Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}\_{Z}(X)$ non-singular? 


The blow-up of a non-singular variety along a non-singular subvariety is well-known to be non-singular, so the real question is ``what happens when $Z$ is singular?" The blow-up can be singular as the case when $X = \mathbb{A}^{2}$and $Z$ is defined by the ideal $(x^2, y)$ shows. On the other hand, the example where $Z$ is defined by the ideal $(x,y)^2$ shows that the blow-up can be non-singular.


**Edit:** Based on the comments of Karl Schwede and VA, I think that it would also be interesting to find non-trivial examples of appropriate $Z$'s. I am splitting this off as a  [separate question.](https://mathoverflow.net/questions/21627/what-are-non-trivial-examples-of-non-singular-blow-ups) In the comments there, the users quim and Karl Schwede say a bit about what can be said about this question using Zariski factorization.



When is a blow-up non-singular?
Craig Huneke told me about this [paper](https://www.tandfonline.com/doi/abs/10.1080/00927879708825958): "On the smoothness of blow-ups" ([MR1446135](https://mathscinet.ams.org/mathscinet-getitem?mr=1446135), [Zbl 0878.13004](https://zbmath.org/?q=an%3A0878.13004), by O'Carroll and Valla). The title alone seems to suggest it might be useful for you.


