Rank jumps and Multisections of elliptic fibrations on K3 surfaces

We consider the countably many families ${\mathcal{L}}_d$, $d\in {\mathbb{N}}_{\ge 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell–Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell–Weil rank jump property for $d\mathrm{\ne }2,3$ and moreover the Mordell–Weil rank jump property for $d\equiv 3\mathrm{m}\mathrm{o}\mathrm{d}4$, $d\mathrm{\ne }3$. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell–Weil rank jump property and the presence of special multisections of the fibration.