  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.
