TITLE: When is a blow-up along a union of subvarieties smooth?
QUESTION [8 upvotes]: 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?

REPLY [5 votes]: 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.