# Finite generation of certain graded sequences of ideals

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 $$\frak{a}_\it{m}\rm:=\mathcal{I}_\it Y^{}$$, where $$\mathcal{I}_Y^{

}$$

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 $$V$$?

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.