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The power-saving Manin-Peyre conjecture for a senary cubic. (English) Zbl 1459.11086

Summary: Using work of the first author [Algebra Number Theory 13, No. 2, 251–300 (2019; Zbl 1470.11217)], 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 V. Blomer et al. [Math. Ann. 370, No. 1–2, 491–553 (2018; Zbl 1437.11048)] 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 A. Chambert-Loir and Y. Tschinkel [Invent. Math. 148, No. 2, 421–452 (2002; Zbl 1067.11036)]). Together with V. Blomer et al. [Proc. Lond. Math. Soc. (3) 108, No. 4, 911–964 (2014; Zbl 1338.11058)] or with work of the second author [Math. Proc. Camb. Philos. Soc. 166, No. 3, 433–486 (2019; Zbl 1419.14026)], this settles the study of the Manin-Peyre conjectures for this equation.

MSC:

11D45 Counting solutions of Diophantine equations
11N37 Asymptotic results on arithmetic functions
11M41 Other Dirichlet series and zeta functions
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