Manin's conjecture for a class of singular hypersurfaces

Let n be a positive multiple of 4 or n = 2. In this talk, we shall show how to establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by [ $x^{3} = (y^2_{1} + ... + y^2_{n})z$ ], by analytic method. This result is new in two aspects: first, it can be viewed as a modest start on the study of density of rational points on those singular cubic hypersurfaces which are not covered by the classical theorems of Davenport or Heath-Brown; second, it proves Manin’s conjecture for singular cubic hypersurfaces $S_n$ defined above. (Joint works with Regis de la Breteche, Jianya Liu & Yongqing Zhao)