[Paper Review] Singular matroid realization spaces

Daniel Corey, Dante Luber|arXiv (Cornell University)|Jul 21, 2023

Retinoids in leukemia and cellular processesBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper establishes that the realization spaces of complex-realizable rank 3 matroids are smooth for ground sets of size 11 or fewer, but become singular at size 12 or more, with nodal singularities appearing. Using theoretical reductions and computer-aided verification via OSCAR, the authors prove that the open Grassmannian $Γ^{∞}(3,n;\mathbb{C})$ is not schön for $n \geq 12$, resolving a key question in tropical geometry and moduli theory.

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 $ ext{Gr}^{\circ}(3,n;\mathbb{C})$ -- the locus of the Grassmannian where all Plücker coordinates are nonzero -- is not schön for $n\geq 12$.

Motivation & Objective

• To determine the smallest ground set size $n$ for which a complex-realizable rank 3 matroid has a singular realization space.
• To extend the study of smoothness and irreducibility to rank 4 matroids with small ground sets.
• To investigate the schön property of the open Grassmannian $\mathsf{Gr}^\circ(3,n;\mathbb{C})$ and its moduli space $\mathcal{R}(3,n)$, particularly for $n \geq 12$.
• To apply matroid realization theory to tropical geometry, specifically to initial degenerations and their singularities.
• To identify conditions under which realization spaces are singular and to explore the range of possible singularity types for fixed $(d,n)$ pairs.

Proposed method

• Use of principal extensions and structural results to reduce the verification of smoothness to smaller matroids, leveraging closure under free extension.
• Employment of the OSCAR computational algebra system to verify smoothness for 50,945 out of 500,957 matroids, significantly reducing the computational burden.
• Application of Mnëv’s universality theorem and the Gelfand-MacPherson correspondence to relate matroid strata to initial degenerations of the Grassmannian.
• Construction of one-parameter families of matrices $E_1(t)$ and $E_2(t)$ with algebraically independent entries to model initial degenerations and analyze tropicalizations.
• Use of inverse limit constructions and morphisms $\pi: \mathsf{Gr}(\mathsf{w}) \to \mathcal{R}(\mathsf{Q}_{\text{sing}};\mathbb{C})$ to analyze the structure of initial degenerations.
• Verification that the tropicalizations $\mathsf{trop}(\mathsf{p}_1(t))$ and $\mathsf{trop}(\mathsf{p}_2(t))$ equal a fixed weight vector $\mathsf{w}$, confirming $\mathsf{w} \in \mathsf{TGr}^\circ(3,n;\mathbb{C})$.

Experimental results

Research questions

• RQ1 What is the minimal $n$ such that there exists a complex-realizable rank 3 matroid on $n$ elements with a singular realization space?
• RQ2 Are the realization spaces of complex-realizable rank 4 matroids smooth for $n \leq 9$?
• RQ3 Is the open Grassmannian $\mathsf{Gr}^\circ(3,n;\mathbb{C})$ schön for $n = 9, 10, 11$?
• RQ4 What singularity types can appear in realization spaces of matroids for fixed $(d,n)$ pairs, and are there obstructions to certain types?
• RQ5 Can the techniques used for $n \geq 12$ be extended to resolve the schön property for $n = 9,10,11$?

Key findings

• The realization spaces of $\mathbb{C}$-realizable rank 3 matroids on 11 or fewer elements are all smooth.
• For $n \geq 12$, there exist rank 3 matroids whose realization spaces have nodal singularities over $\mathbb{C}$, establishing $n=12$ as the minimal such size.
• The realization spaces of $\mathbb{C}$-realizable rank 4 matroids on $n \leq 9$ elements are all smooth.
• The open Grassmannian $\mathsf{Gr}^\circ(3,n;\mathbb{C})$ is not schön for $n \geq 12$, as confirmed by the existence of singular initial degenerations.
• The moduli space $\mathcal{R}(3,n)$ is not schön for $n \geq 12$, due to the equivalence of the schön property under torus factors.
• The inverse limit morphism $\psi_{\mathsf{w}}$ is an isomorphism, confirming the structure of the initial degeneration $\mathsf{Gr}(\mathsf{w})$ as a reduced scheme with two irreducible components meeting transversally.