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