The moduli stack $\mathcal{M}\_{0,4}$ of 4 points on $\mathbb{P}^1$ is isomorphic to $\mathbb{P}^1 - \{0,1,\infty\}$, where $[a,b,c,d]$ is identified with the point $\lambda\_{a,b,c}(d)$, where $\lambda\_{a,b,c}\in\text{Aut}(\mathbb{P}^1)$ sends $a,b,c\mapsto 0,1,\infty$. We will use $\lambda$ to identify $\mathcal{M}\_{0,4}$ and $\mathbb{P}^1 - \{0,1,\infty\}$. Over $\mathcal{M}\_{0,4}$, it has a universal family $\mathcal{M}\_{0,4}\times\mathbb{P}^1$, with four canonical sections $s\_1 = 0,s\_2 = 1,s\_3 = \infty,s\_4 = \Delta$, where $\Delta$ denotes the diagonal. The complement of the four sections is naturally identified with $\mathcal{M}\_{0,5}$.


The Deligne-Mumford compactification of $\mathcal{M}\_{0,4}$ is obtained by adding in the boundary points, so we have $\overline{\mathcal{M}\_{0,4}} \cong\mathbb{P}^1$.


On the other hand [page 5 of these notes](https://anddil.github.io/_pages/CohFT/Lecture1.pdf) indicate that the compactification of $\mathcal{M}\_{0,5}$ should be obtained by blowing up $\overline{\mathcal{M}\_{0,4}}\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$ at 3 points, and that this should now be the universal family over $\overline{\mathcal{M}\_{0,4}}$. Essentially, $\sigma\_4 = \Delta$ intersects $\sigma\_1,\sigma\_2,\sigma\_3$ each at 1 point, the intersection points lying over $\lambda = 0,1,\infty\in\overline{\mathcal{M}\_{0,4}}$ respectively.


This seems odd to me, since my feeling is that the sections $\sigma\_1,\sigma\_2,\sigma\_3,\sigma\_4$ should all be symmetric -- of course their definition is not, but their role in the moduli problem is. However, $\sigma\_1,\sigma\_2,\sigma\_3$ each meet the exceptional divisors at exactly 1 point, whereas $\sigma\_4$ meets them at 3 points, so if I haven't made a mistake we should have $\sigma\_1^2 = \sigma\_2^2 = \sigma\_3^2 = -1$, whereas $\sigma\_4^2 = -3$.


Is this correct? If so, can someone explain why the symmetry present in the moduli formulation is not present in the geometry of the sections?



Self intersections of the canonical sections on $\overline{\mathcal{M}_{0,5}}$
Your calculation of the self-intersection of $\sigma\_4$ is wrong. You're correct that the self-intersection drops by $1$ for each time it intersects the exceptional divisor, but you've assumed the self-intersection before you blow up is $0$ (appropriate for a constant section) and not the correct value of $2$ (the self-intersection of the diagonal of $\mathbb P^1 \times \mathbb P^1$).


