Finite generation of the graded algebra of a delta-minimizing valuation

Determine whether the graded algebra \(\mathrm{gr}_{\nu}R\) is finitely generated over \(k\) for every \(\mathbb{T}\)-invariant valuation \(\nu\) computing \(\delta(X,\Delta;\nu_0)\) on a log Fano cone singularity.

Background

The existence of a minimizing valuation is motivated by stable degeneration results, where finite generation of the associated graded algebra is a central structural property. The paper notes that finite generation is known when ν0\nu_0 is induced by a Reeb vector or when ν\nu minimizes normalized local volume, but leaves the general case of a computing valuation for δ(X,Δ;ν0)\delta(X,\Delta;\nu_0) unresolved.

References

Let $x\in (X:=Spec R,\Delta,\mathbb{T})$ be a log Fano cone singularity, and let $\nu_0\in \mathrm{Val}{\mathbb{T},*}_{X,x}$ be a $\mathbb{T}$-invariant valuation with $A_{X,\Delta}(\nu_0)<\infty$. Let $\nu\in \mathrm{Val}{\mathbb{T},*}_{X,\ni x}$ be a computing valuation of $\delta(X,\Delta;\nu_0)$. Can we say that $\mathrm{gr}_{\nu}R$ is a finitely generated $k$-algebra?

— On the existence of minimizer on a log Fano cone singularity  (2608.23019 - Kim, 24 Aug 2026) in Section 4, “Questions,” final question