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

TL;DR

This paper demonstrates that the moduli space of higher genus curves mapped to low degree hypersurfaces becomes singular for large degrees, providing bounds on the singular locus dimension.

Contribution

It establishes the non-smoothness of moduli spaces for high-degree maps from genus g curves to low degree hypersurfaces and offers bounds on the singular locus.

Findings

Moduli space is singular when degree e is large relative to genus g.

Provides a lower bound for the dimension of the singular locus.

Shows non-smoothness for a broad class of 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.