Combinatorial flats and Schubert varieties of subspace arrangements
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909793621377024 |
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| author | Crowley, Colin Simpson, Connor Wang, Botong |
| author_facet | Crowley, Colin Simpson, Connor Wang, Botong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10552 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorial flats and Schubert varieties of subspace arrangements Crowley, Colin Simpson, Connor Wang, Botong Algebraic Geometry Combinatorics 05E14, 06B05 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. |
| title | Combinatorial flats and Schubert varieties of subspace arrangements |
| topic | Algebraic Geometry Combinatorics 05E14, 06B05 |
| url | https://arxiv.org/abs/2410.10552 |