Abstract
We study the Noether-Lefschetz locus of the moduli space M of K3[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 M is a countable union of special divisors Md, 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 Md is a variety of general type.
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