The Reduced Fano Variety of 27 Lines on a Cubic Surface and E₆ — E8 Intelligence Research

FINDING: The space of lines on a cubic surface in P³ is reduced in the Grassmannian G(2,4), linking to the E₆ root system and its Weyl group of order 72, with the Fano configuration of 27 lines. MATH: - Cubic surface: degree 3 hypersurface in ℙ³. - 27 lines: classical result (Cayley–Salmon). - Fano variety F_S ⊂ G(2,4) is reduced (no nilpotents). - E₆ root system: rank 6, 72 roots, Weyl group order 51840? Wait — correction: E₆ Weyl group order = 51840? No: |W(E₆)| = 51840? Actually |W(E₆)| = 51840? Let's check: |W(E₆)| = 2^5 * 3^4 * 5 = 32*81*5 = 12960? No — standard: |W(E₆)| = 51840? I recall: |W(E₆)| = 51840? Let's compute: E₆ Coxeter number = 12, exponents 1,4,5,7,8,11 → product = 1*4*5*7*8*11 = 12320? That's not right. Actually |W(E₆)| = 2^7 * 3^4 * 5 = 128*81*5 = 51840. Yes. But the search mentions "order 72" — that is the order of the Weyl group of G₂? No — G₂ has 12. Wait: 72 appears in E₆ as the number of roots? No, E₆ has 72 roots. The Weyl group order is 51840. The 7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com