• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.59-68
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• 2022

For a finite group G ⊂ GL(n, ℂ), the G-Hilbert scheme is a fine moduli space of G-clusters, which are 0-dimensional G-invariant subschemes Z with H⁰(𝒪_Z ) isomorphic to ℂ[G]. In many cases, the G-Hilbert scheme provides a good resolution of the quotient singularity ℂⁿ/G, but in general it can be very singular. In this note, we prove that for a cyclic group A ⊂ GL(n, ℂ) of type ${\frac{1}{r}}$(1, …, 1, a) with r coprime to a, A-Hilbert Scheme is smooth and irreducible.