What circles can do for you
-- Will Sawin, November 16, 2018

<p>

In joint work with Tim Browning, we study the moduli space of rational curves on a very low-degree hypersurface in projective space. We calculate its dimension and show it is irreducilble and smooth in high codimension. In addition, we lower bound the codimension of the locus of rational curves that are not very free. These results are proved using the circle method from analytic number theory.