The density of rational points on non-singular hypersurfaces, II

This paper establishes the conjecture that a non-singular projective hypersurface of dimension $r$, which is not equal to a linear space, contains $O(B^{r+\epsilon})$ rational points of height at most $B$, for any choice of $\epsilon>0$. The implied constant in this estimate depends at most upon $\epsilon, r$ and the degree of the hypersurface.