  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.
