Let us consider the space of configurations $\overline{\mathcal{M}}_{0,2,1,(1,1)}$ of an annulus with a marked point on the interior boundary component (let's call it "out") a marked point on the exterior one (say "in") and an interior marked point "z". 

> I'd like to understand better those configurations satisfying a symmetry condition given by reflection along the horizontal axis, let's call them $\overline{\mathcal{M}^{\Bbb Z_2}}_{0,2,1,(1,1)}$


[![enter image description here][1]][1]

I'm thinking of this reflection as a $\Bbb Z_2$-action on the moduli space $\overline{\mathcal{M}}_{0,2,1,(1,1)}$, and the symmetric configurations are by definition the fixed locus of such action.

As far as I understood (following mainly the [PhD thesis of Melissa Liu][2] ) to construct such moduli space for an annulus $(A,\text{in},\text{out},z)$ we start with the moduli space of the double of the annulus $(A_{\Bbb C},\sigma,\text{in},\text{out},z,z')$  were $\sigma$ is the antiholomorphic involution (should be well defined up to iso from the construction of the double). The way I'm thinking about it is to consider the standard torus $S^1\times S^1$ in $\Bbb R^3$ and the plane $(z=0)$ to be the plane of symmetry. $\sigma$ corresponds to reflecting along this plane. In this model the boundaries of the annulus lie on such plane and hence $\sigma(\text{in})=\text{in}$, $\sigma(\text{out})=\text{out}$ and we require $\sigma(z)=z'$.

In this setting my additional symmetry appears as another symmetry along (for example) the plane $(x=0)$. Forcing this kind of symmetry (and taking the compactification) gives me a pentagon as a moduli space, where the two coordinates are basically the conformal parameter of the annulus and the position of the marked point $z$ along the blue segment. The boundary marked points $\text{in}$ and $\text{out}$ are not allowed to move.

[![enter image description here][3]][3]

Is the fact that by direct inspection the moduli space of symmetric configurations appears as a pentagon enough to say that the coordinates given by conformal parameter and position of the point $z$ (along the blue segment) enough to conclude that it's a manifold with corners?


As the tag suggests, my background is symplectic geometry, I would appreciate some reference in case of extensive use of algebraic geometry.


  [1]: https://i.sstatic.net/Br6f0.jpg
  [2]: https://arxiv.org/pdf/math/0210257.pdf
  [3]: https://i.sstatic.net/VE64L.jpg