Newton non-degenerate $μ$-constant deformations admit simultaneous embedded resolutions

28 Jan 2020  ·  Maximiliano Leyton-Álvarez, Hussein Mourtada, Mark Spivakovsky ·

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 will 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 $(W(\mathbb{C}_o)_m)_{red} \rightarrow \mathbb{C}_{o}$ 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. read more

PDF Abstract
No code implementations yet. Submit your code now

Categories


Algebraic Geometry 14B05, 14B07, 14B25, 14E15, 14E25, 32S05, 32S10, 32S15