Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory 47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre 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 et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann. 370 (2018), 491–553] 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 by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc. 108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc. 166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.