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Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.

I feel that I should already know the answer to this, but it never sits quite right in my head.

When dealing with Gromov-Witten theory, one is obviously interested in part in the moduli stack of stable curves, $\overline{M}_{g,n}$ (actually, we probably should really care about the stack of pre-stable curves, but that's not going to be hugely relevant for my case).

When dealing with orbifold Gromov-Witten theory, we instead need to consider the moduli stack of twisted stable curves, which are those curves that may have isotropy at the marked points and nodes. For the problems that I am interested in, it is usually the case that every marked point has the same isotropy---in particular, this is $\mathbb{Z}/2$. Let us denote then the stack of twisted stable curves whose marked points all have $\mathbb{Z}/r$ isotropy as $\overline{M}_{g,n}^{\mathbb{Z}/r}$.

There is an obvious map $\overline{M}_{g,n}^{\mathbb{Z}/r} \to \overline{M}_{g,n}$ given by taking the coarse moduli space.

My question is: What is the relationship between these two stacks?

Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.

I feel that I should already know the answer to this, but it never sits quite right in my head.

When dealing with Gromov-Witten theory, one is obviously interested in part in the moduli stack of stable curves, $\overline{M}_{g,n}$ (actually, we probably should really care about the stack of pre-stable curves, but that's not going to be hugely relevant for my case).

When dealing with orbifold Gromov-Witten theory, we instead need to consider the moduli stack of twisted stable curves, which are those curves that may have isotropy at the marked points and nodes. For the problems that I am interested in, it is usually the case that every marked point has the same isotropy---in particular, this is $\mathbb{Z}/2$. Let us denote then the stack of twisted stable curves whose marked points all have $\mathbb{Z}/r$ isotropy as $\overline{M}_{g,n}^{\mathbb{Z}/r}$.

There is an obvious map $\overline{M}_{g,n}^{\mathbb{Z}/r} \to \overline{M}_{g,n}$ given by taking the coarse moduli space.

My question is: What is the relationship between these two stacks?

Edit further: Some further questions that I feel that are relevant.

The moduli spaces $\overline{M}_{g,n}$ have a lot of structure maps between them. For example, they have forgetful maps $\overline{M}_{g,n+1} \to \overline{M}_{g,n}$, gluing maps, etc. They also have a bunch of canonically defined bundles over them, such as the cotangent line bundles, the Hodge bundles, etc. In particular, the divisor classes of the cotangent line bundles can be computed recursively given a few base cases by pulling them back via the forgetful maps.

How does these work for the moduli spaces $\overline{M}_{g,n}^{\mathbb{Z}/r}$? From what I understand, the $\psi$-classes come from the coarse curve, and so all of the expected relations should still hold. In particular, for $g = 0$ we can describe the $\psi$-classes purely combinatorially in terms of partitioning the marked points among different components of our curve. Is this picture correct?

Perhaps a better question: Is there a good reference which covers this?

I feel that I should already know the answer to this, but it never sits quite right in my head. Let's look at something relatively concrete.

One way you can look at the moduli of hyperelliptic curves is by viewing them as double covers of a $(2g+2)$-marked $\mathbb{P}^1$. Naively, this seems fine; if we look at the dimension of these two spaces, they certainly agree. There is likely an issue regarding a choice of hyperelliptic involution, and an ordering of the marked points, but that's not really my concern here.

My concern is more the following. Which is the correct moduli space to look at?

On the one hand, we have simply the moduli space $\overline{\mathcal{M}}_{0, 2g+2}$ of genus 0, $(2g+2)$-marked points. One can by hand then build a double cover of any given curve of this form to obtain a hyperelliptic curve of genus $g$.

On the other hand, we have the moduli space $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ (for lack of better notation) consisting of those orbifold genus zero curves with $(2g+2)$ marked points each of which has $\mathbb{Z}/2$-isotropy. This definitely yields a map to the moduli space of hyperelliptic curves by taking the pullback via the diagram $$ \begin{align} & pt \\ & \downarrow \\ U \to & B\mathbb{Z}/2 \end{align} $$ where $U$ is any family of curves in $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$.

