Desmic quartic surfaces in arbitrary characteristic

A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the cubic line complex $\text{frakG}$ associated with the desmic tetrahedra introduced by G. Humbert. We prove that $\text{frakG}$ is a rational $\text{bbQ}$-Fano threefold with 34 nodes and the group of projective automorphisms isomorphic to $({\text{frakS}}_4\times {\text{frakS}}_4)⋊2$.