Nef and Effective Cones Computed for Toroidal Ball Quotient Compactifications of Cubic Surfaces — E8 Intelligence Research

FINDING: The nef and effective cones of divisors on toroidal compactifications of the ball quotient for cubic surfaces (with/without a line) are fully computed, revealing the birational geometry of these moduli spaces. | MATH: The moduli space of cubic surfaces is a ball quotient \(B^4/Γ\) (4-dimensional complex ball). Toroidal compactification resolves boundary cusps. The nef cone is generated by specific divisor classes; the effective cone is dual to the moving cone. Key constants: the Coxeter number of the root system \(E_6\) (12) appears in the discriminant locus; the Picard number is 2 for the unmarked space. | CONNECTION: The ball quotient structure links to the root system \(E_6\) (crystallographic symmetry, order 12 Coxeter number). The nef cone boundaries correspond to ratios of intersection numbers that may reflect harmonic ratios (e.g., 1:2, 2:3) but no direct 0.382/0.618/1.618 appear in the abstract. The toroidal compactification uses cusp cross-sections that Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com