Várilly-Alvarado, AnthonyGoldman, Ron2020-08-112020-08-112020-082020-08-10August 202Petok, Jack. "Kodaira dimensions of some moduli spaces of special hyperkähler fourfolds." (2020) Diss., Rice University. <a href="https://hdl.handle.net/1911/109182">https://hdl.handle.net/1911/109182</a>.https://hdl.handle.net/1911/109182We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ 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 $\mathcal{M}$ 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. 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 $\mathcal{M}_d$ 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 $\mathcal{M}_d$ grow fast enough to conclude that $\mathcal{M}_d$ is of general type for all but $40$ values of $d$. We also give information about the Kodaira dimension of $\mathcal{M}_d$ for 6 additional values of $d$, leaving only 34 values of $d$ for which we cannot yet say anything about the Kodaira dimension.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Algebraic geometrynumber theory.Thesis2020-08-11