Matroid and Hypergraph Varieties: An Algebraic and Geometric Approach

In this talk, I will discuss the dimension and irreducible decomposition of several varieties arising from matroids and hypergraphs, including realization spaces, matroid and circuit varieties, and hypergraph varieties. I will begin by introducing the notion of the naive dimension of a matroid realization space and compare it with both the expected and algebraic dimensions, exploring conditions under which these notions agree. I will then present the class of inductively connected matroids and describe the geometry of their realization spaces. Finally, I will outline an efficient method to decompose the circuit variety of a matroid M, based on an algorithm that identifies its minimal extensions. These extensions correspond to the smallest elements above M in the poset defined by the dependency order.

This is joint work with Fatemeh Mohammadi and Rémi Prébet.