International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 48, Pages 2547-2581
doi:10.1155/S0161171204302140
On hypersurface quotient singularities of dimension 4
Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan
Received 18 February 2003
Copyright © 2004 Li Chiang and Shi-Shyr Roan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension n≥4 through the theory of Hilbert scheme of group orbits. For a linear special group G acting on ℂn, we study the G-Hilbert scheme HilbG(ℂn) and crepant resolutions of ℂn/G for G the A-type abelian group Ar(n). For n=4, we obtain the explicit structure of HilbAr(4)(ℂ4). The crepant resolutions of ℂ4/Ar(4) are constructed through their relation with HilbAr(4)(ℂ4), and the connections between these crepant resolutions are found by the “flop” procedure of 4-folds. We also make some primitive discussion on HilbG(ℂn) for G the alternating group 𝔄n+1 of degree n+1 with the standard representation on ℂn; the detailed structure of Hilb𝔄4(ℂ3) is explicitly constructed.