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 moduli 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 higher genus smooth projective curves. In particular, we explain how the geometry of numbers can be understood via the Beauville-Laszlo theorem in terms of vector bundles on curves and their slopes, allowing us to prove a higher genus variant of Davenport's shrinking lemma. As a corollary, we apply this result to show the Fujita invariant of any proper subvariety of a smooth hypersurface of low degree is less than 1.