Combinatorial flats and Schubert varieties of subspace arrangements

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.