Math 274: Tropical Geometry Assignment IV

Exercise 1. [Solution due to Alex Fink] Invoking our fan structure for T (M), we know that a general point x = (x1, . . . , xn) ∈ T (M) has the property that xi = a(min{j : i ∈ Fj}) for some reals a(1) > · · · > a(s), where ∅ = F0 ( · · · ( Fs = [n] is a flag of flats of M of some length. This point x lies in the relative interior of the cone corresponding to the flag {Fi}. In particular, the points x ∈ T (M) with exactly two distinct components arise from a flag of length s = 2, where F1 is a flat other than ∅ or [n]; and F1 is the set of indices i such that xi is the lesser of the two occurring values. This lets us recover all the nonempty flats of M from T (M) (which in particular is enough data to recover M). Now, we have that the bases B of M are exactly those subsets of [n] which can be obtained by choosing a maximal flag of flats