On the Nash modification of a germ of complex analytic singularity

2016/08/26 by Arturo Giles Flores, Flores, Arturo Giles
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1608.07353

arxiv created 2016/08/26 · openalex publication_date 2016/08/26 · arxiv updated 2016/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a germ (X,0) ⊂ (ℂⁿ,0) of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety Z ⊂ ℂⁿ × G(k,n). We give a characterization of the subvarieties of ℂⁿ × G(k,n) that are the Nash modification of its image under the projection to ℂⁿ. This result generalizes the characterization of conormal varieties as Legendrian subvarieties of ℂⁿ × \checkℙ^n-1 with its canonical contact structure. As a by-product we define the d-conormal space of (X,0) for any d ∈ \k, …, n-1\ which is a generalization of both the Nash modification and the conormal variety of (X,0).

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