The valuation of the discriminant of a hypersurface

Abstract

Let $R$ be a discrete valuation ring, with valuation $v:R\twoheadrightarrow {\mathbb{Z}}_{\ge 0}\cup \{\infty \}$ and residue field $k$. Let $H$ be a hypersurface $\mathrm{Proj}(R[x_0,\dots ,x_n]/⟨f⟩)$. 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.

14 pages