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 \notin \{11,35,42,48\}$$ , other several values k up to $$k=97$$ . Our proof based on a systematic study projective models elliptic in $${\mathbb {P}}^n$$ $$3\le n 5$$ containing either union two smooth rational curves or curve an intersecting at one point.