The locus of plane curves in the moduli stack of curves

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 $[U_d\mathrm{/}{\mathrm{PGL}}_3]\to {\mathcal{M}}_g$ is a locally closed embedding.