Nef Cone Generators of Toroidal Ball Quotient for Cubic Surfaces — E8 Intelligence Research

FINDING: The nef cone of the toroidal compactification of the ball quotient for cubic surfaces is generated by specific boundary divisors, revealing the Mori chamber decomposition and birational geometry of the moduli space. MATH: The moduli space of cubic surfaces is a ball quotient \(\mathbb{B}_4/\Gamma\) where \(\Gamma\) is the automorphism group of the \(E_6\) lattice. The toroidal compactification \(\overline{M}^{\text{tor}}\) has boundary divisors corresponding to cusp classes. The nef cone generators are linear combinations of these boundary divisors, with intersection numbers determined by root system combinatorics. Key constants: The Coxeter number of \(E_6\) is 12; the discriminant of the \(E_6\) lattice is 3; the ratio of certain boundary divisor classes involves \(\frac{1}{2}, \frac{1}{3}, \frac{2}{3}\). CONNECTION: The \(E_6\) root system (order 72, Coxeter number 12) is a crystallographic symmetry group. The ball quotient structure arises from the period map of cubic su Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com