Waldschmidt constants and invariant geometry for exceptional rank-four complex reflection groups

arXiv:2611.13108v1 Announce Type: new Abstract: We compute the Waldschmidt constants of the point configurations dual to the reflection arrangements of the exceptional complex reflection groups $G_{29}$, $G_{31}$ and $G_{32}$. They equal $3$ for the forty points of $G_{29}$, $36/11$ for the sixty-point Klein configuration, and $10/3$ for the forty-point Penrose configuration. Together with the known cases $W(F_4)$ and $W(H_4)$, this completes the computation for all exceptional irreducible complex reflection groups of rank four. An invariant dodecic and its sixfold orbit product give extremal surfaces for the first two configurations, and we identify the six dodecics as pullbacks of the coordinate hyperplane sections of the Segre cubic under the Igusa-Segre duality. The matching lower bounds follow from restriction to quadrics and distinguished planes, including a weighted B\'ezout estimate arising from two related Hesse arrangements. As a consequence, we determine the asymptotic resurgences of the three ideals.

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