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PreprintCC BY 4.0arXiv

Lifting negative curves from finite fields and Seshadri constants at ten points

  • Piotr Pokora
  • Tomasz Szemberg
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Abstract

We use finite-field calculations and mixed-characteristic deformation to construct nef divisors on very general complex blow-ups of the plane. A lifting and smoothing criterion converts a reducible nodal divisor over 𝔽127 into a smooth genus-$10$ curve of class 117H−37(E1+⋯+E10), retaining control of its first-order obstruction map. Its normal bundle in a suitable deformation is semistable, so Petrakiev's multiplicity estimate yields the nef class 27379H−8658(E1+⋯+E10) at ten very general points. Their multipoint Seshadri constant is therefore at least $8658/27379$, improving Petrakiev's bound 228/721. The geometric input is an existence criterion in characteristic zero verified by computations in positive characteristic.

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Cite

BibTeXRIS
BibTeX
@misc{pokora2026lifting,
  title = {{Lifting negative curves from finite fields and Seshadri constants at ten points}},
  author = {Piotr Pokora and Tomasz Szemberg},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06225},
  eprint = {2610.06225},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06225},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lifting negative curves from finite fields and Seshadri constants at ten points
AU  - Piotr Pokora
AU  - Tomasz Szemberg
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06225
UR  - https://arxiv.org/abs/2610.06225
AB  - We use finite-field calculations and mixed-characteristic deformation to construct nef divisors on very general complex blow-ups of the plane. A lifting and smoothing criterion converts a reducible nodal divisor over 𝔽_127 into a smooth genus-$10$ curve of class 117H−37(E_1+⋯+E_10), retaining control of its first-order obstruction map. Its normal bundle in a suitable deformation is semistable, so Petrakiev's multiplicity estimate yields the nef class 27379H−8658(E_1+⋯+E_10) at ten very general points. Their multipoint Seshadri constant is therefore at least $8658/27379$, improving Petrakiev's bound 228/721. The geometric input is an existence criterion in characteristic zero verified by computations in positive characteristic.
ER  -