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If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\\mathcal{R}(I)$ of $I$. Hence, many invariants of $\\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\\text{v}$-number, exhibit well behaved asymptotic behavior. 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