Geometric equations for matroid varieties

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