Moduli of codimension two linear sections of subadjoint varieties

Let [Formula: see text] be a simple algebraic group of type [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] with Lie algebra [Formula: see text]. The adjoint variety [Formula: see text] associated to [Formula: see text] is a Fano contact manifold. The subadjoint variety [Formula: see text] consists of lines on [Formula: see text] through a fixed point [Formula: see text], where [Formula: see text] is the contact hyperplane. Vinberg’s theory of [Formula: see text]-groups identifies the GIT quotient space of codimension two linear sections of [Formula: see text] with the weighted projective space [Formula: see text]. In this paper, we investigate the geometric interpretation of this isomorphism. Our main result establishes, for each [Formula: see text] of the above type, a natural open embedding of the GIT quotient space of nonsingular codimension two linear sections of [Formula: see text] into [Formula: see text] whose complement is a fixed hypersurface of degree 12. The construction is uniform, employing a correspondence of Bhargava and Ho that relates this moduli problem to one concerning curves of genus one.