# Geometric Equations for Matroid Varieties

## TL;DR

This paper investigates the algebraic and geometric properties of matroid varieties within Grassmannians, exploring their stratification, singularities, and ideal structures using projective geometry and Grassmann-Cayley algebra.

## Contribution

It introduces new classes of examples of matroid varieties and demonstrates how Grassmann-Cayley algebra can be used to derive non-trivial ideal elements geometrically.

## Key findings

- Matroid strata can have arbitrary singularities.
- Construction of new classes of matroid varieties.
- Application of Grassmann-Cayley algebra to derive ideal elements.

## Abstract

Each point $x$ in Gr$(r,n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in \mathrm{Gr}(r,n) | M_y = M_x\}$ form a stratification of Gr$(r,n)$ with many beautiful properties. However, results of Mn\"ev and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals $I_x$ of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich.