Manin’s conjecture for singular cubic hypersurfaces

Let $S_Q$ denote the cubic hypersurface $x^3=Q(y_1,\dots ,y_m)z$,

where $Q$ is a positive definite quadratic form in $m$ variables with integer coefficients.

This $S_Q$ ranges over a class of singular cubic hypersurfaces as $Q$ varies.

For $S_Q$, we prove that Manin’s conjecture is true if $Q$ is locally determined, and we give an explicit asymptotic formula with a power saving error term; we also show in general that Manin’s conjecture is true up to a leading constant if $m\ge 6$ is even.