A geometric invariant of 6-dimensional subspaces of 4 × 4 matrices

Let R denote a 6-dimensional subspace of the ring M₄(k) of 4 × 4 matrices over an algebraically closed field k. Fix a vector space isomorphism M₄(k) ^∼= k⁴ k⁴. We associate to R a closed subscheme X_R of the Grassmannian of 2-dimensional subspaces of k⁴, where the reduced subscheme of X_R is the set of 2-dimensional subspaces Q ⊆ k⁴ such that (Qk⁴) ∩R = {0}. Our main result is that if X_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P⁵ via the Plücker embedding. We present several examples of X_R that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, X_R is a P¹-bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L of a certain graded algebra determined by R. In that context, it has been an open question for several years to describe such L of minimal dimension, i.e., those L of dimension one. Our main result implies that if dim(L) = 1, then, as a subscheme of P⁵ under the Plücker embedding, deg(L) = 20.