Can one have a nontrivial 'resolution of singularities' of a smooth variety? Suppose $z_1,z_2$ are coordinates on $\mathbb{A}^2$ and $(w_1,w_2)$ homogeneous coordinates on $\mathbb{P}^1$. We can define a subvariety $X \subset \mathbb{A}^2 \times \mathbb{P}^1$ by $w_1z_2 - w_2z_1 = 0$. Then this is smooth (right?) and the projection $\pi:X \rightarrow \mathbb{A}^2$ is an isomorphism away from $(0,0)$, but $\pi^{-1}(0,0)$ is isomorphic to $\mathbb{P}^1$.
So my question is: Is this a valid resolution of singularities, or am I misunderstanding the definition of a resolution of singularities.
For reference: I thought that if $f:X' \rightarrow X$ is a proper map, $X'$ is smooth, and there is a dense open set $U$ of $X'$ such that $f\mid_U$ is an isomorphism, then $f$ is a resolution of singularities.
 A: The answer depends on the definition of resolution of singularities you use. To make that less of a banality, let me mention two possible definitions:


*

*A resolution of singularities for $X$ is a surjective proper birational morphism $\pi: X' \rightarrow X$ from a nonsingular variety $X'$.

*A resolution of singularities for $X'$ is a morphism $\pi$ as above such that the rational map $\pi^{-1}: X \dashrightarrow X'$ is an isomorphism on the smooth locus $X \setminus \operatorname{Sing} X$. 


Your example satisfies 1, but not 2. 
Definition 2 is asking that we change $X$ as little as possible while "making it smooth". This is the more usual definition of resolution of singularities, and this is the kind that Hironaka proved the existence of. (In fact he proved a lot more: see the Wikipedia article for a good summary.)
This more restrictive notion prevents you from doing naughty things like in your example, but as @Hoot mentioned uniqueness is still an issue: if you prove something about $X$ using a resolution, you'd better check it doesn't depend on the resolution you chose. In fact, for surfaces this is not really a problem: it turns out that surfaces have a minimal resolution, which is unique, and we can work with that. In higher dimensions things are more complicated: starting in dimension 3, even minimal resolutions (whatever that now means) need not be unique, because of birational maps called flops. 
