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Singular matroid realization spaces
Daniel Corey, Dante Luber
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We study smoothness of realization spaces of matroids for small rank and ground set. For 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 Gr°(3,n;ℂ) -- the locus of the Grassmannian where all Pl\"ucker coordinates are nonzero -- is not sch\"on for n≥12.
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