Non-smoothness of Moduli Spaces of Higher Genus Curves on Low Degree Hypersurfaces

Abstract

We show that the moduli space of degree $e$ maps from smooth genus $g\ge 1$ curves to an arbitrary low degree smooth hypersurface is singular when $e$ is large compared to $g$. We also give a lower bound for the dimension of the singular locus.

10 pages, comments welcome!