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PreprintCC BY 4.0arXiv

Finite holomorphic pullbacks, Segre numbers, and generic Lê numbers

  • Z. Jelonek
  • V. H. Jorge Perez
  • R. N. Araújo Dos Santos
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Abstract

We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert--Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.

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Cite

BibTeXRIS
BibTeX
@misc{jelonek2026finite,
  title = {{Finite holomorphic pullbacks, Segre numbers, and generic Lê numbers}},
  author = {Z. Jelonek and V. H. Jorge Perez and R. N. Araújo Dos Santos},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07528},
  eprint = {2610.07528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07528},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite holomorphic pullbacks, Segre numbers, and generic Lê numbers
AU  - Z. Jelonek
AU  - V. H. Jorge Perez
AU  - R. N. Araújo Dos Santos
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07528
UR  - https://arxiv.org/abs/2610.07528
AB  - We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert--Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.
ER  -