Combinatorial flats and Schubert varieties of subspace arrangements

The lattice of flats <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℒ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> of a matroid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> is combinatorially well-behaved and, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℒ</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> of “combinatorial flats” of a polymatroid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> . Combinatorially, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℒ</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> exhibits good behavior analogous to that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℒ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> : it is graded, determines <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> is simple, and is top-heavy. When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> is realizable over a field of characteristic 0, we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℒ</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> 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.