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].

中文翻译：

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热带共线点的空间是可贝壳的

空间T _{d ; Ñ}的Ñ在固定热带射影空间热带共线点\（\ mathbb【T} \ mathbb {P} ^ {d - 1} \）相当于的行列式各种等级的矩阵的热带化至多2，其由最多为2的热带或Kapranov等级的实d × n矩阵组成，列的模射等价。我们表明，它等于模空间的图像\（\ mathcal {M} _ {0，N}（\ mathbb【T} \ mathbb {P} ^ {d - 1}，1）\）的Ñ评估图下\（\ mathbb {T} \ mathbb {P} ^ {d-1} \）中标记的热带线。因此，我们得出了T的自然简单扇形结构 _{d ; n}使用\（\ mathcal {M} _ {0，n}（\ mathbb {T} \ mathbb {P} ^ {d-1}，1）\）的简单扇形结构，该结构与d + n分类群上的系统发育树。系统发育树的空间已被Trappmann和Ziegler证明是可剥壳的。使用类似的方法，我们表明T _d；n具有简单的扇形结构，可以计算出起点链接的同源性。T _{d ; n}的可壳性已由Develin在[1]中推测。