International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 48, Pages 2547-2581

On hypersurface quotient singularities of dimension 4

Li Chiang and Shi-Shyr Roan

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.


We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension n4 through the theory of Hilbert scheme of group orbits. For a linear special group G acting on n, we study the G-Hilbert scheme HilbG(n) and crepant resolutions of n/G for G the A-type abelian group Ar(n). For n=4, we obtain the explicit structure of HilbAr(4)(4). The crepant resolutions of 4/Ar(4) are constructed through their relation with HilbAr(4)(4), and the connections between these crepant resolutions are found by the “flop” procedure of 4-folds. We also make some primitive discussion on HilbG(n) for G the alternating group 𝔄n+1 of degree n+1 with the standard representation on n; the detailed structure of Hilb𝔄4(3) is explicitly constructed.