Kodaira dimension of moduli of special $K{3}^{[2]}$-fourfolds of degree 2

Abstract

We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of $K{3}^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the Noether-Lefschetz locus in $\mathcal{M}$ is a countable union of special divisors ${\mathcal{M}}_d$, where the discriminant $d$ is a positive integer congruent to $0,2,$ or $4$ modulo 8. We compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for $d>224$ and for many other small values of $d$, the space ${\mathcal{M}}_d$ is a variety of general type.

arXiv comment25 pages; Lemma 4.3 is now corrected and Section 4 has been altered to reflect the correct statement of this lemma