Equisingularity of sections, (𝑡^{𝑟}) condition, and the integral closure of modules

Gaffney, Terence; Trotman, David; Wilson, Leslie

doi:10.1090/S1056-3911-08-00488-8

Equisingularity of sections, $(t^{r})$ condition, and the integral closure of modules

Authors: Terence Gaffney, David Trotman and Leslie Wilson
Journal: J. Algebraic Geom. 18 (2009), 651-689
DOI: https://doi.org/10.1090/S1056-3911-08-00488-8
Published electronically: December 2, 2008
MathSciNet review: 2524594
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Abstract | References | Additional Information

Abstract: This paper uses the theory of integral closure of modules to study the sections of both real and complex analytic spaces. The stratification conditions, which play a key role, are the $(t^{r})$ conditions introduced by Thom and Trotman. Our results include an algebraic formulation of the $(t^{r})$ in terms of the integral closure of modules, and a new simple proof showing how the $(t^{r})$ conditions improve under Grassmann modification. In the complex analytic case, we characterise $(t^{r})$ in terms of the genericity of the multiplicity of a certain submodule of the Jacobian module, then use the principle of specialisation of integral dependence for modules to give an equimultiplicity criterion for $(t^{r})$. As a consequence we obtain numerical criteria for Verdier equisingularity of families of plane sections in various situations.

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Additional Information

Terence Gaffney
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Address at time of publication: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720-5070
MR Author ID: 70390
ORCID: 0000-0003-3420-0150
Email: gaff@research.neu.edu

David Trotman
Affiliation: LATP (UMR 6632), Centre de Mathématique et Informatique, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille, France
Email: David.Trotman@cmi.univ-mrs.fr

Leslie Wilson
Affiliation: Department of Mathematics, University of Hawaii at Manoa, 2565 The Mall, 96822 Honolulu, Hawaii
Email: les@math.hawaii.edu

Received by editor(s): April 30, 2007
Received by editor(s) in revised form: July 2, 2007
Published electronically: December 2, 2008
Additional Notes: The first author was supported in part by NSF Grant #9803691. The first and third authors were supported in part by the University of Provence (Aix-Marseille 1). The second author was supported in part by the European Real Algebraic and Analytic Geometry project, and by the University of Hawaii.