Desmic–Hessian geometric correspondence

Determine the precise geometric correspondence between the occurrence of the Reye configuration in desmic geometry and the production of the second triple of Schur quartics by binary Hessians, distinguishing an actual desmic equivalence from the weaker Reye/Hessian relation exhibited by the construction.

Background

The Reye configuration appears classically in desmic geometry, while the second triple of Schur quartics in the paper is generated using binary Hessians. The paper identifies a relationship between these phenomena only at the level of a Reye/Hessian shadow.

It remains unresolved whether there is a precise geometric correspondence or a stronger actual desmic equivalence connecting the two constructions.

References

What precise geometric correspondence relates these facts? In particular, one should distinguish an actual desmic equivalence from the weaker Reye/Hessian shadow exhibited here.

— The Reye geometry inside the 64 lines of the Schur quartic  (2609.10751 - Nurowski, 9 Sep 2026) in Section 5, Conclusions and open problems, item 2

Is there a natural algebraic-geometric construction on the corresponding K3 surface---for example a quotient, rational map, correspondence, or resolution---whose induced action on the distinguished lines is precisely this Reye projection?

— The Reye geometry inside the 64 lines of the Schur quartic  (2609.10751 - Nurowski, 9 Sep 2026) in Section 5, Conclusions and open problems, item 3