Resumen de The space of tropically collinear points is shellable

Hannah Markwig, Josephine Yu

• The space $T_{d,n}$ of $n$ tropically collinear points in a fixed tropical projective space $\mathbb{TP}^{d-1}$ is equivalent to the tropicalization of the determinantal variety of matrices of rank at most $2$, which consists of real $d\times n$ 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 $\mathcal{M}_{0,n}(\TP^{d-1},1)$ 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 $\mathcal{M}_{0,n}(\TP^{d-1},1)$ 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 \cite{Dev05}.