Relations among the platonic-family elliptic surfaces

Characterize precisely how the eleven rational elliptic surfaces with three semistable reduced fibres and one reduced Kodaira fibre of type II, III, or IV are related within the proposed platonic family.

Background

The paper identifies three elliptic surfaces arising directly from the tetrahedral, octahedral, and icosahedral reflection groups, and places them together with eight related Herfurtner surfaces in a broader platonic family. Although several pairs are connected by explicit isogenies, the authors do not establish a general relationship among all members of the family. They explicitly single out the precise structure of these relationships as unresolved.

References

We call them the cousins of our platonic elliptic surfaces and we obtain in this way a rather strange platonic family of elliptic surfaces of which it is not clear how the members are related precisely (Table \ref{tab:platonic-family}).

— The platonic elliptic surfaces  (2608.25644 - Gegelia et al., 26 Aug 2026) in Section 2, subsection “The platonic elliptic surfaces”