Upper bound on the multiplicity of rational and Du Bois singularities

TL;DR

The paper proves a sharp multiplicity bound for points on varieties with Du Bois singularities: at a point $x$ of a $d$-dimensional variety with embedding dimension $e$, the multiplicity satisfies $\mathrm{mult}_x X \le \binom{e}{d}$, and this bound recovers the known bound for rational singularities. The authors adapt Helmke's 1-dimensional multiplicity techniques via blow-ups and leverage Koszul complexes tied to sets of Cartier divisors to compare ambient and intrinsic data. For rational singularities, the bound is $\binom{e-1}{d-1}$, while for Du Bois singularities the bound is $\binom{e}{d}$, with a boundary example showing sharpness. The work unifies and extends previous results, connects to related $F$-singularity bounds in positive characteristic, and provides a framework based on explicit complexes to study multiplicities through hyperplane reductions to curves.

Abstract

This paper resolves a question of Huneke and Watanabe by proving a sharp upper bound for the multiplicity of Du Bois singularities: at a point of a $d$-dimensional variety with Du Bois singularities and embedding dimension $e$, the multiplicity is at most $\binom{e}{d}$. Additionally, the result recovers the previously known upper bound for the multiplicity of rational singularities.

Theorems & Definitions (4)

• Theorem 1.1
• Conjecture 1.2
• Lemma 2.1: Helmke97*Lemma 4.2
• proof