Which subvarieties of $P^3$ are isomorphic to $P^2$?

Question

What are all the subvarieties $X$ in the projective space $P^3$ such that $X$ is isomorphic to $P^2$?

Answer 1

I am assuming you want to know which hypersurface $X \subset \Bbb P^3$ is isomorphic to $\Bbb P^2$. First, it is clear that any linear equation $f(x,y,z,w) = 0$ defines such an hypersurface. Conversely, for $d \geq 2$ a surface of degree $d$ is not even diffeomorphic to $\Bbb P^2$ as an easy Euler characteristic computation shows, so any hypersurface isomorphic to $\Bbb P^2$ should be given by a linear equation.