Polymatroids and moduli of points in flags

TL;DR

This paper explores new compactifications of the moduli space of weighted points in flags, connecting them to polymatroids, toric varieties, and known geometric structures, revealing their fibration and quotient relationships.

Contribution

It introduces novel compactifications of moduli spaces using Fulton-MacPherson methods, linking them to polymatroids and toric varieties, and uncovers their fibration and quotient structures.

Findings

Certain compactifications are toric and isomorphic to polypermutohedral and polystellahedral varieties.

These compactifications have a fibration structure with fibers isomorphic to the Losev-Manin space.

The compactifications are related via geometric quotients.

Abstract

We introduce and study different compactifications of the moduli space of $n$ distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.