Unirational moduli spaces of some elliptic K3 surfaces

Abstract

We show that the moduli space of U⊕ ⟨ - 2 k⟩ -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.