Unirational moduli spaces of some elliptic K3 surfaces

Abstract

We show that the moduli space of U ⊕−2k -polarized K3 surfaces is unirational for k ≤ 50 and k ∈ { / 11, 35, 42, 48}, and for other several values of k up to k = 97. Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in Pn for 3 ≤ n ≤ 5 containing either the union of two smooth rational curves or the union of a smooth rational curve and an elliptic curve intersecting at one point.