La conjecture de Manin pour une famille de vari\'et\'es en dimension sup\'erieure

Inspired by a method of La Bret`eche relying on some unique factorisation, we generalize work of Blomer, Br"udern, and Salberger to prove Manin’s conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the projective varieties defined by the following equation $$x_1{y}_2{y}_3\cdots y_n+x_2{y}_1{y}_3\cdots y_n+\cdots +x_n{y}_1{y}_2\cdots y_{n-1}=0.$$ in ${\mathbb{P}}_{\mathbb{Q}}^{2n-1}$ for all $n⩾3$. This paper comes with an Appendix by Per Salberger.