Newton non-degenerate $\hat{I}¼$-constant deformations admit simultaneous embedded resolutions

Abstract

Let ${\mathbb{C}}_o^{n+1}$ denote the germ of ${\mathbb{C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in ${\mathbb{C}}_o^{n+1}$ 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 will prove that $W$ is a $\hat{I}¼$-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 $(W({\mathbb{C}}_o{)}_m{)}_{red}\to {\mathbb{C}}_o$ is flat, where $W({\mathbb{C}}_o{)}_m$ is the relative space of $m$-jets. On the way tothe proof of our main result, we give a complete answer to a question ofArnold on the monotonicity of Newton numbers in the case of convenientNewton polyhedra.

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