Singular matroid realization spaces

TL;DR

This paper determines the smoothness landscape of matroid realization spaces for small rank, proving that all complex realizable rank-$3$ spaces on at most $11$ elements are smooth while constructing singular examples at $n=12$, and showing smoothness for rank-$4$ spaces with up to $9$ elements. It develops a robust framework combining matroid operations, coordinate-ring presentations, and principal extensions to propagate smoothness and irreducibility from smaller ground sets, complemented by extensive computer-aided verification. A central application is that the open Grassmannian $\mathsf{Gr}^{\circ}(3,n;\mathbb{C})$ is not schön for $n\ge 12$, via singular initial degenerations linked to specific rank-$3$ matroids; the work also connects these phenomena to flag varieties and paving matroids through a systematic study of initial degenerations and hypersimplices. Overall, the results illuminate Mnëv-type universality in concrete, small-scale settings and provide computational tools and criteria for detecting singularity and non-schönness in Grassmannian-related 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 Plücker coordinates are nonzero -- is not schön for $n\geq 12$.

Figures (3)

• Figure 2.1: A projective realization of the $(3,9)$--matroid from Example \ref{['ex:3-9']}
• Figure 3.1: A special and a general configuration of points in $\mathbb{P}^{3}$ realizing the matroid from Example \ref{['ex:morphismNotSmooth']}
• Figure 4.1: Projective realization of $\mathsf{Q}_{\mathsf{sing}}$

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Example 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4
• proof : Proof of Proposition \ref{['prop:principalExtension']}