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The Geometry of Noncommutative Singularity Resolutions

By Charlie Beil

Abstract

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, non-conventional resolutions that are hidden if only commutative functions are considered. Geometrically, these non-conventional resolutions result from shrinking exceptional loci to ramified (non-Azumaya) point-like spheres.Comment: 42 pages. Comments welcom

Topics: Mathematics - Algebraic Geometry, High Energy Physics - Theory, Mathematics - Rings and Algebras, Mathematics - Representation Theory
Year: 2011
OAI identifier: oai:arXiv.org:1102.5741
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