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 $\mathbf{ Q}\left( \sqrt{ a} \right)$.
Categories: math.NT math.AG