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Let $k$ be an algebraically closed field and ${\sf G}(2,k^4)$ theGrassmannian of 2-planes in $k^4$. We associate to each 6-dimensional subspace$R$ of the space of 4x4 matrices over $k$ a closed subscheme ${\bf X}_R\subseteq {\sf G}(2,k^4)$. We show that each irreducible component of ${\bfX}_R$ has dimension at least one and when ${\rm dim}({\bf X}_R)=1$, then ${\rmdeg}({\bf X}_R)=20$ where degree is computed with respect to the ambient${\mathbb P}^5$ under the Pl\"ucker embedding ${\sf G}(2,k^4) \to {\mathbbP}^5$. We give two examples involving elliptic curves: in one case ${\bf X}_R$is the secant variety for a quartic elliptic curve, so ${\rm dim}({\bfX}_R)=2$, in the other ${\bf X}_R$ is a curve having 7 irreducible components,three of which are elliptic curves, and four of which are smooth conics.

A geometric invariant of $6$-dimensional subspaces of $4\times 4$ matrices