R i=1 e 0 e n ) 0 = R n (C R n ) = C for all such cycles C. Corollary Let σ L n n k, let x σ and let U S σ = σ L n n k :σ σ (σ ) ri be an open subset of L n n k containing x. Moreover, let F be the open fan F := { x+σ U σ Ln n k } and F the associated tropical fan. Then there are Cartier divisors h i,j on F F such that [ F] = r h i,1...h i,n k [ F F]. i=1 Proof. To obtain the Cartier divisors h i,j we just have to restrict the Cartier divisors h i,j from corollary to the open set U U, translate them suitably and extend them from F F to the associated tropical fan F F. Example The following figure shows two fans associated to open subsets of L 3 2 as in corollary : 64

75 Chapter 3: Tropical intersection products on smooth varieties Lemma Let C Z k (R n ) and D Z l (R n ) be tropical cycles such that there exist representations of the diagonals C and D as sums of products of Cartier divisors on C C and D D, respectively. Then there also exists a representation of C D as a sum of products of Cartier divisors on (C D) 2. Proof. Let C = r i=1 ϕ i,1...ϕ i,k (C C) and D = s i=1 ψ i,1...ψ i,l (D D). Moreover, let π x,π y : (R n ) 4 (R n ) 2 be given by (x 1,y 1,x 2,y 2 ) (x 1,x 2 ) and (x 1,y 1,x 2,y 2 ) (y 1,y 2 ), respectively. Then we have the equation ( r ) ( s ) C D = πxϕ i,1...πxϕ i,k πyψ i,1...πyψ i,l (C D) 2. i=1 i=1 Now we are ready to define intersection products on all spaces on which we can express the diagonal by means of Cartier divisors: Definition (Intersection products) Let C Z k (R n ) be a tropical cycle and assume that there are Cartier divisors ϕ i,j on C C such that r C = ϕ i,1...ϕ i,k (C C). i=1 Moreover, let π : C C C be the morphism given by (x,y) x. Then we define the intersection product of subcycles of C by Z k l (C) Z k l (C) Z k l l (C) (D 1,D 2 ) D 1 D 2 := π ( r i=1 ϕ i,1...ϕ i,k (D 1 D 2 )). We use the rest of this section to prove that this intersection product is independent of the used representation of the diagonal and that it has all the properties we expect at least for those spaces we are interested in: Lemma Let C Z k (R n ) be a tropical cycle, D Z k l (C), E Z k l (C) be subcycles, let ϕ Div(C) be a Cartier divisor and π : C C C the morphism given by (x,y) x. Then the following equation holds: (ϕ D) E = π ϕ (D E). Proof. The proof is exactly the same as for lemma Corollary Let C Z k (R n ) be a tropical cycle that admits an intersection product as in definition , let D Z k l (C), E Z k l (C) be subcycles and let ϕ Div(C) be a Cartier divisor. Then the following equation holds: (ϕ D) E = ϕ (D E). 65

76 3.1 Intersection products on tropical linear spaces Proof. The proof is exactly the same as for lemma Proposition Let D Z l (L n n k ) be a subcycle. Then the equation holds on L n n k. D [L n n k] = [L n n k] D = D Proof. Let π i : L n n k Ln n k Ln n k be the morphism given by (x 1,x 2 ) x i. By remark we get the equation ( D [L n n k ] = (π r 1) h i,1...h i,n k (D [L n n k ( ])) i=1 ( r ) (( ) ( )) k 0 e0 = (π 1 ) h i,1...h i,n k + (D R )) i=1 e 0 e n 0 = (π 1 ) ( R n (D R n )) = (π 1 ) ( D ) = D. Furthermore, let φ : L n k Ln k Ln k Ln k be the morphism induced by (x,y) (y,x). Obviously we have the equality ( r ) ( r ) h i,1...h i,n k [L n n k L n n k] = φ h i,1...φ h i,n k [L n n k L n n k]. i=1 If π ij : (L n n k )4 (L n n k )2 is the morphism given by (x 1,x 2,x 3,x 4 ) (x i,x j ) and ( r ) ( r ) := π13h i,1...π13h i,n k π24h i,1...π24h i,n k i=1 we can conclude by proposition and lemma that ( r ) φ h i,1...φ h i,n k (D [L n n k ( ]) i=1 r ) ) = φ h i,1...φ h i,n k ((D [L nn k ( ]) ([Lnn k Lnn k ]) i=1 r ) ( = φ h i,1...φ h i,n k (π 12 ) ((D ) [L n n k ]) ([Ln n k Ln n k ( ])) i=1 r ) ) = φ h i,1...φ h i,n k (π 12 ) ( D [L n n k ] ( i=1 r ) ) = φ h i,1...φ h i,n k (π 34 ) ( D [L n n k ] i=1 (( r ) = (π 34 ) π34φ h i,1...π34φ h i,n k ((D [L n n k ]) ([Ln n k Ln n k ( ]))) i=1 (( r ) )) = (π 34 ) (D [L n n k ]) φ h i,1...φ h i,n k [L n n k Ln n k ( (( ] i=1 r ) )) = (π 34 ) (D [L n n k ]) h i,1...h i,n k [L n n k Ln n k ( ] i=1 r ) = h i,1...h i,n k (D [L n n k ]). i=1 i=1 i=1 66

77 Chapter 3: Tropical intersection products on smooth varieties Hence we can deduce that This proves the claim. D [L n n k ] = (π 1) ( D ) = (π 2 ) (( ( D ) r ) ) = (π 2 ) h i,1...h i,n k (D [L n n k (( ]) i=1 r ) ) = (π 2 ) φ h i,1...φ h i,n k (D [L n n k (( ]) i=1 r ) ) = (π 1 ) h i,1...h i,n k ([L n n k ] D) i=1 = [L n n k ] D. Remark We can prove in the same way that [L n n k Lm m l ] D = D holds for all subcycles D of L n n k Lm m l and even that [Ln 1 n 1 k 1... L nr n r k r ] D = D holds for all r 1 and all subcycles D of L n 1 n 1 k 1... L nr n r k r. Moreover, restricting the intersection products to open subsets of L n k or Ln 1 n 1 k 1... L nr n r k r, respectively, implies that X D = D also holds for all subcycles D Z l (X) if X {[ F], [ F 1... F r ]} where F, Fi are tropical fans associated to an open subsets of some L n k like in corollary Proposition Let C Z k (R n ) be a tropical cycle that admits an intersection product as in definition and let D,D Z l (C), E Z l (C) be subcycles. Then the following equation holds: (D + D ) E = D E + D E. Proof. The proof is exactly the same as for theorem Proposition Let C Z k (R n ) be a tropical cycle that admits an intersection product as in definition and let D Z l (C) be a subcycle of C. Moreover, let E Z l (C) be a subcycle such that there are Cartier divisors ψ i,j Div(C) with r ψ i,1...ψ i,k l C = E. i=1 If additionally C D = D holds then r ψ i,1...ψ i,k l D = E D. i=1 Proof. The proof is the same as for corollary Remark The meaning of proposition is the following: If X Z k (R n ) is a tropical cycle 67

78 3.1 Intersection products on tropical linear spaces such that the diagonal X can be written as a sum of products of Cartier divisors as in definition and additionally (X X) Y = Y is fulfilled for all subcycles Y of X X then we can apply proposition with C := X X and E := X to deduce that the definition of the intersection product is independent of the choice of the Cartier divisors describing the diagonal. In particular we have well-defined intersection products on L n k, Ln 1 k 1... L nr k r, F and F 1... F r for all tropical fans F, F i associated to an open subset of some L n k like in corollary Theorem Let C Z k (R n ) be a tropical cycle that admits an intersection product as in definition such that additionally (C C) D = D is fulfilled for all subcycles D of C C. Moreover, let E,E Z l (C), F Z l (C) and G Z l (C) be subcycles. Then the following equations hold: (a) E F = F E, (b) (E F) G = E (F G). Proof. The proof is exactly the same as for theorem (a) and (c). We finish this section with an example showing that even curves intersecting in the expected dimension can have negative intersections: Example Let C,D Z 1 (L 3 2) be the curves shown in the figure. We want to compute the intersection C D. By proposition the easiest way to achieve this is to write one of the curves as ψ [L 3 2] for some Cartier divisor ψ on L D C Let F be the refinement of L 3 2 arising by dividing the cones e 1, e 2 R 0 and e 0, e 3 R 0 into cones e 1, e 1 e 2 R 0, e 2, e 1 e 2 R 0 and e 0, e 0 e 3 R 0, e 3, e 0 e 3 R 0, respectively. Then 1 1 ψ :=

79 Chapter 3: Tropical intersection products on smooth varieties defines a rational function on F. As shown in example we have ψ [L 3 2] = C. Hence we can calculate C D = ψ D = ψ 3 + ψ 2 + ψ 1 ψ 0 {0} = ( ) {0} = 1 {0}. 3.2 Intersection products on smooth tropical varieties In this section we use our results from section 3.1 to define an intersection product on smooth tropical varieties, i.e. on varieties with tropical linear spaces as local building blocks: Definition (Smooth tropical varieties) An abstract tropical variety C is called a smooth variety if it has a representative (((X, X ),ω X ), {Φ σ }) such that all the maps Φ σ : S σ = σ X,σ σ (σ ) ri F σ F σ (cf. definition 1.4.4) map into tropical fans F σ = F 1 σ... F r σ σ where the F i σ tropical fans associated to open subsets of some L n σ,i k σ,i as in corollary are Remark Note that the existence of such a representative (((X, X ),ω X ), {Φ σ }) for C implies that all representatives of C have the requested property. Example The following figures show two examples of smooth tropical varieties: Definition Let C be an abstract tropical cycle, D a subcycle of C with representative X and U C an open subset. We denote by X U the open tropical polyhedral complex X U := ({σ U σ X}, X U) 69

80 3.2 Intersection products on smooth tropical varieties and by [X U] its equivalence class modulo refinements. As this class only depends on the class of X we can define D U := [X U]. Remark If we are given an open covering {U 1,...,U r } of C and open tropical polyhedral complexes D 1 U 1,...,D r U r such that D i U i U j = D j U i U j we can glue D 1 U 1,...,D r U r to obtain a cycle D Z (C). Definition (Intersection products) Let C be a smooth tropical variety and let (((X, X ),ω X ), {Φ σ }) be a representative of C like in definition Moreover, let D,E be subcycles of C. We construct local intersection products as follows: For every σ X we can regard (D S σ ) and (E S σ ) as open tropical cycles in F σ via the map Φ σ. Let D S σ and Ẽ S σ be any tropical cycles in F σ restricting to D S σ and E S σ. As we have an intersection product on F σ by remark we can define the intersection ( ) (D σ E) S σ := ( D S σ ) (Ẽ S σ ) S σ. Note that (D σ E) S σ does not depend on the choice of the cycles D S σ and Ẽ S σ. Since {S σ σ X} is an open covering of C and the local intersection products (D σ E) S σ, σ X are compatible by the following lemma we can glue them to obtain a global intersection cycle D E Z (C). Lemma For the local intersection products in definition holds: (D σ E) S σ S σ = (D σ E) S σ S σ. Proof. By definition we have an integer linear map F 1 Φ σ (S σ S σ ) f Φ σ (S σ S σ ) F 2 with integer linear inverse f 1, where F 1, F 2 are the tropical fans generated by Φ σ (S σ S σ ) and Φ σ (S σ S σ ), respectively. Let C 1,C 2 be subcycles of F 1. We have to show that C 1 C 2 = (f 1 ) (f (C 1 ) f (C 2 )). If π is the respective projection on the first factor we obtain by proposition and remark the equation ( )) (f 1 ) (f (C 1 ) f (C 2 )) = (f 1 ) π (f ( F2 (C 1 ) f (C 2 )) ( )) = π (f 1 f 1 ) (f ( F2 (C 1 ) f (C 2 )) ( )) = π (f 1 f 1 ) ((f f) ) (f f) ( F1 (C 1 C 2 ) ( ) = π C F1 1 C 2 = C 1 C 2. 70

