Is there any example (or more ambitiously, classification) of $X$ with following properties? 


* $X$ is a variety over $\mathbb{C}$;
* $X$ is projective and normal;
* $\rho(X) = 1$;
* $X$ is birational to $\mathbb{P}^n$.


Also, I want to hear a result after adding a singularity condition: 
How about when $X$ is $\mathbb{Q}$-factorial?
How about $X$ is non-singular?


I can't find an example which is not isomorphic to $\mathbb{P}^n$.
The only result I know in this direction is Mori's theorem:
If a nonsingular variety $X$ has ample tangent bundle, then $X$ is isomorphic to $\mathbb{P}^n$.



Example of rational projective variety of Picard number 1
Hyperquadrics of dimension at least three.


