标题： 超曲面判别式的估值 显示英文标题

标题： The valuation of the discriminant of a hypersurface

摘要： 设$R$是一个离散赋值环，其赋值为$v \colon R \twoheadrightarrow \mathbb{Z}_{\ge 0} \cup \{\infty\}$，剩余域为$k$。 设$H$是一个超曲面$\operatorname{Proj}(R[x_0,\ldots,x_n]/\langle f \rangle)$。 设$H_k$是特殊纤维，令$(H_k)_{\mathrm{sing}}$是其奇点子概形。 设$\Delta(f)$为$f$的判别式。 我们使用 Zariski 的主要定理和退化论证来证明 $v(\Delta(f))=1$当且仅当$H$是正则的且$(H_k)_{\mathrm{sing}}$在$k$上由一个非退化二重点组成。 我们还给出了当$H_k$具有多个奇点或高维奇点时$v(\Delta(f))$的下界。 显示英文摘要

摘要： Let $R$ be a discrete valuation ring, with valuation $v \colon R \twoheadrightarrow \mathbb{Z}_{\ge 0} \cup \{\infty\}$ and residue field $k$. Let $H$ be a hypersurface $\operatorname{Proj}(R[x_0,\ldots,x_n]/\langle f \rangle)$. Let $H_k$ be the special fiber, and let $(H_k)_{\mathrm{sing}}$ be its singular subscheme. Let $\Delta(f)$ be the discriminant of $f$. We use Zariski's main theorem and degeneration arguments to prove that $v(\Delta(f))=1$ if and only if $H$ is regular and $(H_k)_{\mathrm{sing}}$ consists of a nondegenerate double point over $k$. We also give lower bounds on $v(\Delta(f))$ when $H_k$ has multiple singularities or a positive-dimensional singularity.