On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms

Abstract

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor lies in any of its osculating planes. While the non-generic lines have been classified into $G$-orbits in literature, it has been an open problem to classify the generic lines into $G$-orbits. For a general field $F$ of characteristic different from $2$ and $3$, the twisted cubic determines a symplectic polarity on $\mathbb P^3$. In the Klein representation of lines of $\mathbb P^3$, the tangent lines of $C$ are represented by a degree $4$ rational normal curve in a hyperplane $\mathcal H$ of the second exterior power $\mathbb P^5$ of $\mathbb P^3$. Atiyah studied the lines of $\mathbb P^3$ with respect to $C$, in terms of the geometries of these two curves. Polar duality of lines on $\mathbb P^3$ corresponds to Hodge duality on $\mathbb P^5$, and $\mathcal H$ is the hyperplane of Hodge self-dual elements of $\mathbb P^5$. We show that $\mathcal H$ can be identified in a $PGL_2$-equivariant way with the space of binary quartic forms over $F$, and that pairs of polar dual lines of $\mathbb P^3$ correspond to binary quartic forms whose apolar invariant is a square. We first solve the open problem of classifying binary quartic forms over $GF(q)$ into $G$-orbits, and then use it to solve the main problem.