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The valuation of the discriminant of a hypersurface

Bjorn Poonen, Michael Stoll

TL;DR

The paper develops a precise link between the arithmetic valuation of the discriminant $Δ(f)$ of a hypersurface $H$ over a discrete valuation ring and the geometric nature of the special fiber $H_k$. By combining Zariski's main theorem, degeneration arguments, and a detailed analysis of discriminants for general and special fibers, it proves that $v(Δ(f))=1$ exactly when $H$ is regular and its special fiber has a single nondegenerate double point; it also provides lower bounds on $v(Δ(f))$ when the singular locus has multiple or higher-dimensional components. The work develops a framework for bounding discriminant valuations via the minimal valuation of Δ on coefficient spaces, using restriction of scalars and Greenberg functors to relate arithmetic and geometric data. In addition to the sharp valuation-1 criterion, the authors derive bounds for hypersurfaces with several singularities or positive-dimensional singular loci and give plane-curve refinements, contributing to a deeper understanding of discriminants in arithmetic geometry and their role in degeneration theory.

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 $Δ(f)$ be the discriminant of $f$. We use Zariski's main theorem and degeneration arguments to prove that $v(Δ(f))=1$ 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 $v(Δ(f))$ when $H_k$ has multiple singularities or a positive-dimensional singularity.

The valuation of the discriminant of a hypersurface

TL;DR

The paper develops a precise link between the arithmetic valuation of the discriminant of a hypersurface over a discrete valuation ring and the geometric nature of the special fiber . By combining Zariski's main theorem, degeneration arguments, and a detailed analysis of discriminants for general and special fibers, it proves that exactly when is regular and its special fiber has a single nondegenerate double point; it also provides lower bounds on when the singular locus has multiple or higher-dimensional components. The work develops a framework for bounding discriminant valuations via the minimal valuation of Δ on coefficient spaces, using restriction of scalars and Greenberg functors to relate arithmetic and geometric data. In addition to the sharp valuation-1 criterion, the authors derive bounds for hypersurfaces with several singularities or positive-dimensional singular loci and give plane-curve refinements, contributing to a deeper understanding of discriminants in arithmetic geometry and their role in degeneration theory.

Abstract

Let be a discrete valuation ring, with valuation and residue field . Let be a hypersurface . Let be the special fiber, and let be its singular subscheme. Let be the discriminant of . We use Zariski's main theorem and degeneration arguments to prove that if and only if is regular and consists of a nondegenerate double point over . We also give lower bounds on when has multiple singularities or a positive-dimensional singularity.
Paper Structure (11 sections, 31 theorems, 7 equations)

This paper contains 11 sections, 31 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. The discriminant
  3. Quadratic forms
  4. Nondegenerate double points and ordinary double points
  5. The general singular locus
  6. Commutative algebra
  7. Hypersurfaces with several singularities
  8. Valuations of polynomial values
  9. Minimal valuations of the discriminant
  10. When the discriminant has valuation 1
  11. Hypersurfaces with a positive-dimensional singularity

Key Result

Theorem 1.1

Let $f \in R[x_0,\dots,x_n]$ be a homogeneous polynomial. Let $\Delta(f)$ be its discriminant. Let $H = \mathop{\mathrm{Proj}}\nolimits(R[x_0,\dots,x_n]/\langle f \rangle)$. Then the following are equivalent:

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1: SGA7.1*VI.6
  • Remark 4.2
  • ...and 60 more