Equations of multi-Rees algebras

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Describing defining equations of Rees algebra is a central problem in commutative algebra. More generally, given a family of ideals, one would like to describe the equations of the multi-Rees algebra of these ideals. Indeed, the multi-Rees algebra in question is simply the Rees algebra of the module. There is little work on the defining equations of the multi-Rees algebra compared to the ordinary Rees algebra. Another motivation for investigating the multi-Rees algebra is an illustration of the theory of Rees algebra of modules. In this work first, we describe defining equations of multi-Rees algebras of powers of an ideal of linear type. In the second part we describe defining equations of multi-Rees algebras of a family of ideals, where these ideals are generated by subsets of a regular sequence.