Counting rational points on hypersurfaces and higher order expansions

As an application we derive higher order expansions for the number of rational points of bounded anticanonical height on the projective hypersurface F(x)=0 for forms F(x) in sufficiently many variables. The main term of this expansion is the one predicted by Manin's conjecture. Our new result gives some evidence for how the conjecture could be refined to cover lower order terms in the setting of high-dimensional complete intersections in projective space.