This secondary approach is certainly more natural. As described, it comes with a natural map to the moduli space of marked hyperelliptic curves, as well as a natural map (by taking the coarse moduli space) to $\overline{\mathcal{M}}_{0,2g+2}$

I suppose then the main thrust of the question is: What is the difference (in terms of deformations, bundles, etc.) between the moduli spaces $\overline{\mathcal{M}}_{0,n}^{\mathbb{Z}/r}$ and $\overline{\mathcal{M}}_{0,n}$ where $r$ can be any integer? And what happens for higher genus? For example, I know that the tangent space at a smooth curve $(\Sigma, x_1, \ldots, x_n)$ in the moduli space $\overline{\mathcal{M}}_{0,n}$ is given by $$ H^1\big(\Sigma, T_\Sigma(-x_1 - \cdots - x_n)\big) $$

Side notes: I'm more interested in the case that $r = 2$, but I feel that I should ask more generally. Also, if the isotropy varies from marked point to marked point, then I would expect the issue to be more complicated, so I'm restricting to this simpler case. However, if there is no real difficulty in extending it, then by all means, do tell.

Edit: I realize that what I said about $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ being the something like the moduli space of hyperelliptic curves is not quite correct. What is correct is that $$ \overline{M}_{0,2g+2}(B\mathbb{Z}/2) $$ gives us that, since we actually need the data of the map to get a hyperelliptic curve. However, my main question is about the relationship/differences between $\overline{\mathcal{M}}_{0,2g+2}$ and $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$, so I'll leave that part stand as a "motivating example".

Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.

I feel that I should already know the answer to this, but it never sits quite right in my head.

When dealing with Gromov-Witten theory, one is obviously interested in part in the moduli stack of stable curves, $\overline{M}_{g,n}$ (actually, we probably should really care about the stack of pre-stable curves, but that's not going to be hugely relevant for my case).

When dealing with orbifold Gromov-Witten theory, we instead need to consider the moduli stack of twisted stable curves, which are those curves that may have isotropy at the marked points and nodes. For the problems that I am interested in, it is usually the case that every marked point has the same isotropy---in particular, this is $\mathbb{Z}/2$. Let us denote then the stack of twisted stable curves whose marked points all have $\mathbb{Z}/r$ isotropy as $\overline{M}_{g,n}^{\mathbb{Z}/r}$.

There is an obvious map $\overline{M}_{g,n}^{\mathbb{Z}/r} \to \overline{M}_{g,n}$ given by taking the coarse moduli space.

My question is: What is the relationship between these two stacks?

I feel that I should already know the answer to this, but it never sits quite right in my head. Let's look at something relatively concrete.

One way you can look at the moduli of hyperelliptic curves is by viewing them as double covers of a $(2g+2)$-marked $\mathbb{P}^1$. Naively, this seems fine; if we look at the dimension of these two spaces, they certainly agree. There is likely an issue regarding a choice of hyperelliptic involution, and an ordering of the marked points, but that's not really my concern here.

My concern is more the following. Which is the correct moduli space to look at?

On the one hand, we have simply the moduli space $\overline{\mathcal{M}}_{0, 2g+2}$ of genus 0, $(2g+2)$-marked points. One can by hand then build a double cover of any given curve of this form to obtain a hyperelliptic curve of genus $g$.

On the other hand, we have the moduli space $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ (for lack of better notation) consisting of those orbifold genus zero curves with $(2g+2)$ marked points each of which has $\mathbb{Z}/2$-isotropy. This definitely yields a map to the moduli space of hyperelliptic curves by taking the pullback via the diagram $$ \begin{align} & pt \\ & \downarrow \\ U \to & B\mathbb{Z}/2 \end{align} $$ where $U$ is any family of curves in $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$.

