Hi everybody,

let $\mathcal{P}_g$ be the moduli stack parametrizing pairs $(S,C)$ where $S$ is a K3 surface with a primitive polarization $L$ of genus $g$ and $C \in |L|$ is a smooth curve of genus $g$. Let $\mathcal{M}_g$ be the moduli stack of smooth curves of genus $g$ and let $c_g:\mathcal{P}_g \rightarrow \mathcal{M}_g$ be the morphism defined as $c_g((S,C))=[C]$. Define $\mathcal{K}_g$ as the image of $c_g$ and $\overline{\mathcal{K}}_g$ as its closure inside $\overline{\mathcal{M}}_g$, the moduli stack of stable curves of genus $g$.

Is it something known about the codimension of the singular locus $\text{Sing}(\overline{\mathcal{K}}_g)$ in $\overline{\mathcal{K}}_g$?

Maybe $\overline{\mathcal{K}}_g$ is even smooth as a substack of $\overline{\mathcal{M}}_g$...

I'm particularly interested in the case $g=10$, but all contributions are welcome.

Thank you!