Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds

We study divisors on moduli spaces of cubic fourfolds with simple singularities and of quasi-polarized K3 surfaces of degree $2d$. For the moduli space of cubic fourfolds, we introduce a slope quantity to characterize the effective cone and prove an explicit bound for it. For the K3 moduli spaces, we give an explicit finite presentation of the rational Picard group by showing that it is generated by Noether-Lefschetz divisors of discriminant less than or equal to $4d$. As a byproduct, we obtain two explicit expressions for the Hodge class in terms of Noether-Lefschetz divisors, and we indicate analogous results for higher-codimension Noether-Lefschetz cycles.