ABSTRACT. We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R n .

Abstract. We define tropical Psi-classes on M 0,n (R 2 , d) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psiand evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi-and evaluation conditions.

ABSTRACT. We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked curves Mn and M lab n (∆, R r ). This intersection product can be extended to smooth varieties (whose local models are matroid varieties). We also study pull-backs of cycles and rational equivalence.

For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar\'e duality for integral tropical homology. Comment: 27 pages, 6 figures, published version

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties P 1 , P 2 , P 1 × P 1 and with Psiconditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for P 2 in [MR08]). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).2010 Mathematics Subject Classification. Primary 14T05; Secondary 14N35, 52B20.To be more precise, let Γ ϕ be the graph of ϕ in X × R. It is a polyhedral complex whose polyhedra are in one-to-one correspondence with those of X, but in general Γ ϕ is not balanced. However, it can be completed to a cycle by adding facets in (0, −1)-direction at each ridge of Γ ϕ , equipped with the above weights. Now, if we (imaginary) intersect this tropically completed graph of ϕ with X × {−∞} (i.e. compute the tropical zero locus of ϕ), we obtain the cycle div(ϕ) = ϕ · X of our definition. If ϕ is globally affine (resp. linear), all weights are zero, which we denote by ϕ · X = 0. Let the support of ϕ, denoted by |ϕ|, be the subcomplex of X containing the points x ∈ |X| where ϕ is not locally affine. Then we have |ϕ · X| ⊆ |ϕ|. Furthermore, the intersection product is bilinear (see [AR07, 3.6]). As the restriction of a rational function to a subcycle is again a rational function, we can also form multiple intersection products ϕ 1 · . . . · ϕ l · X. In this case we will sometimes omit "·X" to keep formulas shorter. Note that multiple intersection products are commutative (see [AR07, 3.7]).A morphism of cycles X ⊆ V = Λ ⊗ R and Y ⊆ V ′ = Λ ′ ⊗ R is a map f : |X| → |Y | that is induced by a linear map b Λ to Λ ′ and that maps each polyhedron of X into a polyhedron of Y . We call f an isomorphism and write X ∼ = Y , if there exists an inverse morphism and if for all facets σ ∈ X we have ω X (σ) = ω Y (f (σ)). Such a morphism pulls back rational functions ϕ on Y to rational functions f * (ϕ) = ϕ • f on X. Note that the second condition of a morphism makes sure that we do not have to refine X further. f * (ϕ) is already affine on each cone. The inclusion |f * (ϕ)| ⊆ f −1 (|ϕ|) holds, as the composition of an affine and a linear function is again affine. Furthermore, we can push forward subcycles Z of X to subcycles f * (Z) of Y of same dimension. This is due [GKM07, 2.24 and 2.25] in the case of fans and can be generalized to complexes (see [AR07, 7.3]). We can omit further refinements here if we assume that f...