On Hypersurface Quotient Singularity of Dimension 4

Details On Hypersurface Quotient Singularity of Dimension 4 On Hypersurface Quotient Singularity of Dimension 4

2000-11-21

arxiv.org/abs/math/0011151v4

IJMMS 2004:48 (2004) 2547-2581

Mathematics

Algebraic Geometry

27 pages, Latex, 11 figures, Some reorganizations and improvement of presentations, Typos corrected, Arguments of Theorem 1 of

Scientific paper

10.1155/S0161171204302140

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.

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