Multi-Rees algebras of strongly stable ideals

We prove that the multi-Rees algebra ℛ(I_1 ⊕⋯⊕ I_r) of a collection of strongly stable ideals I_1, … , I_r is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of ℛ(I_1 ⊕⋯⊕ I_r) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of ℛ(I_1 ⊕ I_2) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.