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.

In this thesis, we compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for d>176 and for many other small values of d, the space Md is a variety of general type. The main idea of the proof is to study the Kodaira dimension of the moduli spaces using the "quasi-pullback" trick of Gritsenko-Hulek-Sankaran: by explicitly constructing certain modular forms on the period domain, we can show the plurigenera of a smooth compactification of Md grow fast enough to conclude that Md is of general type for all but 40 values of d. We also give information about the Kodaira dimension of Md for 6 additional values of d, leaving only 34 values of d for which we cannot yet say anything about the Kodaira dimension.