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Multi-Rees algebras of strongly stable ideals

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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.

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The authors sincerely thank the reviewers for their helpful and constructive suggestions which improved the presentation of the manuscript immensely. The first named author was partially supported by the University of South Alabama Arts and Sciences Support and Development Award. Thanks to this award, part of this work was done while the second author visited the University of South Alabama. The second author thanks the Department of Mathematics and Statistics for their hospitality. Many of the computations related to this paper was done using Macaulay2 [16].

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Correspondence to Kuei-Nuan Lin.

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Kara, S., Lin, KN. & Castillo, G.S. Multi-Rees algebras of strongly stable ideals. Collect. Math. 75, 213–246 (2024).

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