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

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 $x_1{y}_2{y}_3+x_2{y}_1{y}_3+x_3{y}_1{y}_2=0$. We thus improve on recent work of Blomer, Brüdern 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.

29 pages (title fixed)

References in corpus (3)

• Tamagawa numbers of polarized algebraic varieties
• On a senary cubic form
• Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups