The valuation of the discriminant of a hypersurface

Abstract

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.