open access

https://doi.org/10.14943/83494

https://hdl.handle.net/2115/69098

Hilbert schemes and simple singularities An and Dn

英語

1996-09-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

348

1-22

For any finite subgroup G of SL(2, C) of order n, we consider a certain subscheme HilbG(A2 ) of Hilbn(A2 ) consisting of G-invariant 0-dimensional subschemes of length n. We prove without using the classification of finite subgroups in SL(2,C) that HilbG(A2 ) is a minimal resolution of a simple singularity A2 /G in the canonical manner. Any point of the exceptional set is a G-invariant 0-dimensional subscheme of A2 with support the origin, to which we associate the ideal sheaf I defining the subscheme. A minimal G-submodule of I generating the O A2-module I is a one dimensional trivial G-module plus one or two nontrivial mutually distinct irreducible G-modules. This gives a bijective correpondence between the set of all the irreducible components of the exceptional set and the set of all the equivalence classes of irreducible G-modules, which turns out to be the ( classical) McKay correspondence.

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