We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène-Sansuc and Harpaz-Skorobogatov-Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.

Abstract. -Following the line of attack from La Bretèche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Châtelet surfaces which are defined as minimal proper smooth models of affine surfaces of the form,] is a polynomial of degree 4 whose factorisation into irreducibles contains two non proportional linear factors and a quadratic factor which is irreducible over. This result deals with the last remaining case of Manin's conjecture for Châtelet surfaces with a = −1 and essentially settles Manin's conjecture for Châtelet surfaces with a < 0.

In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface ofin agreement with the Manin-Peyre conjectures.

Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

Using recent work of the first author [3], 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 P 2 × P 2 with bihomogeneous coordinates [x 1 : x 2 : x 3 ], [y 1 : y 2 , y 3 ] and in P 1 × P 1 × 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 [9] 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 A 1 and three lines (the other existing proof relying on harmonic analysis [18]). Together with [8] or with recent work of the second author [22], this settles the study of the Manin-Peyre's conjectures for this equation.1991 Mathematics Subject Classification. 11D45, 11N37, 11M41.