A compactification of the space of points on the affine line I recently encountered an interesting space.  It is a compactification of the space of $ n$ points in $ \mathbb A^1 $ modulo translation, $ (\mathbb A^1)^n / \mathbb G_a $.
Let $ n \in \mathbb N $ and let $ p([n]) = \{ (i,j) \in \{1, \dots, n\}^2 : i \ne j \} $.  Let $ Y_n $ be the subscheme of $ (\mathbb P^1)^{p([n])} $ defined by the equations
$$ 
\tau_{ik} = \tau_{ij} + \tau_{jk} \quad \tau_{ij} = -\tau_{ji}
$$
If we restrict to the open set $ Y_n^{\circ} $ where $ \tau_{ij} \ne \infty $, then we have an isomorphism $ Y_n^\circ \cong (\mathbb A^1)^n / \mathbb G_a $ where $ \tau_{ij} = x_i - x_j $.
In fact, $Y_n $ has a stratification with strata
$$ (\mathbb A^1)^{S_1} / \mathbb G_a \times \cdots \times (\mathbb A^1)^{S_m} / \mathbb G_a$$
where $ \{1, \dots, n \} = S_1 \sqcup \cdots \sqcup S_m $.
Another interesting thing about $Y_n $ is that it is the degeneration of the permutahedral toric variety (also known as the Losev-Manin space).  This latter space can be realized as a subscheme of $ (\mathbb P^1)^{p([n])} $ by the equations
$$ \alpha_{ij} \alpha_{jk} = \alpha_{ik} \quad \alpha_{ij} = \alpha_{ji}^{-1} $$
Has anyone seen this space before?
 A: Given a linear space $L \subset \mathbb{A}^n$, the closure of $L$ inside of $(\mathbb{P}^1)^n$ is called the Schubert variety of $L$, or of the hyperplane arrangement obtained by intersecting the coordinate hyperplanes with $L$.  Your variety is the Schubert variety of the Coxeter arrangement of type $A_{n-1}$.
The Schubert variety was introduced by Ardila and Boocher https://arxiv.org/pdf/1312.6874.pdf.  The stratification is not explicitly studied there, but it was observed both by Huh and Wang https://arxiv.org/pdf/1609.05484.pdf (who used it to prove the top-heavy conjecture for realizable matroids, which features prominently in Huh's Fileds Medal citation) and by Young, Xu, and myself https://arxiv.org/pdf/1706.05575.pdf.  If you blow up first the point stratum, then the strict transforms of the 1-dimensional strata, and so on, you obtain the augmented wonderful variety https://arxiv.org/pdf/2010.06088.pdf, which is the topic of a lot of current research.  The induced stratification of the affine patch in which all coordinates are nonzero (called the reciprocal plane, or the spectrum of the Orlik--Terao algebra) was first studied much earlier by Speyer and myself https://arxiv.org/pdf/math/0410069.pdf.
Sometimes the Schubert variety of a hyperplane arrangement is referred to as a matroid Schubert variety, since its cohomology ring is a matroid invariant (called the graded Mobius algebra), as is its intersection cohomology module.  But that's kind of sloppy terminology, as the isomorphism type of the variety itself is not determined by the matroid.
In the case that you describe, $Y_n$ can be regarded as the compactification of the configuration space of $n$ labeled points in $\mathbb{A}^1$ up to translation in which the distances between points are allowed to go either to zero or to infinity.  The stratification that you describe comes from asking which collections of points are a finite distance away from each other, which gives a partition of the set $[n]$.  This particular case was studied in detail by Young and myself https://arxiv.org/pdf/1704.04510.pdf.
