The Geometry of Noncommutative Singularity Resolutions

Charlie Beil

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arXiv · 1102.5741

The Geometry of Noncommutative Singularity Resolutions

Charlie Beil

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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.

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BibTeX RIS

Charlie Beil. 2011-02-28. The Geometry of Noncommutative Singularity Resolutions. https://arxiv.org/abs/1102.5741

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