Tangent loci and certain linear sections of adjoint varieties

An adjoint variety X (g) associated to a complex simple Lie algebra g is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X (g) in terms of s-fraktur sign and l-fraktur sign₂-triples. Secondly for a graded decomposition of contact type g = ⊕_-2≤i≤2g_i, we show that the intersection of X (g) and the linear subspace ℙ*(g₁) in ℙ*(g) coincides with the cubic Veronese variety associated to g.