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

DOI10.1007/S00209-022-03152-YarXiv2004.05441MaRDI QIDQ6338540

Agustín Romano-Velázquez

Publication date: 11 April 2020

Abstract: Raynaud and Gruson developed the theory of blowing-up an algebraic variety $X$ along a coherent sheaf $M$ in the sense that there exists a blow-up $X^{\prime }$ of $X$ such that the "strict transform" of $M$ is flat over $X^{\prime }$ and the blow-up satisfies an universal (minimality) property. 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 $mathcalM$ 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 $mathcalM$. Moreover, we prove that if $X$ is Gorenstein and $M$ is special in the sense of Wunram and Riemenschneider (generalized in a previous work by Bobadilla and the author), 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.

Mathematics Subject Classification ID

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) (13H10) Complex surface and hypersurface singularities (32S25) Cohen-Macaulay modules (13C14) Local complex singularities (32S05) McKay correspondence (14E16)

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