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Crepant resolutions of 3-dimensional quotient singularities via Cox rings. (English) Zbl 1433.14011

This paper studies crepant resolutions of quotient singularities of the form \(\mathbb C^3/G\), where \(G\) is a finite subgroup of \(\mathrm{SL}_3(\mathbb C)\). A crepant resolution of \(\mathbb C^3/G\) always exists, but unlike in the surface case, it is usually not unique. All crepant resolutions of \(\mathbb C^3/G\) are related by small \(\mathbb Q\)-factorial modifications.
The authors study and describe the Cox ring of a crepant resolution \( X_0 \to \mathbb C^3/ G\), and obtain information on the structure of the set of all crepant resolutions of \(\mathbb C^3/G\) such as their number, and relations between them. This information is encoded in the chamber decomposition of the movable cone \(\mathrm{Mov}(X_0)\) determined by the Cox ring. Based on previous work starting with [M. Donten-Bury and J. A. Wiśniewski, Kyoto J. Math. 57, No. 2, 395–434 (2017; Zbl 1390.14048)], the authors produce generating sets for the Cox ring of \(X_0\to \mathbb C^3/ G\) when \(G\) is a representation of a dihedral group, when \(G\) is a non-abelian reducible representation and in three cases where \(G\) is an irreducible representation.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14E30 Minimal model program (Mori theory, extremal rays)
14E16 McKay correspondence
14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
14C20 Divisors, linear systems, invertible sheaves

Citations:

Zbl 1390.14048
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References:

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