81 Chapter 3: Tropical intersection products on smooth varieties Remark Lemma also implies that further refinements of the representative (((X, X ),ω X ), {Φ σ }) of C do not change the result D E. Hence the intersection product is well-defined. Our last step consists in proving basic properties of our intersection product: Theorem Let C be a smooth tropical variety, let D,D Z l (C), E Z l (C) and F Z l (C) be subcycles and let ϕ Div(C) be a Cartier divisor on C. Then the following equations hold in Z (C): (a) C D = D, (b) D E = E D, (c) (D + D ) E = D E + D E, (d) (D E) F = D (E F), (e) ϕ (D E) = (ϕ D) E. If moreover D = ( r i=1 ϕ i,1 ϕ i,l ) C for some Cartier divisors ϕ i,j Div(C) then D E = r ϕ i,1 ϕ i,l E i=1 holds. Proof. The statements follow immediately from the definition of the intersection product and the corresponding statements in section Pull-backs of cycles on smooth varieties We will now use the intersection product defined in section 3.2 to introduce pull-backs of tropical cycles along morphisms between smooth tropical varieties. Definition (Pull-back) Let X and Y be smooth tropical varieties of dimension m and n, respectively, and let f : X Y be a morphism of tropical cycles. Moreover, let π : X Y X be the projection onto the first factor and let γ f : X X Y be the morphism given by x (x,f(x)). We denote by Γ f := (γ f ) X the graph of f. For a cycle C Z n k (Y ) we define its pull-back f C Z m k (X) to be f C := π (Γ f (X C)). The easiest non-trivial, but nevertheless important example of a pull-back is the following: 71

82 3.3 Pull-backs of cycles on smooth varieties Example Let C and D be smooth tropical cycles and let p : C D D be the projection on the second factor. We want to calculate the pull-back p E for a cycle E Z k (D): The map γ p from definition is then just given by γ p : C D C D D : (x,y) (x,y,y) and the map π : C D D C D is the projection to the first two factors. Hence we can conclude that Γ p = C D. Moreover, let π 1 : C D D C be the projection to the first and π 2 : C D D D be the projection to the second factor. We obtain by definition 3.3.1: p E = π (Γ p (C D E)) = π ((C D ) (C D E)) = π 1 (C C) π 2 ( D (D E)) = C E. The pull-back has the following basic properties: Theorem Let X,Y and Z be smooth tropical varieties and let f : X Y and g : Y Z be morphisms of tropical cycles. Moreover, let C,C Z (Y ) and D Z (X) be subcycles. Then the following holds: (a) f Y = X, (b) id Y C = C, (c) if C = ϕ 1 ϕ r Y then f C = f ϕ 1 f ϕ r X, (d) C f D = f (f C D), (e) (g f) C = f g C, (f) f (C C ) = f C f C. Proof. Throughout the proof, let π X,π X,π 1,π 1,π Y,π Y,π 2,π 2,π X,Y,π X,Y,π 1,2,π 1,2 and so forth be the projections to the respective factors. (a) and (b): By definition of the pull-back follows and (c): We have f Y = π X (Γ f (X Y )) = π X (Γ f ) = X id Y C = π 1 (Γ idy (Y C)) = π 1 ( Y (Y C)) = Y C = C. f C = π X (Γ f (X (ϕ 1 ϕ r Y ))) = π X (π 2ϕ 1 π 2ϕ r Γ f (X Y )) = π X (π 2ϕ 1 π 2ϕ r Γ f ). By definition of the intersection product (see definitions and 1.5.5) this last line is equal to f ϕ 1 f ϕ r X. 72

83 Chapter 3: Tropical intersection products on smooth varieties (d): Let π X : X Y X be the projection on X. By example we know that π X D = D Y. As the diagonal X can locally be expressed by Cartier divisors we can apply proposition and statement (c) locally to deduce that for all subcycles E of X Y holds This implies that D π X E = π ( 1 X (D π X E)) = π ( 1 X (id π X ) (D E)) = π ((id 1 π X ) ((id π X ) X (D E))) = π ((id 1 π X ) (( X Y ) (D E))) = π 1 (π 1,2 = π 1 (π 1,2 = π 1 ((D Y ) E) = π X (π X D E). (( X Y ) (D E))) ( X Y (D Y E))) f C D = D π X (Γ f (X C)) = π X (π X D Γ f (X C)) = π X ((D Y ) Γ f (X C)) = π X (Γ f (D C)). Moreover, it is easy to check that (f id) Y = Γ f. As above we can conclude that C f D = π 1 ( Y (C f D)) = π 1 ((id f) ((id f) Y (C D))) = f (π X ((id f) Y (C D))) = f (π X ((f id) Y (D C))) = f (π X (Γ f (D C))) = f (f C D). (e): Let Φ : X X Y Z be given by x (x,f(x),g(f(x))). An easy calculation shows that (Γ f Z) (X Γ g ) = Φ X. Hence we can conclude by statement (d) that f g C = π X = π X ( Γf (X π Y (Γ g (Y C)) )) ( π X,Y ((Γ f Z) (X Γ g ) (X Y C)) ) = π X ((Γ f Z) (X Γ g ) (X Y C)) = π X (Φ X (X Y C)) = π X (Γ g f (X C)) = (g f) C. (f): Let Φ : X X Y Y be given by x (x,f(x),f(x)) and let π 1,2,π 1,3 : X Y Y X Y be the projections to the respective factors. An easy calculation shows that (Γ f Y ) (X Γ idy ) = Φ X = π 1,2Γ f π 1,3Γ f. 73

84 3.3 Pull-backs of cycles on smooth varieties Hence we can deduce that f (C C ) = π X (Γ f (X (C C ))) = π X (Γ f (X π (Γ 1 idy C C ))) = π X (Γ f π 1,2 ((X Γ idy ) (X C C ))) = π X (π 1,2 ((Γ f Y ) (X Γ idy ) (X C C ))) = π X (π ( 1,3 ((Γ f Y ) (X Γ idy ) (X C C ))) = π X π 1,3 (π1,2γ f π1,3γ f (X C C )) ) = π X (Γ ( f π 1,3 ((Γ f Y ) (X C C ))) = π X Γf (π X (Γ f (X C)) C ) ) = π X (Γ f (f C C )) = f C f C. We finish the section with another important example: Example Let D be a smooth tropical variety and let C Z k (D) be a smooth tropical subvariety. Moreover, let ι : C D be the inclusion map. We want to calculate the pull-back ι E for a cycle E Z l (D): Let π C : C D C and π D : C D D be the projections to the first and second factor and let γ ι : C C D be given by x (x,x). Hence we can deduce that Γ ι = (γ ι ) C = C and by example that (π D ) E = C E. Thus we can conclude by theorem (d): ι E = π C (Γ ι (C E)) = π C ( C (C E)) = π D ( C (C E)) = π D ( C (π D ) E) = π D ( C ) E = C E, where C E is the intersection product on D. 74

85 4 Weil and Cartier divisors under tropical modifications In this chapter we study contractions (the concept of contractions was introduced by G. Mikhalkin in [M06]) and modifications the inverse operation of contractions of affine tropical cycles. We will prove that under some further assumptions these modifications preserve the 1:1-correspondence of Weil and Cartier divisors that exists, for example, on R n. In particular we can conclude that on the moduli spaces M 0,n,trop of n-marked abstract tropical curves Weil and Cartier divisors agree. Applying our results locally we can moreover prove that there is a 1:1-correspondence of Weil and Cartier divisors on smooth tropical varieties as introduced in chapter Modifications and contractions In this section we introduce modifications and contractions as our main objects of study. Definition (Modifications and contractions) Let C Zm aff (V ) be an m-cycle in V and h K (C), h : C R a rational function on C. Let (X,ω X ) be a representative of C such that h is linear on every face of X, i.e. for every face σ X we have h σ = h σ + c for some h σ Λ and c R. Let V := (Λ Z) Z R. Then we define the modification of X along h to be the tropical fan (Γ X,h,ω ΓX,h ) in V, where Γ X,h := {(id h σ )(σ) σ X} {(id h σ )(σ) + ({0 V } R 0 ) σ X \ X (m) }, ω ΓX,h ((id h σ )(σ)) := ω X (σ) for all σ X, ω ΓX,h ((id h σ )(σ) + ({0 V } R 0 )) := ω h (σ) for all σ X (m 1) (cf. construction 1.2.3). Conversely, X is called the contraction of Γ X,h along h. Note that the equivalence class [Γ X,h ] only depends on C and not on the choice of the representative X. Hence we can define the modification of C along h to be and call C the contraction of Γ C,h along h. Γ C,h := [Γ X,h ] Z aff m (V ) 75

86 4.1 Modifications and contractions D ω = 1 C ω = 1 h = 0 h = 1 h = 0 h = 1 h = 0 h = 0 ω = 1 ω = 1 Figure 4.1: D Z aff m 1(C), but Γ D,h Z aff m 1(Γ C,h ). Definition For C Z aff m (V ) we define a map γ : Z aff m 1(C) Z aff m 1(Γ C,h ) : D Γ D,h. Note that for some cycles C Z aff m (V ) this map is not well-defined as D Z aff m 1(C) does in general not imply that Γ D,h is a subcycle of Γ C,h (the simplest example of such cycles C and D is drawn in figure 4.1). But the following lemma shows that the latter implication in fact is true for the cases we are interested in: Lemma Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C) aff : ϕ ϕ C is an isomorphism. Then holds: (a) γ is well-defined, i.e. Γ D,h Z aff m 1(Γ C,h ) for all D Z aff m 1(C), (b) γ(d 1 + D 2 ) = γ(d 1 ) + γ(d 2 ). Proof. (a): We fix a Weil divisor D Z aff m 1(C) and have to prove that Γ D,h Γ C,h. Therefore it suffices to show that h D h C. By assumption we know that D = ϕ C for some Cartier divisor ϕ Div(C) and hence (b): Is obvious by definition of γ. h D = h ϕ C = ϕ h C h C. Definition Let C Zm aff (V ). The projection p : Λ Z Λ induces a morphism p : Γ C,h C and thus a homomorphism p : Zm 1(Γ aff C,h ) Zm 1(C). aff 76

