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

Let $k$ be an algebraically closed field and $\mathsf{G}(2,k^4)$ the Grassmannian 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 ${\bf X}_R \subseteq {\sf G}(2,k^4)$\mathrm{.}Weshowthateachirreduciblecomponentof${\bf X}_R$hasdimensionatleastoneandwhen${\rm dim}({\bf X}_R)=1$,then${\rm deg}({\bf X}_R)=20$wheredegreeiscomputedwithrespecttotheambient${\mathbb P}^5$underthePl\ddot{u}ckerembedding${\sf G}(2,k^4) \to {\mathbb P}^5$\mathrm{.}Wegivetwoexamplesinvolvingellipticcurves:inonecase${\bf X}_R$isthesecantvarietyforaquarticellipticcurve,so${\rm dim}({\bf X}_R)=2$,intheother${\bf X}_R$ is a curve having 7 irreducible components, three of which are elliptic curves, and four of which are smooth conics.