Pullback of boundary divisors under forgetful maps

Question

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\mathbf{U}}_{0,n}\longrightarrow\overline{\mathbf{M}}_{0,n}$ with the n disjoints sections $\sigma_i: \overline{\mathbf{M}}_{0,n}\longrightarrow \overline{\mathbf{U}}_{0,n}.$ Joachim Kock in "An invitatation to quantum cohomology" Example 1.5.11 gave the relationship between a boundary $\mathbf{F}_n$ of $\overline{\mathbf{M}}_{0,n}$ with the boundary $\mathbf{F}_{n+1}$ of $\overline{\mathbf{M}}_{0,n+1}$ by the following formula $$ \mathbf{F}_{n+1} = \varepsilon^*\mathbf{F}_n + \sum_{i}\sigma_i$$ where $\varepsilon: \overline{\mathbf{M}}_{0,n+1}\longrightarrow \overline{\mathbf{M}}_{0,n}$ is the forgetful maps. Can someone explain me more how to get the formula?

Answer 1

Here are some examples.

1. Consider $\epsilon: \overline{M}_{0,4} \to \overline{M}_{0,3}$. Since $\overline{M}_{0,3}$ is a point, $F_3 = 0$. On the other hand, the boundary of $\overline{M}_{0,4}$ consists of three points $0,1,\infty$. These are exactly the images of the three sections.
2. Consider $\epsilon : \overline{M}_{0,5} \to \overline{M}_{0,4}$ forgetting the $5$th marked point. The boundary of $\overline{M}_{0,5}$ consists of ten divisors $D_{ij,kmn}$. The divisors $D_{i5,jkl}$ are the images of the four sections of $\epsilon$. The remaining 6 divisors look like $D_{ij,kl5}$. Forgetting the marked point $5$, each of these divisors is a stable 4-pointed curve $D_{ij,kl}$, which is exactly a boundary point of $\overline{M}_{0,4}$. So the pullback of the boundary of $\overline{M}_{0,4}$ consists of these 6 divisors. This gives the formula in this case.

The general case is basically the same as the $\overline{M}_{0,5}$ case. All the divisors on $\overline{M}_{0,n}$ look like $D_{I,J}$ where $I \cup J$ is a partition of $\{1,\ldots,n\}$ with $\lvert I \lvert, \lvert J \lvert \geq 2$. The divisors where $I = \{i,n\}$ are the images of the $n-1$ sections of $\overline{M}_{0,n} \to \overline{M}_{0,n-1}$. The images of the remaining divisors on $\overline{M}_{0,n}$ give all the divisors on $\overline{M}_{0,n-1}$.

I would like to echo what David Holmes says in the comments. You could explicitly write out all the boundary divisors and check the statement by hand probably up to $\overline{M}_{0,6}$. (After that point there's probably too many divisors to do by hand, and regardless the result should be clear by then.) I think it's very helpful to do so in order to see what's going on.