Extremal divisors on moduli spaces of K3 surfaces

Abstract

We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective divisors on any moduli space~$\mathcal{F}_{2d}$ of quasi-polarized K3 surfaces of degree $d$, as well as on any normal projective $\mathbb{Q}$-factorial compactification $\overline{\mathcal{F}}_{2d}$ of $\mathcal{F}_{2d}$ lying over the Baily--Borel compactification.