Moduli of codimension two linear sections of subadjoint varieties

Let $G$ be a simple algebraic group of type $F_{4}$, $E_{6}$, $E_{7}$ or $E_{8}$, and let $\mathfrak{g}$ be its Lie algebra. The adjoint variety $X_{ad} \subseteq \mathbb{P} \mathfrak{g}$ is defined as the unique closed orbit of the adjoint action of $G$ on $\mathbb{P}\mathfrak{g}$. $X_{ad}$ is a Fano contact manifold covered by lines in $\mathbb{P} \mathfrak{g}$. The subadjoint variety $S \subseteq \mathbb{P} W$ is denoted by the variety of lines on $X_{ad}$ through a fixed point $x$, where $W \subseteq T_{x}X$ is taken as the contact hyperplane. It follows from a result in representation theory of Vinberg that the GIT quotient space of codimension two linear sections of $S$ is isomorphic to the weighted projective space $\mathbb{P}(1,3,4,6)$. In this note, we investigate the problem of finding a geometric interpretation of the above isomorphism. As a main result, for each $\mathfrak{g}$ of the above type, we construct a natural open embedding of the GIT quotient space of nonsingular codimension two linear sections of $S$ into $\mathbb{P}(1,3,4,6)$ whose complement is a fixed hypersurface of degree 12. The key ingredient of our construction is to apply a correspondence of Bahargava and Ho which relates the above moduli problem to a moduli problem on curves of genus one.