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 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?
