Quartic del Pezzo surfaces over function fields of curves

TL;DR

This paper classifies quartic del Pezzo surface fibrations over the projective line, explores their geometric properties, and discusses implications for arithmetic over finite fields.

Contribution

It provides a classification based on numerical invariants and explicit geometric descriptions, including examples where certain spaces are birational, with applications to arithmetic questions.

Findings

Explicit classification of fibrations for small invariants

Descriptions of spaces of sections and their mappings

Examples of birational cases with arithmetic implications

Abstract

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.