The locus of plane curves in the moduli stack of curves

Abstract

Let $d\ge 4$ and let $U_d$ denote the locus of smooth curves in the Hilbert scheme of degree $d$ plane curves. If the members of $U_d$ have genus $g$, let ${\mathcal{M}}_g$ denote the moduli stack of genus $g$ curves. We show that the natural map $[U_d/{\mathrm{PGL}}_3]\to {\mathcal{M}}_g$ is a locally closed embedding.

Minor edits were made. The paper has been accepted to a special issue of "Algebraic Geometry and Physics" in honor of Fedor Bogomolov