How to show $P^1\times P^1$ (as projective variety by Segre embedding) is not isomorphic to $P^2$? I am a beginner.
This is an exercise from Hartshorne Chapter 1, 4.5. By his hint, it seems this can be argued that there are two curves in image of Segre embedding that do not intersect with each other while in $P^2$ any two curves intersect.
I feel this solution is very special. I would like to know more.
Is there any invariant to detect whether two birational equivalent varieties are iso or not?
Thanks!

We have  for the dimension of the space of sections of the tangent bundle   $$h^0(\mathbb P^2, T_{\mathbb P^2})=8$$This follows from the Euler sequence of vector bundles $$ 0\to \mathcal O_{\mathbb P^2}\to \mathcal O_{\mathbb P^2}(1)^{\oplus 3} \to T_{\mathbb P^2}\to 0$$ by taking the long exact sequence of cohomology and remembering that $H^1(\mathbb P^2, \mathcal O_{\mathbb P^2})=0$.
The corresponding result on $\mathbb P^1\times \mathbb P^1$ however is $$h^0(\mathbb P^1\times \mathbb P^1, T_{\mathbb P^1\times \mathbb P^1})=h^0(\mathbb P^1\times \mathbb P^1, \mathcal O_{\mathbb P^1\times \mathbb P^1}(2,2)=9$$
