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Let $n$ be a positive multiple of $4$. We establish an asymptotic formula forthe number of rational points of bounded height on singular cubic hypersurfaces$S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new intwo aspects: first, it can be viewed as a modest start on the study of densityof rational points on those singular cubic hypersurfaces which are not coveredby the classical theorems of Davenport or Heath-Brown; second, it provesManin's conjecture for singular cubic hypersurfaces $S_n$ defined above.

Manin's conjecture for a class of singular cubic hypersurfaces