What is the moduli space $M _{0,5}$? Let $M_{0,n}$ denote the moduli space which consists of genus 0 non singular projective curves with n distinct marked points upto marked point isomorphism.
So $M_{0,3}$ is a singleton set as any 3 points on $\mathbb P^1$ can be mapped to ${0,1,\infty }$ via an automorphism.
Similarly, $M_{0,4}$ can be identified with $\mathbb P^1-\{{0,1, {\infty \} }}$ as the first 3 marked points can be sent to $0,1,\infty$ and the 4th point can be mapped to any other point.

What is $M_{0,5}$?


What is
$\overline M _{0,5}$? (The Kontsevich Compactification which consists of nodal singular genus 0 projective curves with n distinct marked points upto marked point isomorphism with finite automorphism group)

Here also we can send first 3 points to $0,1,\infty$ but what shall we do about the rest 2 points?
 A: For the moduli space $M_{0, 5}$, you can take a similar approach as for $M_{0, 4}$. Note that any curve $(C, x_1, x_2, x_3, x_4, x_5)$ with $5$ marked points is isomorphic to a curve of the form $(\mathbb P^1, 0, 1, \infty, t_1, t_2)$. It should be clear that two such curves $(\mathbb P^1, 0, 1, \infty, t_1, t_2)$ and $(\mathbb P^1, 0, 1, \infty, t'_1, t'_2)$ are isomorphic as marked curves iff $t_1 = t'_1$ and $t_2 = t'_2$, so that $M_{0, 5} = \{ (t_1, t_2) \in M_{0, 4} \mid t_1 \neq t_2\}$. You can generalize this to any $M_{0, n}$ with $n \geq 4$ in a similar manner.
For the compactification you can refer to Proposition 10 in these lecture notes by A. Di Lorenzo. The space $\overline{M}_{0, n}$ is the universal family $\overline{U}_{0, n - 1}$ over $\overline{M}_{0, n-1}$. In your specific case, we have that $\overline{M}_{0, 5}$ is the blow-up of $\mathbb P^1 \times \mathbb P^1$ at the points $(0, 0)$, $(1, 1)$ and $(\infty, \infty)$.
Judging by your comment, you seem to be interested in cohomology of stable curves. This paper by D. Zvovinke might be interesting. In particular, you can find the same argument for the universal curve of $\overline{M}_{0, 4}$ in Example 1.47.
