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PreprintCC BY 4.0arXiv

Local monodromy, Behrend function and Hilbert scheme of points on ℂ3

  • Yunfeng Jiang
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Abstract

We prove a criteria for the Behrend function values on the Hilbert scheme of n-points on C3 from local monodromy and the Denef--Loeser's Lefschetz trace formula of jet spaces. If the power two, three, or four of the monodromy is unipotent, it reduces respectively to a Hessian parity formula, a cubic-complement formula, or a quartic formula. This gives a Milnor fiber and monodromy based proof of nonconstancy of the Behrend function on Hilbn(C3) after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in Hilb9(C3) and Hilb12(C3) so that the Behrend function is not ± 1, answering a question of Ricolfi.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026local,
  title = {{Local monodromy, Behrend function and Hilbert scheme of points on $\mathbbC^3$}},
  author = {Yunfeng Jiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22469},
  eprint = {2609.22469},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22469},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Local monodromy, Behrend function and Hilbert scheme of points on ℂ^3
AU  - Yunfeng Jiang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.22469
UR  - https://arxiv.org/abs/2609.22469
AB  - We prove a criteria for the Behrend function values on the Hilbert scheme of n-points on C^3 from local monodromy and the Denef--Loeser's Lefschetz trace formula of jet spaces. If the power two, three, or four of the monodromy is unipotent, it reduces respectively to a Hessian parity formula, a cubic-complement formula, or a quartic formula. This gives a Milnor fiber and monodromy based proof of nonconstancy of the Behrend function on Hilb^n(C^3) after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in Hilb^9(C^3) and Hilb^12(C^3) so that the Behrend function is not ± 1, answering a question of Ricolfi.
ER  -