The space of tropically collinear points is shellable

Abstract

The spaceT _d;n ofn tropically collinear points in a fixed tropical projective space$\mathbb{T}\mathbb{P}^{d - 1}$ is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of reald×n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space$\mathcal{M}_{0,n} (\mathbb{T}\mathbb{P}^{d - 1} , 1)$ ofn-marked tropical lines in$\mathbb{T}\mathbb{P}^{d - 1}$ under the evaluation map. Thus we derive a natural simplicial fan structure forT _d;n using a simplicial fan structure of$\mathcal{M}_{0,n} (\mathbb{T}\mathbb{P}^{d - 1} , 1)$ which coincides with that of the space of phylogenetic trees ond +n taxa. The space of phylogenetic trees has been shown to be shellable by Trappmann and Ziegler. Using a similar method, we show thatT _d;n is shellable with our simplicial fan structure and compute the homology of the link of the origin. The shellability ofT _d;n has been conjectured by Develin in [1].