# Pullback of boundary divisors under forgetful maps

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?

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• You write `stable n-pointed smooth'; I think you want to omit the word smooth. As far as answering the question, have you worked out some examples with small n? I think if you work out the cases $n=3$ and $n=4$ by hand you will see what is going on. I think this is more useful than me writing out an argument, but feel free to disagree :-). – David Holmes May 21 at 8:33