Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface

Abstract

Let S be a smooth projective complex surface with irregularity q=h^1,0(S) and geometric genus g=h^2,0(S), and let S^[n] denote its Hilbert scheme of n points. We prove that, for every n≥0, the sequence (h^p,0(S^[n]))_p=0²ⁿ is log-concave if and only if g≤binomq+12. Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for n=2. We also prove that these sequences are unimodal for every n≥0 if and only if q≥1 or g=0, and that this condition is already detected by the sequence for n=1.

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BibTeX RIS

BibTeX

@misc{pahari2026logconcavity,
  title = {{Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface}},
  author = {Anubhab Pahari},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04095},
  eprint = {2609.04095},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04095},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface
AU  - Anubhab Pahari
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04095
UR  - https://arxiv.org/abs/2609.04095
AB  - Let S be a smooth projective complex surface with irregularity q=h^1,0(S) and geometric genus g=h^2,0(S), and let S^[n] denote its Hilbert scheme of n points. We prove that, for every n≥0, the sequence (h^p,0(S^[n]))_p=0^2n is log-concave if and only if g≤binomq+12. Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for n=2. We also prove that these sequences are unimodal for every n≥0 if and only if q≥1 or g=0, and that this condition is already detected by the sequence for n=1.
ER  -

Abstract

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