Topology of real Schläfli six-line configurations on cubic surfaces and in ℝℙ³

Abstract

A famous configuration of 27 lines on a non-singular cubic surface in P 3 \mathbb {P}^3 contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case of real cubic surfaces from a topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property of homogeneity. This property distinguishes them in the list of 11 deformation types of configurations formed by six disjoint lines in R P 3 \mathbb {RP}^3 .