Singular matroid realization spaces

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 $n\geq 12$.