Combinatorial flats and Schubert varieties of subspace 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.

arXiv commentFixed error in the statement of Theorem 3.20