The space of tropically collinear points is shellable
Abstract
The space Td,n of n tropically collinear points in a fixed tropical projective space TPd-1 is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d x 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 M0,n(TPd-1,1) of n-marked tropical lines in TPd-1 under the evaluation map. Thus we derive a natural simplicial fan structure for Td,n using a simplicial fan structure of M0,n(TPd-1,1) which coincides with that of the space of phylogenetic trees on d+n taxa. The space of phylogenetic trees has been shown to be shellable by Trappmann and Ziegler. Using a similar method, we show that Td,n is shellable with our simplicial fan structure and compute the homology of the link of the origin. The shellability of Td,n has been conjectured by Develin in 2005.