Chow quotients of Grassmannians by diagonal subtori

TL;DR

This paper extends the study of torus orbits in Grassmannians to diagonal subtori, introducing Chow quotients related to discrete polymatroids, and explores their connections to moduli spaces and generalized dualities.

Contribution

It generalizes Kapranov's Chow quotient compactifications to diagonal subtori, linking them to discrete polymatroids and extending Gelfand-MacPherson and Gale duality frameworks.

Findings

Chow quotients compactify spaces of parameterized linear subspaces

A birational equivalence to moduli spaces of pointed trees of projective spaces

Open question on extending the Borel transfer principle to Chow quotients

Abstract

The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birational to the Chen-Gibney-Krashen moduli space of pointed trees of projective spaces, and we show that the question of whether this birational map is an isomorphism is a specific instance of a much more general question that hasn't previously appeared in the literature, namely, whether the geometric Borel transfer principle in non-reductive GIT extends to an isomorphism of Chow quotients.