This secondary approach is certainly more natural. As described, it comes with a natural map to the moduli space of marked hyperelliptic curves, as well as a natural map (by taking the coarse moduli space) to $\overline{\mathcal{M}}_{0,2g+2}$

I suppose then the main thrust of the question is: What is the difference (in terms of deformations, bundles, etc.) between the moduli spaces $\overline{\mathcal{M}}_{0,n}^{\mathbb{Z}/r}$ and $\overline{\mathcal{M}}_{0,n}$ where $r$ can be any integer? And what happens for higher genus? For example, I know that the tangent space at a smooth curve $(\Sigma, x_1, \ldots, x_n)$ in the moduli space $\overline{\mathcal{M}}_{0,n}$ is given by $$ H^1\big(\Sigma, T_\Sigma(-x_1 - \cdots - x_n)\big) $$

Side notes: I'm more interested in the case that $r = 2$, but I feel that I should ask more generally. Also, if the isotropy varies from marked point to marked point, then I would expect the issue to be more complicated, so I'm restricting to this simpler case. However, if there is no real difficulty in extending it, then by all means, do tell.

I feel that I should already know the answer to this, but it never sits quite right in my head. Let's look at something relatively concrete.

One way you can look at the moduli of hyperelliptic curves is by viewing them as double covers of a $(2g+2)$-marked $\mathbb{P}^1$. Naively, this seems fine; if we look at the dimension of these two spaces, they certainly agree. There is likely an issue regarding a choice of hyperelliptic involution, and an ordering of the marked points, but that's not really my concern here.

My concern is more the following. Which is the correct moduli space to look at?

On the one hand, we have simply the moduli space $\overline{\mathcal{M}}_{0, 2g+2}$ of genus 0, $(2g+2)$-marked points. One can by hand then build a double cover of any given curve of this form to obtain a hyperelliptic curve of genus $g$.

On the other hand, we have the moduli space $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ (for lack of better notation) consisting of those orbifold genus zero curves with $(2g+2)$ marked points each of which has $\mathbb{Z}/2$-isotropy. This definitely yields a map to the moduli space of hyperelliptic curves by taking the pullback via the diagram $$ \begin{align} & pt \\ & \downarrow \\ U \to & B\mathbb{Z}/2 \end{align} $$ where $U$ is any family of curves in $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$.

This secondary approach is certainly more natural. As described, it comes with a natural map to the moduli space of marked hyperelliptic curves, as well as a natural map (by taking the coarse moduli space) to $\overline{\mathcal{M}}_{0,2g+2}$

I suppose then the main thrust of the question is: What is the difference (in terms of deformations, bundles, etc.) between the moduli spaces $\overline{\mathcal{M}}_{0,n}^{\mathbb{Z}/r}$ and $\overline{\mathcal{M}}_{0,n}$ where $r$ can be any integer? And what happens for higher genus? For example, I know that the tangent space at a smooth curve $(\Sigma, x_1, \ldots, x_n)$ in the moduli space $\overline{\mathcal{M}}_{0,n}$ is given by $$ H^1\big(\Sigma, T_\Sigma(-x_1 - \cdots - x_n)\big) $$

Side notes: I'm more interested in the case that $r = 2$, but I feel that I should ask more generally. Also, if the isotropy varies from marked point to marked point, then I would expect the issue to be more complicated, so I'm restricting to this simpler case. However, if there is no real difficulty in extending it, then by all means, do tell.

Edit: I realize that what I said about $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ being the something like the moduli space of hyperelliptic curves is not quite correct. What is correct is that $$ \overline{M}_{0,2g+2}(B\mathbb{Z}/2) $$ gives us that, since we actually need the data of the map to get a hyperelliptic curve. However, my main question is about the relationship/differences between $\overline{\mathcal{M}}_{0,2g+2}$ and $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$, so I'll leave that part stand as a "motivating example".