Geometric and arithmetic aspects of rational elliptic surfaces

TL;DR

This dissertation explores the geometric and arithmetic properties of rational elliptic surfaces, focusing on conic bundles, intersection numbers of sections, and fibers with high Mordell-Weil ranks.

Contribution

It introduces new results on conic bundle structures, intersection theory, and fiber rank variations in rational elliptic surfaces.

Findings

Classification of conic bundles on rational elliptic surfaces

Bounds on intersection numbers of sections

Existence of fibers with Mordell-Weil rank significantly higher than generic

Abstract

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem regarding the set of fibers whose Mordell-Weil ranks are at least 3 units higher than the generic rank of the elliptic fibration.