The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

Abstract

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U ⊕ ⟨ − 2 k ⟩ -polarized K3 surfaces. Such moduli spaces are proved to be of general type for k ⩾ 220 . The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for k < 11 and for 19 other isolated values up to k = 64 .