Rees Algebras of Diagonal Ideals

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, the kernel of the multiplication map. We prove that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra in some special cases. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.