87 Chapter 4: Weil and Cartier divisors under tropical modifications Remark Note that p γ = id Z aff m 1 (C) if Div(C) Z aff m 1(C). The next proposition shows how the intersection products on C and Γ C,h are related: Proposition Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C). aff Moreover let ϕ Div(C) be a Cartier divisor and let D Zm 1(C) aff be a Weil divisor. The following equation holds: ϕ D = p (p ϕ γ(d)). Proof. Using the projection formula (cf. proposition 1.3.8) we can conclude that ϕ D = ϕ (p γ)(d) = p (p ϕ γ(d)). Proposition Let C Zm aff (V ) be a cycle in V = Λ Z R and let D Zn aff (V ) be a cycle in V = Λ Z R. Moreover let h C K (C) and h D K (D) be rational functions. For simplicity of notation we write h C and h D for the pull-backs of h C and h D along the occurring projection maps as well. Then the following equation holds: Γ ΓC D,hC,h D = Γ C,hC Γ D,hD. Proof. An easy calculation shows that Γ C D,hC = Γ C,hC D. Hence Γ ΓC D,hC,h D = Γ C,hC D,h D = Γ C,hC Γ D,hD. Corollary Let V = Λ Z R and V = Λ Z R. Then let C Zm aff (V R k ) and D Zn aff (V R l ) be cycles that arise from C Zm aff (V ) and D Zn aff (V ) respectively by a finite series of modifications. Then C D Zm+n(V aff V R k+l ) arises from C D by a finite series of modifications. 4.2 Cartier divisors and Weil divisors Our aim in this section is to prove that a 1:1-correspondence between Weil divisors and Cartier divisors on an affine cycle C Zm aff (V ) implies a 1:1-correspondence between Weil divisors and Cartier divisors on many modifications Γ C,h of C. To prove this statement we need some preparations: Definition Let C Zm aff (V ) be a cycle. If C 0 we define the number lcm(c) to be the lowest common multiple lcm { ω X (σ) σ (X ) (m)} Z >0 of all non-zero weights of the facets of X for some (and thus for every) representative (X,ω X ) of C. If C = 0 we define lcm(c) := 0. 77

88 4.2 Cartier divisors and Weil divisors Definition Let C Zm aff (V ) be a cycle. C is called locally irreducible if for some (and thus for every) reduced representative (X,ω X ) of C holds: For every cone τ X (m 1) the equality λ σ u σ/τ = 0 Λ/Λ τ, λ σ Z σ>τ (where u σ/τ denotes the primitive normal vector of σ relative to τ) implies that there exists λ Q such that λ σ = λ ω X (σ) for all σ > τ. Example The following figure shows an example of a tropical cycle C which is not locally irreducible as the condition in definition is not fulfilled around τ. But note that the condition is fulfilled for all other cones of codimension one. τ Figure 4.2: A cycle C which is not locally irreducible. Lemma Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C), aff let h K (C) be a rational function and let D ker(p ) Zm 1(Γ aff C,h ). If m 2 or Γ C,h is locally irreducible then there exists a Cartier divisor ϕ D Div(Γ C,h ) such that ϕ D Γ C,h = lcm(γ C,h ) D. Proof. Note that for m = 1 the statement is trivial. Thus we may assume that m 2. Moreover we can choose representatives X = X of C and Γ Y = Γ Y of D such that p (Γ Y ) = {p(σ) σ Γ Y contained in a maximal cone of Γ Y on which p is injective} and such that Y := p (Γ Y ) X (cf. constructions and 1.3.2). Using these representatives we ensure that above every cone σ Y (m 1) there is a finite number of facets σ 1,...,σ r(σ) (σ) := (id h σ )(σ) + ({0 V } R 0 ) projecting injectively onto σ as shown in figure 4.3, plus possibly some additional facets in the boundary of (σ) on which p is not injective. Our assumption D ker(p ) implies that Γ Y σ X (m 1) (id h σ )(σ) + ({0 V } R 0 ) and thus that, in fact, every facet of Γ Y is of one of those two types mentioned above. Now we construct a piecewise linear function ϕ : Γ Y R as follows: First, we set 78

89 Chapter 4: Weil and Cartier divisors under tropical modifications a (r) 1 e m a (1) 1 a (1) 2 (σ) 0 σ 1 ϕ 0 v 1 a (r) 2 (σ) 1 ϕ 1 v r σ r ϕ r (σ) r σ p Figure 4.3: σ 1,...,σ r (id h σ )(σ) + ({0 V } R 0 ). ϕ (id hσ)(σ) := 0 for all σ X (m) and ϕ (σ) := 0 for all σ X (m 1) \ Y (m 1). Then we fix a cone σ Y (m 1). Choosing a basis b 1,...,b m 1 of Λ σ we obtain a basis b 1,...,b m 1,b m := (0 Λ, 1) of Λ (σ) = Λ σ Z and can hence consider the face (σ) in R m = V (σ) with lattice Zm (see figure 4.3): Let σ 1,...,σ r, r = r(σ), be all facets of Γ Y contained in (σ), let σ 0 := (id h σ )(σ) and let (σ) i denote the m-dimensional cone in (σ) bounded by σ i and σ i+1, respectively the cone below σ r for i = r (note that (σ) 0 might be (m 1)-dimensional if σ 0 = σ 1 ). We will construct a linear function ϕ i on every (σ) i in order to define a piecewise linear function on (σ) as follows: We set ϕ 0 := 0. Then ϕ 1 is determined on σ 1. In order to get ω ϕ (σ 1 ) = lcm(γ C,h ) ω ΓY (σ 1 ) we have to set ϕ 1 (v 1 ) := χ σ ω ΓY (σ 1 ) with χ σ := lcm(γ C,h) ω ΓC,h ( (σ)) and ϕ 1 is entirely determined then. Now ϕ 2 is determined on σ 2. In order to get ω ϕ (σ 2 ) = lcm(γ C,h ) ω ΓY (σ 2 ) we have to set ϕ 2 (v 2 ) := χ σ ω ΓY (σ 2 ) + ϕ 1 (v 2 ) and ϕ 2 is entirely determined. Going on this way we set ϕ i (v i ) := χ σ ω ΓY (σ i ) + ϕ i 1 (v i ) for all i and obtain the function ϕ (σ). Using the same procedure we can construct functions ϕ (σ) for all σ Y (m 1) and it remains to check that all our pieces ϕ (σ), σ X (m 1) glue together to obtain a globally defined rational function ϕ on C. Therefore, for all i = 1,...,r we choose a basis a (i) 1,...,a (i) m 1 of Λ σi Z m and a representative v i Z m of the primitive normal vector of (σ) i relative to σ i. We set ( det a (i) 1 a (i) m 1 e m ) =: αi Z >0, where e j denotes the j-th unit vector in R m = V (σ). Using our assumption D ker(p ) we can conclude that 0 = ω Y (σ) = r ω ΓY (σ i ) α i. ( ) i=1 79

90 4.2 Cartier divisors and Weil divisors As a (i) 1,...,a (i) m 1, e m is a basis of R m we can express v i by a linear combination v i = c (i) 1 a (i) c (i) m 1a (i) m 1 c (i) m e m. It is easily checked that c (i) m = 1 α i. We can use the above linear combination to calculate ϕ i ( e m ): ϕ 1 (v 1 ) = χ σ ω ΓY (σ 1 ) = c (1) 1 ϕ 1 (a (1) 1 ) c (1) m 1ϕ 1 (a (1) m 1) + 1 α 1 ϕ 1 ( e m ) = 1 α 1 ϕ 1 ( e m ) implies ϕ 1 ( e m ) = χ σ ω ΓY (σ 1 )α 1. Similarly, ϕ 2 (v 2 ) = χ σ ω ΓY (σ 2 ) + ϕ 1 (v 2 ) = χ σ ω ΓY (σ 2 ) + c (2) 1 ϕ 1 (a (2) 1 ) c (2) m 1ϕ 1 (a (2) m 1) + 1 α 2 ϕ 1 ( e m ) = c (2) 1 ϕ 2 (a (2) 1 ) c (2) m 1ϕ 2 (a (2) m 1) + 1 α 2 ϕ 2 ( e m ) implies χ σ ω ΓY (σ 2 ) + 1 α 2 ϕ 1 ( e m ) = 1 α 2 ϕ 2 ( e m ) and hence Inductively we obtain and hence using ( ): ϕ 2 ( e m ) = χ σ (ω ΓY (σ 2 )α 2 + ω ΓY (σ 1 )α 1 ). ϕ i ( e m ) = χ σ ϕ r ( e m ) = χ σ i ω ΓY (σ j )α j j=1 r ω ΓY (σ j )α j = 0. j=1 Thus all patches ϕ (σ), σ X (m 1) fit together on the ray {0 V } + ({0 V } R 0 ) and we are done in the case m = 2. Now let m 3. To see that the patches ϕ (σ) glue together everywhere we have to study the consequences of the balancing condition of Γ Y : Let τ Γ (m 1) X,h be a common face of the cones (σ (1) ),..., (σ (s) ). For all i we may choose a representative v i of the primitive normal vector of (σ (i) ) relative to τ such that the equation s ω ΓX,h ( (σ (i) )) v i = 0 Λ i=1 holds (cf. remark 1.1.9). Like above, for i = 1,...,s let σ (i) 1,...,σ r (i) i be all facets of Γ Y contained in (σ (i) ) on which p is injective and which meet in a common face τ 1 τ of dimension m 2 (see figure 4.4). Then let v (i) j be a representative of the primitive ( ) 80

91 Chapter 4: Weil and Cartier divisors under tropical modifications τ τ 1 κ τ 2 τ 3 τr Figure 4.4: The common face τ of (σ (1) ),..., (σ (s) ). normal vector of σ (i) j relative to τ 1. The balancing condition of Γ Y implies that s r i ω ΓY (σ (i) j ) [v (i) j ] = 0 Λ /Λ τ. ( ) i=1 j=1 Choosing a basis u 1,...,u m 2 of Λ τ 1 and extending it by a vector u m 1 to a basis of Λ τ we can express the vectors v (i) j by Z-linear combinations v (i) j = c (i) 1 u c (i) m 1u m 1 + t (i) j v i. Note that the orientations of v i and v (i) j implies i=1 j=1 imply that t (i) j s r i ω ΓY (σ (i) j ) t (i) j [v i] = 0 Λ /Λ τ. > 0. Equation ( ) then As Γ C,h is locally irreducible we can conclude that there exists some λ Q such that r i j=1 ω ΓY (σ (i) j ) t (i) j = λ ω ΓX,h ( (σ (i) )) for all i. Now we fix i {1,...,s} and choose a basis b 1,...,b m 2,b m 1 := v i,b m := (0 Λ, 1) of Λ (σ (i) ) = Z m like above. Note that in general we cannot simply take again the vectors u 1,...,u m 2 we have chosen above because in general u 1,...,u m 2,b m is not a basis of Λ τ. Nevertheless, on the other hand, we can express the vectors v (i) j by an R-linear combination v (i) j = d (i) 1 u d (i) m 2u m 2 + t (i) j v i + d (i) m b m 81

