2020/04/11 by Agustín Romano Velázquez, Velázquez, Agustín Romano · 1 citation
Mathematics · #13C14 #13H10 #14E16 #32S05 #32S25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2004.05441
openalex publication_date 2020/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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' of X such that the "strict transform" of M is flat over X' 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 OX-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 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 M. 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.