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

Finashin, Sergey; Zabun, Remzi̇ye Arzu

doi:10.1090/proc/14340

Topology of real Schläfli six-line configurations on cubic surfaces and in $\mathbb {RP}^3$
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by Sergey Finashin and Remzi̇ye Arzu Zabun PDF

Proc. Amer. Math. Soc. 147 (2019), 3665-3674 Request permission

Abstract:

A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb {P}^3$ 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 $\mathbb {RP}^3$.

References

Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
O. Ya. Viro and Yu. V. Drobotukhina, Configurations of skew-lines, Algebra i Analiz 1 (1989), no. 4, 222–246 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 4, 1027–1050. MR 1027468
V. F. Mazurovskiĭ, Configurations of six skew lines, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 121–134, 191 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2825–2832. MR 964261, DOI 10.1007/BF01099247
B. Segre, The Non-singular Cubic Surfaces, Oxford University Press, Oxford, 1942. MR 0008171
L. Schläfli, An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface, Quart. J. Pure Appl. Math., 2 (1858), 110-120.