92 4.2 Cartier divisors and Weil divisors and can hence calculate using ( ): (ϕ (σ (i) )) κ ( e m ) = χ (i) σ = χ (i) σ = χ (i) σ r i j=1 r i j=1 r i j=1 ω ΓY (σ (i) j ) ( det ũ 1 ũ m 2 ṽ (i) j e m ) ω ΓY (σ (i) j )t (i) j det ( ũ 1 ũ m 2 e m 1 e m ) }{{} =:B ω ΓY (σ (i) j )t (i) j B (here x denotes the representation of x in the basis b 1,...,b m ). Note that B is independent of i. Thus we obtain: r i (ϕ (σ )) (i) κ ( e m ) = χ σ (i) ω ΓY (σ (i) j )t (i) j B = λb lcm(γ C,h) j=1 for all i. This proves that the pieces ϕ (σ) of our map ϕ agree on common faces of codimension one. Moreover, the argument shows that if one of the cones (σ (i) ) does not contain any facet σ (i) j with non-zero weight on which p is injective, then ϕ (σ ) (i) τ = 0 for all i. It remains to check that our pieces of ϕ (σ) glue together on cones of higher codimension as well: Therefore let τ < (σ 1 ), (σ 2 ) be a common face of (σ 1 ) and (σ 2 ), σ 1,σ 2 X (m 1). If σ 1,σ 2 Y (m 1) then ϕ (σ1 ) = 0 = ϕ (σ2 ) and we are done. Hence let σ 1 Y (m 1). We have to distinguish between two cases: If all facets (σ) with (σ) > τ contain facets σ j with non-zero weights on which p is injective then ϕ (σ1 ) τ = ϕ (σ2 ) τ by applying our above argument several times. Otherwise ϕ (σ1 ) τ = 0 = ϕ (σ2 ) τ anyway. Hence ϕ is in fact a rational function on C. The faces of ϕ Γ X,h with non-zero weights on which p is injective are by construction exactly the faces of lcm(γ C,h ) Γ Y on which p is injective and these faces occur with correct weights. As both ϕ Γ X,h and lcm(γ C,h ) Γ Y fulfill the balancing condition it follows that [ϕ Γ X,h ] = [lcm(γ C,h ) Γ Y ] = lcm(γ C,h ) D. Hence ϕ D := [ϕ] Div(Γ C,h ) is our wanted Cartier divisor with ϕ D Γ C,h = lcm(γ C,h ) D. Using this lemma we are able to prove the first part of the promised 1:1-correspondence between Weil divisors and Cartier divisors: Theorem Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C), aff let h K (C) be a rational function and let D Zm 1(Γ aff C,h ). If m 2 or Γ C,h is locally irreducible then there exists a Cartier divisor ϕ D Div(Γ C,h ) with ϕ D Γ C,h = lcm(γ C,h ) D. Proof. D Zm 1(Γ aff C,h ) implies that p D Zm 1(C). aff By assumption there exists a Cartier divisor ϕ D Div(C) with ϕ D C = p D. By proposition we know that p D = ϕ D C = p (p ϕ D γ(c)) = p (p ϕ D Γ C,h ). 82

93 Chapter 4: Weil and Cartier divisors under tropical modifications Hence D p ϕ D Γ C,h ker(p ) and by lemma there exists a Cartier divisor ϕ D Div(Γ C,h) with ϕ D Γ C,h = lcm(γ C,h ) (D p ϕ D Γ C,h). Thus we obtain (ϕ D + lcm(γ C,h ) p ϕ D) Γ C,h = lcm(γ C,h ) D. Now we prove the missing part of the 1:1-correspondence. Note that in contrast to the first part we need no special assumptions here. Theorem Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C), aff let h K (C) be a rational function and let Φ Div(Γ C,h ) with Φ Γ C,h = 0. Then Φ = 0 Div(Γ C,h ). Proof. Let (X,ω X ) be a reduced representative of C and let v 1,...,v k X be a basis of X V. For every cone σ X (m) and all τ < σ, τ X (m 1) we can choose representatives v σ,τ of the primitive normal vector of σ relative to τ such that ω X (σ)v σ,τ = 0 σ>τ by remark Moreover, for every cone τ X we choose a basis x τ 1,...,x τ d τ of Λ τ. For σ X (m) with v σ 1,...,v σ t σ σ, v σ j {v 1,...,v k }, we have linear relations f 1 ({v σ,τ τ < σ,τ X (m 1) } {v σ 1,...,v σ t σ } {x τ j τ σ}) = 0,. f sσ ({v σ,τ τ < σ,τ X (m 1) } {v σ 1,...,v σ t σ } {x τ j τ σ}) = 0. Our assumption Div(C) Zm 1(C) aff then implies that the following system of equations in the variables ψ σ (v σ,τ ), ψ σ (x τ j) and ψ σ (vj σ ) has exactly the solution 0: τ X (m 1) : σ>τ ω X(σ)ψ σ (v σ,τ ) = 0, σ 1,σ 2 X (m) : ψ σ1 (x τ j) = ψ σ2 (x τ j) τ < σ 1,σ 2, j, σ X (m) : f i ({ψ σ (v σ,τ ) τ < σ,τ X (m 1) } {ψ σ (v1),...,ψ σ σ (vt σ σ )} {ψ σ (x τ j) τ σ}) = 0 i, σ X (m) : ψ σ (vj σ ) = 0 j. Now we set ṽ i := (v i,h σ (v i )) for all i, we set x τ j := (x τ j,h τ (x τ j)) for all τ and j and ṽ σ,τ := (v σ,τ,h σ (v σ,τ )) for all τ X (m 1), σ X (m) with τ < σ. Moreover we set τ := (id h τ )(τ) and (τ) := τ +({0 V } R 0 ). Now we choose a representative ϕ K (X) of Φ with ϕ((0 V, 0)) = 0, ϕ(ṽ i ) = 0 for all i and additionally ϕ((0 V, 1)) = 0. Note that Φ Γ C,h = 0 implies that ϕ is linear on every face of Γ X,h. Then we consider the following system of equations: τ X (m 1) : ω ΓX,h ( (τ))ϕ (τ) ((0 V, 1)) + σ>τ ω X(σ)ϕ σ (ṽ σ,τ ) = 0, σ 1,σ 2 X (m) : ϕ σ1 ( x τ j) = ϕ σ2 ( x τ j) τ < σ 1,σ 2, j, σ X (m) : f i ({ϕ σ (ṽ σ,τ ) τ < σ,τ X (m 1) } {ϕ σ (ṽ σ 1),...,ϕ σ (ṽ σ t σ )} {ϕ σ ( x τ j) τ σ}) = 0 i, σ X (m) : ϕ σ (ṽ σ j ) = 0 j, τ X \ X (m) : ϕ (τ) ((0 V, 1)) = 0. 83

94 4.2 Cartier divisors and Weil divisors As this system is fulfilled by the values of our given function ϕ and since we can simplify this system to the one above we have the unique solution ϕ σ (ṽ σ,τ ) = ϕ σ ( x τ j) = ϕ σ (ṽ σ j ) = 0 for all σ, τ and j. Hence Φ = 0 Div(Γ C,h ). Let C Z aff m (V ) with Div(C) = Z aff m 1(C) like above. The preceding theorem enables us to define an intersection product of two cycles in a modification Γ C,h of C if one of them can be expressed by a Cartier divisor: Definition Let C Zm aff (V ) be a cycle such that Div(C) Zm 1(C) aff and let Γ C,h be a modification of C. Let D Zk aff(γ C,h) be any cycle and let E Zm 1(Γ aff C,h ) be a Weil divisor such that there exists ϕ E Div(Γ C,h ) with ϕ E Γ C,h = E. Then we define: E D := ϕ E D. Example The moduli space M 0,n,trop of n-marked abstract tropical curves embedded into the real vector space R (n 2) n as defined in [GKM07, section 3] arises from R n 3 by a finite series of modifications R n 3 = C 0 C 1 C 2 C ( n 2) 2n+3 = M 0,n,trop such that every C i is locally irreducible and lcm(c i ) = 1 for all i (a proof for this fact can be found in [H07, pages 44ff]). Hence we can conclude by theorems and that Div(M 0,n,trop ) = Zn 4(M aff 0,n,trop ) and we can define an intersection product of arbitrary cycles with Weil divisors as given in definition Example The tropical linear space L n k introduced in definition arises from Rk by a finite series of modifications R k L k+1 k L k+2 k L n k via maps max{0,x 1,...,x k },...,max{0,x 1,...,x n 1 }. Moreover, every L k+i k is locally irreducible and lcm(l k+i k ) = 1 for all i. Using the same maps this also holds for tropical fans F associated to open subsets of L n k as in corollary Using proposition we can extend this statement to all tropical fans F = F 1... F r, where all Fi are tropical fans associated to open subsets of L n k as in corollary Hence we can define an intersection product of arbitrary cycles with Weil divisors on these spaces as introduced in definition Moreover, we can deduce the following corollary. Corollary Let C Z n be a smooth tropical variety (cf. definition 3.2.1). Then the map is an isomorphism. Div(C) Z n 1 (C) : ϕ ϕ C 84

95 Chapter 4: Weil and Cartier divisors under tropical modifications Proof. Let (((X, X ),ω X ), {Φ σ }) be a representative of C as in definition and let D Z n 1 (C) be a cycle of codimension one. For all σ X we can regard D S σ as an open tropical cycle in F σ via the map Φ σ. Let D S σ be any tropical cycle in F σ restricting to D S σ. By example and theorem there is a Cartier divisor ϕ σ Div( F σ ) with ϕ σ F σ = D S σ. As all these Cartier divisors ϕ σ are unique by theorem they agree on overlaps S σ S σ and hence define a global Cartier divisor ϕ Div(C). Moreover, ϕ is locally unique and thus globally unique. Hence the claim follows. 85

96

97 5 Chern classes of tropical vector bundles In this chapter we introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in [T09] for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms. 5.1 Tropical vector bundles In this section we will introduce our basic objects such as tropical vector bundles, morphisms of tropical vector bundles and rational sections. Definition (Tropical matrices) A tropical matrix is an ordinary matrix with entries in the tropical semi-ring (T = R { },, ), where a b = max{a,b} and a b = a + b. We denote by Mat(m n, T) the set of tropical m n matrices. Let A Mat(m n, T) and B Mat(n p, T). We can form a tropical matrix product A B := (c ij ) Mat(m p, T) where c ij = m k=1 a ik b kj. Moreover, let G(r s) Mat(r s, T) be the subset of tropical matrices with at most one finite entry in every row. Let G(r) be the subset of G(r r) containing all tropical matrices with exactly one finite entry in every row and every column. Remark Note that a matrix A G(r s) does, in general, not induce a map f A : R s R r : x A x as the vector A x may contain entries that are. To obtain a map f A : R s R r anyway we use the following definition: Let x R s and A x = (y 1,...,y r ) T r with y i = for i I and y i R for i I. Then we define f A (x) := (ỹ 1,...,ỹ r ) R r with ỹ i := 0 for i I and ỹ i := y i for i I. Notation For an element σ of the symmetric group S r we denote by A σ the tropical matrix 87

98 5.1 Tropical vector bundles A σ = (a ij ) Mat(r r, T) given by { a ij := 0, if j = σ(i), else. Moreover, for a 1,...,a r R we denote by D(a 1,...,a r ) the tropical diagonal matrix D(a 1,...,a r ) = (d ij ) Mat(r r, T) given by { ai, if i = j d ij :=, else. Note that every element M G(r) can be written as M = A σ D(a 1,...,a r ) for some σ S r and some numbers a 1,...,a r R. Moreover, G(r) together with tropical matrix multiplication is a group with neutral element E := D(0,...,0). Lemma G(r) is precisely the set of invertible tropical matrices, i.e. G(r) = {A Mat(r r, T) A Mat(r r, T) : A A = A A = E}. Proof. The inclusion G(r) {A Mat(r r, T) A Mat(r r, T) : A A = A A = E} is obvious. Thus, let A,A Mat(r r, T) be given such that A A = A A = E. Assume that A = (a ij ) contains more than one finite entry in a row or column. For simplicity of notation we assume that a 11,a 12. As A A = E we can conclude that the first two rows of A look as follows: A = α... β... * for some α,β R. As moreover A A = E holds, we can conclude from the second line of A and the first column of A that a 11 + β =, which is a contradiction to a 11,β R. We have all requirements now to state our main definition: Definition (Tropical vector bundles) Let X be a tropical cycle. A tropical vector bundle over X of rank r is a tropical cycle F together with a morphism π : F X and a finite open covering {U 1,...,U s } of X as well as a homeomorphism Φ i : π 1 (U i ) = U i R r for every i {1,...,s} such that 88

