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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 descriptionof the same. In this way we determine the algebraic structure of theexceptional divisor, whose compact component is the second Hirzebruch surface$\mathbb F_2$. We determine the explicit K\"ahler geometry of the smoothresolved 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 ofstability parameters (corresponding in gauge theory to the Fayet-Iliopoulosparameters) that are involved in the construction of the desingularizationseither by generalized Kronheimer quotient, or as algebro-geometric quotients.The walls of the chambers correspond to two degenerations; one is a partialdesingularization of the quotient, which is the total space of the canonicalbundle of the weighted projective space $\mathbb P[1,1,2]$, while the other isthe product of the ALE space $A_1$ by a line, and is related to the fullresolution in a subtler way. These geometrical results will be used to look forexact supergravity brane solutions and dual superconformal gauge theories.

Crepant Resolutions of $\mathbb{C}^3/\mathbb{Z}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)