A higher genus circle method and an application to geometric Manin's conjecture

Browning and Vishe used the Hardy-Littlewood circle method to show the space of rational curves on smooth hypersurfaces of low degree is irreducible and of the expected dimension. We reinterpret the circle method geometrically and prove a generalization for fixed smooth projective curves. We then apply this result to show the Fujita invariant of any proper subvariety of a smooth hypersurface of low degree is less than 1.