Extremal divisors on moduli spaces of K3 surfaces

Abstract: We establish numerical criteria for when a Noether-Lefschetz divisor on a moduli space of quasi-polarized K3 surfaces $F_{2d}$, or more generally on an orthogonal modular variety, generates an extremal ray in the cone of pseudoeffective divisors. In particular, for all d, we exhibit many extremal rays of the cone of pseudoeffective divisors of both $F_{2d}$ and any normal projective $\mathbb{Q}$-factorial compactification of $F_{2d}$ lying over its Baily-Borel compactification.