Each point x in $\mathrm{Gr}(r,n)$ corresponds to an $r\times n$ matrix $A_x$ which gives rise to a matroid ${\mathcal{M}}_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y\in \mathrm{Gr}(r,n)|{\mathcal{M}}_y={\mathcal{M}}_x\}$ form a stratification of $\mathrm{Gr}(r,n)$ 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.

每个点x in$\mathrm{GR}（\mathrm{[R}，ñ）$ 对应于 $\mathrm{[R}\times ñ$ 矩阵 ${\mathrm{一种}}_X$ 产生了拟阵 ${\mathcal{中号}}_X$在其列上。Gel'fand，Goresky，MacPherson和Serganova展示了这些场景$\{ÿ\in \mathrm{GR}（\mathrm{[R}，ñ）|{\mathcal{中号}}_{ÿ}={\mathcal{中号}}_X\}$ 形成分层 $\mathrm{GR}（\mathrm{[R}，ñ）$具有许多美丽的属性。但是，Mnëv和Sturmfels的结果表明，这些地层可能非常复杂，特别是可能具有任意的奇点。我们研究理想${\mathrm{一世}}_X$类属动物品种，这些地层的Zariski封闭。我们基于射影几何定理构造了几类示例，并描述了Grassmann-Cayley代数如何用于推导的非平凡元素。${\mathrm{一世}}_X$ 当拟阵的组合足够丰富时，则在几何上。