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We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of$K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett'swork on cubic fourfolds, Debarre, Iliev, and Manivel have shown that theNoether-Lefschetz locus in $\mathcal{M}$ is a countable union of specialdivisors $\mathcal{M}_d$, where the discriminant $d$ is a positive integercongruent to $0,2,$ or $4$ modulo 8. We compute the Kodaira dimensions of thesespecial divisors for all but finitely many discriminants; in particular, weshow that for $d>224$ and for many other small values of $d$, the space$\mathcal{M}_d$ is a variety of general type.

Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2