Exploring Matroid Varieties: A Tropical Perspective on Combinatorial Structures and Intersection Products

Matroid varieties give a wide range of tropicalizations and a complex way to understand their combinatorial structure when applied to classical linear spaces. In this paper, we delve into the intricate connections between matroid theory and tropical geometry, highlighting how matroid varieties inherit a natural fan structure that elegantly organises themselves based on the flats of thematroid they stem from. This connection facilitates a seamless translation of matroid operations into the tropical realm, akin to speaking the same language in a different mathematical dialect. We focus on constructing an intersection product of cycles on matroid varieties, akin to understanding how loops interact within the landscape. By demonstrating the utility and consistency of this operation, we pave the way for further exploration and application in both matroid and tropical geometry.