Let $M$ be a manifold (variety, scheme, your favorite object) and let $N\_1,N\_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N\_1 \cap N\_2 \neq \emptyset$. Denote by $\text{BL}\_{N\_i}M$ the blowup of $M$ along $N\_i, i=1,2$ and let $\text{BL}\_{N\_1}\text{BL}\_{N\_2}M$ be the blowup of $\text{BL}\_{N\_1}M$ along the proper transform $N\_2^\prime$ of $N\_2$. (Define $\text{BL}\_{N\_2}\text{BL}\_{N\_1}M$ vice versa). 



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> Question: Under which conditions does the following hold: $\text{BL}\_{N\_1}\text{BL}\_{N\_2}M \cong \text{BL}\_{N\_2}\text{BL}\_{N\_2}M$ ?
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This should be true if we are in the situation $N\_1 \subseteq N\_2 \subseteq M$, since ``blowups commute with restriction'' but is it in genreal true? If so, is there some reference available? If not, is there a criterium when blow ups commute (e.g. something like transversal intersection or) or/and can you come up with an easy counterexample?



When do blowups ''commute''?
Sándor is exactly right, this generally isn't true. However, there is a related statement that is always true, instead of blowing up strict transforms of $N\_1$ and $N\_2$, you should blow up total transforms.




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Blowing up total transforms
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Suppose that $I\_1$ and $I\_2$ are the ideals defining $N\_1$ and $N\_2$ in $M$. Let $$Y\_1 = Bl\_{I\_1} M = Bl\_{N\_1} M.$$
Now define the *total transform of* $I\_2$, denoted $\overline{I\_{2}}$, to be the ideal formed by extending $I\_2$ to $Y\_1$, in other words $\overline{I\_2} = I\_2 \cdot O\_{Y\_1}$ (note that by the universal property of blowing up $I\_1 \cdot O\_{Y\_1}$ is an invertible sheaf).


$\overline{I\_2}$ need not define a manifold, or even a reduced scheme, but we can still blow it up. Set $$Y\_{1,2} = Bl\_{\overline{I\_2}} Y\_1.$$
I claim: 


**Theorem:** *We have* $Y\_{1,2} = Y\_{2,1}$ *where the second object is obtained in the same way but blowing up* $I\_2$ *first.*


There are at least two ways to see this. 


1. You can do this from the universal property of blowing up.
2. Since you already assumed that $M$ is a manifold, it is integral. Thus the charts of a blowup can be computed as follows. Suppose $M = \text{Spec } A$ is affine for simplicity. If $\langle x\_1, \dots, x\_n\rangle$ generate an ideal $I$, then as $i = 1,...,n$ varies, $Y\_{I,i} = \text{Spec } A[x\_1/x\_i, \dots, x\_i/x\_i, \dots, x\_n/x\_i]$ are affine charts covering the blowup. $Y\_I = Bl\_I Y$.


**EDIT:** As Dustin Cartwright points out in a comment below, blowing up $I\_1$ followed by blowing up $\overline{I\_2}$ is the same as blowing up $I\_1 \cdot I\_2$. 




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A normal crossings example
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I'd like to give you an example in $\mathbb{A}^4 = \text{Spec } k[x,y,u,v]$ showing the difference between blowing up strict transforms and total transforms, even when the $N\_i$ intersect transversally (are in normal crossings). Let $N\_1 = V(x,y)$ be a plane and let $N\_2 = V(y,u,v)$ be a line. 


Suppose we blowup the line $N\_2$ first, then the strict transform and the total transform of $N\_1$ coincide. In particular, $Y\_{2,1}$ is the object you were considering. 


On the other hand, let's blow up $N\_1$ first, in this case the strict transform $\widetilde{N\_2}$ of $N\_2$ is a line, but the total transform $\overline{N\_2}$ of $N\_2$ is two lines. $\overline{N\_2}$ contains the strict transform but also one of the new $\mathbb{P}^1$'s lying over the origin of $N\_1 \subseteq \mathbb{A}^4$. The blowups of the strict and total transform of $N\_2$ on $Y\_1$ are thus manifestly different. 


One can check that $Y\_{1,2} = Y\_{2,1}$ if you did the total transform blowup, and so blowing up the strict transforms does not commute.


**Note:** If I recall correctly, you can also do the example when two planes are *kissing.* $N\_1 = V(x,y)$, $N\_2 = V(u,v)$. The same sort of thing happens, but it's even messier. I've seen this example in various papers on resolution of singularities.




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In the literature
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The difference between strict and total transforms plays a key role in modern resolution of singularities algorithms. Even though conceptually strict transforms are much easier to think about, they are much *harder* to compute or control. Thus modern algorithms tend to use total transforms instead in *SOME* places, and then peel off any copies of the exceptional you can, while possibly leaving embedded components. 


See for example: 


1. Section 7 of [This paper by Bravo-Encinas-Villamayor](http://arxiv.org/pdf/math/0206244v2.pdf)
2. Example 2.3 of [This paper by Howard Thompson](http://arxiv.org/pdf/1006.1915.pdf)


