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 M_n and M_n~(lab)(Δ, ?~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.
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