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Minimal values of multiplicities of ideals have a strong relation with thedepth of blowup algebras. In this paper, we introduce the notion ofGoto-minimal $j$-multiplicity for ideals of maximal analytic spread. In aCohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T.Puthenpurakal, and J. Verma, we study the interplay among this new notion, thenotion of minimal $j$-multiplicity introduced by C. Polini and Y. Xie, and theCohen-Macaulayness of the fiber cone of ideals satisfying certain residualassumptions. We are also able to provide a bound on the reduction number ofideals of Goto-minimal $j$-multiplicity having either Cohen-Macaulay associatedgraded algebra, or linear decay in the depth of their powers.

Artin-Nagata properties, minimal multiplicities, and depth of fiber cones