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arXiv · 2307.11915

Singular matroid realization spaces

Daniel Corey·Dante Luber
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Abstract

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 Pl\"ucker coordinates are nonzero -- is not sch\"on for $n\geq 12$.

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Daniel Corey, Dante Luber. 2023-07-21. Singular matroid realization spaces. https://arxiv.org/abs/2307.11915

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