On the quasi-monomiality of the $α$-and $δ$-invariants

Abstract: In this paper, we show that for any projective klt pair $(X,Δ)$ and ample line bundle $L$, there exist quasi-monomial valuations computing $α(X,Δ,L)$ and $δ(X,Δ,L)$, independently of whether the base field is countable. This also yields an alternative proof of the existence of valuations computing $α(X,Δ,L)$ and $δ(X,Δ,L)$ that was originally proved by Blum-Jonsson over an uncountable field.