Desmic quartic surfaces in arbitrary characteristic

Abstract

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 $\mathfrak{G}$ associated with the desmic tetrahedra introduced by G. Humbert. We prove that $\mathfrak{G}$ is a rational Fano threefold with 34 nodes. The number 34 is the maximum number of nodes on a Fano threefold of degree 6 in ${\mathbb{P}}^5$, and the group of projective automorphisms is isomorphic to ${\mathfrak{S}}_4\wr 2=({\mathfrak{S}}_4\times {\mathfrak{S}}_4)⋊2$.