Geometric Equations for Matroid Varieties

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 Mnëv 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.

Updated Proposition 2.1.3. Added Theorem 2.1.7 and Remark 3.0.3