Pairs \((X,\iota)\) of a \(K3\) surface \(X\) and a non-symplectic involution \(\iota\) on \(X\) are called \(2\)-elementary \(K3\) surfaces, which are classified into \(75\) classes by Nikulin in terms of the main invariant \((r,a,\delta)\). The moduli space \(\mathcal{M}_{r,a,\delta}\) of \(2\)-elementary \(K3\) surfaces with main invariant \((r,a,\delta)\) is described as in a modular variety by Yoshikawa. The aim of the paper under review is to prove the unirationality of moduli spaces \(\mathcal{M}_{r,a,\delta}\).
Together with birational morphisms, claims for unirationality of \(75\) moduli spaces are reduced to those of \(22+5\) spaces. Geometric observations interpret each of the \(22\) discriminant covers of moduli spaces into a moduli space of certain plane sextics, some \((4,4)\)-curves on \(\mathbb{P}^1\times\mathbb{P}^1\), or appropriate \(d\) points on \(\mathbb{P}^2\) with \(5\leq d\leq 8\). The main theorem is induced from the unirationality of the discriminant covers of moduli spaces and individual treatments for other \(5\) spaces. In turn, the moduli space of \(d\) points on \(\mathbb{P}^2\) is described as an arithmetic quotient.
In Appendix by Yoshikawa, the Kodaira dimension of \(\mathcal{M}_{r,a,\delta}\) is studied with automorphic forms.