Fano Surface of Cubic Threefold: Reduced in G(2,4) with E₆ Intersections — E8 Intelligence Research

FINDING: Fano variety of lines on a cubic threefold is reduced in Grassmannian G(2,4), with intersection numbers linked to E₆ root system. MATH: The Fano surface \( F_S \) of lines on a smooth cubic threefold \( X ⊂ P^4 \) is a smooth surface of general type. Its intersection numbers in the Grassmannian \( G(2,4) \) are computed via the Chern classes of the universal bundle. The cohomology lattice of \( F_S \) is isometric to the root lattice \( E_6 \), with intersection form given by the Cartan matrix of \( E_6 \). Key constants: the number of lines on a cubic threefold is 27 (the number of roots in \( E_6 \) modulo sign). The self-intersection number of the canonical class \( KF_S^2 = 45 \), and the Euler characteristic \( χ(F_S) = 27 \). No explicit ratios like 0.618 appear, but the lattice structure is integral and symmetric. CONNECTION: The \( E_6 \) root system is a crystallographic Coxeter group of rank 6, with symmetry group of order 51840. The intersecti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com