Multi-Rees algebras of strongly stable ideals

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