Let $\overline{M_g}$ be the moduli stack of stable curves of genus $g$. Let $H_g$ be the moduli stack of smooth hyperelliptic curves and $\overline{H_g}$ its compactification whose stack structure is given by the Hurwitz stack parameterizing degree two admissible covers in the sense of Harris--Mumford. There is a natural morphism $\overline{H_g} \to \overline{M_g}$ which is a regular embedding at the points corresponding to smooth curves. The question is: what is the behavior at the boundary? In other words, is it still a close embedding? If so, is it a regular embedding?

To put this in another way, consider a family of stable curves $X \to B$ such that it is a Kuranish family around every point of $B$. Let $H_B^\circ$ be the locus on $B$ whose fiber is smooth hyperellitpic and let $H$ be its closure in $B$. Then is the morphism $H \to B$ a regular embedding? What is the normal sheaf of $H \to B$?