99 Chapter 5: Chern classes of tropical vector bundles (a) for all i we obtain a commutative diagram Φ i π 1 (U i ) U i R r π p 1 U i where p 1 : U i R r U i is the projection to the first factor, (b) for all i,j the composition p (i) j Φ i : π 1 (U i ) R is a regular invertible function, where p (i) j : U i R r R : (x, (a 1,...,a r )) a j, (c) for every i,j {1,...,s} there exists a transition map M ij : U i U j G(r) such that Φ j Φ 1 i : (U i U j ) R r (U i U j ) R r is given by (x,a) (x,m ij (x) a) and the entries of M ij are regular invertible functions on U i U j or constantly, (d) there exist representatives F 0 of F and X 0 of X such that F 0 = {π 1 (τ) τ X 0 } and ω F0 (π 1 (τ)) = ω X0 (τ) for all maximal polyhedra τ X 0. An open set U i together with the map Φ i : π 1 (U i ) = U i R r is called a local trivialization of F. Tropical vector bundles of rank one are called tropical line bundles. Remark Let V 1,...,V t be any open covering of X. Then the covering {U i V j } together with the restricted homeomorphisms Φ i π 1 (U i V j ) and transition maps M ij (Ui V k ) (U j V l ) fulfills all requirements of definition 5.1.5, too, and hence defines again a vector bundle. As the open covering, the homeomorphisms and the transition maps are part of the data of definition this new bundle is (according to our definition) different from our initial one even though they are the same in some sense. Hence, in the following we will identify vector bundles that arise by such a construction one from the other: Definition Let π : F X together with open covering U 1,...,U s, homeomorphisms Φ i and transition maps M ij and π : F X together with open covering V 1,...,V t, homeomorphisms Ψ i and transition maps N ij be two tropical vector bundles according to definition We will identify these vector bundles if the vector bundles π : F X with open covering {U i V j } and restricted homeomorphisms Φ i π 1 (U i V j ) respectively Ψ j π 1 (U i V j ) and transition maps M ij (Ui V k ) (U j V l ) respectively N kl (Ui V k ) (U j V l ) are equal. Remark Let π 1 : F 1 X and π 2 : F 2 X be two vector bundles on X. By definition we can always assume that F 1 and F 2 satisfy definition with the same open covering. 89

100 5.1 Tropical vector bundles Remark Let π : F X be a vector bundle with open covering U 1,...,U s and transition maps M ij as in definition On the common intersection U i U j U k we obviously have M ij (x) = M kj (x) M ik (x). This last equation is called cocycle condition. Conversely, given an open covering U 1,...,U s of X and maps M ij : U i U j G(r) such that the entries of M ij (x) are regular invertible functions on U i U j or constantly and the cocycle condition M ij (x) = M kj (x) M ik (x) holds on U i U j U k, we can construct a vector bundle π : F X with this given open covering and transition functions M ij : Take the disjoint union s i=1 (U i R r ) and identify points (x,y) (x,m ij (x) a) to obtain the topological space F. We have to equip this space with the structure of a tropical cycle. As this construction is exactly the same as for tropical line bundles, we only sketch it here and refer to [T09] for more details. Let (((X 0, X 0, {ϕ σ }),ω X0 ), {Φ σ }) be a representative of X. We define F 0 := {π 1 (σ) σ X 0 } and ω F0 (π 1 (σ)) := ω X0 (σ) for all maximal polyhedra σ X 0. Our next step is to construct the polyhedral charts ϕ π 1 (σ) for F 0 : Let σ X 0 be given and let U i1,...,u it be all open sets with non-empty intersection with σ. Moreover, let {V i i I} be the set of all connected components of all σ U ik. Every such set V i comes from a set U j(i) of the given open covering. Hence, for every pair k,l I we have a restricted transition map N kl := M j(k),j(l) Vk V l. This implies that for all k,l I the entries of N kl Φ 1 σ are (globally) integer affine linear functions on V k V l. As σ is simply connected, for every such entry h O (V k V l ) of N kl there exists a unique continuation h O (σ). Hence we can extend all transition maps N kl : V k V l G(r) to maps N kl : σ G(r). Now we choose for every i I a point P i V i and for all pairs k,l I a path γ kl : [0, 1] σ from P k to P l. Let k,l I be given. As the image of γ kl is compact there exists a finite covering V µ1,...,v µc of γ kl ([0, 1]). For x V l we set S(γ kl )(x) := (N µ 1,µ 2 (x)) 1 (N µ c 1,µ c (x)) 1 G(r). Now fix some k 0 I. For all l I we define maps ϕ (l) π 1 (σ) : V l R r = π 1 (V l ) R nσ+r : (x,a) (ϕ σ (x),s(γ k0 l)(x) a). These maps agree on overlaps and hence glue together to an embedding ϕ π 1 (σ) : π 1 (σ) R nσ+r. In the same way we can construct the fan charts Φ π 1 (σ). Then we define F to be the equivalence class F := [ ] (((F 0, F 0, { ϕ π 1 (σ)}),ω F0 ), { Φ π 1 (σ)}). Example Throughout the chapter, the curve X := X 2 from example will serve us as a central example. Recall that X arises by gluing open fans as drawn in the figure: 90

101 Chapter 5: Chern classes of tropical vector bundles X Moreover, recall from definition that the transition functions between these open fans composing X are integer affine linear. This implies that the curve X has a welldefined lattice length L. We can cover X by open sets U 1,U 2,U 3 as drawn in the following figure: X = U 1 U 3 U 2 The easiest way to construct a (non-trivial) vector bundle of rank r on X is fixing a (non-trivial) transition map M 12 : U 1 U 2 G(r) and defining M 23 : U 2 U 3 G(r), M 31 : U 3 U 1 G(r) to be the trivial maps x E for all x. We will see later that in fact every vector bundle of rank r on X arises in this way. Knowing what tropical vector bundles are, there are a few notions related to this definition we want to introduce now: Definition (Direct sums of vector bundles) Let π 1 : F 1 X and π 2 : F 2 X be two vector bundles of rank r and r, respectively, with a common open covering U 1,...,U s and transition maps M (1) ij and M (2) ij, respectively, satisfying definition (see remark 5.1.8). We define the direct sum bundle π : F 1 F 2 X to be the vector bundle of rank r + r we obtain from the gluing data U 1,...,U s M (1) ij M (2) ij : U i U j G(r + r ) : x ( ij (x) M (2) ij (x) M (1) ). Definition (Subbundles) Let π : F X be a vector bundle with open covering U 1,...,U s and homeomorphisms Φ i according to definition A subcycle E Z l (F) is called a subbundle of rank r of F if π E : E X is a vector bundle of rank r such that we have for all i = 1,...,s: Φ i (π E ) 1 (U i ) : (π E ) 1 (U i ) = U i e j1,...,e jr R for some 1 j 1 <... < j r r, where the e j are the standard basis vectors in R r. 91

102 5.1 Tropical vector bundles Remark If π : F X is a vector bundle of rank r with subbundle E of rank r like in definition this implies that there exists another subbundle E of rank r r with Φ i (π E ) 1 (U i ) : (π E ) 1 (U i ) = U i e j j {j 1,...,j r } R and hence that F = E E holds. Definition (Decomposable bundles) Let π : F X be a vector bundle of rank r. We say that F is decomposable if there exists a subbundle π E : E X of F of rank 1 r < r. Otherwise we call F an indecomposable vector bundle. As announced in the very beginning of this section we also want to talk about morphisms and, in particular, isomorphisms of tropical vector bundles: Definition (Morphisms of vector bundles) A morphism of vector bundles π 1 : F 1 X of rank r and π 2 : F 2 X of rank r is a morphism Ψ : F 1 F 2 of tropical cycles such that (a) π 1 = π 2 Ψ and (b) there exist an open covering U 1,...,U s according to definition for both F 1 and F 2 (see remark 5.1.8) and maps A i : U i G(r r) for all i such that Φ F 2 i Ψ (Φ F 1 i ) 1 : U i R r U i R r is given by (x,a) (x,f Ai (x)(a)) (cf ) and the entries of A i are regular invertible functions on U i or constantly. An isomorphism of tropical vector bundles is a morphism of vector bundles Ψ : F 1 F 2 such that there exists a morphism of vector bundles Ψ : F 2 F 1 with Ψ Ψ = id = Ψ Ψ. Lemma Let π 1 : F 1 X and π 2 : F 2 X be two vector bundles of rank r over X. Then the following are equivalent: (a) There exists an isomorphism of vector bundles Ψ : F 1 F 2. (b) There exist a common open covering U 1,...,U s of X and transition maps M (1) ij for F 1 and M (2) ij for F 2 satisfying definition (cf. remark 5.1.8) and maps E i : U i G(r) for i = 1,...,s such that the entries of E i are regular invertible functions on U i or constantly and for all i,j holds E j (x) M (1) ij (x) = M(2) ij (x) E i(x) for all x U i U j. 92

103 Chapter 5: Chern classes of tropical vector bundles Proof. (a) (b): We claim that the maps A i : U i G(r r) of definition are the wanted maps E i. As Ψ is an isomorphism we can conclude that A i (x) is an invertible matrix for all x U i, i.e. that A i : U i G(r). Hence it remains to check that A j (x) M (1) ij Ψ : F 1 F 2 is an isomorphism, the diagram (x) = M(2) ij (x) A i(x) holds for all x U i U j : Let i,j be given. As (U i U j ) R r Φ F 2 Φ F 1 j (Φ F 1 i ) 1 (U i U j ) R r Φ F 2 i Ψ (Φ F 1 i ) 1 j Ψ (Φ F 1 j (U i U j ) R r Φ F 2 j (Φ F 2 i ) 1 ) 1 (U i U j ) R r commutes. Hence A j (x) M (1) ij (x) = M(2) ij (x) A i(x) holds. (b) (a): Conversely, let the maps E i : U i G(r) be given. The equation for all x U i U j ensures that the maps E j (x) M (1) ij (x) = M(2) ij (x) E i(x) U i R r U i R r : (x,a) (x,e i (x) a) on the local trivializations can be glued to a globally defined map Ψ : F 1 F 2. Moreover, this map is a morphism as π 1,π 2 are morphisms and the maps p (i) j Φ F 1 i, p (i) j Φ F 2 i and the finite entries of E i are regular invertible functions (cf. definition 5.1.5). The equation E j (x) M (1) ij E 1 j (x) M (2) ij (x) = M(2) ij (x) E i(x) implies that (x) = M(1) ij (x) E 1 i (x) holds for all x U i U j, where E 1 k (x) := (E k(x)) 1 for all x U k. As the finite entries of E 1 k : U k G(r) are again regular invertible functions we can also glue the maps U i R r U i R r : (x,a) (x,e 1 i (x) a) on the local trivializations to obtain the inverse morphism Ψ : F 2 F 1, which proves that Ψ is an isomorphism. The morphisms we have just introduced admit another important operation, namely the pull-back of a vector bundle: Definition (Pull-back of vector bundles) Let π : F X be a vector bundle of rank r with open covering U 1,...,U s and transition maps M ij as in definition 5.1.5, and let f : Y X be a morphism of tropical cycles. Then the pull-back bundle π : f F Y is the vector bundle we obtain by gluing the patches f 1 (U 1 ) R r,...,f 1 (U s ) R r along the transition maps M ij f. Hence we obtain the commutative diagram f F f π Y f F X π 93

