Rees algebras of truncations of complete intersections

In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I   of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal mm. To describe explicitly the Rees algebra R(I)R(I) in terms of generators and relations we map another Rees ring R(M)R(M) onto it, where M   is the direct sum of powers of mm. We compute a Gröbner basis of the ideal defining R(M)R(M). It turns out that the normal domain R(M)R(M) is a Koszul algebra and from this we deduce that in many instances R(I)R(I) is a Koszul algebra as well.