The valuation of the discriminant of a hypersurface

2025/10/16 by Poonen, Bjorn, Stoll, Michael
#14B05 #14G20 (Secondary) #14J17 (Primary) 11G25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.14434

Abstract

Let R be a discrete valuation ring, with valuation v \colon R \twoheadrightarrow ℤ_{≥ 0} ∪ \∞\ and residue field k. Let H be a hypersurface Proj(R[x₀,…,x_n]/⟨ f ⟩). Let H_k be the special fiber, and let (H_k)_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)_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.

Citations

• The Greenberg functor revisited
• On the structure and ideal theory of complete local rings

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