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

Abstract

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 ${\mathbf{X}}_R\subseteq \mathsf{G}(2,k^4)$. We show that each irreducible component of ${\mathbf{X}}_R$ has dimension at least one and when $\dim ({\mathbf{X}}_R)=1$, then $\mathrm{deg}({\mathbf{X}}_R)=20$ where degree is computed with respect to the ambient ${\mathbb{P}}^5$ under the Plücker embedding $\mathsf{G}(2,k^4)\to {\mathbb{P}}^5$. We give two examples involving elliptic curves: in one case ${\mathbf{X}}_R$ is the secant variety for a quartic elliptic curve, so $\dim ({\mathbf{X}}_R)=2$, in the other ${\mathbf{X}}_R$ is a curve having 7 irreducible components, three of which are elliptic curves, and four of which are smooth conics.

minor revsion after referee comments

References in corpus (1)

• The One-Dimensional Line Scheme of a Certain Family of Quantum ${\mathbb{P}}^3$s

Cited by in corpus (1)

• Exotic Elliptic Algebras of dimension 4 (with an Appendix by Derek Tomlin)