Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models

TL;DR

This paper explores hypergraph varieties through projective geometry and matroid theory, providing explicit decompositions, minimal matroids, and realizability results, with applications to statistical conditional independence models.

Contribution

It introduces explicit irreducible decompositions of hypergraph varieties, characterizes minimal matroids, and connects these to statistical CI models, advancing understanding of their geometric and combinatorial structures.

Findings

Hypergraph varieties decompose into irreducible matroid varieties.

All such matroids are realizable over real numbers.

The decomposition simplifies analysis of related statistical models.

Abstract

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the Mn\"ev--Sturmfels universality theorem. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. As corollaries, we give conceptual decompositions of various, previously-studied, varieties associated with graphs, hypergraphs, and adjacent minors of generic matrices. In particular, our decomposition strategy gives immediate matroid interpretations of the irreducible components of multiple families of varieties associated to conditional independence (CI) models in statistical theory and unravels their symmetric structures which hugely simplifies the computations.