Desmic quartic surfaces in arbitrary characteristic

TL;DR

This paper explores desmic quartic surfaces, their classical geometry, and their analogs in arbitrary characteristic, revealing new properties of associated cubic line complexes and automorphism groups.

Contribution

It introduces the study of desmic quartic surfaces in arbitrary characteristic and analyzes the properties of the associated cubic line complex and automorphism group.

Findings

The cubic line complex $rakG$ is a rational $bQ$-Fano threefold with 34 nodes.

The automorphism group of $rakG$ is isomorphic to $(rakS_4 imes rakS_4) times 2$.

Classical geometry of desmic quartic surfaces is extended to 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 $\bbQ$-Fano threefold with 34 nodes and the group of projective automorphisms isomorphic to $(\frakS_4\times \frakS_4)\rtimes 2$.