Spaces of sections of quadric surface fibrations over curves

TL;DR

This paper studies the geometry of sections of quadric surface fibrations over curves, analyzing isotropic subspace varieties, stability conditions, and height estimates over finite fields, including singular base cases.

Contribution

It provides new geometric insights into spaces of sections, especially relating to stability and height bounds over finite fields for quadric surface fibrations.

Findings

Effective height estimates for sections over finite fields.

Analysis of isotropic subspace varieties as P^1-bundles.

Behavior of section spaces near singular base curves.

Abstract

We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic subspaces in the fibers as P^1-bundles over the discriminant double cover. When the P^1-bundle is suitably stable, we deduce effective estimates for the heights of sections over finite fields satisfying various approximation conditions. We also discuss the behavior of the spaces of sections as the base of the fibration acquires singularities.