Birational morphisms of surfaces

Question

I read that in the category of projective smooth models of a function field $K = k(x,y)$ ($k$ algebraically closed), there is at most one morphism between two such models (the morphism being a morphism of schemes, acting identically on $K$). Why is that precisely ?

Answer 1

I do not know how much surface theory you know. Here is a sketch.

If $X,Y$ are two such smooth models and $f,g:X\to Y$ morphisms as you demand, then $K_X-f^*K_Y=K_X-g^*K_Y=\sum n_iE_i$, with $n_i>0$ and $f(E_i)=g(E_i)$ are points in $Y$. So, both $f,g$ are just contracting the same divisors $E_i$ and hence must be the same.