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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 $H \to B$?

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The tangent space at a point should be the deformation space, or $H^1(X,T_X)$. For curves, by Serre, this is dual to $H^0(X,K_X^2)$. For hyperelliptic curves it shouldn't be too hard to write down this vector space in a canonical way. The tricky part is to find what subspace corresponds to the deformations that remain hyperelliptic. I don't immediately see a good way of describing this. – Will Sawin Apr 1 '13 at 3:55
For smooth curves, this is not hard. What's tricky is the singular part... – marker Apr 1 '13 at 14:06
up vote 1 down vote accepted

At least in characteristic not equal to $2$ (so that double covers are tamely ramified), there is a nice stack parameterizing "twisted stable maps" introduced by Abramovich, Corti and Vistoli (and studied further by Abramovich, Olsson and Vistoli in positive characteristic). The deformation theory is discussed in those articles.

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