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.

ABSTRACT. We use piecewise polynomials to define tropical cocycles generalising the well-known notion of Cartier divisors to higher codimensions. We also introduce an intersection product of cocycles with tropical cycles and prove that this gives rise to a Poincaré duality in some cases.

Abstract. We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked abstract rational tropical curves M n .