On a conjecture of Wooley and lower bounds for cubic hypersurfaces

Let X ⊂ PQⁿ⁻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) Bⁿ⁻⁹ if n ≥ 39, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.