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

Chirvasitu, Alex; Smith, S.; Vancliff, Michaela

doi:10.1090/proc/14294

A geometric invariant of $6$-dimensional subspaces of $4\times 4$ matrices
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Proc. Amer. Math. Soc. 148 (2020), 915-928 Request permission

Abstract:

Let $R$ denote a 6-dimensional subspace of the ring $M_4(\Bbbk )$ of $4 \times 4$ matrices over an algebraically closed field $\Bbbk$. Fix a vector space isomorphism $M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4$. We associate to $R$ a closed subscheme ${\mathbf X}_R$ of the Grassmannian of 2-dimensional subspaces of $\Bbbk ^4$, where the reduced subscheme of ${\mathbf X}_R$ is the set of 2-dimensional subspaces $Q \subseteq \Bbbk ^4$ such that $(Q \otimes \Bbbk ^4) \cap R \ne \{ 0\}$. Our main result is that if ${\mathbf X}_R$ has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of $\mathbb {P}^5$ via the Plücker embedding.

We present several examples of $\mathbf X_R$ that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, ${\mathbf X}_R$ is a $\mathbb {P}^1$-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 $\mathbf X_R$ appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme $\mathcal {L}$ of a certain graded algebra determined by $R$. In that context, it has been an open question for several years to describe such $\mathcal {L}$ of minimal dimension, i.e., those $\mathcal {L}$ of dimension one. Our main result implies that if $\dim (\mathcal {L}) = 1$, then, as a subscheme of $\mathbb {P}^5$ under the Plücker embedding, $\deg (\mathcal {L}) = 20$.

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