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

Authors: Alex Chirvasitu, S. Paul Smith and Michaela Vancliff
Journal: Proc. Amer. Math. Soc. 148 (2020), 915-928
MSC (2010): Primary 15A72
Published electronically: December 30, 2019
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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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Additional Information

Alex Chirvasitu
Affiliation: Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900

S. Paul Smith
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Michaela Vancliff
Affiliation: Department of Mathematics, Box 19408, University of Texas at Arlington, Arlington, Texas 76019-0408

Keywords: Subspaces, $4 \times 4$ matrices, Grassmannian, line scheme, minors, non-commutative analogues of $\mathbb{P}^3$
Received by editor(s): February 4, 2018
Received by editor(s) in revised form: June 14, 2018
Published electronically: December 30, 2019
Additional Notes: The first author was partially supported by NSF grant DMS-1565226.
The third author was partially supported by NSF grant DMS-1302050.
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society