104 5.1 Tropical vector bundles where f and π are locally given by f : f 1 (U i ) R r U i R r : (y,a) (f(y),a) and π : f 1 (U i ) R r f 1 (U i ) : (y,a) y. To be able to define Chern classes in the second section we need the notion of a rational section of a vector bundle: Definition (Rational sections of vector bundles) Let π : F X be a vector bundle of rank r. A rational section s : X F of F is a continuous map s : X F such that (a) π(s(x)) = x for all x X and (b) there exist an open covering U 1,...,U s and homeomorphisms Φ i satisfying definition (cf. definition 5.1.7) such that the maps p (i) j Φ i s : U i R are rational functions on U i for all i,j, where p (i) j : U i R r R is given by (x, (a 1,...,a r )) a j. A rational section s : X F is called bounded if the above maps p (i) j Φ i s are bounded for all i,j. Remark Let π : L X be a line bundle and s : X L a rational section. By definition, the map p (i) Φ i s is a rational function on U i for all i. Moreover, on U i U j the maps p (i) Φ i s and p (j) Φ j s differ by a regular invertible function only. Hence s defines a Cartier divisor D(s) Div(X). There is a useful statement on these Cartier divisors D(s) in [T09] that we want to cite here including its proof: Lemma Let π : L X be a line bundle and let s 1,s 2 : X L be two bounded rational sections. Then D(s 1 ) D(s 2 ) = h for some bounded rational function h K (X), i.e. D(s 1 ) and D(s 2 ) are rationally equivalent. Proof. Let U 1,...,U s be an open covering of X with transition maps M ij and homeomorphisms Φ i according to definition such that for all i both s (i) 1 := p (i) Φ i s 1 and s (i) 2 := p (i) Φ i s 2 are rational functions on U i (cf. definition ). We define h i := s (i) 1 s (i) 2 K (U i ). As we have s (i) 1 s (j) 1 = s (i) 2 s (j) 2 = M ij O (U i U j ) for all i,j these maps h i glue together to h K (X). Hence we have D(s 1 ) D(s 2 ) = [{(U i,s (i) 1 )}] [{(U i,s (i) = [{(U i,s (i) 1 s (i) 2 )}] = [{(U i,h i )}] = [{( X,h)}]. 2 )}] Remark Lemma implies that we can associate to any line bundle L admitting a bounded rational section s a Cartier divisor class D(F) := [D(s)] that only depends on the bundle L and not on the choice of the rational section s. 94

105 Chapter 5: Chern classes of tropical vector bundles Combining both the notion of a morphism of vector bundles and the notion of a rational section we can define the following: Definition (Pull-back of rational sections) Let π : F X be a vector bundle of rank r and f : Y X a morphism of tropical varieties. Moreover, let s : X F be a rational section of F with open covering U 1,...,U s and homeomorphisms Φ 1,...,Φ s as in definition Then we can define a rational section f s : Y f F of f F, the pull-back section of s, as follows: On f 1 (U i ) we define f s : f 1 (U i ) f 1 (U i ) R r : y (y, (p i Φ i s f)(y)), where p i : U i R r R r is the projection on the second factor. Note that for y f 1 (U i ) f 1 (U j ) the points (y, (p i Φ i s f)(y)) and (y, (p j Φ j s f)(y)) are identified in f F if and only if (f(y), (p i Φ i s f)(y)) and (f(y), (p j Φ j s f)(y)) are identified in F. But this is the case as (f(y), (p i Φ i s f)(y)) = (Φ i s)(f(y)) (Φ j s)(f(y)) = (f(y), (p j Φ j s f)(y)). Hence we can glue our locally defined map f s to obtain a map f s : Y f F. We finish this section with the following statement on vector bundles on simply connected tropical cycles which will be of use for us later on: Theorem Let π : F X be a vector bundle of rank r on the simply connected tropical cycle X. Then F is a direct sum of line bundles, i.e. there exist line bundles L 1,...,L r on X such that F = L 1... L r. Proof. We show that every vector bundle of rank r 2 on X is decomposable. Let U 1,...,U s be an open covering of X and let M ij (x) = D(ϕ (1) i,j,...,ϕ(r) i,j )(x) A σ ij (x) =: D ij (x) A σij (x), x U i U j with ϕ (1) i,j,...,ϕ(r) i,j O (U i U j ) and σ ij (x) S r be transition functions according to definition We only have to show that it is possible to track the first coordinate of the R r -factor in U 1 R r consistently along the transition maps: Let γ : [0, 1] X be a closed path starting and ending in P U 1. Decomposing γ into several paths if necessary, we may assume that γ has no self-intersections, i.e. that γ [0,1) is injective. As γ([0, 1]) is compact we can choose an open covering V 1,...,V t of γ([0, 1]) such that for all j we have V j U i for some index i = i(j), P V 1 = V t U 1, all sets V j and all intersections V j V j+1 are connected and all intersections V j V j for nonconsecutive indices are empty. For sets V j and V j with non-empty intersection we have restricted transition maps M Vj,V j (x) = D Vj,V j (x) A σvj induced by the transition,v j maps between U i(j) V j and U i(j ) V j. Note that the permutation parts A σvj of,v j the transition maps do not depend on x as all intersections V j V j are connected and the permutations have to be locally constant. We define I γ := σ Vt 1,V t... σ V1,V 2 (1). We have to check that I γ = 1 holds. First we show that I γ does not depend on the choice of the covering V 1,...,V t. Hence, let V 1,...,V t be another covering as above. We may 95

106 5.2 Chern classes assume that all intersections V j V j are connected, too. Between any two sets A,B {V 1,...,V t,v 1,...,V t } with non-empty intersection we have restricted transition maps M A,B (x) = D A,B (x) A σa,b as above. Moreover, let 0 = α 0 < α 1 <... < α p = 1 be a decomposition of [0, 1] such that for all i we have γ([α i,α i+1 ]) V j V j for some indices j,j. Let i 0 be the maximal index such that γ([α i0,α i0 +1]) V a V b and σ Va 1,V a... σ V1,V 2 = σ V b,v a σ V b 1,V b... σ V 1,V 2 is still fulfilled. Assume that i 0 < p 1. Let γ([α i0 +1,α i0 +2]) V a V b. Hence γ(α i0 +1) V a V b V a V b and we can conclude using the cocycle condition: σ Va,V a σ Va 1,V a... σ V1,V 2 = σ Va,V a σ V b,v a σ V b 1,V b... σ V 1,V 2 = σ Va,V a σ V σ b V,Va b,v σ b V b 1,V b... σ V 1,V 2 = σ V b,v a σ V b,v b σ V b 1,V b... σ V 1,V a contradiction to our assumption. Hence i 0 = p 1 and we can conclude that I γ is independent of the chosen covering. If γ and γ are paths that pass through exactly the same open sets U i in the same order, then we can conclude that I γ = I γ holds as exactly the same transition functions are involved. Hence, a continuous deformation of γ does not change I γ. As X is simply connected we can contract γ to a point. This implies I γ = I γ0, where γ 0 is the constant path γ 0 (t) = P for all t. Thus I γ = I γ0 = 1. This proves the claim. There is a related theorem in [T09] which we want to state here. As we will not need the result in this work, we will omit the proof and refer to [T09] instead. Theorem Let π : L X be a line bundle on the simply connected tropical cycle X. Then L is trivial, i.e. L = X R as a vector bundle. Combing both theorem and theorem we can conclude the following: Corollary Let π : F X be a vector bundle of rank r on the simply connected tropical cycle X. Then F is trivial, i.e. F = X R r as a vector bundle. 5.2 Chern classes In this section we will introduce Chern classes of tropical vector bundles and prove basic properties. To be able to do this we need some preparation: Definition Let π : F X be a vector bundle of rank r and let s : X F be a rational section with open covering U 1,...,U s as in definition We fix a natural number 1 k r and a subcycle Y Z l (X). By definition, s ij := p (i) j Φ i s : U i R is a rational function on U i for all i,j. Hence, for all i we can take local intersection products (s (k) Y ) U i := s ij1 s ijk (Y U i ). 1 j 1 <...<j k r 2, 96

107 Chapter 5: Chern classes of tropical vector bundles Since s i j = s iσ(j) + ϕ j on U i U i for some σ S r and some regular invertible map ϕ j O (U i U i ), the intersection products (s (k) Y ) U i and (s (k) Y ) U i coincide on U i U i and we can glue them to obtain a global intersection cycle s (k) Y Z l k (X). Lemma Let π : F X be a vector bundle of rank r, fix k {1,...,r} and let s : X F be a rational section. Moreover, let Y Z l (X) be a cycle and let ϕ K (Y ) be a bounded rational function on Y. Then the following equation holds: s (k) (ϕ Y ) = ϕ (s (k) Y ). Proof. The claim follows immediately from the definition of the product s (k) Y. Lemma Let π : F X and π : F X be two isomorphic vector bundles of rank r with isomorphism f : F F. Moreover, fix k {1,...,r}, let s : X F be a rational section and let Y Z l (X) be a cycle. Then the following equation holds: s (k) Y = (f s) (k) Y Z l k (X). Proof. Let U 1,...,U s be an open covering of X satisfying definition for both F and F and let s ij := p (i) j Φ i s : U i R and (f s) ij := p (i) j Φ i f s : U i R as in definition By lemma the isomorphism f can be described on U i R r by (x,a) (x,e i (x) a) with E i (x) = D(ϕ 1,...,ϕ r ) A σ for some regular invertible functions ϕ 1,...,ϕ r O (U i ) and a permutation σ S r. Hence (f s) ij = s iσ(j) + ϕ j on U i and thus s ij1 s ijk (Y U i ) = (f s) ij1 (f s) ijk (Y U i ), 1 j 1 <...<j k r which proves the claim. 1 j 1 <...<j k r To be able to prove the next theorem, which will be essential for defining Chern classes, we first need some generalizations of our previous definitions: Definition (Infinite tropical cycle) We define an infinite tropical polyhedral complex to be a tropical polyhedral complex according to definition but we do not require the set of polyhedra X to be finite. In particular, all open fans F σ have still to be open tropical fans according to definition Then an infinite tropical cycle is an infinite tropical polyhedral complex modulo refinements analogous to definition Definition (Infinite rational functions and infinite Cartier divisors) Let C be an infinite tropical cycle and let U be an open set in C. As in definition an infinite rational function on U is a continuous function ϕ : U R such that there exists a representative (((X, X, {m σ } σ X ),ω X ), {M σ } σ X ) of C, which may now be an infinite tropical polyhedral complex, such that for each face σ X the map ϕ m 1 σ is locally integer affine linear (where defined). Analogously it is possible to define infinite 97

