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 compactification of the space of points on the affine line
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>.


