Diagonal subalgebras of residual intersections

Let \mathsf {k} be a field, let S be a bigraded \mathsf {k}-algebra, and let S_\Delta denote the diagonal subalgebra of S corresponding to \Delta = \{ (cs,es) \; \vert \; s \in \mathbb{Z} \}. It is known that the S_\Delta is Koszul for c,e \gg 0. In this article, we find bounds for c,e for S_\Delta to be Koszul when S is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul and Cohen-Macaulay properties of the diagonal subalgebras of their Rees algebras.