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

TL;DR

This paper introduces a geometric invariant associated with 6-dimensional subspaces of 4x4 matrices, analyzing the structure and degree of related subschemes within the Grassmannian, with examples involving elliptic curves.

Contribution

It defines a new geometric invariant for 6-dimensional subspaces of matrices and studies the properties and degrees of associated subschemes in the Grassmannian.

Findings

Each irreducible component of ${f X}_R$ has dimension at least one.

When ${ m dim}({f X}_R)=1$, its degree is 20.

Examples include secant varieties of quartic elliptic curves and complex curves with elliptic and conic components.

Abstract

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