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 μk(f)=dimℂ𝒪n/mkJ(f). Our main result concerns an ordinary semi-homogeneous singularity f=fr+fr+1+⋯, where the projective hypersurface defined by fr is smooth. We prove that μk(f)=μk(fr) for every k≥0. 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 μk and determine the sharp stabilization index of mkJ(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≥2 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.
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BibTeX
@misc{liu2026jacobian,
title = {{Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity}},
author = {Zhiwen Liu and Stephen S. -T. Yau and Mingyi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32523},
eprint = {2610.32523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32523},
}
RIS
TY - UNPB
TI - Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity
AU - Zhiwen Liu
AU - Stephen S. -T. Yau
AU - Mingyi Zhang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32523
UR - https://arxiv.org/abs/2610.32523
AB - 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_ℂ𝒪_n/m^kJ(f). Our main result concerns an ordinary semi-homogeneous singularity f=f_r+f_r+1+⋯, where the projective hypersurface defined by f_r is smooth. We prove that μ_k(f)=μ_k(f_r) for every k≥0. 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 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≥2 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.
ER -