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On the blow-up of a normal singularity at maximal Cohen–Macaulay modules

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Abstract

In Raynaud and Gruson (Invent Math 13:1–89, 1971) and Raynaud (Compos Math 24:11–31, 1972) developed the theory of blowing-up an algebraic variety X along a coherent sheaf M . However, not much is known about the singularities of the blow-up. In this article, we prove that if X is a normal surface singularity and M is a reflexive \({\mathcal {O}}_{X}\)-module, then such a blow-up arises naturally from the theory of McKay correspondence. We show that the normalization of the blow-up of Raynaud and Gruson is obtained by a resolution of X such that the full sheaf \(\mathcal {M}\) associated to M (i.e., the reflexive hull of the pull-back of M) is globally generated and then contracting all the components of the exceptional divisor not intersecting the first Chern class of \(\mathcal {M}\). Moreover, we prove that if X is Gorenstein and M is special in the sense of Wunram (Math Ann 279(4):583–598, 1988) and Riemenschneider (Compos Math 24:11–31, 1972) (generalized in Fernández de Bobadilla and Romano-Velázquez (Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli, arXiv:1812.06543, 2018)), then the blow-up of Raynaud and Gruson is normal. Finally, we use the theory of matrix factorization developed by Eisenbud, to give concrete examples of such blow-ups.

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Acknowledgements

We would like to thank Javier Fernández de Bobadilla and Ananyo Dan for helpful discussions during the course of this work. We also thank Takehiko Yasuda for his comments in a previous version of this article. We thank the anonymous referee for the careful reading of our paper.

Funding

The author is partially supported by European Research Council Executive Agency 615655 NMST Consolidator Grant, Consejo Nacional de Ciencia y Tecnología CB 2016-1 Num. 286447, Consejo Nacional de Ciencia y Tecnología 253506, Fondo Institucional de Fomento Regional para el Desarrollo Científico, Tecnológico y de Innovación 265667 and by Tata Institute of Fundamental Research Visiting Fellow.

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  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India

    Agustín Romano-Velázquez

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The author is partially supported by European Research Council Executive Agency 615655 NMST Consolidator Grant, Consejo Nacional de Ciencia y Tecnología CB 2016-1 Num. 286447, Consejo Nacional de Ciencia y Tecnología 253506, Fondo Institucional de Fomento Regional para el Desarrollo Científico, Tecnológico y de Innovación 265667 and by Tata Institute of Fundamental Research Visiting Fellow.

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Romano-Velázquez, A. On the blow-up of a normal singularity at maximal Cohen–Macaulay modules. Math. Z. 303, 6 (2023). https://doi.org/10.1007/s00209-022-03152-y

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