Combinatorial Foundations for Algebraic Geometry via Matroid Theory

The purpose of this paper is to investigate the connection between matroid theory and algebraic geometry. It demonstrates how matroids offer a unified approach to the study of geometric and combinatorial features. The approach is substantiated by the formulation of four key theorems that establish connections between matroids and divisor classes, rank functions, tropical geometry, and dual complexes in algebraic varieties. Matroids provide essential insights and tools for developing both the theory and practice of algebraic geometry, as demonstrated in this work through the use of thorough proofs and examples.