Rational points on a class of cubic hypersurfaces

Let r >= 3 be an integer and Q any positive definite quadratic form in r variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S-Q defined by x(3) = Q(y(1), . . . ,y(r))z. This confirms Manin's conjecture for any S-Q. Our proof is based on analytic methods, and uses some estimates for character sums and moments of L-functions. In particular, one of the ingredients is Siegel's mass formula in the argument for the case r = 3.