Rank jumps and Multisections of elliptic fibrations on K3 surfaces

Abstract

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 $d\ne 2,3$ and moreover the Mordell--Weil rank jump property for $d\equiv 3\mathrm{mod}4$, $d\ne 3$. 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.

22 pages; V2: corrected a mistake in Lemma 4.1 of v1; definitive version, to appear in Forum of Mathematics Sigma

References in corpus (1)

• The Kodaira dimension of some moduli spaces of elliptic K3 surfaces