Browsing by Author "Petok, Jack"

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    Kodaira dimensions of some moduli spaces of special hyperkähler fourfolds
    (2020-08-10) Petok, Jack; Várilly-Alvarado, Anthony; Goldman, Ron
    We 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.