Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with bihomogeneous coordinates $[x_1:x_2:x_3],[y_1:y_2,y_3]$ and in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with multihomogeneous coordinates $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ defined by the same equation $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$. We thus improve on recent work of Blomer, Br\"udern and Salberger \cite{BBS} and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type $\mathbf{A}_1$ and three lines (the other existing proof relying on harmonic analysis \cite{CLT}). Together with~\cite{Blomer2014} or with recent work of the second author \cite{Dest2}, this settles the study of the Manin-Peyre's conjectures for this equation.

中文翻译：

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SENARY CUBIC 的节能 MANIN-PEYRE 猜想

使用第一作者最近的工作~\cite{Bet}，我们证明了 Manin-Peyre 猜想的强版本，在 $\mathbb{P}^2 中分别对两个变体具有完全渐近和省电误差项\times \mathbb{P}^2$ 具有双齐次坐标 $[x_1:x_2:x_3],[y_1:y_2,y_3]$ 和 $\mathbb{P}^1\times \mathbb{P}^1 \次 \mathbb{P}^1$ 具有多重齐次坐标 $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ 由相同的方程 $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$ 定义。因此，我们改进了 Blomer、Br\"udern 和 Salberger \cite{BBS} 的最近工作，并基于在具有奇点类型的 6 次 del Pezzo 曲面的情况下对猜想的通用托量的下降提供了不同的证明$\mathbf{A}_1$ 和三行（另一个依赖谐波分析的现有证明 \cite{CLT}）。