Modular Interpretation Of A Non-Reductive Chow Quotient
This work is a preprint that is freely available from arXiv.org at arXiv:1509.03608, courtesy of Cambridge University Press and Edinburgh Mathematical Society.
The final publication version can be freely accessed courtesy of Cambridge Core Share.
The space of n distinct points and adisjoint parametrized hyperplane in projective d-space up to projectivity – equivalently, configurations of n distinct points in affine d-space up to translation and homothety – has a beautiful compactification introduced by Chen, Gibney and Krashen. This variety, constructed inductively using the apparatus of Fulton–MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of (ℙd)n by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen–Gibney–Krashen space Td, n. This is a non-reductive analogue of Kapranov's famous quotient construction of , and indeed as a special case we show that is the Chow quotient of (ℙ1)n−1 by an action of �m ⋊ �a.