On Hypersurface Quotient Singularity of Dimension 4

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme, $\hl^G(\CZ^n)$, and crepant resolutions of $\CZ^n/G$ for $G$=the $A$-type abelian group $A_r(n)$. For $n=4$, we obtain the explicit structure of $\hl^{A_r(4)}(\CZ^4)$. The crepant resolutions of $\CZ^4/A_r(4)$ are constructed through their relation with $\hl^{A_r(4)}(\CZ^4)$, and the connections between these crepant resolutions are found by the "flop" procedure of 4-folds. We also make some primitive discussion on $\hl^G(\CZ^n)$ for the $G$= alternating group ${\goth A}_{n+1}$ of degree $n+1$ with the standard representation on $\CZ^n$; the detailed structure of $\hl^{{\goth A}_4}(\CZ^3)$ is explicitly constructed.