Classification of singular loci of matroid varieties

Classify the singular loci of matroid varieties in general, extending the explicit description established in the paper for rank-2 matroid varieties.

Background

Matroid varieties are Zariski closures of realization spaces determined by the vanishing pattern of Pl0cker coordinates. Their singularities can be substantially more complicated than those of the rank-2 examples analyzed in the paper, and Mn39v19s Universality Theorem indicates the broad complexity of realization spaces.

The paper gives an explicit description for rank-2 matroid varieties: when there are at least three parallel classes, the singular locus is a union of matroid varieties obtained by turning a parallel class of size at least two into loops. A general classification for matroid varieties of arbitrary rank remains unresolved.