Multi-Rees Algebras of Strongly Stable Ideals

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