The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic

TL;DR

This paper establishes sharp upper bounds for the quotient of Milnor and Tjurina numbers in hypersurface singularities across arbitrary characteristic, advancing understanding of their relationship and confirming conjectures.

Contribution

It introduces new bounds using multiplicity concepts and constructs examples showing these bounds are optimal, addressing open problems and conjectures.

Findings

Derived upper bounds for μ/τ in positive and zero characteristic

Constructed hypersurface families approaching the bounds

Partially proved a conjecture of P. Almirón for surface singularities

Abstract

In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface singularities of any dimension in positive characteristic. Using this result, we also derive an upper bound for the quotient of the Milnor numbers $\mu$ and the Tjurina numbers $\tau$ for isolated hypersurface singularities of any dimension in characteristic zero. In particular, as a corollary, we obtain that for an isolated surface singularity $(f,0) \subset (\mathbb{C}^3,0)$, $\frac{\mu(f)}{\tau(f)}\leq \frac{3}{2}$, which partially answers a conjecture of P. Almir\'{o}n, replacing the original strict inequality $<$ by $\leq$. This is also a weak version of Durfee's conjecture. We have also constructed a family of hypersurface singularities of any dimension for which $\frac{\mu}{\tau}$ tends to the bound we get, which means that the bound is sharp, and at the same time answers an open problem raised by P. Almir\'{o}n.