The intersection data of an arrangement of hyperplanes is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this talk, we will similarly characterize the combinatorial structure underlying certain arrangements of subvarieties by defining a class of geometric posets and a generalization of matroids called matroid schemes. We will discuss some notions from matroid theory which extend to this setting and touch on the topological implications of this framework.