Third-Order Weierstrass Points on Plane Quartics

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 ${\text{SL}}_3$-covariant form ${\mathrm{\Omega }}_3$ 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 ${\mathrm{\Omega }}_3$ for the Fermat, Komiya–Kuribayashi, Klein and Picard quartics; for the Klein quartic ${\mathrm{\Omega }}_3\equiv {\mathrm{\Phi }}_6^2{\mathrm{\Phi }}_{21}(3{\mathrm{\Phi }}_{14}^3+21962{\mathrm{\Phi }}_6^7)$ 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 $\mathtt{\text{SINGULAR}}$ scripts.