When is a blow-up along a union of subvarieties smooth? Let $V_1$ and $V_2$ be two distinct smooth subvarieties of the smooth variety $X$ which are regularly embedded. I would like to find a reasonable criteria which guaranties the smoothness of the Blow-up $\widetilde{X}$ of $X$ along the union of $V_1$ and $V_2$. 
For example, is $\widetilde{X}$ smooth if $V_1$ and $V_2$ meet transversally? 
 A: Appearently, the blow-up IS smooth if $V_1$ and $V_2$ intersect transversally. In this case we have that 
$$
Bl_{V_1 \cup V_2} X = Bl_{\bar{V_1}}Bl_{V_2}X =Bl_{\bar{V_2}}Bl_{V_1}X 
$$where $\bar{V_i}$ denotes the proper transform of $V_i$. This is essentially Proposition 2.9 in Kiem and Moon's article http://arxiv.org/abs/1002.2461. 
A: In general the answer is no.
Take $X=\mathbb{A}^3$ with coordinates $x,y,z$ and let $V_1$ and $V_2$ be two lines meeting in one point, for instance 
$V_1:=\{x=y=0\}, \quad V_2:= \{x=z=0\}$. 
Then the ideal of $V_1 \cup V_2$ is $I=(x, yz)$ and the equation of the blow-up $\widetilde{X}$ of $X$ along $I$ are given in $X \times \mathbb{P}^1$ by
$\lambda x - \mu yz=0$,
where $[\lambda : \mu]$ are homogeneous coordinates in $\mathbb{P}^1$. In the chart $\mu=1$ the blow-up is then given by 
$\textrm{Spec }k[x,y,z, \lambda]/(\lambda x - yz)$,
hence it has an isolated singularity at the origin.
The other chart $\lambda=1$ is instead smooth, so this is actually the unique singular point of $\widetilde{X}$. 
A: Let $V$ be a finite-dimensional complex vector space, let $A$ be a subspace, and let $\alpha:V\to V/A$ be the projection.  The blow-up of $PV$ along $PA$ can then be identified with 
$\{(L,M)\in PV\times P(V/A): L\leq \alpha^{-1}(M)\}$
Now suppose we have another subspace $B$.  I think that the blow-up along $PA\cup PB$ is just the fibre product of the blow-ups along $PA$ and $PB$, namely
$X=\{(L,M,N)\in PV\times P(V/A)\times P(V/B) : L\leq \alpha^{-1}(M)\cap\beta^{-1}(N)\}$
We have $PA\cap PB=P(A\cap B)$, and this intersection is transverse iff $A+B=V$.  If so, then the map $(\alpha,\beta):V\to V/A\times V/B$ is surjective, so the spaces $\alpha^{-1}(M)\times\beta^{-1}(N)=(\alpha,\beta)^{-1}(M\times N)$ form a vector bundle over $P(V/A)\times P(V/B)$, whose associated projective bundle is $X$; this shows that $X$ is smooth.  Thus, the question has an affirmative answer at least for linear subspaces of projective space.
