K3 surfaces of degree six arising from desmic tetrahedra

TL;DR

This paper investigates special K3 surfaces of degree six with unique configurations of skew lines, exploring their geometric properties, automorphisms, and rational curves, linking classical desmic structures with modern algebraic geometry.

Contribution

It introduces new models and detailed lattice and automorphism group computations for these K3 surfaces arising from desmic tetrahedra.

Findings

Identification of two sets of 12 skew lines on the surfaces.

Explicit computation of the Picard lattice.

Description of rational curves of low degree.

Abstract

We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the pencil of desmic quartic surfaces introduced by George Humbert and recently studied by the second and third authors. We discuss alternative birational models of the surfaces, compute the Picard lattice and a group of projective automorphisms, and describe rational curves of low degree on the general surface.