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

Abstract

Let R R denote a 6-dimensional subspace of the ring M 4 ( k ) M_4(\Bbbk ) of 4 × 4 4 \times 4 matrices over an algebraically closed field k \Bbbk . Fix a vector space isomorphism M 4 ( k ) ≅ k 4 ⊗ k 4 M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4 . We associate to R R a closed subscheme X R {\mathbf X}_R of the Grassmannian of 2-dimensional subspaces of k 4 \Bbbk ^4 , where the reduced subscheme of X R {\mathbf X}_R is the set of 2-dimensional subspaces Q ⊆ k 4 Q \subseteq \Bbbk ^4 such that ( Q ⊗ k 4 ) ∩ R ≠ { 0 } (Q \otimes \Bbbk ^4) \cap R \ne \{ 0\} . Our main result is that if X R {\mathbf X}_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P 5 \mathbb {P}^5 via the Plücker embedding. We present several examples of X R \mathbf 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 {\mathbf X}_R is a P 1 \mathbb {P}^1 -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 \mathbf X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L \mathcal {L} of a certain graded algebra determined by R R . In that context, it has been an open question for several years to describe such L \mathcal {L} of minimal dimension, i.e., those L \mathcal {L} of dimension one. Our main result implies that if dim ⁡ ( L ) = 1 \dim (\mathcal {L}) = 1 , then, as a subscheme of P 5 \mathbb {P}^5 under the Plücker embedding, deg ⁡ ( L ) = 20 \deg (\mathcal {L}) = 20 .