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Rank jumps and Multisections of elliptic fibrations on K3 surfaces

We consider the countably many families $\mathcal{L}_d$, $d\in\mathbb{N}_{\geq 2}$, 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 $d\neq 2,3$ and moreover the Mordell--Weil rank jump property for $d\equiv 3\mod 4$, $d\neq 3$. 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.