Counting rational points on hypersurfaces

Let $F(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ be a form of degree $d\ge 2$, which produces a geometrically irreducible hypersurface in ${\mathbb{P}}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever $n<6$, or whenever the hypersurface is not a union of lines, we obtain estimates that are essentially best possible and that are uniform in $d$ and $n$.