Example of a nontrivial finite endomorphisms of projective space $\mathbb{P}_k^n$ I'm looking for an example of a nontrivial finite endomorphism of projective space $\mathbb{P}^n$. By nontrivial, I mean that the morphism $f:\mathbb{P}^n\to\mathbb{P}^n$ should not simply restrict to some very obvious affine morphism $f:V_+(x_i)\to V_+(x_i)$, though of course these examples are the easiest to come up with, making my task a bit more difficult.
More specifically, I'm looking for some finite endomorphism on $\mathbb{P}_k^n$ the intuitively spreads out the affine opens to cover a larger part of $\mathbb{P}_k^n$ then before. Is there some easy example of this?
As a slightly related question, if I replace "finite" with "surjective", what examples are there?
 A: 1) A morphism $f:\mathbb P^n\to \mathbb P^n$ is either constant or finite. Strange, eh?      
2) If $f$ is not constant it is of the form $f(x)=(f_0(x):\cdots : f_n(x))$ where the $f_i\in k[T_0:\cdots:T_n]$ are homogeneous polynomials of  the same degree $d$.
An arbitrary choice of the $f_i$'s however does not necessarily yield  a morphism:
For example if $n=2$ the Cremona transformation $\mathbb P^2 \to \mathbb P^2:(x_0:x_1:x_2)\mapsto (x_0x_1:x_1x_2:x_2x_0)$ is a rational map which is not  a morphism because it is not defined at the three points $(1:0:0), (0:1:0),(0:0:1)$.     
3) The $f_i$'s will define a morphism $f$ if and only if the hypersurfaces $V(f_i)\subset \mathbb P^n$ have empty intersection $\cap^n_{i=0} V(f_i)=\emptyset $, just as Mohan wrote in his comment to the question.
If this is the case the morphism $f=(f_0:\cdots:f_n):\mathbb P^n\to \mathbb P^n$ will have degree $d^n$.
The simplest example of such a morphism of degree $d$ is $f(x_0:\cdots:x_d)=(x_0^d:\cdots:x_n^d)$ . 
4) As to your request, it can indeed happen that arbitrarily small open sets in the domain   of $f$ get spread out in the sense that their image  is the whole codomain.
For example if $f$ is the map $$f:\mathbb P^1(\mathbb C)\to \mathbb P^1(\mathbb C):(x_0:x_1)\mapsto (x_0^2:x_1^2)$$ 
the image of the open subset $\mathbb P^1(\mathbb C)\setminus\{(1:1),(1:2)\cdots (1:2017)\}$ of $\mathbb P^1(\mathbb C)$ is all of $\mathbb P^1(\mathbb C)$.
(Like good wine this example will improve with the passing of years).
