Random conic bundle surfaces satisfy the Hasse principle

TL;DR

This paper proves that all conic bundle surfaces over the projective line over rationals satisfy the Hasse principle, confirming the principle for a broad class of algebraic surfaces.

Contribution

It demonstrates that 100% of conic bundle surfaces over  over  satisfy the Hasse principle, a significant advance in understanding rational points on these surfaces.

Findings

Proves the Hasse principle holds for 100% of conic bundle surfaces over .

Establishes a density result for rational points on conic bundle surfaces.

Provides a framework for analyzing rational points on algebraic surfaces.

Abstract

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.