Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity

Abstract

Let $f$ be an isolated hypersurface singularity and let $J(f)$ be its Jacobian ideal. We study the higher Milnor numbers $μ_k(f)=\dim_{\mathbb C}\mathcal O_n/\mathfrak m^kJ(f).$ Our main result concerns an ordinary semi-homogeneous singularity $f=f_r+f_{r+1}+\cdots$, where the projective hypersurface defined by $f_r$ is smooth. We prove that $μ_k(f)=μ_k(f_r) \qquad\text{for every }k\geq0.$ This gives the equal-weight case of a conjecture of Shi, Yau, and Zuo. The main algebraic input is the common-order case of the Valabrega--Valla criterion. From it, we obtain the graded Jacobian module and the exact intersection formula. As a consequence, we obtain a closed formula for $μ_k$ and determine the sharp stabilization index of $\mathfrak m^kJ(f)$. We also determine the optimal uniform range in which the Tjurina formula holds for all ordinary semi-homogeneous germs with fixed $n$ and $r$. We next consider the log-concavity property. For ordinary semi-homogeneous singularities, the higher Milnor sequence has an initial log-convex part and a log-concave tail. For an arbitrary isolated hypersurface singularity in $n\geq2$ variables, both the higher Milnor and higher Tjurina sequences are eventually strictly log-concave. In the plane curve case, we prove that the log-concavity defects change sign at most once, and we characterize the sequences which are log-concave at every positive index. Finally, we give examples showing that the higher Milnor sequence does not determine the contact-equivalence class.