Rank jumps on elliptic surfaces and the Hilbert property

Daniel Loughran, Cecília Salgado

Published 2019-07-03Version 1

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.