I am trying to show that the the forgetful map $\pi:\overline{\mathcal{M}}_{0,5}\to\overline{\mathcal{M}}_{0,4}$ is complex analytic. The description that I have of this map is purely geometrical:
- it forgets the last point on the non-nodal curves with 5 marked points,
- it does the same on nodal curves if the result is stable;
- if the result is not stable, then if the last point is on a bubble between nodes, we leave out the last point and the bubble, joining the two nodes, and if the last point is on a bubble which has another marked point $x_i$ on it and one node, the last point and the bubble go and the node gets the label $x_i$.
I have no idea how one would go about proving that this is a complex analytic map. My first thought was to look at how this map looks on $\mathcal{M}_{0,5}\to\mathcal{M}_{0,4}$, but since $(\mathcal{M}_{0,5}\cong\mathbb{C}P^1\setminus\{0,1,\infty\})^2\setminus\Delta$ and $\overline{M}_{0, 4}\cong\mathbb{C}P^1$, this is just the projection which is indeed complex analytic, but doesn't help me to say anything more about the map $\pi$. Any thoughts or hints are appreciated.