The space of tropically collinear points is shellable

TL;DR

This paper proves that the space of n tropically collinear points in tropical projective space is shellable, providing a new simplicial fan structure and connecting it to phylogenetic trees, confirming a conjecture from 2005.

Contribution

It establishes the shellability of T_{d,n} and constructs a natural simplicial fan structure using tropical moduli spaces, linking tropical geometry and phylogenetics.

Findings

T_{d,n} is shellable with a natural simplicial fan structure.

The homology of the link of the origin in T_{d,n} is computed.

The shellability confirms a conjecture by Develin from 2005.

Abstract

The space T_{d,n} of n tropically collinear points in a fixed tropical projective space TP^{d-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 M_{0,n}(TP^{d-1},1) of n-marked tropical lines in TP^{d-1} under the evaluation map. Thus we derive a natural simplicial fan structure for T_{d,n} using a simplicial fan structure of M_{0,n}(TP^{d-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 T_{d,n} is shellable with our simplicial fan structure and compute the homology of the link of the origin. The shellability of T_{d,n} has been conjectured by Develin in 2005.