The power-saving Manin-Peyre's conjectures for a senary cubic

TL;DR

This paper proves a strong version of the Manin-Peyre's conjectures with full asymptotics and power-saving error terms for specific algebraic varieties defined by a cubic equation, improving previous results and providing a new proof method.

Contribution

It establishes the conjectures for two varieties using a descent on the universal torsor, offering a different approach from harmonic analysis and settling the conjectures for this case.

Findings

Proved strong asymptotics with power-saving error terms.

Validated Manin-Peyre's conjectures for the specified varieties.

Provided a new proof method via universal torsor descent.

Abstract

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.