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

Commutative Algebra