In this chapter we investigate the higher rank varieties. They are the analogues of determinantal varieties for more complicated representations LλE. They were first considered in the paper [Po] of Porras.

In section 7.1 we look at the general case. We prove that higher determinantal varieties have rational singularities, and we find equations defining them set-theoretically. We also classify the rank varieties whose defining ideals are Gorenstein.

In section 7.2 we investigate the rank varieties for symmetric tensors of degree bigger than two. We prove that in this case the defining equations described in section 7.1 generate the radical ideal. We also analyze the cases of tensors of rank one, which correspond to the cones over multiple embeddings of projective spaces.

In section 7.3 we look at rank varieties for skew symmetric tensors of degree bigger than two. An interesting feature is that the normality of these rank varieties depends on the characteristic of the base field. We pay particular attention to the special case of syzygies of Plücker ideals defining the cones over Grassmannians embedded into projective space by Plücker embeddings.

Basic Properties

Let λ be a partition. Let E be a vector space of dimension n over K. Consider the representation X = Kλ′E* as an affine space over K.