Tropical schemes, tropical cycles, and valuated matroids

Abstract

We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed in [GG16], so there is a well-defined Hilbert–Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal $I$ we show that the Giansiracusa tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces $I_d$ for $d\ge 0$. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.