108 5.2 Chern classes regular invertible functions on U. A representative of an infinite Cartier divisor on C is then a set {(U i,ϕ i ) i I}, where {U i } is an open covering of C and ϕ i is an infinite rational function on U i. An infinite Cartier divisor on C is then a representative of an infinite Cartier divisor modulo the equivalence relation given in definition Remark Using these basic definitions it is possible to generalize many other concepts to the infinite case. In particular, as our infinite objects are locally finite, it is possible to perform intersection theory as before. Definition (Tropical vector bundles on infinite cycles) Let X be an infinite tropical cycle. A tropical vector bundle over X of rank r is an infinite tropical cycle F together with a morphism π : F X such that properties (a) (d) given in definition are fulfilled with the difference that the open covering {U i } of X may now be infinite. Now we are ready to prove the announced theorem: Theorem Let π : F X be a vector bundle of rank r and s 1,s 2 : X F two bounded rational sections. Then s (k) 1 Y and s (k) 2 Y are rationally equivalent, i.e. holds for all subcycles Y Z l (X). [s (k) 1 Y ] = [s (k) 2 Y ] A (X) Proof. Let p : X X be the universal covering space of X. We can locally equip X with the tropical structure inherited form X and obtain an infinite tropical cycle X according to definition Moreover, pulling back F along p, we obtain a tropical vector bundle p F on X according to definition As X is simply connected we can conclude by lemma that p F = L 1... L r for some infinite tropical line bundles L 1,...,L r on X. Hence, the bounded rational sections p s 1 and p s 2 correspond to r infinite tropical Cartier divisors as in definition each, which we will denote by ϕ 1,...,ϕ r and ψ 1,...,ψ r, respectively. By lemma we can conclude that for all i these Cartier divisors differ by bounded infinite rational functions only, i.e. ϕ i ψ i = h i for some bounded infinite rational function h i on X. In particular, ( ) ϕ j1 ϕ jk ψ j1 ψ jk X = h ξ 2 ξ k X 1 j 1 <...<j k r 1 j 1 <...<j k r with a bounded infinite rational function h and infinite Cartier divisors ξ i. Then we can define a rational function h, which is then also bounded, and Cartier divisors ξ i on X as follows: Let U X and Ũ X be open subsets such that p Ũ : Ũ U is bijective with inverse map p : U Ũ. Then we locally define h U := (p ) h Ũ and ξ i U := (p ) ξi Ũ. Note that h and ξ i are well-defined as the Cartier divisors ϕ i and ψ i, 98

109 Chapter 5: Chern classes of tropical vector bundles respectively, are the same on every possible set Ũ = U. As we locally have ( ) (s (k) 1 Y ) U = p ϕ j1 ϕ jk (p ) (Y U) and we can conclude that which proves the claim. (s (k) 2 Y ) U = p ( 1 j 1 <...<j k r 1 j 1 <...<j k r ) ψ j1 ψ jk (p ) (Y U) (s (k) 1 s (k) 2 ) Y = h ξ 2 ξ k Y, Now we are ready to give a definition of Chern classes: Definition (Chern classes) Let π : F X be a vector bundle of rank r admitting bounded rational sections. For k {1,...,r} we define the k-th Chern class of F to be the endomorphism c k (F) : A (X) A (X) : [Y ] [s (k) Y ], where A (X) = i A i(x) and s : X F is any bounded rational section. Note that the map c k (F) is well-defined by lemma and independent of the choice of the rational section s by theorem Moreover, we define c 0 (F) : A (X) A (X) to be the identity map and c k (F) : A (X) A (X) to be the zero map for all k {0,...,r}. To stress the character of an intersection product of c k (F) we usually write c k (F) Y instead of c k (F)(Y ) for Y A (X). Remark Note that lemma implies that isomorphic vector bundles have the same Chern classes. As announced in the beginning we finish this section with proving some basic properties of Chern classes: Theorem (Properties of Chern classes) Let π : F X and π : F X be vector bundles of rank r and r, respectively, admitting bounded rational sections. Moreover, let f : X X be a morphism of tropical cycles. Then the following holds: (a) c i (F) = 0 for all i {0,...,rank(F)}, (b) c i (F) (c j (F ) Y ) = c j (F ) (c i (F) Y ) for all Y A (X), (c) f (c i (f F) Y ) = c i (F) f (Y ) for all Y A ( X), (d) c i (f F) f (Y ) = f (c i (F) Y ) for all Y A (X) if X and X are smooth varieties, (e) c k (F F ) = i+j=k c i(f) c j (F ) 99

110 5.3 Vector bundles on an elliptic curve (f) c 1 (F) Y = D(F) Y for all Y A (X) if r = rank(f) = 1, where D(F) is the Cartier divisor class associated to F. Proof. Properties (a) and (e) follow immediately from definition 5.2.9, property (b) follows from the fact that the intersection product is commutative and property (f) follows from remark (c): The projection formula implies f (c i (f F) Y ) = f ([(f s) (i) Y ]) = [s (i) f Y ] = c i (F) f Y, where s is any bounded rational section of F. (d): Applying theorem (c) and (f) we obtain c i (f F) f Y = [(f s) (i) f Y ] = [f (s (i) Y )] = f [s (i) Y ] = f (c i (F) Y ), where s is again any bounded rational section of F. Remark In classical algebraic geometry even the following, generalized version of property (e) is true: Let 0 F F F 0 be an exact sequence of vector bundles, then c k (F) = i+j=k c i(f ) c j (F ). In the tropical world it is not entirely clear what an exact sequence of tropical vector bundles should be. Nevertheless, in some sense the classical statement is true in tropical geometry as well: Let π 1 : F 1 X and π 2 : F 2 X be tropical vector bundles of rank r 1 and r 2, respectively, and let U 1,...,U s be an open covering of X such that all requirements of definition are fulfilled for F 1 and F 2 simultaneously. Moreover, let f : F 1 F 2 be an injective morphism of tropical vector bundles such that (Φ F 2 i f (Φ F 1 i ) 1 )(U i R r 1 ) = U i e i1,...,e ir1 R for all i, i.e. such that the image of F 1 under f is a subbundle F of F 2 (cf. definition ). Then we can conclude by remark that F 2 is decomposable into F 2 = F F = F1 F for some other subbundle F of F 2. Hence we can conclude by theorem that c k (F 2 ) = i+j=k c i(f 1 ) c j (F ). 5.3 Vector bundles on an elliptic curve In this section we will give a complete classification of all vector bundles on an elliptic curve up to isomorphism. One characteristic to distinguish different bundles will be the following: Definition (Degree of a vector bundle) Let X := X 2 be the curve from example and let π : F X be a vector bundle of rank r. We define the degree of F to be the number deg(f) := deg(c 1 (F) X). As already advertised in example vector bundles on the elliptic curve X can be described by a single transition function. We will prove this fact in the following lemma: 100

111 Chapter 5: Chern classes of tropical vector bundles Lemma Again, let X := X 2 be the curve from example and let π : F X be a vector bundle of rank r. Then F is isomorphic to a vector bundle π : F X that admits an open covering U 1,...,U s and transition maps M ij such that at most one transition map is non-trivial. Proof. Let U 1,...,U s be the open covering with transition maps M ij for F according to definition We may assume that all sets U i are connected and that for all i,j the intersections U i U j are connected as well. Moreover, we may assume that the sets U i are numbered consecutively as shown in the figure. For simplicity of notation we will consider our indices modulo s. X = U 1 U 2 U 3 U 5 U4 We can write every map M i,i+1, i = 1,...,s, as M i,i+1 (x) = D(ϕ (1) i,i+1,...,ϕ(r) i,i+1 )(x) A σ i,i+1 =: D i (x) P i for some regular invertible functions ϕ (k) i,i+1 O (U i U i+1 ) and permutations σ i,i+1 S r. We will show that we can replace successively all the transition maps M i,i+1 but one by the constant map M i,i+1 : U i U i+1 G(r) : x E and the resulting vector bundle F is isomorphic to F: Choose j 0 {2,...,s}. Note that if we are given a regular invertible function ϕ O (U i U j ) there is a unique regular invertible function ϕ O (U i ) such that ϕ Ui U j = ϕ. As they are regular invertible functions, too, we can extend in exactly the same way the finite entries of the matrix D j0 along the chain U j0 1,U j0 2,...,U i+1 to any set U i+1 for i {2,...,j 0 1}. By abuse of notation we will denote this continuation of D j0 as well by D j0. Now, we take U i := U i for all i = 1,...,s and { M i,i+1(x) Pj0 D := j0 (x) M i,i+1 (x) D j0 (x) 1 P 1 j 0, if i {2,...,j 0 1} M i,i+1 (x), if i {j 0 + 1,...,s}. Moreover, we set M 12(x) := P j0 D j0 (x) D 1 (x) P 1 and M j 0,j 0 +1(x) := E. To check that the vector bundle F we obtain from this gluing data is isomorphic to F we apply lemma : We set { Dj0 (x) P E i (x) := j0, if i {2,...,j 0 } E, else, and get (D j0 P j0 ) (D 1 P 1 ) = (D j0 P j0 D 1 P 1 ) E (D j0 P j0 ) (D 2 P 2 ) = (D j0 P j0 D 2 P 2 D 1 j 0 P 1 j 0 ) (D j0 P j0 ).. E (D j0 P j0 ) = E (D j0 P j0 ). This finishes our proof. 101

112 5.3 Vector bundles on an elliptic curve To classify all vector bundles on our elliptic curve X we give now a non-redundant parametrization of all indecomposable vector bundles on X. Arbitrary vector bundles are then just direct sums of these building blocks. Theorem (Vector bundles on elliptic curves) Let X := X 2 be the curve from example Then the set of indecomposable vector bundles of rank r and degree d is in natural bijection with gcd(r,d) X, i.e. with points of the curve X stretched to gcd(r,d) times the original length. Proof. Let π : F X be an indecomposable vector bundle of rank r with open covering U 1,...,U s and transition maps M ij according to definition Again, we may assume that all sets U i are connected, that for all i,j the intersections U i U j are connected as well and that the sets U i are numbered consecutively. Moreover, by lemma we may assume that M 12 is the only non-trivial transition map. Let M 12 (x) = D(ϕ 1,...,ϕ r )(x) A σ =: D(x) A σ for some regular invertible functions ϕ 1,...,ϕ r O (U 1 U 2 ) and a permutation σ S r. As F is indecomposable σ must by a single cycle. Hence there exists S r such that σ 1 = (12...r). We will apply lemma to show that we can replace M 12 (x) by M 12(x) := A D(x) A 1 A (12...r) without changing the isomorphism class of F: We set E i (x) := A for all x and all i and obtain A (D(x) A σ ) = (A D(x) A 1 A (12...r) ) A A E = E A.. A E = E A. Hence we may assume that σ = (12...r). Our next step is to apply lemma to show that we may replace D(x) = D(ϕ 1,...,ϕ r ) by D (x) = D(ϕ, 0,...,0) for some ϕ O (U 1 U 2 ) without changing the isomorphism class of F. For i = 1,...,r let α i be the slope of ϕ i and let L be the (lattice) length of our curve X. For i = 2,...,r we set δ i := r j=i (j i + 1) α j. Moreover, we define ϕ := ϕ ϕ r δ 2 L. Note that if we are given a regular invertible function ψ O (U i U j ) there is a unique regular invertible function ψ O (U i ) such that ϕ Ui U j = ϕ. Hence we can extend our regular invertible functions ϕ 1,...,ϕ r along the chain U 2,U 3,...,U s,u 1 to any of the sets U 1,...,U s. Note that on U 1 U 2 the extension of ϕ i to U 2 and the extension of ϕ i to U 1 differ exactly by α i L. We use these continuations to define the maps E i : E i (x) := D( ϕ ϕ r δ 2 L, ϕ ϕ r δ 3 L,..., ϕ r δ r L, 0), where for entries of E i the map ϕ j denotes the continuation of ϕ j to U i. Hence we obtain on U 1 U 2 : E 2 M 12 = D( ϕ ϕ r δ 2 L,..., ϕ r δ r L,0) (D(ϕ 1,...,ϕ r ) A σ ) = D(ϕ ϕ r δ 2 L,...,ϕ r δ r L,0) (D(ϕ 1,...,ϕ r ) A σ ) = D(ϕ ϕ r δ 2 L, ϕ ϕ r δ 3 L,...,ϕ r 1 + ϕ r δ r L, ϕ r ) A σ 102

