Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2

TL;DR

This paper investigates the Kodaira dimension of special divisors within the moduli space of certain hyperkähler fourfolds, revealing that many are of general type for large discriminants.

Contribution

It computes the Kodaira dimensions of Noether-Lefschetz divisors in the moduli space of $K3^{[2]}$-fourfolds, extending understanding of their geometric properties.

Findings

Most divisors with discriminant > 224 are of general type.

The Kodaira dimensions are computed for all but finitely many discriminants.

The work generalizes previous results on cubic fourfolds to hyperkähler fourfolds.

Abstract

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. 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.