Unirational moduli spaces of some elliptic K3 surfaces

We show that the moduli space of \(U\oplus \langle -2k \rangle\)-polarized K3 surfaces is unirational for \(k \le 50\) and \(k \notin \{11,35,42,48\}\), 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 \(\mathbb{P}^n\) for \(3\le n \le 5\) containing either the union of two rational curves or the union of a rational and an elliptic curve intersecting at one point.