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

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.