# Rank jumps on elliptic surfaces and the Hilbert property

## TL;DR

This paper investigates the distribution of fibers with higher Mordell-Weil rank in elliptic surfaces over number fields, demonstrating that such fibers form a non-thin set under certain conditions, with applications to specific algebraic surfaces.

## Contribution

It proves that fibers with increased Mordell-Weil rank are not thin in elliptic surfaces, extending understanding of rank jumps and their distribution.

## Key findings

- Fibers with higher Mordell-Weil rank are not thin in elliptic surfaces.
- Results apply to quadratic twist families and degree 1 del Pezzo surfaces.
- Provides new insights into the distribution of rank jumps in algebraic geometry.

## Abstract

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 $1$.