I will report on joint work with Matthew Hase-Liu that shows that the moduli space of genus $g$ curves of degree $e$ on a smooth hypersurface of low degree only has terminal singularities, provided $e$ is sufficiently large with respect to $g$. Using a spreading argument together with a result of Mustata, we reduce the problem to counting points over finite fields on the jet schemes of these moduli spaces. We solve this counting problem by developing a suitable version of the circle method.
Régis de la Bretèche
Cathy Swaenepoel