I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point.
My idea is to ignore the stack structure at first, construct a moduli space and take the stack structure finally into account by displaying the stack as an algebraic groupoid $[M/G]$, with the action of $G$ on $M$ counting in automorphisms of the domain if necessary. A naive first approach is to describe the branching locus together with the marked point via $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$
, letting the first and second component describe the branch points and the third component describe the marked point.


So far I ignore the fact that the marked point is only on one sheet of the domain, is it legit?


Now when it comes to taking care of automorphisms I struggle. The generic case (three distinct points) should only possess a trivial automorphism group, but degenerate cases (e.g. the marked point coinciding with one of the branch points) hold automorphisms (e.g. the automorphism swapping the sheets of the domain).


Are there further automorphisms to be considered and how do I arrive at a global presentation of the form $[M/G]$
(is it even possible)? Last but not least I am not sure if I have to adjust the unnatural labels of the branch points by a $\mathbb{Z}\_2$ action arriving at something like $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1/\mathbb{Z}\_2$.


I am thankful for any advice.



moduli stack of double covers of $\mathbb{P}^1$ with one marked point
It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but you should always at least have a precise idea of what the field-valued points are.


For instance, you don't specify how many branch points / what genus you want the covers to have. I guess you want two branch points, and genus zero.


You also have to define the stability condition for a cover. The most reasonable choice is probably Kontsevich's definition of stable maps, where double cover is defined to be any stable map of degree 2, but you could also take a double cover to literally be a two-to-one map with no constant components. This is no problem in the genus zero case because there are no stable covers with constant component.


Your construction is completely valid for the coarse moduli space, although of course you are right that you have to quotient by $\mathbb Z/2$ switching the two marked points. However, taking that quotient in stacks gives the wrong answer, because it implies you have extra automorphism when the two points coincide. Assuming you interpret this as a (stable) nodal curve, that extra automorphism is meaningless. Instead it's better to use the isomorphism $(\mathbb P^1 \times \mathbb P^1 ) / (\mathbb Z/2) = \mathbb P^2$, which you can see explicitly by viewing a degree $2$ branch divisor as a line inside the space of sections of $\mathcal O(2)$.


The moduli stack of double covers of $\mathbb P^1$ is not quite $\mathbb P^2$, because we have to account for the hyerelliptic involution. In fact, it is the quotient of $\mathbb A^3 - \{0\}$ by $\mathbb G\_m$, where $\mathbb G\_m$ acts via scalar multiplication by the square. To see this, represent a double cover as the relative Spec of $\mathcal O\_{\mathbb P^1} + \mathcal O\_{\mathbb P^1}(-1)$, with multiplication given by $(a\_1,b\_1)\times(a\_2,b\_2) \to (a\_1a\_2 + f b\_1b\_2, a\_1b\_2 + a\_2b\_1)$ for $f$ a nonzero section of $\mathcal O\_{\mathbb P^1}(2)$. This defines a double cover, every double cover can be defined this way, and equivalences between the double covers defined by $f\_1$ and $f\_2$ are parameterized by $\lambda \in \mathbb G\_m$ with $f\_2=\lambda^2 f\_1$. Locally, this is equal to the quotient of $\mathbb P^2$ by $ \pm 1 \subset \mathbb G\_m$, but not globally (as I earlier claimed).


To construct the moduli stack in this setting, you have to take the universal family of double covers over this stack. In other words, you take the universal family of double covers over $\mathbb A^3 - \{0\}$, constructed by exactly this relative Spec, and quotient by this $\mathbb G\_m$ action. This works because in general, the moduli stack of varieties with one marked point is the universal family over the moduli stack of varieties. For curves, this is true with Deligne-Mumford or Kontsevich stability conditions even if the marked point touches a singularity, although this case requires blowing up the universal family of curves over the moduli space of curves with marked points to produce the universal family of curves with marked points.


With greater numbers of branch points, this won't quite work, as when three or more branch points coincide the double cover is no longer stable, and you will need to do some blow-up, or other tricks, to deal with it.


