A dodecic surface with 320 cusps
Cédric Bonnafé
TL;DR
This work constructs a degree-$12$ invariant of the complex reflection group $G_{29}$ and uses it to produce a dodecic surface with $320$ cusps of type $A_2$, improving the known lower bound for $\mu_{A_2}(12)$. By detailing the fundamental invariants $f_1,f_2,f_3,f_4$ and the associated degree-$12$ invariant $F$, the authors define a two-parameter family of invariant surfaces $\mathcal{S}_{12}^{\lambda,\mu}$ that are generically irreducible of degree $12$. Through Magma-based computations, they identify the singular locus in parameter space, isolating a component that yields the $320$ cusps as a single $G_{29}$-orbit on $\mathcal{S}_{12}^{+}$ and $\mathcal{S}_{12}^{-}$ (defined over $\mathbb{Q}(\sqrt{3})$). The paper also explores additional singular dodecics arising from the same invariants, including maximal $D_4$, $A_3$, and novel singularities, and shows how some singularities degenerate to $A_2$ under parameter specialization, contributing to broader lower bounds in the landscape of surface singularities.
Abstract
We construct a degree $12$ homogeneous invariant of the complex reflection group $G_{29}$ (in Shephard-Todd's notation) whose associated surface has 320 singularities of type $A_2$, improving previous records for dodecic surfaces.