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 $\frakG$ associated with the desmic tetrahedra introduced by G. Humbert. We prove that $\frakG$ 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 $\bbP^5$, and the group of projective automorphisms is isomorphic to $\frakS_4\wr 2 = (\frakS_4\times \frakS_4)\rtimes 2$.