There is no general criterion, as far as I know, it is all try and see.

Any proper projective birational morphism $f:X\to Y$ between varieties is the blowup of some sheaf of ideals $I$ on $Y$, so you can see that anything can happen.

There is no general criterion, as far as I know, it is all try and see.

Any proper birational morphism $f:X\to Y$ between varieties is the blowup of some sheaf of ideals $I$ on $Y$, so you can see that anything can happen.