The power-saving manin-peyre conjecture for a senary cubic

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 ď2 × ď2 with bihomogeneous coordinates [x1 : X2 : X3], [y1 : Y2, y3] and in ď1 × ď1 × ď1 with multihomogeneous coordinates [x1 : Y1], [x2 : Y2], [x3 : Y3] defined by the same equation x1y2y3 + x2y1y3 + x3y1y2 = 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 A1 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.