Compact moduli of K3 surfaces

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces, in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface $(X,L)\in F_{2d}$. The main new notion is that of a recognizable divisor $R$, a choice which can be consistently extended to all central fibers of Kulikov models. We prove that any choice of recognizable divisor leads to a semitoroidal compactification of the period space, at least up to normalization. Finally, we prove that the rational curve divisor is recognizable for all degrees.