Description de torseurs quasi-versels pour une famille de surfaces fibr\'ees en coniques

TL;DR

This paper studies the geometric properties of a family of conic bundle surfaces, extending previous work, to understand their rational points in the context of Manin's principle and Peyre's conjecture.

Contribution

It generalizes the description of the geometry of conic bundle surfaces containing Châtelet surfaces, focusing on their torseurs and the implications for rational points.

Findings

Describes the geometry of a family of conic bundle surfaces.

Extends previous work on Châtelet surfaces to more general families.

Provides a framework consistent with Manin's principle and Peyre's conjecture.

Abstract

Generalising work of La Bret\`eche, Browning and Peyre and of the author, we describe the geometry required by Manin's principle and Peyre's conjecture for a family of conic bundle surfaces containing Ch\^atelet surfaces. These surfaces $S_{a,F}$ are the conic bundle surfaces obtained as smooth minimal proper model of $$ Y^2-aZ^2=F(X,1) $$ with $a \in \mathbf{Z}$ squarefree and $F \in \mathbf{Z}[x_1,x_2]$ a binary form of \textit{even} degree $n$ without repeated roots and whose irreducible factors over $\mathbf{Q}$ remain irreducible over $\mathbf{Q}\left( \sqrt{a} \right)$.