On a conjecture of Wooley and lower bounds for cubic hypersurfaces

Let X Pₐ^n-1 be a cubic hypersurface cut out by the vanishing of a non-degenerate rational cubic form in n variables. Let N (X, B) denote the number of rational points on X of height at most B. In this article we obtain lower bounds for N (X, B) for cubic hypersufaces, provided only that n is large enough. In particular, we show that N (X, B) B^n-9 if n 39, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.