Irreducibility, Smoothness, and Connectivity of Realization Spaces of Matroids and Hyperplane Arrangements

Abstract: We study the realization spaces of matroids and hyperplane arrangements. First, we define the notion of naive dimension for the realization space of matroids and compare it with the expected dimension and the algebraic dimension, exploring the conditions under which these dimensions coincide. Next, we introduce the family of inductively connected matroids and investigate their realization spaces, establishing that they are smooth, irreducible, and isomorphic to a Zariski open subset of a complex space with a known dimension. Furthermore, we present an explicit procedure for computing their defining equations. As corollaries, we identify families of hyperplane arrangements whose moduli spaces are connected. Finally, we apply our results to study the rigidity of matroids. Rigidity, which involves matroids with a unique realization under projective transformations, is key to understanding the connectivity of the moduli spaces of the corresponding hyperplane arrangements.