Singular matroid realization spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916651965874176 |
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| author | Corey, Daniel Luber, Dante |
| author_facet | Corey, Daniel Luber, Dante |
| contents | We study smoothness of realization spaces of matroids for small rank and ground set. For $\mathbb{C}$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer elements, and there are singular realization spaces for $12$ and greater elements. For rank $4$ and $9$ or fewer elements, we prove that these realization spaces are smooth. As an application, we prove that $\text{Gr}^{\circ}(3,n;\mathbb{C})$ -- the locus of the Grassmannian where all Plücker coordinates are nonzero -- is not schön for $n\geq 12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11915 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singular matroid realization spaces Corey, Daniel Luber, Dante Algebraic Geometry Combinatorics 14B05 (primary) 05E14, 14T90, 52B40 (secondary) We study smoothness of realization spaces of matroids for small rank and ground set. For $\mathbb{C}$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer elements, and there are singular realization spaces for $12$ and greater elements. For rank $4$ and $9$ or fewer elements, we prove that these realization spaces are smooth. As an application, we prove that $\text{Gr}^{\circ}(3,n;\mathbb{C})$ -- the locus of the Grassmannian where all Plücker coordinates are nonzero -- is not schön for $n\geq 12$. |
| title | Singular matroid realization spaces |
| topic | Algebraic Geometry Combinatorics 14B05 (primary) 05E14, 14T90, 52B40 (secondary) |
| url | https://arxiv.org/abs/2307.11915 |