2019/04/23 by Dmitry Kerner, Kerner, Dmitry
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1904.10790
openalex publication_date 2019/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider matrices with entries in a local ring, Mat(R). Fix a group action, G on Mat(R), and a subset of allowed deformations, Σ. The traditional objects of study in Singularity Theory and Algebraic Geometry are the tangent spaces T(Σ,A), T(GA,A), and their quotient, the tangent module to the miniversal deformation, T1(Σ,G,A). This module plays the key role in various deformation problems, e.g., deformations of maps, of modules, of (skew-)symmetric forms. In particular, the first question is to determine the support/annihilator of this tangent module. In [Belitski-Kerner.1] we have studied this tangent module for various R-linear group actions. In the current work we study the support of the module T1(Σ,G,A) for group actions that involve automorphisms of the ring. (Geometrically, these are group actions that involve the local coordinate changes.) We obtain various bounds on localizations of T1(Σ,G,A) and compute the radical of the annihilator of T1(Σ,G,A), i.e., the set-theoretic support. This brings the definition of an (apparently new) type of singular locus, the "essential singular locus" of a map/sub-scheme. It reflects the "unexpected" singularities of a subscheme, ignoring those imposed by the singularities of the ambient space. Unlike the classical singular locus (defined by a Fitting ideal of the module of differentials) the essential is defined by the annihilator ideal of the module of derivations.