Let $X$ be a smooth scheme over a field $k$, and let $Z\subset X$ be a closed sub-scheme of codimension $2$. Let $Bl\_Z(X)$ denote the blow-up of $X$ at $Z$, and let $\pi\colon Bl\_Z(X)\to X$ denote the projection. Suppose I have a section $\sigma\colon Z\to Bl\_Z(X)$ of $\pi$ over $Z$ (i.e. $\pi\sigma=1\_Z$). 


Question: (when) is the map $Bl\_Z(X)\setminus\sigma(Z)\to X$ a weak equivalence?


The references to blow-up theorems which I have found (e.g. Voevodsky's Seattle lectures) suggest that it becomes an equivalence after suspension, but I'd like to avoid suspending if possible. 


I'm also happy to restrict the choice of $Z$ and $X$. The case I'm most interested has $X$ being an iterated blow-up of affine space at (proper transforms) of linear sub-spaces, and $Z$ being a (proper transform of a) linear sub-space.



Blow-ups in Motivic Homotopy Theory
I assume that $Z$ is also smooth over $k$. Then the base change of the map


$$Bl\_Z(X)-\sigma(Z)\to X$$


to $Z$ is the map 


$$\mathbb{P}(N\_{X,Z})-\sigma(Z) \to Z$$


where $\mathbb{P}(N\_{X,Z})$ is the bundle of lines in the normal bundle $N\_{X,Z}$. The second map is an equivalence iff $Z$ has codimension $2$. If not, the two sides have different $\ell$-adic cohomology after pulling back to a geometric point, for example. So the first map can only be a weak equivalence when the codimension is exactly $2$.


But if the codimension is $2$, I don't know if your map an equivalence. I haven't heard of any progress towards removing the suspension in Voevodksy's blow-up theorem...


