We prove that the multi-Rees algebra \({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\) of a collection of strongly stable ideals \(I_1, \ldots , I_r\) 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 \({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner 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.

我们证明了一组强稳定理想\(I_1, \ldots , I_r\)的多重 Rees 代数\({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\ )是纤维类型的. 特别是，我们为其定义理想提供了一个 Gröbner 基，作为其特殊纤维和二项式合集的 Gröbner 基的并集。我们还根据与集合相关的参数研究\({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\)的 Koszulness。此外，我们建立了\({\mathcal {R}}(I_1 \oplus I_2)\)的定义理想的二次 Gröbner 基，其中每个强稳定理想都有两个二次 Borel 生成元。因此，我们得出结论，这个多 Rees 代数是 Koszul。