  As a continuation of a general program started in two previous publications,
in the present paper we study the K\"ahler quotient resolution of the orbifold
$\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description
of the same. In this way we determine the algebraic structure of the
exceptional divisor, whose compact component is the second Hirzebruch surface
$\mathbb F_2$. We determine the explicit K\"ahler geometry of the smooth
resolved manifold $Y$, which is the total space of the canonical bundle of
$\mathbb F_2$. We study in detail the chamber structure of the space of
stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos
parameters) that are involved in the construction of the desingularizations
either by generalized Kronheimer quotient, or as algebro-geometric quotients.
The walls of the chambers correspond to two degenerations; one is a partial
desingularization of the quotient, which is the total space of the canonical
bundle of the weighted projective space $\mathbb P[1,1,2]$, while the other is
the product of the ALE space $A_1$ by a line, and is related to the full
resolution in a subtler way. These geometrical results will be used to look for
exact supergravity brane solutions and dual superconformal gauge theories.
