Finite generation of higher rank quasi-monomial valuations via the extended Rees algebra

In the algebraic theory of K-stability, one of the most challenging problems is to show the graded algebra associated with certain higher rank quasi-monomial valuations are finitely generated. In the global case of Fano varieties and local case of klt singularities, the finite generation has been proved for quasi-monomial valuations on models of qdlt Fano type. In this paper, we generalize these results using a different argument, by studying the extended Rees algebra via a more algebraic approach. As consequences, our results apply to fibrations of Fano type with singularities worse than qdlt, and graded algebras coming from the multi-section ring of arbitrary divisors.