McKay correspondence and G-Hilbert schemes

The observation of McKay relates exceptional curves in the minimal resolution of quotient singularities \A²_\C/G for finite subgroups $G\subset\SL(2,\C)$ to the representation theory of the group $G$: the representation graph of the group $G$ is isomorphic to the intersection graph of the exceptional curves, both are graphs of ADE type (classical McKay correspondence). The McKay correspondence in a broader sense describes the geometry of resolutions of quotient singularities $X/G$ in terms of the $G$-equivariant geometry of $X$. A method to construct resolutions of quotient singularities is the $G$-Hilbert scheme $\GHilb X$ for a scheme $X$ with $G$-operation. It parametrises $G$-clusters, these are $G$-stable finite closed subschemes $Z\sub X$, whose coordinate ring as a representation is isomorphic to the regular representation of $G$. In this work we consider McKay correspondence over fields that are not necessarily algebraically closed and for finite group schemes instead of simply finite groups. Let $G\subset\SL(2,K)$ be a finite subgroup scheme over a field $K$ of characteristic $0$. Over non algebraically closed $K$ there may exist both representations of $G$ and components of the exceptional divisor in the minimal resolution of \A²_K/G that are irreducible over $K$ but split over the algebraic closure. We show that these two kinds of splittings that arise by extending the ground field are linked and formulate a McKay correspondence relating nontrivial irreducible representations to exceptional prime divisors over arbitrary fields $K$ of characteristic $0$. With the aim to generalise the McKay correspondence, we generalise the $G$-Hilbert scheme construction to finite group schemes. Further, we introduce relative $G$-Hilbert schemes associated to a scheme with $G$-operation over another scheme and vary the base scheme. This allows to construct the $G$-Hilbert scheme without using the Hilbert scheme of $n$ points. This new construction works under more natural hypotheses, moreover, it yields additional information about the morphism from the $G$-Hilbert scheme to the quotient, which is interpreted as the structure morphism of a relative $G$-Hilbert scheme.

Author

• Blume, Mark

Identifier

hdl:10900/49066

OAI record oai:identifier

https://publikationen.uni-tuebingen.de/oai/request?verb=GetRecord&metadataPrefix=mets&identifier=oai%3Apublikationen.uni-tuebingen.de%3A10900%2F49066