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 $\mathbb{P}^n$.