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Newton non-degenerate $\mu$-constant deformations admit simultaneous embedded resolutions

Part of: Local theory Birational geometry Singularities

Published online by Cambridge University Press:  11 August 2022

Maximiliano Leyton-Álvarez ,
Hussein Mourtada  and
Mark Spivakovsky
Maximiliano Leyton-Álvarez
Affiliation:
Instituto de Matemáticas, Universidad de Talca, Camino Lircay S\N, Campus Norte, 3460787, Talca, Chile mleyton@utalca.cl leyton@inst-mat.utalca.cl
Hussein Mourtada
Affiliation:
Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France hussein.mourtada@imj-prg.fr
Mark Spivakovsky
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219 du CNRS, Université Paul Sabatier, CNRS, 118 route de Narbonne, 31062 Toulouse, France mark.spivakovsky@math.univ-toulouse.fr LaSol, UMI 2001, Instituto de Matemáticas, Unidad de Cuernavaca, Av. Universidad s/n Periferica, Cuernavaca, 62210 Morelos, Mexico
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Abstract

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W( {\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W( {\mathbb {C}}_{o})}_{m}$ is the relative space of $m$-jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra.

Keywords

algebraic geometrysingularities of algebraic varietiesdeformations of singularitiessimultaneous embedded resolutionsMilnor numberNewton numberm-jet spaces

MSC classification

Primary: 14B05: Singularities 14B07: Deformations of singularities 14B25: Local structure of morphisms 14E15: Global theory and resolution of singularities 14E25: Embeddings 32S15: Equisingularity (topological and analytic)
Secondary: 32S05: Local singularities 32S10: Invariants of analytic local rings
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 6 , June 2022 , pp. 1268 - 1297
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The first author is partially supported by Projects ANID FONDECYT 1170743 and 1221535. The second author is partially supported by Projet ANR LISA, ANR-17-CE40-0023.

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