Third-Order Weierstrass Points on Plane Quartics

Abstract

For a smooth plane quartic C, the $3$-Weierstrass points are the $300$ points (counted with weights) at which some plane cubic has contact order at least ten with C. Revisiting the work of Kamel and Elshareef, we write the third Veronese Wronskian as an SL₃-covariant form Ω₃ of degree $75$, given by a determinant of iterated Hamiltonian derivatives, and explain the special $3$-Weierstrass points through contact cubics and homologies preserving C. We determine Ω₃ for the Fermat, Komiya--Kuribayashi, Klein and Picard quartics; for the Klein quartic Ω₃≡Φ₆² Φ₂₁ (3Φ₁₄³+21962 Φ₆⁷) modulo the curve, up to a non-zero scalar. Finally, the dectactic points of the Fermat quartic are not $4$-Weierstrass points, which disproves a conjecture of Kamel and Elshareef. All computations are supported by \texttt{SINGULAR} scripts.

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BibTeX RIS

BibTeX

@misc{nawara2026thirdorder,
  title = {{Third-Order Weierstrass Points on Plane Quartics}},
  author = {Ewelina Nawara},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30992},
  eprint = {2610.30992},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30992},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Third-Order Weierstrass Points on Plane Quartics
AU  - Ewelina Nawara
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30992
UR  - https://arxiv.org/abs/2610.30992
AB  - For a smooth plane quartic C, the $3$-Weierstrass points are the $300$ points (counted with weights) at which some plane cubic has contact order at least ten with C. Revisiting the work of Kamel and Elshareef, we write the third Veronese Wronskian as an SL_3-covariant form Ω_3 of degree $75$, given by a determinant of iterated Hamiltonian derivatives, and explain the special $3$-Weierstrass points through contact cubics and homologies preserving C. We determine Ω_3 for the Fermat, Komiya--Kuribayashi, Klein and Picard quartics; for the Klein quartic Ω_3≡Φ_6^2 Φ_21 (3Φ_14^3+21962 Φ_6^7) modulo the curve, up to a non-zero scalar. Finally, the dectactic points of the Fermat quartic are not $4$-Weierstrass points, which disproves a conjecture of Kamel and Elshareef. All computations are supported by \texttt{SINGULAR} scripts.
ER  -

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