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 we show that the Giansiracusa tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces for . We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.
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Diane Maclagan, Felipe Rincón, Tropical schemes, tropical cycles, and valuated matroids. J. Eur. Math. Soc. 22 (2020), no. 3, pp. 777–796DOI 10.4171/JEMS/932