Moduli of codimension two linear sections of subadjoint varieties

Let G be a simple algebraic group of type F₄, E₆, E₇ or E₈, and let g be its Lie algebra. The adjoint variety Xad ⊆ P g is defined as the unique closed orbit of the adjoint action of G on Pg. Xad is a Fano contact manifold covered by lines in P g. The subadjoint variety S ⊆ P W is denoted by the variety of lines on Xad through a fixed point x, where W ⊆ TₓX 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 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 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 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.