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Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\frak{a}}_\it{m}\}$ be a graded sequence of ideal sheaves on $U$ given by $\frak{a}_\it{m}\rm:=\mathcal{I}_\it Y^{<cm>}$, where $\mathcal{I}_Y^{<p>}$ is the $p$-th symbolic power of $\mathcal{I}_Y$.

If the log canonical threshold $\text{LCT}_o(\frak{a}_{\bullet})=\rm 1$ at $o$, how can we establish that there exists an open neighborhood $V\subset\subset U$ of $o$ such that the Rees algebra of $\frak{a}_{\bullet}$ is a finitely generated $\mathcal{O}_U$-algebra on $V$?

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The best (algebraic) advice I can give is to look at section 2.2.5 of https://arxiv.org/pdf/1812.03538.pdf ( Uniqueness of K-polystable degenerations of Fano varieties by Blum and Xu) (eventually, this follows from BCHM)

When the sequence $\mathfrak{a}_m$ is the induce by an exceptional divisor which is an extraction, finite generation follows from ampleness.

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