Is there a geometric model for the universal cover $\tilde{X}$ of the space $X=\mathbb{CP}^2\text{-Riemann surface}$?

The Riemann surface is given by $\{(x,y)\mid P(x,y) = 0\}$ for generic polynomial $P$, in local coordinates.

I am looking for 2d analogy of 1d situation. In 1d $\mathbb{CP}^1-\{x_i\}$ is uniformized by upper half plane without points $a_i=0,...,n$.

I think this is some well studied space, probably well covered in textbooks, I just do not know the name.