The normalized local volume of a non-closed point equals an expression built from the normalized local volumes of closed points.
On the quasi-monomiality of the $\alpha$-and $\delta$-invariants
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abstract
In this paper, we show that for any projective klt pair $(X,\Delta)$ and ample line bundle $L$, there exist quasi-monomial valuations computing $\alpha(X,\Delta,L)$ and $\delta(X,\Delta,L)$, independently of whether the base field is countable. This also yields an alternative proof of the existence of valuations computing $\alpha(X,\Delta,L)$ and $\delta(X,\Delta,L)$ that was originally proved by Blum-Jonsson over an uncountable field.
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On the normalized local volume of a non-closed point
The normalized local volume of a non-closed point equals an expression built from the normalized local volumes of closed points.