Combinatorial flats and Schubert varieties of subspace arrangements

TL;DR

This paper introduces the concept of combinatorial flats for polymatroids, showing they behave well and can be modeled by Schubert varieties of subspace arrangements when realizable, extending previous work on hyperplane arrangements.

Contribution

It defines combinatorial flats for polymatroids and demonstrates their geometric modeling via Schubert varieties of subspace arrangements, generalizing prior results.

Findings

Combinatorial flats are graded and top-heavy.

When realizable, combinatorial flats correspond to Schubert varieties of subspace arrangements.

The geometry of these Schubert varieties is more complex than hyperplane arrangements.

Abstract

The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "Schubert variety of hyperplane arrangement". In contrast, the lattice of flats of a polymatroid exhibits many combinatorial pathologies and admits no similar geometric model. We address this situation by defining the lattice $\mathcal L_P$ of "combinatorial flats" of a polymatroid $P$. Combinatorially, $\mathcal L_P$ exhibits good behavior analogous to that of $\mathcal L_M$: it is graded, determines $P$ when $P$ is simple, and is top-heavy. When $P$ is realizable over a field of characteristic 0, we show that $\mathcal L_P$ is modeled by "the Schubert variety of a subspace arrangement". Our work generalizes a number of results of Ardila-Boocher and Huh-Wang on Schubert varieties of hyperplane arrangements; however, the geometry of Schubert varieties of subspace arrangements is noticeably more complicated than that of Schubert varieties of hyperplane arrangements. Many natural questions remain open.