Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.

Let $H_{X,d}$ be the Hilbert scheme of smooth hypersurfaces of degree $d$ in $X$.

Note that $H_{X,d}$ is a quasi-projective scheme over $\mathbb C$ equipped with an action of the automorphism group $G:=\mathrm{Aut}(X)$ of $X$.

Is the action of $G$ on $H_{X,d}$ proper? That is, is the DM-stack $[G\backslash H_{X,d}]$ separated?

There is a natural representable morphism of stacks

$$[G\backslash H_{X,d}] \to \mathcal M_g$$

What are some of its abstract properties? Is it quasi-finite? Does it induce an open immersion on coarse spaces?

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    $\begingroup$ There exist fourfolds $X$ such that the action of $G$ on the Hilbert scheme of smooth curves is non-proper, e.g., project a canonical curve in $\mathbb{P}^4$ to a hyperplane. I know of no example with $X$ a threefold. There are no examples with $X$ a surface: the specialization curve must be contained in a proper closed subvariety of $X$ that is invariant under an infinite subgroup of $G$, and that must be a rational curve if $X$ is a surface. Of course the map to $M_g$ need not be finite, e.g., curves of bidegree $(2,e)$ on $\mathbb{P}^1 \times \mathbb{P}^1$ with $e>2$ (count moduli). $\endgroup$ – Jason Starr May 27 '16 at 16:24
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    $\begingroup$ I just saw the condition about "no curves of degree $d$ and genus $g<2$". So for the $\mathbb{P}^1\times \mathbb{P}^1$ example, take a re-embedding by the complete linear system of curves of type $(2,3)$, and let $e$ in the comment above be $6$. The degree $d$ of these curves of type $(2,6)$ equals $18$, curves of type $(1,n)$ have degree $2n+3\neq 18$, and curves of type $(m,1)$ have degree $3m+2\neq 18$. Of course curves of type $(2,2)$ have degree $10\neq 18$. $\endgroup$ – Jason Starr May 27 '16 at 17:10

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