论文标题
牛顿非脱位$μ$ $ - 恒定变形同时嵌入了分辨率
Newton non-degenerate $μ$-constant deformations admit simultaneous embedded resolutions
论文作者
论文摘要
令$ \ mathbb {c}^{n+1} _o $ $表示$ \ mathbb {c}^{n+1} $的细菌。令$ v $为$ \ mathbb {c}^{n+1} _o $和$ w $的hypersurface germ，$ v $ over $ \ mathbb {c} _ {o} _ {o}^{m} $的变形$ v $。根据$ w $是牛顿非脱位变形的假设，在本文中，我们将证明$ w $是$μ$ constant的变形，并且仅当$ w $允许同时嵌入分辨率。该结果提供了有关$ w $的大量信息，例如，家庭$ w $的拓扑琐事以及自然的形态$（w（\ mathbb {c} _o）_m）_ {red} \ rightarrow \ rightarrow \ mathbb {c} _ {c} _ {c} $ withy $ w n $ withy $ $ m $ -Jets。在我们的主要结果证明的途中，我们对牛顿数字单调性的问题给出了一个完整的答案。
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 $μ$-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.