113 Chapter 5: Chern classes of tropical vector bundles and M 12 E 1 = (D(ϕ ϕ r δ 2 L,0,...,0) A σ ) D( ϕ ϕ r δ 2 L,..., ϕ r δ r L,0) = (D(ϕ ϕ r δ 2 L,0,...,0) A σ ) D(ϕ ϕ r δ 3 L,...,ϕ r δ r 1 L,0) = D(ϕ ϕ r δ 2 L, ϕ ϕ r δ 3 L,...,ϕ r 1 + ϕ r δ r L, ϕ r ) A σ. The other conditions are trivially fulfilled as E i Ui U i+1 = E i+1 Ui U i+1 for all i 1. Hence we may assume that M 12 (x) = D(x) A σ = D(ϕ, 0,...,0)(x) A (12...r). As F is a vector bundle of degree d the affine linear map ϕ must have slope d. Thus, the transition map M 12 is determined by the isomorphism class of F up to translations of ϕ. To prove the claim it remains to show that two vector bundles F and F as above with transition maps M 12 (x) = D(ϕ, 0,...,0)(x) A (12...r) and M 12(x) = D(ϕ + cl, 0,...,0)(x) A (12...r) are isomorphic if and only if c is an integer multiple of gcd(r,d): By lemma F and F are isomorphic if and only if for all i = 1,...,s there exists a map E i : U i G(r) such that for all i the equation E i+1 (x) M i,i+1 (x) = M i,i+1(x) E i (x) holds for all x U i U i+1. As M i,i+1 is trivial for all i 1 these equations imply E i Ui U i+1 = E i+1 Ui U i+1 for all i 1. Hence F and F are isomorphic if and only if there exist a permutation τ S r and regular invertible functions ψ 1,...,ψ r O (U 1 U 2 ) with continuations ψ 1,..., ψ r to all sets U 1,...,U s along the chain U 2,U 3,...,U s,u 1 such that (D( ψ 1,..., ψ r ) A τ ) (D(ϕ, 0,...,0) A σ ) = (D(ϕ+cL,0,...,0) A σ ) (D( ψ 1,..., ψ r ) A τ ) holds on U 1 U 2. In particular, the last equation implies A τ A σ = A σ A τ and hence τ = σ k for some k Z. Thus F and F are isomorphic if and only if there exist k Z and ψ 1,...,ψ r as above such that D( ψ 1,..., ψ k, ψ k+1 + ϕ, ψ k+2,..., ψ r ) A σ k+1 = D(ϕ + cl + ψ r, ψ 1,..., ψ r 1 ) A σ k+1. Let α i be the slope of ψ i. Then on U 1 U 2 the continuation of ψ i to U 2 and the continuation of ψ i to U 1 differ exactly by α i L. Hence we obtain the system of equations ψ 1 ψ 2 = ϕ + cl + ψ r + α r L = ψ 1 + α 1 L.. ψ k = ψ k 1 + α k 1 L ψ k+1 + ϕ = ψ k + α k L ψ k+2 = ψ k+1 + α k+1 L.. ψ r = ψ r 1 + α r 1 L. In particular, we can conclude that α 1 =... = α k and α k+1 =... = α r. Hence F and F are isomorphic if and only if there exist α 1,α r,k Z such that c = (r k) α r + k α 1 and α 1 = d + α r, or equivalently if and only if there exist α r,k Z with This finishes the proof. c = rα r k d. 103

114 5.3 Vector bundles on an elliptic curve Remark Note that the claim of theorem coincides with the equivalent result in classical algebraic geometry (see [A57, theorem 7]). 104

115 Bibliography [A06] Mohammed Abouzaid, Morse Homology, Tropical Geometry, and Homological Mirror Symmetry for Toric Varieties, preprint arxiv:math.ag/ [A09] [A57] [AR07] [AR08] [B71] [BG84] [EKL04] Lars Allermann, Tropical intersection products on smooth varieties, preprint arxiv: Michael Francis Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc., Volume s3-7, Number 1, 1957, pp (39). Lars Allermann, Johannes Rau, First steps in tropical intersection theory, Mathematische Zeitschrift (to appear); also at arxiv: Lars Allermann, Johannes Rau, Tropical rational equivalence, preprint arxiv: George M. Bergman, The logarithmic limit set of an algebraic variety, Trans. AMS, Volume 157, 1971, pp (11). Robert Bieri and John R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math., Volume 347, 1984, pp (28). Manfred Einsiedler, Mikhail Kapranov, Douglas Lind, Non-archimedean amoebas and tropical varieties, preprint arxiv:math.ag/ [F84] William Fulton, Intersection theory, Springer Verlag, [FS97] [G07a] [G07b] William Fulton, Bernd Sturmfels, Intersection Theory on Toric Varieties, Topology, Volume 36, Number 2, March 1997, pp (19); also at arxiv:alg-geom/ Walter Gubler, Tropical varieties for non-archimedean analytic spaces, Invent. Math., Volume 169, 2007, pp (56); also at arxiv:math.nt/ Walter Gubler, The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math., Volume 169, 2007, pp (24); also at arxiv:math.nt/

116 Bibliography [GKM07] Andreas Gathmann, Michael Kerber, Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, preprint arxiv: [GM08] Andreas Gathmann, Hannah Markwig, Kontsevich s formula and the WDVV equations in tropical geometry, Advances in Mathematics, Volume 217, Issue 2, January 2008, pp (24); also at arxiv:math.ag/ [H07] [IKS03] [IKS04] [IKS09] [J08] [K08] [K09] [KM06] Matthias Herold, Intersection Theory of the Tropical Moduli Spaces of Curves, Diploma thesis, TU Kaiserslautern, Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, Welschinger invariant and enumeration of real plane rational curves, Intern. Math. Research Notices, Volume 49, 2003, pp (15); also at arxiv:math.ag/ Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, Logarithmic equivalence of Welschinger and Gromov-Witten invariants, Uspehi Mat. Nauk, Volume 59, 2004, pp (26); also at arxiv:math.ag/ Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces, Commentarii Mathematici Helvetici, Volume 84, 2009, pp (40); also at arxiv:math.ag/ Michael Joswig, Tropical Convex Hull Computations, Proceedings of the International Conference on Tropical and Idempotent Mathematics (to appear); also at arxiv: Eric Katz, A Tropical Toolkit, preprint arxiv:math.ag/ Eric Katz, Tropical intersection theory from toric varieties, preprint arxiv: Michael Kerber, Hannah Markwig, Counting tropical elliptic plane curves with fixed j-invariant, Commentarii Mathematici Helvetici (to appear); also at arxiv:math.ag/ [M05] Grigory Mikhalkin, Enumerative tropical algebraic geometry in R 2, J. Amer. Math. Soc., Volume 18, Number 2, 2005, pp (65); also at arxiv:math.ag/ [M06] [PS04] Grigory Mikhalkin, Tropical Geometry and its applications, Proceedings of the ICM, Madrid, Spain, 2006, pp (26); also at arxiv:math.ag/ Lior Pachter, Bernd Sturmfels, Tropical Geometry of Statistical Models, Proceedings of the National Academy of Sciences, USA, Volume 101, 2004, pp (6); also at arxiv:q-bio/

117 Bibliography [R08] Johannes Rau, Intersections on tropical moduli spaces, preprint arxiv: [RGST05] [S05] [ST08] [T09] Jürgen Richter-Gebert, Bernd Sturmfels, Thorsten Theobald, First steps in tropical geometry, Idempotent Mathematics and Mathematical Physics (G.L. Litvinov and V.P. Maslov, eds.), Proceedings Vienna 2003, American Mathematical Society, Contemp. Math., (377), 2005; also at arxiv:math.ag/ David Speyer, Tropical Geometry, PhD thesis, University of California, Berkeley, 2005; available at Bernd Sturmfels, Jenia Tevelev, Elimination theory for tropical varieties, Mathematical Research Letters, Volume 15, Number 3, March 2008, pp (20); also at arxiv: Carolin Torchiani, Line bundles on tropical varieties, Diploma thesis, TU Kaiserslautern, work in progress. [Z95] Günter M. Ziegler, Lectures on Polytopes, Springer Verlag,

118

119 Appendix: Pictures of tropical surfaces Tropical varieties arise in a natural way as images of algebraic varieties under valuation maps. Let us make this precise: Definition (Tropicalization of algebraic varieties) Let K be the field of Puiseux series with complex coefficients, i.e. { } K := a q t q a q C and {q R a q 0} R is bounded below and has no accumulation points q R with the usual addition and multiplication of power series. This field K is algebraically closed and admits a non-archimedean valuation val : K R : q R a q t q min{q R a q 0} R. We use this valuation to define the map Val : K n T n : (x 1,...,x n ) ( val(x 1 ),..., val(x n )). Let X K n be an algebraic variety. Then the set Trop(X) := Val(X (K ) n ) R n is called the tropicalization of X. Remark The tropicalization Trop(X) of an algebraic variety X carries a natural structure of a tropical variety in R n. More details on this fact can be found, for example, in [S05]. A special case of this process is the case where X is a hypersurface, i.e. the zero locus of a single polynomial over K. In this situation it is much easier to describe the tropicalization of X. Definition (Tropicalization of polynomials) Let f = i 1,...,i n a i1...i n z i 1 1 zn in K[z 1,...,z n ] be a polynomial over the field K. Then we call Trop(f) := max { val(a i1...i n ) + i 1 x i n x n } i 1,...,i n the tropicalization of f. Obviously, the function Trop(f) : R n R is piecewise integer linear. We denote by V (Trop(f)) the corner locus of Trop(f), i.e. the set of all points x R n where Trop(f) is not differentiable. 109

120 Appendix: Pictures of tropical surfaces With these definitions we get the following statement: Proposition Let f K[z 1,...,z n ] be a polynomial over K and V (f) its zero locus. Then the following equation holds: Trop(V (f)) = V (Trop(f)). Proof. A proof of this fact can be found in [EKL04]. In fact, every tropical hypersurface, i.e. every tropical variety of codimension one in R n, arises as a tropicalization of a hypersurface V (f) K n (see for example [M05]) and every tropical cycle of codimension one in R n is a difference of tropical hypersurfaces. As the corner locus of a tropical polynomial is relatively easy to calculate, tropical varieties of the form V (Trop(f)) are an important source of examples in tropical geometry. For some purposes the easiest possible case of curves in R 2 cannot provide sufficient examples and one has to deal with hypersurfaces in 3-space. This was the reason for creating a computer program to calculate the corner locus of a tropical polynomial in 3 variables: TropicalSurfaces. The program is written in Delphi and binary versions as well as the source code are available on The usage of TropicalSurfaces is easy: Type a tropical polynomial in the text box and press the button labeled with Zeichne. 110

121 Appendix: Pictures of tropical surfaces A picture of the corner locus of the given polynomial is drawn. Click with the left mouse button into the drawing area and move the cursor to rotate the picture in 3-space. If you press the right mouse button within the picture, a context menu appears. Here you can zoom in and out, change transparency settings, save the displayed tropical surface as a binary or text file, load tropical surfaces from binary or text files, export the displayed picture as a pixel graphic, POV-Ray file or scalable vector graphic, change other settings, e.g. the color scheme and the clipping area. Moreover, you can create random polynomials of degrees 2 and 3 via the context menu and display the corresponding tropical surfaces. 111